Brian J Morton

Michael Batty

Keywords: LUTI, Tranus, land use, calibration, optimisation. vii LUTI, Tranus, usage de sol, calibration, optimisation. x

In the first place, I would like to thank my director of theses, Peter Sturm. To work under your guidance it's a breeze, you always guided the "aprenti -chercheur" in each phase all along the thesis, with a "Zen" and precise look, exactly what one has need of. I would also like to thank you for your availability and your commitment to all the challenges of this work. Finally, I am grateful of your human qualities and I am proud to have found in you a friend.

I would like to thank my co-director Arthur Vidard. Even if you were not on the site of INRIA, I could always count on you. Particularly, I found extremely useful and reassuring your commitment on the numerical / mathematical issues during the thesis. I am convinced that a great deal of reformulation results are possible thanks to the afternoons working on Tranus equations. Your good humour and availability were very appreciated.

I would like to thank my colleague and friend Laurent Gilquin. We shared a great part of the thesis together, and even if you are not very good in ping-pong

Introduction Introduction (Français) 1 State of the art and background material

Introduction

Most of today's population lives in cities and urbanised areas. Much of the planet's energy consumption, pollution, waste generation etc. happens there, which makes it important to consider urban areas in efforts aiming at sustainable development. The latter is, among others, addressed by transportation and land use planning, where land use here loosely refers to the spatial distribution of economic and other activities. Transportation and land use planning were traditionally carried out in a decoupled manner: although land use is naturally a main input for transportation planning, the impact of changes in transportation infrastructure or policies, on land use, was often ignored. One typical such impact is urban sprawl, whose causes include the dynamic feedbacks between transportation and land use. Neglecting such feedbacks in modelling systems that assist decision making, may lead to incorrect assessments of transportation plans for instance. LUTI (land-use and transportation integrated) models aim at representing the complex interactions between land use and transportation offer and demand within a territory. They are mainly used to evaluate different alternative planning scenarios, by simulating their tendential impacts on patterns of land use and travel behaviour. Since the early 60's LUTI modelling has attracted researchers that aimed to model the complex economical relations in urban areas; a good overview of the evolution and history of LUTI modelling can be found in [START_REF] Wegener | Overview of Land-Use Transport Models[END_REF]). Setting up a LUTI model requires the estimation of several types of parameters to reproduce as closely as possible, observations gathered on the studied area (socio-economic data, transport surveys, etc.). The vast majority of available calibration approaches are semi-automatic, estimating one subset of parameters at a time, without a global integrated estimation. Automatic calibration of LUTI model is not a common practice; an exception has been proposed for the Meplan model [START_REF] Abraham | Parameter Estimation in Urban Models: Theory and Application to a Land Use Transport Interaction Model of the Sacramento, California Region[END_REF].

We consider Tranus (de la Barra 1982; de la [START_REF] De La Barra | Integrated Land Use and Transport Modelling[END_REF], an open source LUTI model that is widely used. Tranus is a classical LUTI models, with two separated modules: the activity module and the transport module. The activity module, is an equilibrium type model based on micro-economic principles that balance the offer and demand of the different economical sectors that interact at each level. Economical sectors are considered in the broad sense, amongst them we have: land, goods, salaries, housing, transportation demand, etc. Also, the price paid for each economical sector has to be balanced with respect to offer and demand, thus there are two equilibria that have to be achieved, offer versus demand and (production) cost versus prices. The transportation module, computes the costs of transportation and assigns the demand to the network. Both modules interact back and forth until a general equilibrium is achieved.

The calibration process is usually done by an expert modeller who iteratively tunes a group of parameters to reproduce as closely as possible the observations gathered in the area of study. This process is usually done manually, with little to no automation, adjusting the different economical parameters (for example, the demand curves for different goods in a specific geographical zone). At the same time, Tranus computes internally a set of adjustment coefficients (called shadow prices in Tranus) that correct the utilities and account for unmodelled effects. These endogenous variables help the model achieve a better response and fit more precisely to the observed data.

In this thesis we address several shortcomings of the classical approach of calibration used in Tranus. We propose the reformulation of the heuristic calibration algorithm used in the land use and activity module as an optimisation problem. Later, we extend this approach by having a closer look at the inner loop that computes the shadow prices and propose an efficient methodology for their estimation by decoupling the calibration in smaller problems.

To be able to do this, we have to carefully investigate the system of equations that are computed in the activity module. We also introduce auxiliary variables, which enables a closed form computation instead of an iterative one. This in turn makes it possible to use sophisticated numerical optimisation methods and opens the door to the simultaneous estimation of different parameter types of the model. The ultimate goal of this approach is to simultaneously calibrate the various parameters of Tranus' inner and outer loops.

Overview of the dissertation

To be able to formulate a semi-automatic calibration of Tranus, it was first necessary to construct a literature review of the state-of-the-art in urban modelling. Chapter 1 describes the various operational LUTI models available and the corresponding calibration approaches.

It also builds a theoretical background on the numerical methods and discrete choice models utilised all along this work.

Chapter 2 describes Tranus' mathematical formulation, particularly for the land use and activity module (from now on, land use module). We expose the various equations involved in the calibration of the land use, also the demand functions and discrete choice models are presented.

Chapter 3 is all about calibration of the land use module, first we present the traditional calibration approach to estimate the so-called "shadow prices" which are endogenous parameters of the model, and latter the reformulation of the calibration as an optimisation problem. This chapter is the core of the thesis, and particular detail is given for the different types of sectors. At the end of the chapter a comprehensive numerical example is given to illustrate our methodology. We also present a detailed methodology for the construction of synthetic scenarios based on real calibrated study areas. These synthetic scenarios have a perfect fit without the need of shadow prices (usually we set their value to zero), enabling us to validate our optimisation algorithms knowing the ground truth values of the shadow prices. A simple example is presented to understand the problematic of synthetic scenario generation and the corresponding equilibrium prices problem. Finally, we question the rationale of usual calibration approaches for Tranus (and other LUTI models), which consists in estimating parameters for which the model reproduces observations exactly. In Tranus, this is achieved by enriching the underlying macro-economic model with the already mentioned auxiliary variables, the shadow prices. While this allows to correct for unavoidable un-modeled effects, it also bears the risk of over-parameterisation/overfitting. We propose a model selection scheme, aiming at a compromise between model complexity (here, number of shadow prices) and goodness of fit to observations, reducing the risk of overfitting and increasing the likelihood of achieving good predictions with a model. After the reformulation of the computation of the shadow prices as an optimisation problem, we are able to include in the optimisation scheme other parameters (than the shadow prices).

In Chapter 4, we deal with the calibration of other Tranus parameters. First we propose a semi-automatic calibration for the penalising factors. These parameters aim to represent the preferences of residential choice of the various household types of the model. The main idea is to include external data (when available) to guess a good starting point for the optimisation which then improves them. This is possible after examining the equations that create the interactions between households and housing, and decoupling the optimisation in smaller problems. Finally, we present a sensitivity analysis to identify influential parameters for transportable sectors. Once the influential sectors are selected, an optimisation algorithm finds the parameters values that improve the calibration.

The last chapter, Chapter 5 presents the methodology applied to real scenarios. We first

Introduction (Français)

La plupart de la population mondiale vit dans des villes et zones urbaines. Par conséquent, la plus grande partie de la consommation d'énergie, de la pollution, de la génération de déchets, de la planète, s'y concentre, d'où l'importance de considérer les zones urbaines dans les efforts visant au développement durable. Celui-ci doit être pris en compte, entre autres, par le transport et l'aménagement du territoire, c'est-à-dire à la répartition spatiale des activités économiques et autres. Jusqu'à présent, la planification du transport et de l'aménagement du territoire a été menée en pratique de manière découplée: bien que l'aménagement du territoire soit logiquement une contribution principale à la planification des transports, l'impact des changements dans les infrastructures ou les politiques de transport sur l'aménagement du territoire a souvent été ignoré. Un impact typique de ce type est l'étalement urbain, dont les causes incluent les réactions dynamiques entre le transport et l'aménagement du territoire.

En négligeant ces rétroactions dans les systèmes de modélisation qui assurent la prise de décision, cela peut conduire à des évaluations incorrectes des plans de transport par exemple.

Les modèles LUTI (aménagement et transport intégrés) visent à représenter les interactions complexes entre l'aménagement du territoire et l'offre et la demande de transport et la demande sur un territoire. Ils sont principalement utilisés pour évaluer différents scénarios de planification alternatifs, en simulant leurs impacts tendanciels sur les modèles d'aménagement du territoire et les comportements de déplacement. Depuis les années 60, la modélisation LUTI a attiré des chercheurs qui visaient à modéliser les relations économiques complexes dans les zones urbaines; Un bon aperçu de l'évolution et de l'histoire de la modélisation LUTI se trouve dans [START_REF] Wegener | Overview of Land-Use Transport Models[END_REF]). La mise en place d'un modèle LUTI nécessite l'estimation de plusieurs types de paramètres pour reproduire le plus près possible, les observations recueillies dans la zone étudiée (données socio-économiques, enquêtes sur les transports, etc.). La grande majorité des approches de calibration disponibles sont semi-automatiques, c'est-àdire estimant un sous-ensemble de paramètres à la fois, sans une estimation globale intégrée.

La calibration automatique des modèle LUTI n'est pas une pratique courante; Une exception a été proposée pour le modèle Meplan [START_REF] Abraham | Parameter Estimation in Urban Models: Theory and Application to a Land Use Transport Interaction Model of the Sacramento, California Region[END_REF].

Nous considérons Tranus (de la Barra 1982; de la Barra 1989), un modèle LUTI open source largement utilisé. Tranus est un modèle LUTI classique, avec deux modules séparés: le module d'activité et le module de transport. Le module d'activité est un modèle de type "à équilibre" basé sur des principes micro-économiques qui équilibrent l'offre et la demande des différents secteurs économiques qui interagissent à chaque niveau. Les secteurs économiques sont considérés au sens large, parmi lesquels nous avons: le sol, les biens, les salaires, le logement, la demande de transport, etc. De plus, le prix payé pour chaque secteur économique doit être équilibré par rapport à l'offre et à la demande, il existe donc deux équilibres qui doivent être atteints: offre par rapport à la demande et coûts de production par rapport aux prix. Le module de transport, calcule les coûts de transport et attribue la demande au réseau. Les deux modules interagissent l'un après l'autre jusqu'à ce qu'un équilibre général soit atteint.

Le processus de calibration est habituellement réalisé par un modélisateur expert qui ajuste de manière itérative un groupe de paramètres pour reproduire aussi précisément que possible les observations recueillies dans le domaine d'étude. Ce processus se fait généralement manuellement, avec peu ou pas d'automatisation, en ajustant les différents paramètres économiques (par exemple, les courbes de demande pour différents produits dans une zone géographique spécifique). Parallèlement, Tranus calcule en interne un ensemble de coefficients d'ajustement (appelés prix sombres dans Tranus) qui corrigent les utilites et représentent des effets non modélisés. Ces variables endogènes aident le modèle à obtenir une meilleure réponse et s'adapter plus précisément aux données observées.

Dans cette thèse, nous abordons plusieurs lacunes de l'approche classique de calibration utilisée dans Tranus. Nous proposons la reformulation de l'algorithme de calibration heuristique utilisé dans le module usage des sols et activités en tant que un problème d'optimisation.

Par ailleurs, nous étendons cette approche en examinant de plus près la boucle interne qui calcule les prix sombres (shadow prices) et proposons une méthodologie efficace pour leur estimation en découplant la calibration en petits problèmes. Pour pouvoir le faire, nous devons étudier attentivement le système d'équations qui sont calculées dans le module d'activité.

Nous introduisons également des variables auxiliaires, ce qui permet un calcul de forme fermé au lieu d'un itératif. Cela permet à la fois d'utiliser des méthodes d'optimisation numérique sophistiquées et ouvre la voie à l'estimation simultanée de différents types de paramètres du modèle. Le but ultime de cette approche est de calibrer simultanément les différents paramètres des boucles interne et externe de Tranus.

Résumé de la dissertation

Pour pouvoir formuler une calibration semi-automatique de Tranus, il fallait d'abord construire une bibliographie de l'état de l'art dans la modélisation urbaine. Le chapitre 1 décrit les différents modèles opérationnels LUTI disponibles et les approches de calibration correspondantes. Il génère également un historique théorique sur les méthodes numériques et les modèles de choix discrets (discrete choice) utilisés tout au long de ce travail.

Le chapitre 2 décrit la formulation mathématique de Tranus, en particulier pour le module d'usage des sols et activités (à partir de maintenant, module d'usage des sols). Nous exposons les différentes équations impliquées dans la calibration de l'usage des sols, ainsi que les fonctions de demande et les modèles discrets sont présentés.

Le chapitre 3 porte sur la calibration du module d'utilisation des sols, nous présentons d'abord l'approche de calibration traditionnelle pour estimer les «prix sombres», qui sont des paramètres endogènes du modèle, et la reformulation de la calibration comme problème d'optimisation . Ce chapitre est le noyau de la thèse, et des détails particuliers sont donnés pour les différents types de secteurs. À la fin du chapitre, un exemple numérique complet est donné pour illustrer notre méthodologie. Nous proposons également une méthodologie détaillée pour la construction de scénarios synthétiques basés sur des zones d'étude calibrées réelles. Ces scénarios synthétiques ont un ajustement parfait sans avoir besoin de prix sombres (en général, nous mettons leur valeur à zéro), nous permettant de valider nos algorithmes d'optimisation en connaissant les valeurs réelles des prix sombres. Un exemple simple est présenté pour comprendre la problématique de la génération de scénarios synthétiques et le problème des prix d'équilibre correspondants. Enfin, nous interrogeons la logique des approches de calibration habituelles pour Tranus (et d'autres modèles LUTI), qui consiste à estimer les paramètres pour lesquels le modèle reproduit exactement les observations. Dans Tranus, cela se réalise en enrichissant le modèle macroéconomique sous-jacent avec les variables auxiliaires déjà mentionnées, les prix sombres. Bien que cela permette de

State of the art and background material

"Not all economic models that are computationally challenging, interesting, and important conform to linear, quadratic, or other standard nonlinear programming formulations. Rather, such models require the solution of highly nonlinear equations systems using nonstandard and innovative, iterative algorithms that exploit the special features of those equations." A.

Anas, 2007

In this chapter we propose a brief review on LUTI models, particularly focusing on the calibration. We are interested in how calibration is performed in the various available LUTI models, specially the ones that perform this with optimisation tools. Then, we review the basic numerical optimisation algorithms used in this thesis, we formulate this methods adapted to the quantities that we need to optimise in latter sections. Finally, we propose a quick review and properties of logit discrete choice models. We explore the basic properties needed for the computation of our Tranus equations.

LUTI models literature review

A fundamental goal of Land Use -Transport Interaction (LUTI) models is to capture the strong interplay between land use and transportation in metropolitan areas or other territories. Inherently, sector-specific models, transport and urban alike, cannot take this interaction into account and thus miss one side of the story. LUTI models aim to fill this gap, and ul-timately to provide better decision helping tools for urban and regional long term planning.

Lowry was the first to build a computable sound LUTI model [START_REF] Lowry | A Model of Metropolis[END_REF], based on gravity theory. In the 60's data collection and computers were not powerful enough to handle more complex dynamics, leading to a partial abandoning of urban models (see [START_REF] Lee | Requiem for large scale models[END_REF], for a discussion on these points). During the period 1970-1990 there were many developments in micro economic theories, mainly in discrete choice models [START_REF] Mcfadden | Conditional logit analysis of qualitative choice behaviour[END_REF][START_REF] Ortuzar | Fundamentals of discrete multimodal choice modelling[END_REF][START_REF] Train | Discrete Choice Methods with Simulation[END_REF]; Ortuzar and Willumsen 2011) to spark a new generation of models. In 1994, Wegener [START_REF] Wegener | Operational Urban Models State of the Art[END_REF][START_REF] Wegener | Overview of Land-Use Transport Models[END_REF]) lists twelve operational LUTI models and later in 2004 upgrades the list to twenty and classifies them according to a number of measures (Comprehensiveness, Model Structure, Theory, Modelling Techniques, Dynamics, Data Requirements, Calibration and Validation, Operationality and Actual Applications). Also driven by the US government and the Clean Air Act, the US Department of Energy commissioned an evaluation of vehicle travel reduction strategies to a consulting firm [START_REF] Southworth | A Technical Review of Urban Land Use-Transportation Models as Tools for Evaluating Vehicle Travel Reduction Strategies[END_REF].

This study describes many of the models described in Wegener's but work in a more detailed way. It discusses many issues related to policy analysis, for instance the overlapping of model validation with calibration. It also provides performance analysis and discusses practical issues that would help a wider application of LUTI tools. However, interest in LUTI models has risen again in the 2000s and their number and complexity have been growing steadily ever since. This goes hand in hand with increasing expectations from end users as well as with new theoretical developments and a drastic increase in computational capacities, the latter enabling for instance the development of micro-simulation models. Another very detailed review on LUTI models is [START_REF] Simmonds | Review of land-use/transport interaction models[END_REF]. In this work, three families of LUTI models are distinguished; static models (DSCMOD, IMREL, MUSSA), spatial economics models (MEPLAN, TRANUS, PECAS, RUBMRIO) and activity based models (UrbanSim, Delta, IRPUD). Static models are models based originally upon the analogies with statistical mechanics ("entropy") pioneered by Alan Wilson in the 1970s. Spatial economic models propose an aggregated approach based on equilibrium principles, while activity based models focus on the system dynamics aiming at more detailed representation of the different processes of change affecting the activities considered and the space which they occupy. Another article that present a brief description of the main Land Use models available is [START_REF] Timmermans | Modelling Land Use and Transportation Dynamics: Methodological Issues, State-of-Art, and Applications in Developing Countries[END_REF]). Timmermans' work covers many urban models and LUTI models, for Tranus he gives a very good insight.

How is Calibration done in some LUTI models?

We are interested in operational LUTI models, models that have been applied to study areas and more importantly, we are interested in the calibration associated techniques. Back in 1973, Lee in his article "Requiem for large scale models" [START_REF] Lee | Requiem for large scale models[END_REF] claimed that it was one of the fundamental flaws of large-scale models that there did not exist reliable and efficient techniques for calibrating their parameters, i.e. determining those values of the parameters of their equations that yielded the best correspondence of the model results with observations from reality. This article was mostly criticising the black box approach and the difficulty to validate a model to assess if it is really doing what we want them to do. Calibration is very hard when one can not understand the effect of the parameters on the model output. From the same author, twenty years later [START_REF] Lee | Retrospective on Large-Scale Urban Models[END_REF]) he still advocates for transparency, replicability and pragmatic evaluation (to make possible to conclude that a LUTI model is better than alternative ones). Even if many progresses have been made in econometrics, optimisation and computer algorithms, the problem still exists, as Wegener's put in 2004: "There has been almost no progress in the methodology to calibrate dynamic or quasi-dynamic models.

In the face of this dilemma, the insistence of some modellers on 'estimating' every model equation appears almost an obsession. It would probably be more effective to concentrate instead on model validation, i.e. the comparison of model results with observed data over a longer period."- [START_REF] Wegener | Overview of Land-Use Transport Models[END_REF]) It is still very expensive to perform calibration of a LUTI model, and validation is often forgotten. In this issues, [START_REF] Prados | To make LUTI models operational toolsfor planning[END_REF] gives a good insight on how we could make LUTI models operational.

The majority of papers published about LUTI models and their applications do not explicitly explain the calibration procedure. We can also say the same about the models, they mostly only give guidelines to calibration, even if this task takes months or years, and enormous resources, very little detail is given as how do we instantiate one of these models. In this thesis we are interested in automatic or semi-automatic calibration of LUTI models, in this section we will try to assess for which models such techniques have been used or developed. Optimisation has been used extensively as an econometric technique to calibrate "externally" parts of these models (sometimes called submodels/submodules), for instance max-likehood optimisation is a recurrent technique to calibrate the discrete choice submodels that many LUTI share. But, we are looking for a more integrated approach, where optimisation is utilised to automatically calibrate the response of the model, or at least part of it. Computer power has grown immensely in the last 10 years, and as Michael Batty said in 1976: "The trial and error method of searching for best-parameter values by running the model exhaustively through a range of parameter values or combinations thereof represents a somewhat blunt approach to model calibration. The process of calibrating an urban model of this kind involves the use of techniques to find parameter values which optimise some criterion measuring the goodness of fit of the model's predictions to the real situation. For example, it may be decided that by minimising the sum of the squared deviations between predictions and observations, the best parameter values can be found" - [START_REF] Batty | Urban Modelling[END_REF]) -we are looking for this type of framework.

Here we present some of the most popular operational LUTI models and published calibration procedures, this list is not exhaustive and is based on the one listed by Hunt [START_REF] Hunt | Current operational urban land-use-transport modelling frameworks: A review[END_REF].

The MEPLAN model has been the result of various works in urban and regional planning for the last 40 years under the direction of Marcial Echenique [START_REF] Echenique | The MEPLAN models of Bilbao, Leeds and Dortmund[END_REF]). It is This approach is effective when for instance one disposes of disaggregate data or when the submodels may be used separately. They also propose a simultaneous calibration approach, highlighting the advantages and disadvantages of each approach. Finally, they expose a sequential calibration, mostly for nested logit parameters for the location choices. For example, in [START_REF] Abraham | Specification and Estimation of a Nested Logit Model of Home. Workplaces and commuter Mode Choices by Multiple Owker Households[END_REF] where each parameter has an economic interpretation. The model consists of a series of non-linear fixed point equations where the solution is obtained with an iterative approach based on gradient descent techniques. The calibration is performed with microeconomic techniques, for instance it utilises maximum likelihood techniques for the estimation of the willingness to pay [START_REF] Lerman | Hedonic theory, bid rents, and willingness-to-pay: some extensions of Ellickson's results[END_REF]. There are no automatic or semi automatic calibration techniques developed for MUSSA.

Similar issues can be found in other type of models than LUTI. For instance, for travel demand models, it is usual to maximise a likelihood function. Transforming the model equations to linear form, and then performing linear regression over the parameters [START_REF] Ortuzar | Modelling Transport[END_REF]. For unconstrained spatial interaction models, (Chisholm and O'Sullivan 1973) utilises this technique, the unconstrained case is non-linear, so the method of the least squares can be used to estimate the model parameters. For other urban and transportation application of least squares minimisation, see chapter "calibration as non-linear optimisation" in [START_REF] Batty | Urban Modelling[END_REF]). This review of LUTI models tries to illustrate examples where optimisation techniques have been used to calibrate or validate LUTI models (or parts of them). There are many other LUTI models around, but we have decided to particularly look at models that have details about the calibration and how the models work internally. The main idea behind the calibration as an optimisation problem is introducing a statistical measure of model's performance, and estimate the parameters such that they optimise this quantity. The techniques vary, depending on the type of problem.

In the next section we will introduce the optimisation algorithms utilised in this thesis.

Local Optimisation

In this section we will briefly introduce the non-linear optimisation techniques used in our work. A large part of our proposed approach on calibration of Tranus is based on numerical optimisation. In general terms, we will utilise numerical optimisation to find the parameters that make our model to reproduce as closely as possible the observed data. Our analysis will be carried out with a goodness-of-fit function called chi-squared error function with weights generally set to 1. Even if we have not introduced yet the quantities utilised by Tranus, we will denote the quantity to be fitted as X 0 (we will later see that X 0 represents the base year's production) and we will develop our analysis with respect to this quantity). The general case consists of a non-linear vector function as follows (also called response function):

X : R n -→ R m σ σ σ -→ X(σ σ σ) = (X 1 (σ σ σ), ..., X m (σ σ σ)) .
Here, σ σ σ is the vector of parameters and the response function X(σ σ σ) depends on the value of these parameters. Also, we consider a set of observations (points): X 0 = {X k 0 , k = 1, ..., m} and a set of weights: W = {w k , k = 1, ..., m}. The quantity that one would like to minimise is the chi-square function χ 2 :

χ 2 (σ σ σ) = m k=1 X k (σ σ σ) -X k 0 w k 2 .
From the latter equation we can see that if a value of σ σ σ is found such that χ 2 (σ σ σ) = 0, then for all k we have X k (σ σ σ) -X k 0 = 0. This means that our response function reproduces the observations perfectly. We can also write the χ 2 function in vector form:

χ 2 (σ σ σ) = (X(σ σ σ) -X 0 ) T W(X(σ σ σ) -X 0 ) (1.1)
here, (•) T denotes the transpose operator.

In Tranus, we are handling high dimensional non-linear response functions, so minimisation of χ 2 (σ σ σ) has to be carried out with numerical methods. We will present a quick review of the most common iterative methods. All the methods presented try to find a way of perturbing an initial value of σ σ σ to reduce the value of χ 2 . The quantity X(σ σ σ) -X 0 is called the vector of residuals.

Gradient Descent

This method, as the name says it, carries out a downhill exploration of the surface of the function to find the lowest value. The direction chosen to update the parameters is the opposite of the gradient of the function. This method works well on simple functions and for large problems, this method is sometimes the only viable option. We can compute the gradient of χ 2 with respect to the parameters σ σ σ (in vector form):

∂χ 2 ∂σ σ σ = 2(X(σ σ σ) -X 0 ) T W ∂X(σ σ σ) ∂σ σ σ
∂X(σ σ σ) ∂σ σ σ is the jacobian matrix of productions with respect to parameters σ σ σ. We will denote from now on this matrix as J.

The gradient descent method updates the parameters in the direction of the steepest descent, by a step of length λ, the perturbation for the gradient descent method is given by the quantity:

∆ GD = λJ T W (X(σ σ σ) -X 0 ) .
where ∆ GD is the update for the gradient descent method.

Gauss-Newton

The Gauss-Newton algorithm can only be used to minimise the sum of squared function values (least squares problems) and unlike the Newton's method, it has the advantage that second derivatives, which can be challenging to compute, are not required. Gauss-Newton takes into consideration the first order Taylor polynomial approximation of the response to update the step. Let us suppose that X can be approximated by:

X(σ σ σ + ∆) ≈ X(σ σ σ) + J ∆ (1.2)
Inserting (1.2) in the objective function; gives the following approximation:

χ 2 (σ σ σ + ∆) ≈ X T W X + X T 0 W X 0 -2X T W X 0 -2(X -X 0 ) T W J∆ + ∆ T J T W J∆ (1.3)
following the assumption that X has a linear approximation near σ (cf. equation (1.2)) and that the residuals are small, we obtain that χ 2 is approximately quadratic in the perturbation ∆. Also, we can identify the quadratic term of equation (1.3) as an approximation of the hessian matrix for χ 2 , given by J T W J. With this in mind, the optimal value for ∆ that minimises χ 2 can be computed imposing ∂χ 2 ∂∆ = 0. Hence:

∂χ 2 ∂∆ (σ σ σ + ∆) ≈ -2(X -X 0 ) T W J + 2∆ T J T W J = 0 ,
obtaining the normal equations for Gauss-Newton method:

[J T W J]∆ GN = J T W (X -X 0 ) .
As the reader can realise, the update of the step requires inverting a linear system.

Levenberg-Marquardt

This is a combination of both gradient descent and Gauss-Newton, taking both types of parameter updates into consideration:

[J T W J + λI]∆ LM = J T W (X -X 0 ) .
If λ = 0, the method is purely Gauss-Newton, if λ is large, the method moves towards gradient descent. The initial values for λ are usually large, starting the algorithm with small steps in the steepest descent direction. As the solution improves, the value of λ is decreased, approaching the Gauss-Newton method, accelerating the solution to the local minimum.

Broyden-Fletcher-Goldfarb-Shannon

In numerical optimisation, the Broyden-Fletcher-Goldfarb-Shannon (BFGS) algorithm is an iterative method for solving unconstrained nonlinear optimisation problems [START_REF] Broyden | The convergence of a class of double-rank minimization algorithms[END_REF]).

The BFGS method approximates Newton's method replacing the objective function by a quadratic model, the key difference with Newton's is that the Hessian of the cost function is approximated by a matrix B that is not updated in each iteration (similar to what is done in Gauss-Newton). However, BFGS has proven to have good performance even for non-smooth

Stochastic optimisation: EGO algorithm

The stochastic optimisation procedure presented in this section corresponds to the Efficient Global Optimisation (EGO) algorithm introduced in (Jones, Schonlau, and Welch 1998).

The main idea underlying the EGO algorithm is to fit a response surface, often denoted by metamodel, to data collected by evaluating the complex numerical model at a few points.

The metamodel is then used in place of the numerical model to optimise the parameters.

The metamodel used in the EGO algorithm is a Gaussian process defined as follows:

g : R d → R x = (x 1 , . . . , x d ) → z = g(x) = µ(x) + (x)
where x are the parameters selected with the sensitivity analysis, z a scalar output of the numerical model, d the dimension of the input space, µ the model trend and is a centered stationary Gaussian process (x) ∼ N(0, K χ ). χ denotes the structure of the covariance matrix K χ of . Let x i , x j denote two points of R d , χ = {r, θ, σ} with (K χ ) i,j = σ 2 r θ (x i -x j ) where:

• r θ (.) is the correlation function chosen here to be the Matèrn 5/2 function,

• σ 2 is the variance of g,

• θ are the hyperparameters of r .

The parameters µ, σ and θ are estimated by maximum likelihood. In the following, Z denotes the random variable modelling the output z.

Expected Improvement Once the metamodel is fitted, it is used by the algorithm to search for a minimum candidate. The EGO algorithm uses a searching criterion called "expected improvement" that balances local and global search. Let x be a candidate point, the expected improvement evaluated at x writes as follows:

EI χ (x) = E[max(z min -Z, 0)],
where z min is the current minimum of the metamodel. A numerical expression of EI χ (x) can be derived. Let Z denote the BLUE (Best Linear Unbiased Estimator) , see (Jones, Schonlau, and Welch 1998) of Z and σ Z its standard deviation, the following expression for EI χ (x) is obtained: For the choice of the stopping criterion, on can look at the value of the expected improvement.

EI χ (x) = (z min -z(x))φ N z min -z(x) σ z + σ z f N z min -z(
Indeed, a value of the expected improvement close to zero indicates that the input space has been sufficiently explored. Thus, a lower bound on the expected improvement can be selected as the stopping criterion. Here, we set the lower bound equal to 10 -5 . Thus, the stopping criterion writes:

EI χ (x) ≤ 10 -5
To ensure that the EGO algorithm finishes, we also fix a maximum number of iterations equal to 200. The two R packages "DiceOptim" and "DiceDesign" developed by (Roustant, Ginsbourger, and Deville 2012) are used to implement the EGO algorithm.

The Logit model

As Tranus uses logit models as fundamental micro-economic tools to model discrete choices, we found important to add a small introduction to readers that are not familiar with this type of theory. The scope of this section is only limited to the basic notion needed to understand this thesis. We encourage the reader to read [START_REF] Train | Discrete Choice Methods with Simulation[END_REF]; Ortuzar and Willumsen 2011) for a comprehensive overview of discrete choice theory and transport modelling.

In this section we will review the classical logit random utility theory and some of useful properties. The original logit formulation stems from Luce [START_REF] Luce | Individual choice behavior: a theoretical analysis[END_REF]). It makes assumptions about the characteristics of the choice probabilities and the independence of irrelevant alternatives (IIA). The latter means that the ratio of probabilities of choosing between two alternatives namely i and k only depends on the attributes of alternatives i and k, no matter what other alternatives are available.

We will utilise the same notation as K. Train in his book [START_REF] Train | Discrete Choice Methods with Simulation[END_REF].

Let us consider an individual n facing a choice among J alternatives. Each of the alternatives j ∈ J has an associated net utility U n j . The utility that the modeller observes can be decomposed in two parts, (1) a measurable part known by the modeller and (2) an unknown random part which reflects the tastes and characteristics of the individual. Together they form:

U n j = V n j + n j . (1.4)
This functional form permits that two individuals with apparently the same attributes and facing the same choice could choose different alternatives, and that some individuals may select an option that maybe is not the best. We have to assume some homogeneity in the population to be able to do such a decomposition, that's why we often segment the market, for instance in Tranus we do so by population categories or socio-economic categories. This enables the groups to face the same sets of alternatives sets and have the same constraints.

The premise of the rational choice model comes from the idea that the individual n will choose the alternative j that gives him the higher satisfaction (utility), this translates in:

U n j ≥ U n i , ∀i ∈ J
and with the decomposition proposed in (1.4):

V n j -V n i ≥ n i -n j , ∀i ∈ J . (1.5)
The logit formulation comes from assuming that the error terms n j are independently, identically distributed extreme values (also called Gumbel and type I extreme values), where the density for each term is given by:

f ( n j ) = e -n j e -e -n j (1.6)
and the cumulative distribution of an extreme value random variable is:

F ( n j ) = e -e -n j (1.7)
The clever part is that the difference between two extreme values follows a logistic distribution, i.e. if we set: δ n ij = n i -n j , then:

F (δ n ij ) = e δ n ij 1 + e δ n ij (1.8)
Equation (1.8) is often utilised to reference the binomial (2 choice) logit formulation.

The shape of the distribution is not as important as the assumption of independent error terms. This means that the random part of one choice does not affect the random part of another alternative, it is a fairly restrictive assumption (other models that lift this assumption are described in chapters 4-6 of [START_REF] Train | Discrete Choice Methods with Simulation[END_REF]). The researcher has to find the good specification of V n j (find the good combination of parameters to put in the observed utility for each population type) to make irrelevant the error term of another alternative. If the observed utility is specified well, the error term can be considered just as white noise.

Following [START_REF] Mcfadden | Conditional logit analysis of qualitative choice behaviour[END_REF], the probability of the individual n choosing alternative j is:

P n j = P(V n j + n j > V n j + n i : ∀i = j) = P( n i < n j + V n j -V n i : ∀i = j) (1.9)
the latter expression is the cumulative distribution of n i evaluated at n j + V n j -V n j . Replacing equation (1.7) in (1.9) we can derive the conditional probabilities:

P n j | n j = i =j e -e -( n j +V n j -V n i ) (1.10)
integrating over all n j :

P n j = ∞ -∞ (P n j | n j ) • f n j d n j (1.11) = ∞ -∞   i =j e -e -( n j +V n j -V n i )   • e -n j e -e -n j d n j (1.12)
noting that V i -V i = 0, we can include the j term inside the product:

P n j = ∞ -∞ i e -e -( n j +V n j -V n i ) • d n j = ∞ -∞ e -i e -( n j +V n j -V n i ) • d n j = ∞ -∞ e -n j i e -(V n j -V n i ) • d n j = 1 i e -(V n j -V n i ) = e V n j i e V n i (1.13)
thus obtaining the classic multinomial logit formulation in equation (1.13). The observed utility is usually considered to be linear in the parameters (in Tranus, all logit models are specified as being linear in the utility), assuming a simple expression for the observed utility:

V n j = k θ kj z n jk , where θ kj represents the parameter for attribute k for choice j, and the vector z n jk is the observed variable for individual n, for attribute k and choice j, then the probabilities are as follows:

P n j = e k θ kj z n jk i e k θ ki z n ik (1.14)
here the θ parameters are assumed constant among all individuals of the homogeneous cluster but may vary across alternatives. This assumption is fairly practical, as it makes the loglikehood function concave [START_REF] Mcfadden | Conditional logit analysis of qualitative choice behaviour[END_REF], so the calibration via numerical maximisation of the log-likehood function is very efficient with softwares such as Biogeme [START_REF] Bierlaire | PythonBiogeme: a short introduction[END_REF] or R (Roustant, Ginsbourger, and Deville 2012).

Consumer Surplus

One of the attractive features about logit models is that the computation of the expected consumer surplus is very simple. By definition, the consumer surplus is the utility in monetary terms that the person receives in the choice situation. The rational individual chooses the alternative that gives the maximum utility, CS n = 1/λ n max i U n i , where λ n is the marginal utility of income for person n, so the division by λ n translates the utility in money terms.

As stated above, the researcher does not observe U n i , so he has to calculate the expected consumer surplus using the observed utilities V n i :

E(CS n ) = 1/λ n [max i V n i + n i ]
if the utilities are linear with respect to income, and the error terms are iid extreme values,

Williams [START_REF] Williams | On the formation of travel demand models and economic evaluation measures of user benefit[END_REF] showed that the latter expression can be re-written as:

E(CS n ) = 1/λ n log( i e V n i ) + C (1.15)
where C is a constant, representing the fact that the absolute value of the utility can not be identified. This constant is irrelevant as the policy makers will be interested in evaluating the change in expected consumer surplus. The consumer surplus of logit models is used extensively in Tranus, often called "composite cost", when utilities are negative.

Properties of Logit models

Discrete choice models in general have many properties, we encourage the reader to review the book "Discrete Choice Methods with Simulation" from Kenneth Train [START_REF] Train | Discrete Choice Methods with Simulation[END_REF] to have a complete overview of discrete choice theory in general. We will only list some algebraic properties that are needed to do some of the computations of our optimisation approach for Tranus. Thus, derivatives of logit models are particularly important for us. If the observed utility for an individual of type n choosing j changes with respect to a parameter z n j , we can write this change as:

∂P n j ∂z n j = ∂ ∂z n j e V n j / i e V n i = e V n j i e V n i ∂V n j ∂z n j - e V n j ( i e V n i ) 2 e V n j ∂V n j ∂z n j = ∂V n j ∂z n j (P n j -(P n j ) 2 ) (1.16)
and for z n l , l = j:

∂P n j ∂z n l = ∂ ∂z n l e V n j / i e V n i = - e V n j ( i e V n i ) 2 e V n l ∂V n j ∂z n l = ∂V n j ∂z n l P n j P n l (1.17)
Another interesting property is that adding a constant to all alternatives does not alter the value of the choice probabilities. Suppose we have a set of observed utilities {V n j , j ∈ J} and we add a constant K to all alternatives, redefining V n j = V n j + K, then:

P n j ( V ) = e V n j i e V n i = e V n j +K i e V n i +K = e K e K • e V n j i e V n i = P n j (V )
Hence, utilities are only defined up to an additive constant.

Chapter 2

Description of Tranus

"Tranus simulates the location of activities in space, land use, the real estate market and the transportation system. It may be applied to urban or regional scales. It is specially designed for the simulation of the probable effects of projects and policies of different kinds in cities and regions, and to evaluate the effects from economic, financial and environmental points of view. The most worthy characteristic of the TRANUS system is the way in which all components of the urban or regional system are closely integrated, such as the location of activities, land use and the transport system. These elements are related to each other in an explicit way, according to a theory that was developed for this purpose. In this way the movements of people or freight are explained as the results of the economic and spatial interactions between activities, the transport system and the real estate market. In turn, the accessibility that results from the transport system influences the location and interaction between activities, also affecting land rent. Economic evaluation is also part of the integrated modeling and theoretical formulation, providing the necessary tools for the analysis of policies and projects." -(de la Barra 1999)

In this chapter we will present the Tranus LUTI model. First we present a brief description of the general structure of the model. Secondly, a detailed description of the land use and activity module is provided, we present all the equations that are necessary to construct our calibration methodology. The land use and activity module simulates a spatial economic system by modelling the locations of activities and the interactions between economic sectors for a specific time period.

General structure of the model

The transportation module, on the other hand, dispatches the travel demand induced by the activity model and assigns it to the transport supply.

Both modules are linked together, serving both as input and output for each other. In this way the movements of people or freight are explained as the results of the economic and spatial interaction between activities, the transport system and the real estate market. In turn, the accessibility that results from the transport system influences the location and interaction between activities, also affecting land rent. The two modules use discrete choice logit models [START_REF] Mcfadden | Conditional logit analysis of qualitative choice behaviour[END_REF][START_REF] Mcfadden | Mixed MNL Models for Discrete Response[END_REF], linked together in a consistent way. This includes activity-location, land-choice, and multi-modal path choice and trip assignment.

First, the land use module needs to achieve equilibrium between offer and demand, and equilibrium between the price paid and the cost of producing each economic sector. This is done at current transportation costs and disutilities. Secondly, the transportation module takes as input the transport demand and equilibrates the transportation network to satisfy the given demand.

Both modules are run iteratively until a general equilibrium status is found. This is achieved when neither land use nor transportation, evolve anymore, as illustrated in Figure 2.2.

The land use and activity module

In this thesis we only work with the land use and activity module, (from now on land use module). Our main goal is to improve this module by making the calibration of the parameters involved easier. We consider the input needed (for the calibration of the land use and activity module) from the transport module as data readily available. This technique of "freezing" the transportation system is already used by Tranus modellers for the calibration of floorspace sectors and land. To do so, we have to make the distinction between two types of economic sectors: transportable and non-transportable sectors. The main difference between these, is that transportable sectors can be consumed in a different place from where they were produced. As an example, the demand for coal from a metal industry can be satisfied by a mining industry located in another region. On the other hand, a typical non-transportable sector is floorspace: land is consumed where it is "produced". commerce. The standard approach for the consumption chain is as follows: Industry has a demand for labour (households) and service businesses. Households also consume services, and services also require labour, thus "consume households". Finally, all businesses and households consume land. For instance, households will locate in residential zones, and the feedback of household and business "consumption" will induce home-to-work trips (see [START_REF] Lowry | A Model of Metropolis[END_REF]. This process results in economic exchanges, sometimes inducing flux (transportable sectors) and sometimes in-place consumption (land). The offer and demand is equilibrated and a set of equilibrium prices for each economic sector is attained.

The land use module's objective is to find an equilibrium between the production and demand of all economic sectors and zones of the modelled region. To attain the equilibrium, various parameters and functions are used to represent the behaviour of the different economic agents. Among these parameters are demand elasticities, attractiveness of geographical zones, technical coefficients, etc. In the following, we introduce the parts of the terminology, parameters and equations used in Tranus that are relevant to this paper. See (de la Barra 1999) for a complete description.

• Productions: X n i expresses how many "items" of an economic sector n are present/produced in a zone i .

• Demands: D mn i expresses how many items of a sector n are demanded by the part of sector m located in zone i .

• Prices: p n i defines the price of (one item of) sector n located in zone i .

It is important to realise that "price" in the case of land, is the actual rent, whereas the price of a household is derived from the salary.

Productions, demands and prices form part of a dynamic system of equations. These equations depend on one another, and are linked by a list of equations that need to be computed one after another. This is detailed in (de la Barra 1999). A graphical representation of this feedback is represented in Figure 2.3. For instance, demand induces production and vice-versa. The iteration scheme is as follows: prices of a current iteration translate into intermediate variables (that will not be detailed here) which enables the computation of demand and consumption costs (noted as c in Figure 2.3). This is done based on the current transportation costs and disutilities. Once demand and costs are known, the current production is computed and fed back to compute a new set of prices, for a next iteration.

The process is bottom-up, starting with land use prices and exogenous production and demand up to the production (destined for exportation outside the study area) and prices of transportable sectors. All the above computations are repeated until convergence is attained in productions X and prices p at the same time (convergence in these two sets of variables implies convergence in all others).

In the following, we only show those model equations that are relevant to this work.

Demand is computed for all combinations of zone i , demanding (consuming) sector m and demanded sector n:

D mn i = (X * m i + X m i ) a mn i S mn i (2.1)
D n i = D * n i + m D mn i (2.2)
where X * m i is the given exogenous production (for exports), X m i the induced endogenous production obtained in the previous iteration (or initial values), and D * n i exogenous demand.

D n i in (2.
2) then gives the total demand for sector n in zone i .

The coefficient a mn i is the demand function of sector n by sector m in the zone i , as an example: if m is a household sector and n a housing type, it represents how many square meters of n are needed by m in zone i, see section 2.2.1. The coefficient S mn i is the substitution proportion of sector n when consumed by sector m in zone i (explained in detail in section 2.2.2).

In parallel to demand, one computes the utility of all pairs of production and consumption zones, j and i :

U n ij = λ n (p n j + h n j ) + t n ij . (2.3)
Here, λ n is the marginal utility of income for sector n and t n ij represents transport disutility. Since utilities and disutilities are difficult to model mathematically (they include subjective factors such as the value of time spent in transportation), Tranus incorporates adjustment parameters h n j , so-called shadow prices, amongst the model parameters to be estimated.

From utility, we compute the probability that the production of sector n demanded in zone i , is located in zone j. Every combination of n, i and j is computed:

P r n ij = A n j e -β n U n ij l A n l e -β n U n il . (2.4)
Here, l ranges over all zones, A n j represents attractiveness of zone j for sector n and β n is the dispersion parameter for the multinomial logit model expressed by the above (see 1.3 for the logit model definition). We will consider a standard formulation of the logit model, and not a scaled version, more details about this in 2.3.

From these probabilities, new productions are then computed for every combination of sector n, production zone j and consumption zone i :

X n ij = D n i P r n ij . (2.5)
Total production of sector n in zone j, is then:

X n j = i X n ij (2.6) = i D n i P r n ij . (2.7)
Given the computed demand and production, consumption costs are computed as

cn i = j X n ij p n j + tm n ij D n i (2.8)
where tm n ij is the monetary cost of transporting one item of sector n from a production zone j to a consumption zone i . These finally determine the new prices:

p m i = V A m i + n a mn i S mn i cn i (2.9)
where V A m i is value added by the production of an item of sector m in zone i , to the sum of values of the input items.

The above represent the main equations of the land use module. We will detail each quantity as we need them in the rest of the work.

The demand functions

The demand functions that are present in equations (2.1) and (2.9) are a substantial part of the land use module. Their main role consists in assessing how many units of a certain sector will be consumed at a given price. We will give the formal definition of these functions and then illustrate with an example. The general form of the demand function is:

a mn i (p n i ) = min mn + (max mn -min mn ) exp(-δ mn p n i ) (2.10)
Where min mn and max mn represent the minimum and maximum values of consumption of economic sector n by sector m, and δ mn is the elasticity to price of sector m when consuming sector n. We will often call the difference:

gap mn = max mn -min mn (2.11)
The parameter δ mn acts as a sensitivity to price of sector n. To illustrate the behaviour of such functions we give the following example. Let us consider the demand for apartments by three different socio economic groups {high_income, medium_income,low_income} with corresponding elasticities {0.1, 0.2, 0.3}. We will assume that the values min mn = 30 and max mn = 100 are the same for all household types.

The demand functions are given by equation (2.12).

a m,apartment = 30 + 70e -δ m p , m ∈ {high_income, medium_income,low_income} (2.12)

From figure 2.4 we can observe that at any given price p, each household type (represented by their elasticities) will demand a different size of apartment. These functions are also known as unitary consumption.

The coefficient a mn i exposed in equation (2.10) is one of the fundamental variables of the land use and activity module, it is an important part in the calibration to estimate correctly the demand curves for residential floorspace.

From the demand curves, one can compute the total expenditure by multiplying the demand by the price: The expenditure by socio-economic sector is monotonic with prices (see figure 2.5 for the expenditures in the above example). This quantity is used for computing the substitution sub model. The calibration of the demand functions is generally done externally, using available data and various econometric techniques. For instance, in (Lo Feudo 2014) the mi n and max parameters are estimated using the floorspace consumption and the available surface in each geographical zone, with an optimisation procedure in Excel.

E mn i (p) = p n i • a mn i = p n i • (min mn + (max mn -min mn )e -δ mn p n i ) (2.

Substitution Probabilities

From equation (2.1) we have that the demand for sector n by sector m is computed as the product of the production of sector m by the demand function of sector m consuming sector n for each given zone.This result is then multiplied by the proportion of sector m that actually consumes n. These proportions are the substitution probabilities. The substitution probabilities are computed with equation:

S mn i = W n i exp(-σ m ω mn a mn i (p n i + h n i ) • [p n i + h n i ]) l∈K m W l i exp (-σ m ω ml a ml i (p l i + h l i ) • [p l i + h l i ]) . (2.14)
Here, K m represents the set of substitutes that sector m has access to. Using Tranus terminology, W n i is an "attractor", a parameter that represents attributes of floor space sector n other than cost (utility); it is specified (and potentially calibrated) for each zone in which sector n is present. The demand function (2.10) is evaluated in the adjusted price p + h, so the demand coefficient a mn i is also a function of the prices and shadow prices. It is important to remember that prices are considered an input for the land sectors. ω mn is the penalising factor of sector n, it indicates the preference of consumption (lower values indicate higher preferences) and σ m is the logit dispersion parameter for household type m.

From the expenditure equation (2.13), one can derive the product ω mn • a mn i

• (p n i + h n i ).
This quantity is called penalised expenditure, and it is the utility term that drives the substitution probabilities logit formula.

To fix ideas, let us say m is a low-income socio-economic sector that can consume three types of housing: {small apartments, mobile homes, detached houses}. These housing types are substitutes to one another, and the low-income household will have to choose between them. Let us assume that in zone i there are 100 low income households. We will consider that the shadow prices h are equal to zero and the attractors W are equal to 1. As the product of σ * ω is not easily identifiable, in the rest of this work, we will set σ m = 1 and only consider ω mn as the parameter to calibrate. In section 4.2 we will develop further on the calibration of the substitution sub-model.

Location Probabilities and Logit scaling issues

The location probabilities are the link between the transport module and the activity module.

Through this variables, the transport costs and disutilities impact the location of commerce, households and industry. From equation (2.3), we see that the drivers of the choice probability are corrected prices (via shadow prices) and the transport disutilities t n ij . The choice probability is in the form of a logit discrete choice formulation among all the possible zones (c.f. section 1.3). As transport disutilities are also computed from a composite cost from a logit choice, this is a consistent way of defining the location probabilities.

In Tranus, usually a scaled logit model is used. This means that instead of the one proposed in (2.4), the utility term U n ij is replace by their normalised version:

Ûn ij = U n ij min l U n il (2.15)
Replacing the utilities by their normalised counterpart completely changes the behaviour of the probabilities. This simple example with only two choices illustrates the behaviour of scaled utilities. This modification is there to smooth the model and make it behave closer to an inversely proportional distribution. We understand that the idea behind this choice is to make the model more stable. The problem we see with this is that we loose the simplicity of the logit probabilities, and the underlying behaviour that only sees the differences in utilities. One could argue that in the case of transportation costs or distances the later approach makes more sense, as the user will not be able to differentiate a route of length 1005 over 1010.

Another inconvenient of this approach is that the composite cost, can not be computed with the traditional log-sum formula [START_REF] Mcfadden | Conditional logit analysis of qualitative choice behaviour[END_REF]. Also, it makes the utility functions non differentiable, and for our optimisation approach on calibration, been able to compute partial derivatives of the cost function is essential. In the rest of this thesis we always consider non scaled "standard" logit models (for the substitution and location probabilities).

Anyway, we propose an elegant solution that could attain the same effect in the probabilities would be to consider a multiplicative error term for the utilities (instead of additive).

Multiplicative error discrete choice model have been used before, and a good literature review on non linear error terms in random utility models can be found in [START_REF] Matzkin | Nonparametric identification[END_REF]).

The most recent work in multiplicative error discrete choice models is from [START_REF] Fosgerau | Discrete choice models with multiplicative error terms[END_REF]). The later, gives a detailed mathematical description of the formulation and calibration of these type of models. These models have proven to be effective and as simple to calibrate that the standard multinomial logit. The advantage of these type of models, is that we get the desired behaviour in the distribution, and keep a closed form expression that is differentiable. Another simple solution, is to choose the dispersion parameters accordingly, to reduce the relative difference between the utilities. The latter, is what we did in this work.

Chapter 3

Calibration of the Tranus land use module: shadow price estimation

In this chapter we will present the estimation of the endogenous variables called the shadow prices. These quantities are very important to a Tranus model, and act as correcting terms of the utility functions, helping the model to reproduce the observed data.

First, we will present the different parameters involved in the calibration and how the calibration is done in Tranus. Secondly, we will reformulate the calibration as an optimisation problem, proposing different techniques for non-transportable and transportable economical sectors. After explaining the separation of the problem, a simple but detailed example is presented.The latter will help the reader to grasp the functioning and the order in which equations are computed. Then, we present a methodology to create synthetic scenarios based on real ones. This methodology permits us to create a model that has a known equilibrium point with known optimal shadow prices and prices. This is important if one wants to assess the performance of our algorithms, and identify how the convergence is performing. We also give a brief example of why this problem is relevant, exposing that even for small problems, the solution is not obvious. A discussion of some numerical issues encountered during the optimisation is presented. Finally we present a model selection methodology developed for the shadow prices variables. We compare a model with reduced number of shadow prices against the standard model with the whole set of shadow prices, this is done with respect to model fit.

Calibration as currently done in Tranus

In the domain of LUTI models, usually calibration comprises the whole construction of the model, i.e. defining the economic sectors, gathering the data, defining the zoning of the study area, etc. In this work, calibration is the process of estimating the model parameters only, once the model definition has been made by the modeller.

The calibration process consists in adjusting the model parameters so as to be able to reproduce a base year's data in the study area. Obtaining a good calibration is a long process, that is usually performed by experts and can take months. A mix of tools are used to estimate the various parameters of the model. Econometrical, ad-hoc procedures and interactive trialand-error can be counted among the tools used by experts to obtain a good fitting model.

For the calibration phase, parameters are separated in three sets:

i. Parameters that are computed externally using the appropriate data and econometrical techniques.

ii. The adjustment parameters h n j of the utilities (2.3), known as shadow prices.

iii. The remaining parameters (for example the penalisation factors and logit dispersion parameters).

After computing the external parameters (set i), and giving initial values to set iii, the model iterates until convergence. The iteration process is constructed in such a way, that the shadow prices will be adjusted to force the productions to reproduce the observed productions X 0 in the study area. These variables will "try" to compensate for the other parameters to have a perfect fit; they act as correction terms to compensate for parts of the utility that are not represented by the model. One wants to make the values of the shadow prices as small as possible. This process of parameter calibration is done repeatedly until the expert modeller is satisfied with the parameters and the values of the shadow prices.

The computation of the shadow prices is automatically done as follows at the end of each iteration (cf. figure 2.3 and the equations exposed in section 2.2):

h n,t+1 i = (h n,t i + p n,t i ) X n,t i X n 0,i -p n,t+1 i . (3.1)
The shadow prices for the next iteration t + 1 increase proportionally to the excess of computed, as compared to observed, productions. The actual computation is a little more complicated than this, it relies on a smoothing factor to reduce the rate of change of the 3.2. Reformulating calibration as an optimisation problem prices in each iteration, averaging the last two iterations. The details about this technique can be found in the Appendix A Our main motivations are to replace the sequential calibration process outlined above by a process that rigorously estimates as many parameters as possible, taking into account all available constraints and assumptions in a systematic manner, to automatise as much as possible the calibration process, and to make it more reproducible. We believe that a natural way of achieving these goals is to explicitly formulate the calibration process in terms of a cost function (or possibly, as a multi-criteria decision problem) that is to be minimised or maximised, with respect to a set of constraints, when given. This is the case in the existing approach, where the estimation of shadow prices and other parameters is done without a definition of a cost function. Formulating calibration via explicit cost functions enables to use the rich variety of optimisation algorithms existing in the literature and in numerical libraries.

A first step in this direction concerns the estimation of shadow prices, a second step deals with the automatic estimation of both shadow prices and other parameters; these two steps are described in the following.

Reformulating calibration as an optimisation problem

It is important to notice that a calibration of the land use module involves the estimation of all the parameters of the model to make productions as close as possible to the base year data.

To reformulate the calibration as an optimisation problem, we must compute shadow prices that make the productions as similar as possible to the observed productions. This can be written as an optimisation problem:

min h X(h) -X 0 2 . (3.2)
Here, h is a vector containing all shadow prices, X 0 the vector of observed productions, and X(h) the vector of productions computed by the model, after convergence of the iterative process shown in figure 2.3. The dependency of X(h) on the shadow prices is visible from equations (2.3) to (2.7). Each evaluation of the productions X(h) involves the convergence of the dynamic system exposed in Figure 2.3. Each evaluation of the cost function involves the convergence of the dynamic system in productions as well as prices.

This double convergence problem can be avoided by including the prices amongst the variables to be optimised, instead of leaving them as endogenous variables. Moreover, one can compute directly productions that are in equilibrium for a given set of shadow prices and prices. To do so, we observe that the computation of demand and production involves a set of linear equations (2.1), (2.2), (2.5), and (2.7). If we re-organise these equations, knowing that only productions are needed in our cost function, we may only need to compute these.

To do so, we substitute D n i in equation (2.5) using equations (2.1) and (2.2), giving:

X n ij = D * n i + m (X * m i + X m i ) a mn i S mn i P r n ij . (3.3)
Upon substituting this into (2.7), we obtain the following linear system in X n j :

∀j, n X n j = i D * n i + m X * m i a mn i S mn i P r n ij + i m a mn i S mn i P r n ij X m i . (3.4)
If presented in matrix form, this correspond in genral to a matrix of size M • N where M is the number of zones, and N is the number of sectors. By construction, the solution of this linear system represents an equilibrium of production and demand: solving the system of equations (3.4) for all productions (all sectors n and all zones j) and then computing demands using equation (2.2), gives a set of productions and demands that are consistent with one another.

The most usual optimisation methods require the computation of partial derivatives of the cost function [START_REF] Nocedal | Numerical Optimization. 2nd Edition[END_REF]. This is still difficult for the cost function (3.2).

Each evaluation of the productions involves solving a linear system of the type (3.4). An analytical solution seems out of reach even for models with few sectors and zones. Estimating the gradient numerically via finite differences, is possible but rather costly. It would require at least one evaluation of X(h) per shadow price to estimate, each evaluation requiring the solution of the linear system (3.4). Moreover, even if productions computed this way are in equilibrium, the prices p still need to iterate until convergence is obtained. Indeed, convergence in prices is only obtained when consumption costs equal corresponding prices (cf. equations (2.8) and (2.9) as well as figure 2.3).

These remaining difficulties can be solved as follows. First, for a successful calibration, we want to have the computed productions equal to the observed base year productions X 0 .

This correspond to the usual rationale for Tranus models 1 . Hence, we can simply impose this condition by replacing productions in the right hand side of (3.4), with the observed base 3.3. Land use sectors (non transportable sectors) year productions. This approach enables us to compute the productions directly, without the need to solve a linear system. Similarly, this simplifies the analytical computation of the cost function's derivatives.

To address the second problem (equilibrium of prices), we add the prices explicitly to the set of parameters to be optimised. We use the current values of prices, and compare them against the prices computed by the model in the next iteration, cf. (2.9). The difference between the current prices and the ones computed by the model trough equations (2.3) to

(2.8), is added to (3.2), in order to form a new cost function:

min h,p X(h, p, X 0 ) -X 0 2 + p(h, p, X 0 ) -p 2 . (3.5)
Here, p is the vector of prices computed by the model using (2.9) and the notation X(h, p, X 0 )

shows that modelled productions are computed as explained above by substituting observed productions X 0 into the right-hand side of (3.4).

The above cost function has a closed-form that permits us to compute the derivatives directly. No more iterations or waiting for convergence is required in this approach. The cost function (3.5) is of (non-linear) least squares type, meaning that any least squares optimisation approach can be used; in our work we apply the Levenberg-Marquardt method [START_REF] Levenberg | A Method for the Solution of Certain Non-Linear Problems in Least Squares[END_REF]).

Let us also note that other choices than the L 2 norm would of course be possible to define the cost function of (3.5). We may also weight the two terms differently, in order to favour equilibrium in production over that in prices or vice-versa in cases where a global equilibrium cannot be reached.

So far, we have not used any specificities of activity sectors in the outlined approach. This is done in the following two sections, first for non-transportable and then for transportable sectors.

Land use sectors (non transportable sectors)

Land is a very peculiar economical sector, it must be consumed where it is produced. By "land", we understand here floorspace sectors or housing sectors. Moreover, land does not consume other economical sectors and the amount of available land is fixed. For the calibration purpose, the prices for land sectors are known, this means the p n i variables for the calibration year are considered as input and do not need to be computed. This translates into a simplified set of equations for the computation of production of land. We have to detail two extra equations to understand how this enters our optimisation scheme. First, as land is non-transportable, the location probability (2.4) vanishes, so equation (2.7) can be re-written as:

X n i = D * n i + m (X * m 0i + X m 0i ) a mn i S mn i . (3.6)
If we consider that the demand functions parameters (min mn , max mn and δ mn ) are calibrated externally, and since the price of land is known, a mn i is only a function of the shadow price h n i . Also, the substitution probability S mn i is only a function of shadow prices associated to the same zone i . As land prices are known, we can clearly see that the production of land

X n
i is only a function of the shadow prices of the land sectors of the same geographical zone i (there is no dependency on other zones). Of course we have interactions with other economical sectors in the same zone, but in practice, the number of economical sectors is much smaller than the number of zones, which leads to optimisation problems (one per zone) that are very small, with the number of variables equal to that of land sectors. We can re-write the optimisation problem (3.5) as one optimisation problem for each geographical zone i :

∀i min h i X i (h i , X 0 ) -X i0 2 (3.7)
Just as an example, for the North-Carolina-1 model (see later), there are only 3 land sectors: apartments, mobile-homes, detached houses.

Once the optimisation is done for each geographical zone and the shadow prices for land are computed, we can proceed to computing the optimal shadow prices of the transportable sectors (see next section). We will further exploit this feature of the model to obtain an automated calibration of the substitution parameters.

From equation (3.7), we need to compute partial derivatives for the production with respect to shadow prices, thus enabling the solution via gradient based methods (see section 1.2). We will compute the partial derivatives for the productions X n i .

Derivative estimation:

In the following, we will note the penalised expenditure as U mn i = -ω mn a mn i (p n i + h n i ). Let us consider m and m as consuming sectors, n as land use sector and q ∈ K m (the set of possible substitutes for sector m, see 2.10). The partial derivatives that we need to compute, 3.4. Transportable sectors are given in the following. First:

∂a mn i ∂h q i =    -δ mn gap mn e -δ mn (p n i +h n i ) q = n 0 q = n (3.8)
The well known logit derivatives for S mn i are:

∂S mn i ∂h q i =              ∂U mn i ∂h n i S mn i -S mn i 2 q = n - ∂U mq i ∂h q i S mn i S mq i q = n (3.9)
if q = n, ∂U mn i ∂h q i = 0, then for q = n:

∂U mn i ∂h n i = -ω mn [ ∂a mn i ∂h n i (p n i + h n i ) + a mn i ] = -ω mn [a mn i -δ mn gap mn (p n i + h n i )e -δ mn (p n i +h n i ) ] (3.10)
With these results, we can compute the partial derivatives of the production function exposed in (3.6):

∂X n i ∂h q i = ∂ m (X m 0i + X * m 0i ) a mn i S mn i ∂h q i = m (X m 0i + X * m 0i ) ∂ ∂h q i [a mn i S mn i ] = m (X m 0i + X * m 0i ) [ ∂a mn i ∂h q i S mn i + a mn i ∂S mn i ∂h q i ] (3.11)
Based on this analytical computation of partial derivatives and eventually, of the gradient of our cost function (3.7), we can optimise the latter using gradient-based optimisers (or others), such as gradient descent, LM, etc. (cf. section 1.2).

Transportable sectors

Transportable sectors, are economical sectors that consume (and can be consumed) in a different location from where they are produced. Housing and commerce are examples of such sectors. For this type of sector, it is common practice in Tranus to consider the demand functions a mn i as constant and to use substitution probabilities S mn i = 1, i.e., there is no substitution considered between transportable economic sectors. Let us look at the total demand for a transportable sector n in zone i , under these assumptions:

D n i = D * n i + m a mn i S mn i 1 (X m i + X * m i )
This implies that the total demand D n i is not a function of the prices or the shadow prices, and enables the computation of the total demand D n i (2.2) for each transportable sector n and geographical zone i . As we want to compute the demand for the base year production, we replace the X m i from above by the base year production X m 0i , thus transforming the demand into a constant that does not change between iterations (it depends neither on prices nor on shadow prices):

D n i = D * n i + m a mn i (X m 0i + X * m i )
If we come back to the initial computations of induced production X from equation (2.7),

X n j = i D n i P r n ij ,
we only need to determine the values of the location probabilities P r n ij . If we go back to the definition of the location probabilities (2.4) and the underlying utilities (2.3), we realise that the utility makes no distinction between the price and shadow price part, so if we set a new variable:

φ n j = p n j + h n j (3.12)
the location probability can be computed as a function of φ. Instead of posing the induced production as a function of (h, p, X 0 ), we can look at the induced production X(φ, X 0 ) as a function of φ. Obtaining the optimal values of φ that minimise the difference between computed and observed productions, is the solution to the following problem:

min φ X(φ) -X 0 2 . (3.13)
Since the location probability P r n ij is a function only of the φ n variables for the same sector n, we get one optimisation problem for each economical sector n. Each of these optimisation problems is relatively simple and small to moderate in size, there are as many variables as geographical zones. The gradient of the cost function can be computed analytically using the well known derivatives of the logit probability P r :

∂P r n ij ∂φ n k =    -λ n β n P r n ij -P r n ij 2 k = j λ n β n P r n ij P r n ik k = j (3.14)
We use the Levenberg-Marquardt method to solve each optimisation problem (3.13) for each sector n. Once all the optimal values of φ have been computed, we can compute the prices by solving the linear system (2.9) for prices. Doing so allows us to finally recover the shadow prices from φ, subtracting the prices from the respective optimal φ values.

One consideration that one has to deal with, is that the location probabilities P r follow a logit formulation, so the utilities can only be identified up to a constant per economical sector (as shown in section 1.3.2). This is a known property of logit models. As the prices are obtained from equation (2.9), this approach is considerably simpler and more stable than solving the double-objective optimisation approach proposed in (3.5), moreover, it exploits every little detail of the formulation of each function of the model. It also permits to calibrate incrementally, starting by the land use sectors and then obtaining the calibration of the transportable sectors. From the mathematical point of view it is also simpler, because the large optimisation problem in (3.5) is now decoupled into smaller ones, with fewer variables, allowing the modeller to finish the calibration of one set of variables before moving to the next stage.

Summary of proposed approach and a numerical example

To summarise, in the case of land use (non-transportable) sectors, there is one small optimisation problem to be solved for each geographical zone, whereas for the transportable sectors, we have one optimisation problem per economic sector.

We encountered some numerical issues relied to the fact the the location probability would vanish for large values of the utility function; this behaviour is explained in 3.5.2.

So far, we have presented in sections 3.3 and 3.4 an optimisation methodology to compute the shadow prices for non transportable and transportable sectors. As the shadow prices are the endogenous adjustment factors computed internally in the calibration phase to obtain a good model fit, it may seem complex for someone who is not familiarised with Tranus to understand how these quantities influence the results. To build an automatic calibration of

Tranus, one has to address this issue first to be able to obtain convergence in the observed quantities, namely induced productions and prices. Once this is achieved, and a robust way to compute the endogenous shadow prices is found, one can expand the methodology to include other parameters in the optimisation scheme (see next section). We now give a concrete numerical example for a simple model, to illustrate setp-by-step, the workings of the proposed calibration approach.

Example of shadow price estimation with the optimisation approach (Example C)

In this section we will present a step by step computation of Tranus equations and our optimisation approach applied to a simple scenario. This is the basic scenario for testing

Tranus functionality and it is readily available from Tranus website.

The scenario "Example C" has 5 economical sectors, and 3 geographical zones. A brief description of the respective economical sectors is presented in table 3.1. There is only one land use sector, this model doesn't have substitution. Traditionally, the substitution in Tranus is only used between land use sectors. In table 3.2 we present a summary of the demand functions. Only elastic demand functions are considered for land consumption. This is also a standard practice in Tranus modelling. Table 3.2 presents the parameters of the demand functions (2.10).

Land (sector 5) shadow prices calibration

As there is only one floorspace sector, we can easily write explicitly the equations corresponding to the production of this sector. For this model exogenous demand is zero for all sectors, so from equation (3.6) we have: here Xk i = X k i + X * k i (induced and exogenous production). The following table presents the base year's data, reproducing this data is the goal of our calibration. For the land sector (sector 5) we have to fit the productions to 66, 110 and 128 respectively. As one can see from Table 3.3, a sector has Exogenous or Induced Production, but not both. So from now on, we will drop the hat from X and just write X. So the calibration of sector 5 can be written as 3 optimisation problems, one per geographical zone:

X 5 i = 4 k=1 a k5 i S k5 i Xk i , i = 1, 2, 3,
min h 5 i X 5 i (h 5 i ) -X 5 0i 2 , i = 1, 2, 3 (3.15)
with only one variable per problem, the corresponding shadow price. In fact, the quantity X 5 i can be simplified, as there is only one land sector, there is no substitution, hence S mn i = 1:

X 5 i = 4 k=1 a k5 i X k i , i = 1, 2, 3,
So, for instance the explicit equation for the production of land in zone 1 comes from:

Transportable sectors (2,3,4) shadow prices calibration

In this section we will present the methodology exposed in 3.4 to compute the prices and shadow prices of transportable sectors. As one can see from table 3.3, we don't know the prices of sectors 1,2,3 and 4. Also, from table 3.2 we can notice that for transportable sectors the demand functions a mn i are constant (or set to their min value). If we go back to the equations of how total demand is computed (2.2):

D n i = D * n i + m a mn i (X m 0i + X * m i )
Table 3.8 presents the final shadow price values, these values are centered (we subtracted the median per economical sector). Subtracting a constant from all terms entering a logit does not change its value (in our case, the location probabilities) logit does not change the probabilities.

Numerical aspects

Local optimisation may converge to local minima. We observed this in practice, depending on the setting of the parameter β and on the starting point for the φ. An observation that seemed strange at first sight, was as follows. When estimating the φ for one sector, after convergence, the residuals of the cost function were all non-zero (besides for two zones for which observed production was zero). Further, all these residuals, besides for one zone, were exactly equal to one another and the residuals summed up to approximately zero. This seemingly strange behaviour has an explanation, as follows.

First, it must be noted that the sum of computed productions, does not depend on the values of the φ:

j X n j = j i D n i P r n ij = i D n i j P r n ij 1 = i D n i
If the data is consistent, then the sum of computed productions must equal that of observed ones:

j X n j = j X n 0j
Hence, the sum of residuals must be equal to zero, as was observed in practice.

The other issue concerned the fact that all non-zero residuals but one, were exactly equal to one another. This can be explained as follows. For one zone j, the value of φ j was sufficiently large at some stage of the estimation, so that the computed probabilities P r ij effectively became equal to zero, for all i : the absolute value of the argument of the exponential exp(-β(λφ j +t ij )) became so large that the exponential effectively got evaluated to zero. This in turn means that the computed production for that zone, also was computed as zero since

j X n j = j i D n i P r n ij 0 = 0
Hence, the residual for zone j is non-zero, and actually equal the (oposite of the) observed production X 0j . Since the sum of residuals over all zones must equal zero, as shown above, we must have:

k =j (X k -X 0k ) = -X 0j
Remember that the cost function to be minimised is the sum of squared residuals; as for the zones other than j, this means:

min k =j (X k -X 0k ) 2
It can be shown that given the constraint that the sum of residuals must equal a known value, the cost function is a minimum if that known value is equally apportioned to the residuals, i.e. if all the residuals are equal to that value, divided by the number of residuals:

∀k = j : X k -X 0k = - X 0j k =j 1
This explains the observation made in practice, described above. This problem can be avoided by choosing a different starting point for the optimisation algorithm.

In the previous sections, we proposed and optimisation reformulation of the land use module for Tranus. We still need to validate this methodology against the current Tranus implementation in practice. In the next section we propose to address this by comparing both methodologies.

Testing the proposed calibration methodology against the one implemented in Tranus

One of the main challenges with Tranus calibration is to make the model converge. Even for experienced modellers, obtaining a set of parameters that produces a successful output for the land use and activity module is very time consuming. The iterative approach utilised by Tranus to estimate the endogenous parameters (namely shadow prices) is somewhat deficient, and fails to converge most of the time if the other parameters are not near the "good" values. Even for reasonably good parameters, where production and demand are in equilibrium, sometimes the iterative approach oscillates and does not converge. Without convergence, the output produced by Tranus has no sense and it is often very difficult to identify why the convergence was not attained. So far, we have proposed an optimisation methodology that replaces the traditional iterative approach (see section 3.2) and always produces an output. The idea of this section is to compare how our approach fares against the standard Tranus program.

Generation of synthetic scenarios for performance assessment

The evaluation of a LUTI calibration is a difficult process, mainly due to the noise in the data and the fact that obtaining ground truth information is almost impossible. Our optimisation scheme needs as input the base years' productions and parameters (X 0 , par ameter s). Then, the calibration is done against this information. We could think of a model that does not need the shadow prices to attain a perfect fit, hence, create a synthetic scenario where a "perfect" fit is achieved with shadow prices set to zero. To generate this "perfect fit" scenario, we have to solve a subproblem of the original calibration optimisation problem exposed in (3.5), where we do not consider the observed productions. We only need to obtain equilibrium in prices, and compute the values of the induced productions afterwards. To do so, we replace the consumption cost equation (2.8) in the prices (2.9), and by identifying the location probability as P r n ij = X n ij /D n i , we obtain the following system:

p m i = V A m i + n a mn i S mn i j P r n ij (h, p) • p n j + tm n ij p(h,p) . (3.18)
The dependence of the location-choice probability makes this system hard to solve even for small models. Our approach to solve this fixed-point equation is to solve the following optimisation problem (see section 3.6.3):

min p p(h, p) -p 2 . (3.19)
We have to make sure that the solution of (3.19) is a set of prices that are in equilibrium, that is for which p = p. After obtaining convergence in the prices, we compute the productions and then use them as observed base year productions in our synthetic scenario. This methodology produces a scenario where the optimal value of the shadow prices is zero (by construction)

and that reproduces the base years' productions perfectly. We could also set the shadow prices value to any other value than zero here. The generation of such a synthetic scenario enables us to test our calibration methodology and optimisation algorithms against a known optimal value (shadow prices equal to zero).

For testing purpose, we propose a way of generating data sets that correspond to productions X equal to the input productions X 0 for a given set of shadow prices h 0 .

If we consider a fixed set of shadow prices h 0 , we need to obtain prices that are in equilibrium. To do so, we solve (2.9) until convergence is reached for the given set of shadow prices h 0 . We iterate until the value of X and p from one iteration to the next one remains constant, obtaining a pair ( X, p) that has attained convergence. It is important to notice that given h 0 fixed, when p attains an equilibrium status, the production X is in equilibrium too. Doing so, is equivalent to solving the optimisation problem exposed in (3.19) Our first approach was for a given scenario,replace the value of X 0 with the output X attained on the convergence of the system. Doing so does not work, as the output value of X is calculated using the input value X 0 . More specifically, the attractor A n i of the logit formulation for the location probability (2.4) depends explicitly X 0 , as shown by the following equation:

A n i = W n i k b n k X k 0i α n (3.20)
where W n i is the input attractor variable for zone i and sector n, b n k and α n are technical coefficients. As the computed production depends on the location probabilities, the computed X is a function of X 0 , the base year production. We need to modify the values of W n i as follows:

Ŵ n i = A n i k b n k Xk i α n
So, to set X as the equilibrium production, we rewrite both X 0 and W :

X 0 ← X W ← Ŵ
Doing this, we can have a scenario that perfectly reproduces the base year's production for a given set of shadow prices h 0 . In practice we use h 0 = 0.

Examples of synthetic scenario generation

We present two synthetic scenarios created with the methodology exposed above. The first model is the classic Tranus example C (see 3.5.1) and the second model is based on a real scenario for the Mississippi region.

Example C:

We consider the same example exposed previously in section 3.5.1, a simple model that allows to illustrate the methodology for generating synthetic scenarios with perfect fit (with "ground truth" shadow prices equal to zero). We applied our approach to the Example_C model from Tranus website2 , a small model with 3 zones and 5 sectors. First, we generated synthetic data from that model as described just above, with shadow prices h n i = 0 for each sector n and zone i . As expected, the cost function is zero at h = 0, and increases its value when we get away from the optimum. The cost function appears to be locally convex near the optimal value, cf. figure 3.3. If we consider for example sector 1 and zone 1, we can plot a "slice" of the cost function (3.5) near the optimal value h 1 1 = 0, p 1 1 = 2.676 as shown in figure 3.2 and figure 3.3. Here we can observe that as the shadow price gets larger the cost increases up to a plateau state (X 1 1 (h) → 0). In the case of the price p, if we move away from the optimal value p = 2.676, the cost increases quadratically.

We tested the robustness of the optimisation scheme with 1,000 random initial sets of shadow price values; the optimisation procedure outlined in section 3.2 always converged to the ground truth solution. The initial values of shadow prices in these random trials were generated from a uniform distribution in [-10, 10], which is a stringent test (prices are in the interval [0, 4] and nearly all shadow prices of a model are in practice smaller than the corresponding prices). ) near the optimal value (0, 2.676).

Real life model: Mississippi model

We also applied the same procedure to a Tranus model for Mississippi (consisting of 102 zones and 12 economical sectors) modified by our synthetic data methodology. After setting the desired value for the shadow prices to h = 0, we tried 10,000 random initial sets of shadow prices values; and the algorithm proved to converge to the correct shadow prices for every single starting point. As for the calibration procedure implemented in Tranus' release, it failed to converge when starting values were too far away from the zero vector. We considered initial shadow prices uniformly distributed in the interval

[ • -p max , • p max ],
where is a parameter in [0, 1] and p max is the maximum observed price. As increases, the initial shadow prices can take values further away from the optimal solution h * = 0. These initial values are representative of the expected values of shadow prices as one would like that shadow prices do not exceed prices. Table 3.9 presents the convergence status for each value of (1,000 random values where taken for each ). We observe that the iterative approach of Tranus fails to converge as the initial values get further away from the true solution.

Reducing the number of shadow prices, early results

Figure 3.5: The graphs show ratios of calibrated shadow prices over prices. Left: when all shadow prices are estimated (in this case, the fit of computed to observed productions is perfect). Right: here only one third of shadow prices are estimated, the others are set to zero. The fit is not perfect but good (3%, see text). Note that the scales of the two graphs are different. One can observe that on the right-hand side, shadow price to price ratios are not much larger than on the left-hand side, another indicator that it is plausible to exclude many shadow prices from the calibration.

We propose a model selection scheme to reduce the number of shadow prices needed to have a model reproduce the observed data. The classical approach is to iteratively modify the parameters until a perfect fit is achieved (with near zero cost function), when this is achieved the modeller will look at the values of the shadow prices as a quality test for the calibration. If the shadow prices are large, it means that the model has to compensate the various effects of the other economical parameters to attain a perfect fit. Thus, the modeller will tweak economical parameters (such as dispersion parameters) to maintain the perfect fit but with smaller shadow prices. A calibration will be completed when the model reproduces the observed data perfectly and the values of the shadow prices are small (for some economical sectors we will ask their variance to be small instead). As there are as many shadow prices as observations we are trying to fit (there is one shadow price per observed production) the risk of overfitting is possible, which will in turn undermine the predictive capabilities of the model. What we propose, is sacrificing the perfect fit of the cost function, in order to lower the number of parameters to calibrate, particularly shadow prices. To find the optimal trade-off between how many shadow prices we keep in the model and the desired value of the cost function is something that will have to be discussed with the community of modellers. What we propose here, is a simple model selection scheme that instead of having one shadow price per economical sector and zone, we keep only one shadow price per geographical zone. Doing so enables us to exploit the fact that we have independent optimisation problems for each geographical zone for land (non transportable sectors).

For the Mississippi model (see section 5.2 for more details), as there are only 3 economical sectors for land, we reduce by two thirds the complexity of the model. The calibration of the remaining third of shadow prices gave rise to a residual fit of the cost function of only 3%

(ratio of residuals over observed productions). Selecting the shadow prices to be kept in the model is easily done for the land sectors (non-transportable), because the prices (rents) are known. We achieved this, by computing the optimal shadow prices from (3.2) and setting to zero the "small" shadow prices. Followed by a re-calibration of the remaining shadow prices.

One can adjust the threshold used to declare shadow prices as being small to a desired level in the cost function, thus keeping more or less shadow prices. We also see in Figure 3.5 that the values of the shadow prices relatively to the prices have not seen a large increase.

Chapter 4

Optimisation of other parameters than shadow prices

In the previous chapter we reformulated the calibration of Tranus as an optimisation problem.

This was mainly done to solve the issues encountered with the estimation of the endogenous parameters called shadow prices. We replaced the original iterative scheme, by a proper optimisation problem with a cost function. In this chapter we will explore how to add other parameters (beside the shadow prices) to the optimisation formulation. We will be particularly interested in the parameters involved in the calibration of the non-transportable sectors (namely land and floorspace) as they seem to be the hardest to calibrate in practice.

First we propose a simultaneous estimation of the substitution probabilities and the shadow prices. These parameters are the drivers of the land use module, as they shape the way in which the households consume housing in the study area. We present this methodology that was originally developed with Brian J. Morton for the North Carolina Model and present a two phase technique to estimate the penalising factors of the substitution model.

Then, we extend this technique to include observed consumption constraints to reproduce more accurately the choices of housing observed in the population.

Finally, building on the idea of simultaneous estimation of parameters, we present a sensitivity analysis to other relevant parameters this time for the transportable sectors of the model (dispersion parameters and attractor weights). We construct a stochastic optimisation technique to improve the values of the parameters identified as the most sensitive ones.

Finally, we present an analytical estimation of the marginal utilities of income as a function of the parameters estimated by this stochastic optimisation.

Parameters to Calibrate

In the previous chapter, we presented a classification of the different types of parameters that are involved in the calibration of Tranus. We also presented a reformulation of the calibration as an optimisation problem, mainly to solve the issue of the calibration of the endogenous parameters called shadow prices. Now that the calibration of the shadow prices is done explicitly with an optimisation methodology, we can propose to include other parameters in the optimisation to be estimated at the same time. Table 4.1 presents all the parameters that need to be calibrated in the land use module, showing where in the computations they are involved, a brief description of their meaning and the type of calibration commonly used to obtain their value (these are the same 3 sets as exposed in section 3.1) . Even if many parameters are estimated externally with available data, very frequently some adjusting is done in Tranus afterwards. Note how the shadow prices are present in three of the intermediate variables.

Here m and n are both sectors (if both occur then m refers to a sector consuming n) whereas i and j refer to zones. Penalising factor (iii)

h n i Shadow price (ii) A n i b n k
Attractor weight for sector k by sector n (iii)

After concertation with modellers, mainly with Brian J. Morton (a senior modeller in Tranus) and Tomás de la Barra, we decided that the first parameter set that needed some type of automatic calibration concerned the substitution probabilities (S mn i ). These probabilities are the main drivers of the land use and activity model, and are generally very hard to estimate.

In the following, we will present a methodology to estimate these quantities.

Simultaneous estimation of shadow prices and land use substitution parameters

As stated in the introduction, one would like to have a simultaneous estimation of the whole set of parameters. In this section we present one step in this direction: we have constructed a two-phase algorithm that permits the estimation of the shadow prices and the substitution parameters within the same problem formulation. We have chosen the penalising factors in the substitution sub model because these are very hard to calibrate parameters, as relevant data are not readily available.

The functionality of substitution models is rather broad, encompassing goods and services other than floor space and agents other than households. In practice, substitution models typically apply to households' consumption of land for residential purposes, businesses' consumption of floor space for offices and factories, and construction companies' consumption of land for building sites. For instance, rich people prefer detached housing, but could also live in apartments if they are well located.

The scheme proposed exploits the fact that the substitution sub model is used for land sectors, where we have already a simplified computation of the productions, as explained in section 3.3. Tranus models include a discrete choice sub-model that represents the households' ability to choose among different types of residential buildings (i.e., floor space). The model is driven by the substitution probabilities (cf. equation (2.14):

S mn i = W n i exp(-ω mn a mn i • (p n i + h n i )) l∈K m W l i exp (-ω ml a ml i • (p l i + h l i ))
.

Here, K m represents the set of substitutes that sector m has access to, for example, for "rich" households m, this could be K m = {condos, detached houses}. Using Tranus terminology, W n i is an "attractor", a parameter that represents attributes of floor space sector n other than cost (utility); it is specified (and potentially calibrated) for each zone in which sector n is present. From equation (2.10) we can see that the demand coefficient a mn i is also a function of the prices and shadow prices. It is important to remember that prices are known for land sectors. This sub model, has two parts to be estimated, first the demand functions a mn i and the substitution probabilities S mn i . The first, is generally estimated externally, using data from land use consumption per socio-economic category delivering good results in general.

The latter, is much more complicated to estimate, because the substitution preferences are aspects of the model which can not be directly associated with observed data.

We propose a hybrid and multiphase process for calibrating substitution models. In the first phase, certain parameters' initial values are estimated with a multinomial logistic regression [START_REF] Train | Discrete Choice Methods with Simulation[END_REF]). In the subsequent phases, mathematical optimisation is used to fine-tune the estimated parameters and to calibrate the other substitution model parameters. With our proposed approach, the process of determining parameter values is fast, replicable, and entirely transparent. Another important benefit is that substitution models are less likely to be overfitted, which is a hazard with the current and universally used calibration practice that sets floor space and land "attractors" to the value of base production (see below).

1. Phase 1: estimating parameters' initial values with multinomial logistic regression. The substitution model's parameters are estimated with multinomial logistic regression [START_REF] Mcfadden | Conditional logit analysis of qualitative choice behaviour[END_REF]. The data that are essential for estimation are household level observations on floor space consumption, housing expenditure, and the Tranus sector to which the household belongs. The dependent variable in the regression is the choice of floor space sector, and the independent variable is the housing expenditure.

The regressions are conducted separately for each household sector, and they yield estimates of -ω mn (the negative of the penalising factor) for each combination of floor space sector n and household sector m.

A constant is not included in the regressions 1 . Assuming that the coefficients on expenditure have the expected negative sign, the absolute values of the coefficients are the penalising factors' initial values.

Phase 2: fine tuning the penalising factors. The penalising factors estimated in

Phase 1 probably still need to be fine tuned to reduce the differences between the predicted production of floor space and the observed production of floor space. Fine tuning probably would also be necessary to achieve reasonable values of the floor space sectors' shadow prices.

If we consider all of Tranus' parameters fixed except the penalising factors ω, and include these parameters in the optimisation problem presented in (3.5), we obtain the following cost function:

f (h, ω) = X(X 0 , h, ω) -X 0 2 . (4.1)
We would like to find values of ω that reduce the corresponding shadow prices as much as possible (refer to section 3.1 for the rationale of doing so). We propose to solve the following equation:

min ω∈Ω f (h = 0, ω) (4.2)
where Ω is a set of bounds on the penalising factors ω. We use a gradient based algorithm to solve this problem (see section 1.2), and the starting point for the optimisation is the set of values obtained from the Multinomial Logistic regression of Phase 1. If we call ω * the solution of (4.2), then the final values for the shadow prices for the land use sectors are:

h * = arg min h f (h, ω * ) .

Derivative estimation:

For an efficient optimisation, we provide analytical estimates of the partial derivatives of the cost function to the optimiser. Following the derivative estimation exposed for the non-transportable sectors in section 3.3, we can compute the necessary derivatives for the optimisation problem proposed above. Let us consider m and m as consuming sectors, n as land use sector and q ∈ K m . From equation (3.6), one can compute the derivatives with respect to the penalising factor ω as follows:

∂X n i ∂ω m q = m∈K n (X m 0i + X * m 0i )a mn i ∂S mn i ∂ω m q = (X m 0i + X * m 0i )a m n i ∂S m n i ∂ω m q (4.3)
where:

∂S mn i ∂ω mq =    -a mn i (p n i + h n i ) S mn i -(S mn i ) 2 q = n a mq i (p q i + h q i )S mn i S mq i q = n (4.4)
replacing (4.4) in Equation ( 4.3) we finally obtain the derivatives necessary for the Phase 2 optimisation algorithm:

∂X n i ∂ω mq =    -(X m 0i + X * m 0i )(a mn i ) 2 (p n i + h n i ) S mn i -(S mn i ) 2 q = n (X m 0i + X * m 0i )a mn i a mq i (p q i + h q i )S mn i S mq i q = n
Results for this methodology are presented for two real case scenarios in the next section.

Observed consumption preferences

Implementing a 2 phase optimisation process as exposed in 4.2 is sometimes not possible, as observations of choices by individuals or individuals households are not always available.

This type of data cross referencing between socio-economic categories and housing choice (as needed for the Phase 1 logistic regression) is not always available. However, aggregated consumption data can be obtained, for instance the INSEE data base has this type of survey for France. As one can observe from these graphs, the consumption is presented in two graphs, one per type of housing (mp, am, ...) and a second one per socio economic category (10, 12, ...), details about this can be found in the next section. Both of the graphs are presented in percentages relative to the total consumption of each household type. For instance, the left graph shows that the preferred type of housing is sector am ("appartement moyen") and only sector 14 prefers housing of type ap ("appartement petit"). The data is aggregated over the whole territory. As a modeller, one would like to have a calibration whose parameters (penalising factors, and others) reflect these relations at some level. For this example, one would like that estimated parameters reflect the observation that sector 14's preferred choice 2

INSEE is the French National Institute for Statistics and Economic Research

was housing type ap.

We will consider this survey data as constraint to our Phase 2 optimisation scheme proposed in the previous section. We took different approaches to do so. The first idea was to construct some initial parameters that represented the behaviour observed in the surveys. For the penalising factors, one can construct relations between parameters for each household type and induce the type of consumption desired in the population. For instance, we could rank the consumption for a household type and then optimise the parameters around these initial values. The optimisation is carried out with intervals around these initial values.

Suppose we called the initial values {ω mn 0 } m,n , then the optimisation is as follows:

min ω∈Ω X(X 0 , h, ω) -X 0 2 (4.5)
in the restricted set Ω = {[ω mn 0 -mn , ω mn 0 + mn ], ∀m, n}, where mn dictates how far away from the initial values ω mn 0 one can explore. Such constraints are called box constraints, and are suitable to be solved using the BFGS-B algorithm exposed in 1.2. This technique works very well, and the optimisation preserves the consumption preferences observed to a certain level. In the next chapter we present results of this methodology for the Grenoble model.

Secondly we propose an optimisation problem with hard constraints on the observed consumption. To do so, we need to look at consumptions and how this translates to our Tranus demand equations. We can easily compute the number of housing units of each type consumed per each household type per zone, by multiplying the number of households of the considered type in the considered zone, X m i , with the proportion of housing of type n that this household consumes, i.e. S mn i . As we know from the observed data how many households of each type we have in each zone, we can replace X m i by X m 0i . It is just like equation (3.6). If we call C mn 0 the consumptions from the survey (such as those shown in 4.1), we can write down the constraints as:

i S mn i * X m 0i -C mn 0 = 0, ∀m, n (4.6)
We can add this term to the objective function exposed in Phase 2 (4.1) with a weight parameter α to obtain the following modified cost function:

f (h, ω, α) = X(X 0 , h, ω) -X 0 2 + α i S mn i * X m 0i -C mn 0 2 . (4.7)
We would like to find the values of ω that reduce the corresponding shadow prices. We propose to solve the following problem:

min ω∈Ω f (h = 0, ω, α) (4.8)
The derivatives for the constraints (4.6) can be computed analytically using the derivatives of the logit formulation (cf. equation (4.4)) as follows:

∂ ∂ω mq [ i S mn i * X m 0i ] = i ∂S mn i ∂ω mq * X m 0i (4.9)

Sensitivity analysis and simultaneous calibration of shadow prices and marginal utility of income

In this section we present the work exposed in [START_REF] Arnaud | Sensitivity Analysis and Optimisation of a Land Use and Transport Integrated Model[END_REF]) where a sensitivity analysis is performed over the Tranus land use module. First a brief introduction to the methodology is presented, technical details about the sensitivity analysis theory can be found in the appendix C. Secondly, a simultaneous estimation of the shadow prices of transportable sectors and their corresponding marginal utility of income is developed. This framework is then tested on the Mississippi Tranus Model.

Sensitivity Analysis

Sensitivity analysis studies how the uncertainty on an output of a mathematical model can be attributed to sources of uncertainty among the inputs. There are two main classes of sensitivity analyses called local and global sensitivity analysis. The former addresses sensitivity relatively to a nominal value of a given parameter. The latter examines sensitivity on the whole set of variations of the parameter. Here, the focus is put on global sensitivity analysis with the aim of identifying the most influential parameters of the land use module of Tranus.

Among the large number of available approaches to perform a global sensitivity analysis, we review the generalisation of the variance-based method introduced by Sobol' [START_REF] Sobol | Sensitivity estimates for non linear mathematical models[END_REF] that relies on the estimation of generalised Sobol' indices [START_REF] Gamboa | Sensitivity analysis for multidimensional and functional outputs[END_REF]).

After introducing the successful semi-automatic calibration techniques for estimating the penalising factors in previous sections, we decided to identify how we could improve the calibration of parameters associated with transportable sectors. Transportable sectors are mainly driven by the logistic location probabilities (2.4):

P r n ij = A n j exp -β n U n ij l A n l exp -β n U n il .
the parameters relevant to this equation are A n j , β n and U n ij . Parameters A n j represent the attractiveness of zone j for sector n, and are governed by the following equation (equation

(3.20)): A n j = W n j k b n k X k 0i
where W n j is not calibrated in Tranus and taken as input (actually, as a common practice it is set equal to the base's year production or simply set to 1), so the only parameters left to calibrate are the cross-consumption parameters b n k . These parameters reflect the influence of other economical sectors on the location of activities, for instance, households of a certain type, could be attracted to live in zones where commerce is present. Often, the matrix b n k is set to the identity matrix for convenience and simplicity, but cross interactions exist in reality.

At the same time, the dispersion coefficients β n of the logit probabilities are also calibrated.

As we have already shown in previous sections, the measure of calibration of Tranus land use module is the shadow price parameter. Given initial values of the parameters for a sector n ∈ N , the land use and activity module estimates the adjustment parameters h n = (h n i ) i∈Z of the utilities (2.3), known as shadow prices. They compensate the utilities to replicate the base year production X 0 . To compute the shadow prices for transportable sectors we solve the optimisation problem exposed in equation (3.2):

ĥn = arg min h n X(h n ) -X 0 2 .
This problem is solved as exposed in section 3.4. The input parameters fall into different categories:

• logit dispersion parameters β n are involved in Equation (2.4),

• cross-consumption parameters b n l , l = n, l ∈ N , are involved in Equation (3.20)

We will also consider the parameter λ n involved in the computation of the utilities U n ij , this parameter is the marginal utility of income, and sets the tradeoff between prices and transport disutilities:

• parameter λ n is involved in Equation (2.3).
The outputs considered are built upon a new quantity called normalised shadow price. The normalised shadow price hn i corresponds to the percentage of the price p n i corrected by the shadow price h n i , that is:

hn i = 100 × h n,i p n,i , ∀i (4.10)
We set hn = ( hn i ) i , the vector of normalised shadow prices relative to the sector n. This is the typical Tranus evaluation quantity called "adjust factor" that one obtains after the calibration with the Tranus software module LCAL. The two outputs considered are the following: i) the variance of the normalised shadow prices: Var[ hn ]. Here the variance is taken across all zones, so we have the variance for each economical sector n.

ii) the maximum of the normalised shadow prices:

max i | hn i |
For each sector n, a good calibration would be one that results in small values of the normalised shadow prices particularly in term of variance. Minimising the variance of the normalised shadow prices is a general consensus reached by both modellers and users of Tranus. This is due to the nature of the logit probabilities for the transportable sectors (the probability is invariant to an additive constant, as explained in section 1.3.2).

Once the sensitivity analysis identifies the parameters that are influential on the outputs exposed above, we proceed to compute optimal values for these using the stochastic optimisation algorithm EGO (see section 1.2.5).

Obtaining the λ n parameters

Once we have found the optimal values for the selected parameters β n and b n k , we can compute the value of the λ n parameter analytically. The basic idea is to start from a good guess of the logit dispersion parameters (β n ) and from there, find the optimal values of the marginal utilities (λ n ) to minimise the variance of the shadow prices for the corresponding economic sector. This methodology explicits the dependency of λ n on β n , showing that the optimal value of λ n is a function of β n .

The parameter λ n is involved in the location probabilities equation (Equation 2.3)

U n ij = λ n (p n j + h n j ) + t n ij , ∀(i , j) (4.11)
The optimal value of λ n cannot be retrieved directly from equation (4.11) as the quantity

(p n j + h n j
) is estimated as a whole during the internal optimisation of the shadow prices. To overcome this problem, we introduce an auxiliary variable, similar as what we did in section 3.4:

φ n j = λ n (p n j + h n j ), ∀j
With this new variable, equation (4.11) can be rewritten as follows:

U n ij = φ n j + t n ij , ∀(i , j) (4.12)
Recall that the shadow prices are price-correcting additive factors that are calibrated to obtain a small variance. From equation (4.12), we can express the optimal value of λ n that minimises the variance of the shadow prices. We set φ n = (φ n j ) j∈Z with all other parameters fixed, in particular the parameters estimated with the EGO algorithm (see section 1.2.5).

The corresponding calibration problem can be written as:

φ n * = arg min φ n X(φ n ) -X 0 2 . (4.13)
Recall that we have p n = (p n j ) j and h n = (h n j ) j the vectors of prices and shadow prices. Once the optimal value φ n * is obtained, the equilibrium prices p n * can be computed solving a linear system (analogously to what was shown in section 3.4). Then, the shadow prices are expressed as follows:

h n = φ n * λ n -p n * .
From this, the following problem can be posed:

min λ n Var φ n * λ n -p n *
where the analytical solution leads to:

λ n * = Var(φ n * ) Cov(φ n * , p n * ) . (4.14)

Summary of the proposed calibration process

The following pseudo-algorithm illustrates how the combination of sensitivity analysis and subsequent optimisation of the most influential parameters, proceeds:

Algorithm 1 Calibration procedure for the land use and activity module 1: for each transportable sector n do 2:

Set:

λ n(0) ← λ n 0 3:
Run sensitivity analysis with inputs: β n , {b n l } n =l and outputs: V ar [ hn ], max i∈Z hn i 4:

Instantiate:

• ρ n(0) ← set of most influent parameters,

• k ← 1 5: while |λ n(k) -λ n(k-1) | ≥ do 6:
Given λ n(k-1) , estimate ρ n(k) with the EGO algorithm 7:

Given ρ n(k) , estimate λ n(k) with the analytical optimisation (cf. equation (

Return optimal values ρ n * and λ n * Once a sector n is selected, the sensitivity analysis presented in Section 4.3.1 is performed on the parameters β n and b n,l , l = n. The outputs considered for the sensitivity analysis are both the variance and the maximum of the normalised shadow prices hn (cf. equation (4.10)). The set of influent parameters selected is denoted by ρ n .

Following the sensitivity analysis, an iterative optimisation is conducted. This optimisation comprises two stages. At iteration k, the EGO algorithm presented in section 1.2.5 is applied to find optimal values for the parameters in the set ρ n(k) given λ n(k-1) . Then, an analytical optimisation of λ n(k) is performed taking as inputs the optimal values found for the set ρ n(k) .

The process is iterated until an equilibrium is reached for λ n . At the end of the iterations optimal values ρ n * and λ n * are returned.

For the stochastic optimisation in step 6 of the above algorithm, we chose to conserve only one output to perform the EGO algorithm: the variance of the normalised shadow prices:

V ar [ hn ].
Finally, we applied this methodology to the Mississippi model in section 5.2.1 with good results.

Chapter 5

Experimental results on real scenarios

In this chapter we present the main results of our methodology to real Tranus models. For all tested scenarios we have utilised our optimisation methodology to compute parameters for the model calibration. In parallel with our methodology we have utilised the Tranus software, always verifying that the results produced are the same. As the models are real scenarios in current projects, we need to ensure that the parameters we found with our optimisation approaches can be utilised directly in Tranus.

The first two scenarios were constructed by Brian J. Morton and helped to develop the methodology to simultaneously estimate the shadow prices and substitution probabilities (see section 4.2). These scenarios were already carefully calibrated and what we propose here is an improvement on a model that was already performing well. The third scenario is for the Grenoble urban area, and is developed by Fausto Lo Feudo and Brian J. Morton. This scenario

is not yet fully calibrated and has been a good test ground to improve our methodology to adapt it to newly created Tranus models.

North Carolina Tennessee (NCT) model

The North Carolina -Tennessee model comprises 38 geographical zones and 12 economical sectors. This model was made available by our partner and friend Brian Morton, with whom we have collaborated to develop the integrated calibration proposed above. We will only describe the model and methodology relevant to our work, the details can be found in the technical report [START_REF] Morton | SMALL COMMUNITIES AND RURAL AREAS : Prepared for the NCHRP Transportation Research Board Of The National Academies[END_REF]).

, for the NCT model we do have substitution between the three housing sectors (31, 32 and 33). So the substitution probabilities S mn i are relevant and have to be calibrated. The substitution probabilities are given by a logit formula as shown in equation (2.14), we present the equation here again:

S mn i = W n i exp(-ω mn a mn i • (p n i + h n i )) l∈K m W l i exp (-ω ml a ml i • (p l i + h l i ))
.

The coefficients W n i represent attractors of sectors n in zones i and are set to the base's year production of the corresponding sector. In this section we present the application of the two phase approach exposed in section 4.2. This technique consists in tuning the values of the penalising factors (ω's) to improve the model fitting. To assess the quality of the fit we look at the values of the corresponding shadow prices. One wants to make these as small as possible. This is basically done by computing the productions of housing sectors without shadow prices (setting their value to zero) and adjusting the penalising factors to make productions as close as possible to base year's data.

As a baseline, we compute the shadow prices when the penalising factors are set to 1, (ω mn = 1, ∀m, n) to have an initial value to compare against. Usually, when no information is available to adequately estimate initial penalising factors, the values are set to 1 (or all to the same value) to represent the absence of preferences in housing choices. Here, shadow prices are estimated using the method of section 3.3, based on the cost function shown in equation ). This manual technique is very tricky, as modifying the value of a single penalising factor, affects the choices for all other sectors. For this model we had data on actual choices of households, so a logistic regression can be performed to estimate an initial value for the whole set of penalising factors.

Phase 1: estimating parameters' initial values with multinomial logistic regression.

To do this, we recognise the term in the utility function a mn i • p n i as the expenditure of household of type m, consuming housing type n in zone i . There is one multinomial logit per household type, and as we don't have data independently per zone, we estimated the penalising factors for all zones. The data utilised for the regression is as shown in Table 5 One would want the value of price plus shadow price (p + h) to be as close as possible to p, meaning that shadow prices are small. As we can see, for sector 31 the green curve is now between the blue and red one (strictly better), for sector 32 (second graph) the green curve is closer than the initial guess, is somehow better. In sector 33, we get the same behaviour as for sector 31, so is also better than the initial guess. We can also look at the statistics, and comparing both tables 5.2 and 5.5 we can assess huge gains in the mean and standard deviation tabs.

Phase 2: Tuning the penalising factors. The penalising factors exposed in table 5.4 are fine tuned. The problem we are solving is exposed in equations (4.1)-(4.2). This methodology takes as input the penalising factors estimated with the logistic regression from phase 1, and fine tune them to obtain a better fitting. We can limit the search space to avoid to go too far away from the penalising factors estimated with actual data. To do this we consider an interval around the penalising factor estimated with the regression, and limit the search to a percentage of the original value. This means that for a particular penalising factor ω we limit the search to the interval ((1 -δ) * ω, (1 + δ) * ω). We present results with δ equal to 10% and 20%. Figure 5.4 presents the comparative results between phase 1 and the optimisation constrained to 10%. As we can see, the results are strictly better. The greatest gain is done by sector 31, now the green curve is even closer to the red one. With this methodology in mind, we skip directly to figure 5.5 where the search space is enlarged to 20% and the fit is almost perfect. Figure 5.6 further shows that shadow prices are small for almost all zones and sectors. It is normal to have some zones where the fit is bad, as the model is a simplification of reality such that one can not expect that the housing choice is only reflected by prices. Computing optimal penalising factors without logistic regression (without phase 1).

Often, data to build a logistic regression is not available when constructing a LUTI model, we wanted to verify that the optimisation algorithm provides good results even without this starting point. Figure 5.7 shows the results of our algorithm starting from the default values ω = 1 against the values obtained after the two stage optimisation (phase 1 and 2). The blue curve represents the previous sections phase 1 -phase 2 estimation results and the green curve is obtained after applying only the phase 2 optimisation with initial values set to ω = 1.

We can see that the results are similar, with the blue curve (phase 1 -phase 2 estimation) almost identical to green curve (optimisation without prior logistic regression). We also present the statistics for this technique in table 5.7. In figure 5.8 we have the resulting prices and shadow prices for the 3 floorspace sectors.

0
The green curves are the result of our methodology, the blue dotted curves are the values obtained after setting the penalising factors to one (ω = 1). We can observe that the penalising factors estimated with the optimisation algorithm produce an excellent fit, almost positioning the green curves all along the prices (red curves In the next part, we will see that more has to be taken into account than fitting base year's productions and reducing shadow prices if one wants a good model in the sense that it reproduces plausible output.

Sensitivity Analysis Results for the MS model

In section 4.3 we presented a sensitivity analysis methodology to identify influent parameters of the land use module. To discover the relations between different economical sectors, the sensitivity analysis is performed, thus giving the relations that are more relevant to calibration. We will consider the MS Mississipi model exposed in 5. A total of 7 sensitivity analyses are performed, one for each transportable sector. For each sensitivity analysis, the 12 following parameters are considered:

• The logit dispersion parameter β n (cf. equation (2.4) ).

• The 11 parameters b n l , for all sectors l = n (cf. equation (3.20)).

In Table 5.9 are listed the distribution of each parameter used as support for the sensitivity analyses. These distributions were selected by expertise, as each model is different, a good parameters labels distributions 0.0 0.1 0.2 0.3 0.4 0.5 0.6 q q q q q q q q q q q q S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 S 11 S 12 q first-order indices confidence intervals qq q q q qq q q q q qq q q q q q q q q q q q q q q qq q q q q qq q q q q q qq q qq q q q q q q q q qq q q q q q qq qq q q 0.00 0.10 0.20 0.30 S 1,2 S 1,8 S 2,4 S 2,10 S 3,7 S 4,5 S 4,11 S 5,10 S 6,10 S 7,11 S 9,10 q q q second-order indices confidence intervals The second order indexes are given in the following order: {(1, 2), ..., [START_REF] Abraham | Parameter Estimation in Urban Models: Theory and Application to a Land Use Transport Interaction Model of the Sacramento, California Region[END_REF][START_REF] Ciuffo | A Sensitivity-Analysis-Based Approach for the Calibration of Traffic Simulation Models[END_REF], (2, 3), ..., [START_REF] Abraham | Parameter Estimation Strategies for Large-Scale Urban Models[END_REF][START_REF] Ciuffo | A Sensitivity-Analysis-Based Approach for the Calibration of Traffic Simulation Models[END_REF], [START_REF] Abraham | Specification and Estimation of a Nested Logit Model of Home. Workplaces and commuter Mode Choices by Multiple Owker Households[END_REF][START_REF] Anas | The economics of cordon tolling: General equilibrium and welfare analysis[END_REF], ....} range of a priori possible values needs to be explored. The Sobol' index S k will refer to the parameter labeled by k, this is standard practice in the field of sensitivity analysis, so for instance, S 1 will always correspond to the logit dispersion parameter β n of the sector under analysis, and S 2 , ...S 12 the corresponding cross relations b n l parameters. (without including the case l = n)

β n 1 U(2,
The outputs considered are the variance and the maximum of the normalised shadow prices, as introduced in Section 4.3. The approach proposed is to use the replication procedure presented in appendix C to estimate first-order and second-order generalised Sobol' indices of these parameters. Asymptotic confidence intervals can be computed for first-order Sobol' indices [START_REF] Tissot | A Randomized Orthogonal Array-based procedure for the estimation of first-and second-order Sobol' indices[END_REF].

Before presenting the main results, we propose to illustrate the selection procedure of the influent parameters for a sector. Figures 5.9 For the estimation of first-order indices, a size N = 5 × 10 3 was chosen for the two replicated Latin Hypercubes required by the replication procedure (details of the replication procedure can be found in the appendix C). Looking at the results, the parameters β 4 and b 4 10 are the most influent (cf. figure 5.10). Since the sum of the first-order indices is less than 75% it is interesting to study the second-order indices.

For the estimation of second-order indices, a size N = 47 2 The same procedure is performed for the other sectors. For each sector n, the set comprised of the most influent parameters selected by the sensitivity analysis is listed in

Results of the subsequent iterative optimisation

Following the results of the above sensitivity analysis, for each transportable sector n, we proceed to find the set of parameters (β n , b n k , λ n ) minimising the variance of hn . The initial value λ n 0 (Step 3 of Algorithm 1, see section 4.3) instantiating the parameter λ n is obtained by expertise, or just set to 1. The results obtained in terms of variance and maximum of the normalised shadow prices are compared to those obtained with a former ad hoc procedure (the parameters calibrated with a classical calibration approach, without optimisation or automatic calibration). The number of initial evaluations performed to fit the metamodel for the set of parameters (β n , b n k , λ n ) of each sector n is the following:

• 21 evaluations for sector 2,

• 51 evaluations for sector 4 to 6.

• 41 evaluations for sector 7 to 9.

The quality of the fitting is assessed by diagnostic plots (fitted values against response values, standardised residuals, Q-Q plots of standardised residuals) based on leave-one-out cross validation results (see (Roustant, Ginsbourger, and Deville 2012) for further details).

For each sector n, the evaluations include the one for the optimal set of parameters obtained with the ad hoc procedure.

Table 5.11 summarises the results obtained with both the ad hoc procedure and our iterative optimisation. The "Optimal params." tab gives the estimated values of the parameters exposed in table 5.10 and λ n * denotes the optimal values of the parameters obtained at the end of both approaches for each transportable sector n. The column gain represents the improvement (in percentage) of the variance obtained with our iterative estimation relatively to the one obtained with the ad hoc procedure conducted by experts.

Looking at the results, we observe that the values of the variance and maximum of the normalised shadow prices obtained with the ad hoc procedure are heterogeneous. Furthermore, the value of the maximum is quite high for some sectors (up to 20% of the price). The results obtained with our iterative optimisation are relatively homogeneous except for sectors 2 and 8. The discrepancy observed for these two sectors comes from the quality of their respective datasets. Indeed, the data relative to commercial business (sector 2) are easy to collect thus of high quality and quantity. At the opposite, data relative to the 65 years and older households (sector 8) are quite complex to collect and often lacking precision. The main observation is that an improvement in terms of both variance and maximum of the normalised shadow prices is observed for all sectors when using our approach. Figure 5.11 gives an illustration of this improvement. The black bars represent the values obtained with the ad hoc procedure, the grey bars those obtained with our iterative approach. A significant diminution for both the variance and maximum criteria is observed. Furthermore and most importantly our approach is drastically faster than the ad hoc procedure conducted by experts. Our calibration procedure requires a few hours compared to several days (up to weeks) for the ad hoc procedure.

As a final remark, we decided here to conserve only the best set of parameters model fit. Figure 5.12 presents the shadow prices before (ω = 1) and after optimisation (Phase 2). The results show better fitting in all sectors besides sectors 100 and 105 where we think prices may have been underestimated, also some zones have very high shadow price values, increasing the variance considerably. From figure 5.14 we observe that the driver of the housing market is sector 10 (active population), seconded by sector 20 (retired) and then sector 16 (inactive). One would want this behaviour to be preserved after the optimisation of the ω parameters. We will add the consumption preferences to the analysis of the model fitting.

How do our results from the optimisation compare to the INSEE data?

The standard output proposed by Tranus is the demands D mn i (see equation (2.2)), these demands are in square meters, so if one wants to have the actual units consumed we have to compute them using the substitution probabilities S mn i . These probabilities are a distribution over all possible consumed sectors, so for each household sector m and each zone i , we have a logit formulation with as many choices as housing possibilities. For our case, we have 5 consuming sectors (10, 12, 14, 16 and 20) (as explained before). We can easily compute the number of housing units of each type consumed by each household type by multiplying the number of households in each zone (X m i ) with the proportion of housing of type n that this household type consumes (S mn i ). It is just like equation (3.6), but without the demand functions that transform the results in square meters. sector 10 prefers medium apartments (am), so the associated penalising factor is the lowest possible, hence to 1. Conversely, sector 10 dislikes small houses (mp) so the penalising factor is set highest, hence to 4. We propose this ranking scheme to have a better initial guess for the penalising factors and also to preserve the order in which each household type consumes housing. This should produce results that are closer to the observed consumption.

After this initial guess, the optimisation procedure is applied. Figure 5.16 presents results of the optimisation procedure with initial guess from the ranking exposed above. We also present the shadow prices before and after the optimisation. From the consumptions exposed in figure 5.16 we observe that the consumption preferences are still close to the INSEE data, and not very different from the starting guess. Also, figure 5.17 shows that even if the initial guess shadow prices were bad (blue dotted line) the results after the optimisation are very close to desired values (green line close to red).

We have presented a trade-off between a pure optimisation of penalising factors and balancing between actual revealed preferences from available data. This is another tool for modellers that have access to this type of consumption preferences instead of actual observations allowing a logistic regression. Compared to MS and NCT models, the Grenoble model is in a developing stage while both American models were almost finished and with very good calibration already. For the American models, improving the values of shadow prices was easier because the starting point was already very good and stable. The strategies proposed for the Grenoble area gives insight into actual calibration of penalising factors for a new model.

We are currently working on implementing the constrained optimisation exposed in section 4.2.1, that includes observed housing preferences as constraints in the optimisation of the penalising factors.

fine tuning, but is a first step in what we consider a promising direction. We believe that the model "as it is" with one shadow price per observation bears a risk of overfitting. Determining which shadow prices have to be removed may not be completely automisable, and the expert eye of the modeller has to have the last call.

The simultaneous calibration of different parameter types is a potentially very powerful tool. The results that we have for both American models and the undergoing Grenoble model have proven to be useful and saved many trial and error sessions. For the American models the improvements were easier to obtain as both models were already calibrated and produced good results. Also, the available data for the logistic regression permit us to establish the two phase algorithm proposed in section 4.2. Here, we have taken models that were already calibrated, and improved their calibration directly. On the other side, the tools developed for the Grenoble model, a model that is in early development, have proven to help the calibration of the substitution parameters enabling modellers to obtain results that are plausible and reproduce the observed data.

Another important point about the methodology to calibrate the penalising factors, is that it is possible to apply these techniques via a program completely outside the Tranus software, adjusting the parameters and then feeding them to Tranus. As much as one would like that the Tranus software would include the optimisation methodology, this approach enables us to already work with real-life operational models. The full integration of this methodology in Tranus would need the re-implementation of the land use module, to include the whole computation of shadow prices as an optimisation problem. Doing so, would have many benefits, for instance we could apply this idea of simultaneous optimisation to other "hard" to calibrate parameters. A fully integrated and automatic calibration is our dream.

We also presented a sensitivity analysis for Tranus, based on the generalisation of the replication procedure to select the most influential parameters of the model. An application to the study area of Mississippi was presented where our methodology was compared to a former ad hoc calibration procedure in terms of variance and maximum of the normalised adjustment shadow prices. Our approach showed a significant improvement on both criteria reducing the value of the variance by a large margin with a drastic gain of time. These results have proven that our methodology is useful to outperform calibration of such models. The next step would consist in verifying if the optimal values found for the parameters ensure better predicting capabilities when evaluating alternative planning scenarios.

Future possibilities we would like to reproduce observed data is a question for a whole other thesis, but tools such as sensitivity analysis could be used to identify which shadow prices can be removed safely, as it is probable that the set of shadow prices varies from model to model. One possible methodology to test the model selection scheme would be to compare the model outputs for two time periods, to determine if noise in the base year productions could really propagate to the shadow prices and undermine the predictive capabilities of the model.

Another issue encountered during this thesis is related to the formulation of the logit probabilities in Tranus. As a common practice, Tranus uses scaled utilities as exposed in section 2.3. Scaled utilities are used to trick the logit formulation to ignore one of their fundamental properties, the invariance of the probabilities to an additive constant (cf. section 1.3.2). If this behaviour is not desired, maybe it would be a wise idea to migrate from the logit formulation to another discrete choice model, for instance, as proposed in section 2.

  corriger des effets non modélisés inévitables, il risque également de sur-paramétrer le modèle. Nous proposons un schéma de sélection de modèle (model selection), visant à a faire un compromis entre la complexité du modèle (ici, le nombre de prix sombres) et la qualité de l'ajustement aux observations, en réduisant le risque d'overfit et en augmentant la probabilité d'obtenir de bonnes prédictions avec le modèle. Après la reformulation du calcul des prix sombres en tant que problème d'optimisation, nous pouvons inclure dans le schéma d'optimisation d'autres paramètres (que les prix sombres). Au chapitre 4, nous traitons la calibration d'autres paramètres de Tranus. D'abord, nous proposons une calibration semi-automatique pour les facteurs de pénalisation (penalising factors). Ces paramètres visent à représenter les préférences du choix résidentiel des différents types de ménages du modèle. L'idée principale est d'inclure des données externes (lorsqu'elles sont disponibles) pour estimer un bon point de départ pour l'optimisation qui les améliore ensuite. Ceci est possible après avoir examiné les équations qui créent les interactions entre les ménages et le logement, et le découpage de l'optimisation en de plus petits problèmes. Enfin, nous présentons une analyse de sensibilité pour identifier les paramètres influents pour les secteurs transportables. Une fois que les secteurs influents sont sélectionnés, un algorithme d'optimisation trouve calcule les valeurs de paramètres qui améliorent la calibration. Le dernier chapitre, chapitre 5, présente la méthodologie appliquée aux scénarios réels. Nous appliquons d'abord notre méthodologie d'optimisation à deux modèles nord-américains, en particulier pour améliorer les facteurs de pénalisation et, ensuite, sur un modèle de la zone urbaine de Grenoble. Chapter 1

  a commercial software sold by Marcial's company ME&P. MEPLAN sets the interaction of two different markets: land use and transportation. The model was used by Hunt and Abraham to model the Sacramento area in the U.S. introducing automatic and semi-automatic tools for the calibration, mostly based on least squares optimisation. In (Abraham and Hunt 2000) they proposed a submodel calibration approach, utilising extra data during calibration.

  they estimated various zone specific constants in this way. The details of the methodology can be found in Abraham's PhD thesis (Abraham 2000), we found much inspiration from his work. The PECAS model is developed by the consulting firm HBA-SPECTO (HBA Specto Incorporated 2007; Hunt and Abraham 2003). PECAS takes as inspiration MEPLAN, and the methodology developed for Sacramento. It utilises as calibration technique least squares minimisation with analytic formulation of derivatives (Zhong, Hunt, and Abraham 2007). There is almost no documentation of PECAS and the user base is limited. The Pirandello model is a french LUTI developed mainly by Jean Delons for the Vinci company. Various econometric techniques are used for the calibration of the different modules, but optimisation is used for the calibration of the firm location module (Delons, Coulombel, and Leurent 2008), gradient descent is systematically used to calibrate parameters or groups of parameters, allowing a model's partial adjustments 1 . Some insights in these procedures are available in the Calibration Report from Vinci (Delons and Chesneau 2013). One recent 1 Private communication with Jean Delons application in France can be found in (Nguyen-Luong 2012) but no detail is given about how these optimisation techniques are applied to the calibration. Alex Anas (Anas and Liu 2007) proposes a LUTI model called RELU-TRAN where all parameters have economic significance, so estimation should be possible using available techniques from the literature, for instance, elasticities have been estimated for a number of relations (location demand with respect to commuting time, housing demand with respect to rent, labor supply with respect to wage, etc). Calibration is a mixture of fixing parameters at reasonable values within ranges found in literature (Anas and Hiramatsu 2013) and tweaking to fit the model to data as closely as possible. Internally, the model finds an equilibrium of the 656 equations using the Newton's algorithm. In the 2007 article, testing of convergence and robustness is mentioned, mostly to evaluate the stability of the equilibrium solution. UrbanSim is a highly popular agent-based model developed by Paul Waddell (Waddell 1998a; Waddell 1998b; Waddell 2002) that also includes demographic change modelling and household formation. It is not strictly a LUTI model, because it relies on an external transportation model to complete the integration and it is widely used in this way. (Kakaraparthi and Kockelman 2011; Deymier and Nicolas 2005; Waddell, Franklin, and Britting 2003), are three recent applications. It is a highly disaggregated model compared to other operational models. For instance, the Eugene-Springfield implementation has 111 household types and could be run using a large weighted sample of observed households. UrbanSim is open source now and has a large community of users. Calibration requires use of standard regression techniques for bid functions, and multinomial or nested logit for the location choice models. An approach to assess model sensitivity and calibrate the whole model is presented in (Ševčíková, Raftery, and Waddell 2007). The ITLUP (Putman 1994) framework has been developed and applied by Stephen Putman at the University of Pennsylvania, Philadelphia, USA, over 25 years. ITLUP consists of two modules: DRAM and EMPAL, and has over a dozen US applications (Putman 1997), although over 40 calibrations have been performed across the USA and elsewhere. In (Duthie et al. 2007) a comparative analysis between Telum (a particular version of ITLUP) and Ur-banSim is presented for Austin, Texas. Detailed information on Telum calibration, mentioning the gradient descent method for the original model and Nelder-Mead's simplex method for the modified version developed in this paper. It is also noted that the ITLUP equations are non-linear, so a global optimal solution cannot be guaranteed. Also from this study, in (Krishnamurthy and Kockelman 2007) a sensitivity analysis by Monte Carlo simulation is presented with very little detail about the calibration. The model is treated as a black box. Tranus is an open-source widely used LUTI created by Tomás de la Barra (de la Barra Chapter 1. State of the art and background material 1982; de la Barra 1989; de la Barra 1998; de la Barra 1999) that has a very active community of users. There have been many applications of Tranus in America (North, central and south) and Europe. Some applications of Tranus are; for the city of Belo Horizonte in Brazil (Pupier 2013) but very little detail is given on how the calibration was done, for Lille in France where most of the calibration was done by Fausto Lo Feudo (Lo Feudo 2014), where ad-hoc procedures and econometric techniques were utilised. Another expert in Tranus calibration is Brian Morton, researcher at University of North Carolina at Chapel Hill, who has developed large scale Tranus models for the Mississippi region and the North Carolina -Tennessee region (Morton, Poros, and Huegy 2012; Morton, Song, et al. 2014). Most of the calibration is done with various econometric procedures and ad-hoc submodule calibration. Some optimisation is used with simple solvers to find better parameter estimation. Besides these cases, most of the calibration is done by experts from consulting firms (Modelistica and Stratec are very experienced 2 ). There is very little detail on how the calibration is done and if there is some automatic or at least sequential calibration performed. Curve fitting and max-likehood estimation is sometimes performed to calibrate certain parameters. There is no complete and standard automatic or sequential calibration methodology developed for Tranus until now. MUSSA (Modelo de Uso de Suelo para Santiago, Chile) is an operational land use model developed by Francisco Martínez and Pedro Donoso from University of Chile (Martínez 1996; Martínez and Donoso 2010). It connects to the transportation model ESTRAUS to obtain a fully connected LUTI model. Recently the model has been acquired by the company Cube and it is a module for the Cube platform called Cube-Land (Martínez 2011). It is a random bid and supply model with a rigorous application of microeconomic theory (like RELUTRANS),

  optimisations. The variant called BFGS-B (Byrd, Lu, and Nocedal 1995) can handle box constraints and it what we use to solve most of our constrained optimisation. For a comprehensive survey on non-linear optimisation techniques we suggest the reader to refer to the book (Nocedal and Wright 2006).

  Tranus is an integrated land use and transportation (LUTI) modelling software developed by Modelistica, the consulting firm of Tomas de la Barra, (de la Barra 1982; de la Barra 1989; de la Barra 1998; de la Barra 1999). It provides a framework for modelling land use and transportation in an integrated manner. It can be used at urban, regional or even national scale. The area of study is divided in spatial zones and economical sectors; the basic concepts of the original input-output model (see[START_REF] Leontief | Multi-Regional Input-Output Analysis[END_REF] have been generalised and given a spatial dimension. The concept of sectors is more general than in the traditional definition. It may include the classical sectors in which the economy is divided (agriculture, manufacturing, mining, etc.), factors of production (capital, land and labour), population groups, employment, floorspace, land, energy, or any other that is relevant to the spatial system being represented. Tranus combines two main modules: the land use and activity and the transportation modules. The main components of both modules are shown in Figure2.1. Within each subsystem a distinction is made between demand and supply elements that interact to generate a state of equilibrium.

Figure 2 . 1 :

 21 Figure 2.1: Main elements of the land use-transport system -from the mathematical description of Tranus (de la Barra 1999)

Figure 2 . 2 :

 22 Figure 2.2: Schematic overview of Tranus.

  Transportable sectors generate flux, that induces transport demand, which ultimately influences transportation costs. Non-transportable sectors, on the other hand, neither require transportation nor generate fluxes. Usually, three types of economic sectors are classified: land or floorspace, households and businesses. Land is usually composed of two or three types of residential floorspace (e.g. detached houses, apartments, mobile homes), and commercial floorspace of offices and stores. Households are usually classified by socio-economic level, based on income or the household composition. Business sectors comprise industries (whose output is mainly destined for exportation), services (schools, universities, recreational) and

Figure 2 . 3 :

 23 Figure 2.3: Sketch of computations in the land use and activity module.

Figure 2 . 4 :

 24 Figure 2.4: Demand curves for different elasticity values.

Figure 2 . 5 :

 25 Figure 2.5: Expenditure curves for different elasticity values.

Example 2 .

 2 If we consider U 1 = 5 and U 2 = 10, with the simplest logit formula, P 1 = e -5 e -5 + e -10 = 99.3%, P 2 = e -10 e -5 + e -10 = 0.7% and for the scaled logit, Û1 = 5/5 = 1 and Û2 = 10/5 = 2 P1 = e -1 e -1 + e -2 = 73.1%, P2 = e -2 e -1 + e -2 = 26.9% that is closer to the proportional distribution. Changing the utilities to U 1 = 1005 and U 2 = 1010 does not affect the classic logit (see 1.3: Properties of Logit models) : P 1 = e -1005 e -1005 + e -1010 = 99.3%, P 2 = e -1010 e -1005 + e -1010 = 0.7% and the scaled logit: P1 = 50.1%, P2 = 49.9%

5 )Figure 3 . 1 :

 531 Figure 3.1:Objective functions (3.15) near their optimal values h 5 = (-0.062, -0.271, -0.388)

Figure 3 . 2 :

 32 Figure 3.2: Slices of the cost function along h 1 i , for each zone i ∈ [1, 2, 3].

Figure 3 . 3 :

 33 Figure 3.3: Plot of cost function for a given pair (h 1 1 , p 1 1) near the optimal value (0, 2.676).

Figure 4 .

 4 1 shows a consumption representation from the INSEE 2 data base for the Grenoble study area.

Figure 4 . 1 :

 41 Figure 4.1: INSEE: Consumption of floorspace (as percentages of total household consumptions), see text for explanation.

Figure 4 .

 4 2 gives a scheme of the inputs and outputs considered for each transportable sector n.

Figure 4 . 2 :

 42 Figure 4.2: Inputs and outputs of the land use and activity module for the sector n generating flux.

Figure 5 . 1 :

 51 Figure 5.1: Shadow prices as percentage of prices for sectors 31, 32 and 33, for all zones.

Figure 5 . 2 :

 52 Figure 5.2: Shadow prices as percentage of prices for sectors 31, 32 and 33 for all zones. After phase 1.

Figure 5 .

 5 Figure 5.3 compares the initial values of the ω (equal to 1) and the results from the logistic regression. The red lines are the prices, the blue (dotted) lines are the prices plus shadow prices resulting for ω values set to one and the green lines (solid) are the resulting prices plus shadow prices after the logistic regression. We present one graph per housing sector (from top to bottom sectors 31, 32 and 33). Prices and shadow prices are absolute

Figure 5 . 4 :

 54 Figure 5.4: Phase 1 compared against the optimisation tuning of phase 2 with 10% search range.

Figure 5 . 5 : 2 0Figure 5 . 6 :

 55256 Figure 5.5: Values from phase 1 compared against the optimisation tuning of phase 2 with a 20% search range. Note how the green lines (prices + shadow prices) almost coincide with the red one (prices) is almost over the prices (red line) after phase 2

Figure 5 . 8 :

 58 Figure 5.8: Values from phase 2 compared against the optimisation tuning starting from base values (without prior logistic regression).

Table 5 . 9 :

 59 Distributions of the 12 parameters, U(a, b) stands for the uniform distribution with support [a, b].

Figure 5 . 9 :

 59 Figure 5.9: First-order indices estimation for the 12 parameters of sector 4. S 1 corresponds to β 4 , S 2 to b 4 1 , S 3 to b 4 2 and so on.

Figure 5 . 10 :

 510 Figure 5.10: Second-order indices estimation for the 12 parameters of sector 4.The second order indexes are given in the following order: {(1, 2), ...,[START_REF] Abraham | Parameter Estimation in Urban Models: Theory and Application to a Land Use Transport Interaction Model of the Sacramento, California Region[END_REF][START_REF] Ciuffo | A Sensitivity-Analysis-Based Approach for the Calibration of Traffic Simulation Models[END_REF], (2, 3), ...,[START_REF] Abraham | Parameter Estimation Strategies for Large-Scale Urban Models[END_REF][START_REF] Ciuffo | A Sensitivity-Analysis-Based Approach for the Calibration of Traffic Simulation Models[END_REF],[START_REF] Abraham | Specification and Estimation of a Nested Logit Model of Home. Workplaces and commuter Mode Choices by Multiple Owker Households[END_REF][START_REF] Anas | The economics of cordon tolling: General equilibrium and welfare analysis[END_REF], ....}

  and 5.10 show the results obtained for the estimation of first-order and second-order indices relative to Sector 4. The dashed line represents the threshold value used for selecting the influent parameters.

  was selected for the two replicated randomised orthogonal arrays required by the replication method. The two black points of figure 5.10 correspond to the two most influent interactions: β 4 * b 4 10 and β 4 * b 4 11 . The number of bootstrap replications used to compute the confidence intervals equals 1000. In conclusion, only 3 of the 12 parameters of the sector 4 are significantly influent either directly by their main effects or through their second-order interactions: β 4 , b 4 10 and b 4 11 .

β 9 * b 9 10 β 9 , b 9 10 93Table 5 . 10 :

 10910510 Most influent parameters selected by the sensitivity analysis based on main effects and second-order interactions.These results fall within our range of expectations. The parameter β n is a dispersion parameter of a multinomial logit function (see Equation (2.4)). A slight variation of this parameter leads to a significant change in the calculation of the probabilities of localisation.Both parameters b n 10 and b n 11 act as weights in the attractiveness for sector n. These two parameters are more prone to be influent than the other b n l since sectors 10 and 11 correspond to the two main floorspace types.

Figure 5 . 11 :

 511 Figure 5.11: Variance (left figure) and maximum (right figure) of the normalised shadow prices obtained with both the ad hoc procedure (referred to as ad hoc) and our iterative optimisation (referred to as iterative)

20 20 Figure 5 . 13 :

 2020513 Figure 5.13: INSEE: Consumption of floorspace (as percentages of total household consumptions).

Figure 5 . 14 :

 514 Figure 5.14: INSEE: Consumption of floorspace (total housing units consumed of each housing type).

Figure 5 .

 5 Figure 5.15 presents the consumption preferences after the optimisation results presented at the beginning of this section (Housing sectors are aggregated to be compared to INSEE sector types). The results are similar to the INSEE preferences, only sector 14 has a different behaviour, consuming more medium apartments than small ones. This inverted behaviour could be explained by the apartment sharing of students in the Grenoble area, even if the model has not implemented this specifically (allowing partial consumption of housing, for instance rooms) the data is telling us something.

Figure 5 . 15 :

 515 Figure 5.15: Consumption of floorspace after optimisation of penalising factors.

Figure 5 . 16 :Figure 5 . 17 :

 516517 Figure 5.16: Consumption of floorspace, after optimisation (as percentages of total household consumptions).

  3, to a multiplicative error term discrete choice model. From our point of view, moving Tranus to this type of model would not be complicated, and all the optimisation framework developed in this thesis would still work, as these functions are differentiable and have a very concise formulation. Indeed, their formulation is very similar to the logit model, with the same type of properties for the derivatives. A good article about this type of models is[START_REF] Fosgerau | Discrete choice models with multiplicative error terms[END_REF]).ConclusionsLe modèle Tranus est un outil très puissant et les possibilités de modélisation sont infinies.Cependant, la complexité de tels modèles à grande échelle ne doit pas être sous-estimée, car elle rend la calibration et l'utilisation de ces outils très coûteux.Danscette thèse, nous avons contribué à une reformulation du module d'usage des sols qui simplifie le processus de calibration. Pour cela, nous avons dû présenter d'une manière différente les équations impliquées dans le calcul du module d'usage des sols, en exploitant les bases fondamentales des mathématiques derrière les modèles microéconomiques utilisés, en établissant une fonction de coût (et les dérivées) nécessaires pour les algorithmes d'optimisation. Beaucoup de temps et d'efforts ont permis de comprendre les différentes interactions entre les paramètres pour pouvoir diviser la calibration en petits problèmes, conduisant ainsi à l'optimisation en deux étapes finalement proposée au chapitre 3 (pour les secteurs non transportables et ensuite transportables). Par exemple, la reformulation pour les secteurs transportables exposés dans la section 3.4 n'est pas évidente, et n'est possible qu'après avoir travaillé avec les équations analytiquement, comme indiqué dans l'exemple numérique dans la section 3.5.1. L'approche d'optimisation est plus stable et plus claire que l'approche classique et permet d'utiliser des algorithmes d'optimisation puissants actuellement disponibles, résolvant les problèmes occasionnels de non convergence de l'approche précédente. Pour pouvoir tester notre approche d'optimisation pour la calibration des prix sombres, nous avons proposé une procédure simple et directe "first for Tranus" pour générer des données synthétiques, ce qui nous permet d'évaluer nos méthodologies. Cette procédure pourrait également être utilisée pour effectuer la validation du modèle, en créant un scénario "présent" ainsi qu'un scénario "futur" synthétique pour comparer les capacités prédictives de Tranus. Ce dernier n'est pas aussi simple qu'il le semble et pourrait être considéré comme une perspective pour des travaux futurs. En outre, en mode prédictif, la reformulation du module d'usage des sols que nous proposons dans cette thèse devrait être ajustées, puisque les productions ne seront plus égales aux productions de l'année de base, voir section 3.2 et équation (3.5)). La méthodologie proposée pour réduire le nombre de prix sombres nécessite un ajustement précis, mais constitue une première étape dans ce que nous considérons comme une direction prometteuse. Nous croyons que le modèle «en tant que tel», avec un prix sombre par observation, risque d'être surparamétré. Déterminer quels prix sombres doivent être supprimés peut ne pas être entièrement automatisable, et l'oeil expert du modélisateur doit avoir le dernier appel. La calibration simultanée de différents types de paramètres est un outil potentiellement très puissant. Les résultats que nous avons pour les modèles américains et le modèle de Grenoble en cours se sont révélés utiles et ont évité des nombreuses sessions d'essai et d'erreur. Pour les modèles américains, les améliorations ont été plus faciles à obtenir, car les deux modèles étaient déjà calibrés et produisaient des bons résultats. De plus, la disponibilité des données pour la régression logistique nous a permis d'établir l'algorithme en deux phases proposé dans la section 4.2. Ici, nous avons pris des modèles déjà calibrés et amélioré leur calibration directement. D'autre part, les outils développés pour le modèle de Grenoble, un modèle qui est en cours de développement, ont prouvé leur aide pour la calibration des paramètres de substitution permettant aux modélisateurs d'obtenir des résultats plausibles et de reproduire les données observées. Un autre point important de la méthodologie pour calibrer les facteurs de pénalisation est qu'il est possible d'appliquer ces techniques via un programme complètement en dehors du logiciel Tranus, en ajustant les paramètres puis en les alimentant vers Tranus. Bien qu'il serait idéal que le logiciel Tranus inclue la méthodologie d'optimisation, cette approche nous permet déjà de travailler avec des modèles opérationnels réels. L'intégration complète de cette méthodologie à Tranus nécessiterait la réintégration du module d'usage des sols, afin d'inclure tout le calcul des prix sombres comme un problème d'optimisation. Cela pourrait avoir de nombreux avantages, par exemple, nous pourrions appliquer cette idée d'optimisation simultanée à d'autres paramètres «difficiles» à calibrer. Une calibration entièrement intégrée et automatique bien évidemment optimale.Nous avons également présenté une analyse de sensibilité pour Tranus, en fonction de la généralisation de la procédure de "réplication" pour sélectionner les paramètres les plus influents du modèle. Une application à la zone d'étude du Mississippi a été présentée où notre méthodologie a été comparée à une ancienne procédure de calibrage ad hoc en termes de variance et au maximum des prix sombres normalisé. Notre approche a montré une amélioration significative sur les deux critères réduisant la valeur de la variance par une large marge avec un gain de temps drastique. Ces résultats ont prouvé que notre méthodologie est utile pour surpasser l'étalonnage de ces modèles. La prochaine étape consisterait à vérifier si les valeurs optimales trouvées pour les paramètres garantissent une meilleure prédiction des capacités lors de l'évaluation de scénarios de planification alternatifs. Implementation L'implémentation actuelle de Tranus est modulaire, avec un programme par sous-module: la calibration d'usage des sols (LCAL), les coûts de transport (COST), le module prédictif (LOC), le module de transport (TRANS), etc. La communication entre chacun des modules sont effectués via des fichiers texte et des fichiers binaires. En fonction de cela, nous avons décidé de remplacer le module de calibration d'usage des sols LCAL par notre implémentation basée sur les techniques d'optimisation. Pour la mise en oeuvre de notre module d'usage des sols, nous avons utilisé Python (Rossum and Drake 2006) avec les bibliothèques scientifiques et numériques: numpy et scipy. Ces deux bibliothèques permettent de remplacer une grande partie de l'approche itérative du calcul des prix sombres par des opérations matricielles et des fonctions vectorisées. Cela se traduit par un code rapide et facile à lire, qui ressemble beaucoup aux équations répertoriées dans la section 2.2. Comme indiqué ci-dessus, notre code peut fonctionner parallèlement au logiciel Tranus, permettant aux modélisateurs de calibrer partiellement certains paramètres et puis de passer sur Tranus pour continuer leur travail. Possibilités futures L'application la plus évidente de ce travail serait que Tranus accepte la reformulation du module d'usage des sols en tant que problème d'optimisation. Cela pourrait être fait, d'abord en séparant le module d'usage des sols en deux parties: transportable et non transportable.Ensuite, le sous-module non transportable pourrait être implémentée avec les techniques de calibration des facteurs de pénalisation exposées dans cette thèse. En outre, l'intégration de la calibration des fonctions de demande pourrait être ajoutée en même temps. Même si les fonctions de demande sont actuellement calibrés à l'extérieur, on pourrait penser à une calibration qui intègre à la fois des facteurs de pénalisation et des fonctions de demande.

Table 2 .

 2 1: Substitution probabilities for dispersion parameter σ = 0.01

	Type of housing	Price	Demand [a mn i ]	Expenditure [p • a mn i ]	σ	ω	σ • ω• expenditure	S
	small app.	10	22	220	0.01	2	4.4	55%
	mobile home	7	30	210		3	6.3	8%
	detached house	12	40	480		1	4.8	37%

Table 2 .

 2 From table 2.1, we have that 55 low-income households will consume small appartements, 8 mobile homes and 37 detached housing. We can change the value of the dispersion parameter σ and obtain different proportions, but the preferences are maintained. See table 2.2.

	Type of housing	Price	Demand [a mn i ]	Expenditure [p • a mn i ]	σ	ω	σ • ω• expenditure	S
	small app.	10	22	220	0.02	2	8.8	67%
	mobile home	7	30	210		3	12.6	2%
	detached house	12	40	480		1	9.6	31%

2: Substitution probabilities for dispersion parameter σ = 0.02

Table 3 .

 3

	Number Name	Type
	1	Basic Employment	Exogenous
	2	Service Employment	Transportable
	3	Low Income Household	Transportable
	4	High Income Household Transportable
	5	Land	Non transportable

1: Example C: Economical sectors description

Table 3 .

 3

		2: Demand functions parameters
	m n Min	Max	Elasticities δ mn
	1	3 1.998969	1.998969	0.0
	1	4 1.248126	1.248126	0.0
	1	5 0.004	0.01	-7e-01
	2	3 1.609238	1.609238	0.0
	2	4 1.448615	1.448615	0.0
	2	5 0.003	0.009	-8e-01
	3	2 0.1203459 0.1203459	0.0
	3	5 0.003	0.008	-7e-01
	4	2 0.1532743 0.1532743	0.0
	4	5 0.005	0.012	-6e-01

Table 3 .

 3 3: Base year's data

	Sector Zone ExogProd X * 0	InducedPro X0 Price
	1	1	5000	0	N/A
	1	2	800	0	N/A
	1	3	1100	0	N/A
	2	1	0	3500	N/A
	2	2	0	700	N/A
	2	3	0	900	N/A
	3	1	0	4000	N/A
	3	2	0	13000	N/A
	3	3	0	5000	N/A
	4	1	0	1500	N/A
	4	2	0	3000	N/A
	4	3	0	11500	N/A
	5	1	0	66	2.5
	5	2	0	110	1.2
	5	3	0	128	1.8

Table 3 .

 3 8: Shadow Prices: computed values and, in brackets, percentage of these values relative to prices for the same sector and zone

			Zone
	Sector	1	2	3
	2 -2.493 (-36.7)	1.069 (24.2)	1.424 (33.9)
	3	0.656 (75.9)	-0.390 (-44.9) -0.266 (-31.1)
	4 1.251 (113.3) -1.014 (-91.5) -0.238 (-21.7)

Table 3 .

 3 9: Comparison of calibration algorithms for the Mississippi model. Shown are the percentage of random trials for which the algorithms converged to the correct solution.

	value:	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
	Tranus	100%	100%	63%	3%	0%	0%	0%	0%	0%	0%
	Our algorithm	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%

Table 4 .

 4 1: Parameters to Calibrate

	Intermediate model variables Parameter Description	Type of Calibration
	a mn i	min mn	The minimum consumption	(i)
		max mn	The maximum consumption	(i)
		δ mn	Demand Elasticity	(i)-(iii)
		h n i	Shadow price	(ii)
	P r n ij	λ n	Marginal utility of Income	(i)-(iii)
		β n	Logit dispersion parameter	(iii)
		h n i	Shadow price	(ii)
	S mn i	σ n	Logit dispersion parameter	(iii)
		ω mn

Table 5

 5

	.1 describes the various economical

For this model, we only focus on the non-transportable sectors, as the transportable part

Table 5 .

 5

1: NCT: Economical sectors description was already very well developed. This was our first model to test the methodology to optimise the penalising factors of the substitution sub-model presented in 4.2. Going back to equation (

Table 5 .

 5 2: Statistics of shadow prices computed over their initial values.(3.7). As this model has three housing sectors(31, 32 and 33) we will present the results by three graphs, one per sector. Figure5.1 presents the shadow prices of each housing sector for all zones. Shadow prices are presented as a percentage of the corresponding price for each zone, as this is the standard Tranus practice. In Tranus terminology, these values are called "adjusts".The corresponding mean and standard deviation computed across the zones are presented in table 5.2. From figure 5.1 and table 5.2 we can observe that sector 31 is overpriced (the shadow prices are negative, pushing expenditures down) compared to the other two. The standard practice for Tranus would be to tune the value of ω m,31 to make the sector 31 more attractive, in this case, to decrease their value and to reduce the penalised expenditure

	Sector	Mean	Std	Min	Max
	31	-17.518 3.611 -25.830	-6.739
	32	6.937 9.838 -15.807 37.354
	33	30.265 9.851	0.000 60.830

Table 5 .

 5 .3.The column "Weight" represents the population weight coefficient, Hhtype is the household 3: Example of data available for a logistic regression of penalising factors ω, for the NCT model.

		Weight Hhtype Housing Choice Expenditure
		3943.3 12	33	203
		2400.8 11	33	582
		1912.3 11	32	521
		2269.8 15	31	1705.7
		. . .	. . .	. . .	. . .
		Sector 11 Sector 12 Sector 13 Sector 14 Sector 15 Sector 16
	31	-1.123	-1.103	-0.709	-0.488	-1.152	-1.588
	32	-1.303	-1.549	-1.050	-0.763	-1.363	-1.703
	33	-1.343	-1.317	-0.899	-0.684	-1.459	-1.761

Table 5 .

 5 4: Penalising factors after phase 1 (computed by logistic regression). The results of the logistic regressions are presented in table 5.4. The regression was made without sign, so the values presented are for (-ω mn

	type, Housing Choice is the actual choice and Expenditure is the corresponding monthly
	expenditure (a mn i	• p n i ).

  ). Prices and shadow prices are absolute Figure5.3: Original ω values against phase 1 values. Red curves show prices, blue curves prices plus shadow prices with ω's set to 1, green is h + p after the logistic regression on the ω values. Top to bottom represent sectors 31, 32 and 33 for all zones. We have opted for line plots instead of bar plot because for comparison purposes this is visually better, even if data are discrete.

		Sector	Mean	Std	Min	Max
		31	-7.562 2.916 -16.872	-1.881
		32	-2.321 8.411 -22.768 20.023
		33	14.181 6.422	0.000 34.491
	Table 5.5: Statistics of shadow prices, after phase 1 (logistic regression of penalising factors).
	values (not percentages).
	$
	0.6591
	0.4848
	$	0		19		38 zones
	0.9426
	0.0000
	$	0		19		38 zones
	0.8673
						p h+p (omega=1) h+p (phase 1)
	0.0000
		0		19		38 zones

Table 5

 5

	.6 presents the statistics of this last phase, average

Table 5 .

 5 6: Phase 2 shadow price statistics. (20% search range)

	Sector	Mean	Std	Min	Max
	31	-0.048 1.719	-3.748	4.517
	32	-1.705 6.149 -16.919 10.825
	33	0.103 4.389 -12.603	9.599

Table 5 .

 5 7: Shadow price statistics without previous logistic regression. This Tranus model of the Mississippi region comprises the Chickasaw, Lee, Pontotoc, and Union Counties, including the areas of the four largest towns, which are Houston, Tupelo, Pontotoc, and New Albany. The analysis zones are census block groups, of which there are 103; there are 12 economic sectors divided in 3 types of employment, 6 socio economic categories of households and 3 types of floorspace (land). We have the same distribution of economic sectors as int the NCT model, exposed in table 5.1.After the successful application of our substitution probabilities optimisation for the NCT model, we applied the same methodology to the MS model. For this model we did not have access to survey data to establish a logistic regression (phase 1), but as we could observe for the NCT model, starting from penalising factors equal to 1 yields similar results as starting from a logistic regression (see previous section).

	19	38 zones

Table 5 .

 5 ). It is a clear improvement over the default values (blue curves). Statistics on the computed shadow prices are given in table5.8. We can conclude that this strategy of tuning the penalising factors has an immense impact on the values of the shadow prices, proving to be useful way to fine tune the calibration of floorspace sectors. 8: Shadow prices statistics for MS model after optimisation (phase 2).

	Sector Mean	Std	Min	Max
	31	-0.029 1.749	-5.118	4.131
	32	-1.122 9.363 -32.436 33.994
	33	1.362 9.768 -15.524 84.807

Table 5 .

 5

	sector	first-order	second-order	selected parameters: ρ variance explained (in percentage)

10. The last column of the table gives the proportion of the model's variance explained by the selected parameters. This proportion is calculated by multiplying the sum of the generalised Sobol' indices of the first two columns by 100. Looking at the results, only 3 parameters appear to be overall the most influent: β n , b n 10 and b n 11 , n ∈ {2, 4, 5, 6, 7, 8, 9}.

Table 5 .

 5 11: Variance and maximum of the normalised shadow prices hn obtained with both ad hoc procedures and our iterative optimisation.

	sector n procedure Optimal params.	λ n *	variance hn max hn gain
	2	ad hoc	2	0.005	0.32	2.95	98%
		iterative	4.03	0.43	7 × 10 -3	0.11
	4	ad hoc	(2, 0, 0)	0.001	13.66	24.95 83%
		iterative	(6.49, 0.38, 0)	0.001	2.26	7.63
	5	ad hoc	(2, 0, 0)	0.001	5.35	14.83 47%
		iterative (2.50, 0.02, 0.79) -0.013	2.85	8.88
	6	ad hoc	(2, 0, 0)	0.001	5.90	16.65 63%
		iterative (6.64, 0.05, 0.79) -0.003	2.18	7.72
	7	ad hoc	(2, 0)	0.001	8.73	19.67 61%
		iterative	(9.17, 1)	0.001	3.40	8.23
	8	ad hoc	(2, 0)	0.001	9.50	20.58 15%
		iterative	(5.72, 0.97)	0.001	8.08	15.03
	9	ad hoc	(2, 0)	0.001	7.36	17.6	64%
		iterative	(9.29, 0.95)	0.001	2.66	6.82

  Comme le montre la section 4.2, les fonctions et les prix de la demande sont la base du calcul des dépenses pénalisées qui définissent les probabilités de substitution. En outre, les fonctions de demande sont strictement liées aux prix observés du marché du logement et à la segmentation socioéconomique des ménages, de sorte que l'élaboration d'outils pour aider les modélisateurs à estimer les prix et les fonctions de demande en même temps serait très appréciée.Nous pensons également qu'il est essentiel d'explorer davantage les problèmes potentiels d'overfit liés au nombre de prix sombre présents dans l'état actuel de Tranus. La génération du scénario synthétique pourrait être utilisée pour tester cette hypothèse, en validant la performance des capacités prédictives de Tranus lorsque certains prix fictifs sont supprimés. Les prix sombres ont d'abord été ajoutés pour corriger les effets non modélisés dans les utilitaires, ce qui permet au modèle de s'adapter parfaitement aux données observées. Déterminer quels sont les prix sombres qui sont vraiment nécessaires, et à quel niveau d'ajustement nous aimerions reproduire les données observées est une question pour toute une autre thèse, mais des outils tels que l'analyse de sensibilité pourraient être utilisés pour identifier les prix sombres pouvant être supprimés en toute sécurité. Il est probable que l'ensemble des prix sombres varie d'un modèle à l'autre. Une méthodologie possible pour tester le schéma de sélection du modèle serait de comparer les résultats du modèle pour deux périodes afin de déterminer si le bruit dans les productions de l'année de base pourrait vraiment se propager aux prix sombres et nuire aux capacités prédictives du modèle. Chapter A. Details on Tranus' shadow price iteration scheme Algorithm 2 Shadow prices computation algorithm 1: procedure newPrices(zone i , sector n, λ n ) as damp = 1/(1+ ), where is chosen by the user in the interface (called smoothing factor). Users usually starts with values of around 2 or 3 (damp = 1/3, 1/4) and gradually reduces the value to end up with values close to = 1 (damp = 1/2).Table B.1: Demand functions parameters for the Grenoble Model

	2: 3:	if X n i = X n 0i then new P r i ce = (p n i + h n i )	X n i X n 0i
	4:	else
	5:	new P r i ce = (p n i + h n i )
	6: 7: 8: 9:	new P r i ce = (p n i + h n i )λ n + (1 -λ n ) • new P r i ce new P r i ce = (p n i + h n i ) • (1 -damp) + damp • new P r i ce p n i = new P r i ce -h n i return p n i , h n i
		m	n	Min Max Elast.	m	n	Min Max Elast.
		10 100 20 10 101 40 10 102 55 10 103 100 10 104 111 10 105 18 10 106 55 10 107 44 10 108 100 12 100 20 12 101 40 12 102 55 12 103 100 12 104 111 12 105 18 12 106 55 12 107 44 12 108 100 14 100 20 14 101 40 14 102 55 14 103 100 14 104 111	39 97 91 270 200 42 88 80 235 39 97 91 270 175 42 88 80 235 39 97 91 270 175	0.0222 0.0154 0.0201 0.0857 0.0633 0.0314 0.0568 0.0305 0.1055 0.02775 0.01925 0.025125 0.107125 0.079125 0.03925 0.071 0.038125 0.131875 0.0555 0.0385 0.05025 0.21425 0.15825	14 105 18 14 106 55 14 107 44 14 108 100 16 100 20 16 101 40 16 102 55 16 103 100 16 104 111 16 105 18 16 106 55 16 107 44 16 108 100 20 100 20 20 101 40 20 102 55 20 103 100 20 104 111 20 105 18 20 106 55 20 107 44 20 108 100	41 88 83 225 39 97 91 270 175 41 88 83 225 39 97 91 270 200 42 88 80 235	0.0785 0.142 0.07625 0.26375 0.0444 0.0308 0.0402 0.1714 0.1266 0.0628 0.1136 0.061 0.211 0.0296 0.02053333 0.0268 0.1142667 0.0844 0.04186667 0.07573333 0.04066667 0.1406667

Modelistica: http://modelistica.com, Stratec: http://stratec.be

Achieving perfect equality between observed productions and productions generated by the model, is in general possible since there are as many shadow prices to adjust, as there are observed productions. In section 3.6.4, we discuss ideas for alternative rationales.

http://www.tranus.com/tranus-english

If the attractors W n i are different from 1, the constant in the logistic regression could account for some of their value.

INSEE is the French National Institute for Statistics and Economic Research.

Acknowledgements: The writing of this thesis took place within the STEEP team at INRIA Grenoble, in offices G104, E101 (also a little bit in the office of Jean-Yves and Luciano).

I could never have done this research without the support of all the people I met here.

apply our optimisation methodology to two North-American models, particularly to improve the penalising factors, and later, to a model of the Grenoble urban area.

we can completely compute demands as constants that do not depend on prices nor shadow prices, see table 3.4. From demands, one can compute productions X n j using equation (2.7) as follows:

The logit P r n ij comes from equation (2.4) and is a function of the location utility defined in equation (2.3): U n ij = (p n j + h n j ) + t n ij . We will utilise the same notation as in section 3.4, let's define φ n j = h n j + p n j . We will rewrite the utility function as U n ij = φ n j + t n ij . The transportation disutilities t n ij come from the transportation module and they are given as input to the land use module (table 3.5, left). The location probabilities are a function of our new variable φ, each location probability is dependent on the same sector and all zones' φ variable, P r n ij (φ n k : ∀k), this comes from the fact that P r is a logit probability, and the denominator has all possible choices, hence, involves all other zones. We will denote φ n • = {φ n k } k the vector for all zones. The optimisation problem we must solve is as follows:

As for the non-transportable sectors, we have separated optimisation problems, but instead, one per economical sector. However, these problems are larger, with as many variables as zones (3 for this example). The optimal solution is presented in the following table (table 3.6): These values represent the sum of p + h, so to identify both parameters separately we must look at the equations that impose equilibrium in prices. These equations are equation (2.8) and (2.9). First, in equation (2.8) we can replace X n ij /D n i by P r n ij using equation (2.5). As we have found the optimal values of φ, the location probabilities are determined by these values. So prices have to satisfy the following linear system:

The coefficients tm n ij represent the monetary costs of traveling (these values come from the transportation module) and are presented in the right side of table 3.5.

Finally we solve the linear system in prices (3.17) to obtain: and the corresponding shadow prices are obtained by subtracting the prices from the φ.

Equilibrium prices in synthetic scenario: 1 economical sector, 2 zones

In this section we present a simple example that shows that the equilibrium pricing problem necesary to construct synthetic scenarios exposed in section 3.6.1 can be complicated, and that uniqueness of the solution is not guaranteed. Let us consider only one economical sector m = 1 (we will just drop the exponent m in the following) and two geographical zones i, j ∈ {1, 2}. Let us consider no substitution, i.e. S 11 1 = S 11 2 = 1. The equilibrium condition (3.18) can be re-written by two equations:

It is important to notice that tm 11 = tm 22 = 0. This simple case is very sensitive to the values of the different parameters. We managed to find combinations of the different parameters (V A, a i and tm ij ) that give rise to multiple solutions, one solution or no solution at all, for the prices, see Figure 3.4. The curves where very sensitive to the demand coefficient a i . This example shows that modifying a certain parameter can shift the whole set of prices to a different equilibrium, and that the modeller has to be aware of this behaviour. From left to right: no solution, one solution and multiple solutions.

We were curious to know if this problem had multiple solutions, and even in this simple case it had proven to be complex. This gives us a starting point to further investigate the problem of potential existence of multiple fixed points for our calibration problems, even though in practice we have not observed problems of convergence to wrong local minima.

Grenoble model

The Grenoble model is developed by Fausto Lo Feudo, Brian J. During a model conception, the first thing to do for the non-transportable sectors is to estimate the prices for the housing market. The latter is done at the same time as the estimation of the demand functions, particularly the demand for land. We will not discuss here how these intermediate variables are estimated and we will consider them as input for our methodology.

Calibration of substitution sub-model

For this model we did not have individual data on floorspace consumption, so a logistic regression was not possible. We applied the same methodology as for the Mississippi model, starting from penalising factors set to one and optimising with our scheme to obtain a better

Conclusions

The Tranus LUTI framework is a very powerful tool and the modelling possibilities are endless. However, the complexity of such large scale models is something that can not be underestimated, making the calibration and utilisation of these tools very expensive.

In this thesis, we have contributed with a reformulation of the land use module that simplifies the calibration process. To do this reformulation, we had to present in a different way the equations involved in the computation of the land use module, exploiting the very basics of the mathematics that are behind the microeconomic models used, establishing the needed cost function (and derivatives) for the optimisation algorithms to succeed.

A lot of time and effort has been put to understand the various interactions between the parameters to be able to divide the calibration in smaller problems, thus leading to the two stage optimisation finally proposed in chapter 3 (for non-transportable and then transportable sectors). For instance, the reformulation for the transportable sectors exposed in section 3.4

is not evident, and only became possible after working with the equations analytically as shown in the numerical example in section 3.5.1. The optimisation approach is more stable and clear than the classical approach, and enables the use of powerful optimisation algorithms currently available, solving the occasional non convergence issues of the previous approach.

To be able to test our optimisation approach for the calibration of the shadow prices,

we have proposed a "first for Tranus" procedure for generating synthetic data that is simple and straightforward, enabling us to try and benchmark our methodologies. This procedure could also be used to perform validation of the model, creating a "present" as well as a synthetic "future" scenario to compare the predictive capabilities of Tranus. The latter, is not as simple as it sounds, and could be considered as one perspective for future work. Also, in predictive mode, the land use module reformulation that we proposed in this thesis would need to be adjusted, as the technique of exploiting the base's year production (cf. section

Future possibilities

The most obvious application of this work would be for Tranus to embrace the reformulation of the land use module as an optimisation problem. This could be done, first by separating the land use module in two parts: transportable and non-transportable. Then the non-transportable sub-module could be implemented with the penalising factors calibration techniques exposed in this thesis. Also, the integration of the calibration of the demand functions could be added at the same time. Even if the demand functions are currently calibrated externally, one could think of an integrated calibration of both the penalising factors and the demand functions. As shown in section 4.2, the demand functions and prices are the input to computing the penalised expenditure which defines the substitution probabilities. Also, the demand functions are strictly related to the observed prices of the housing market and the socio-economic segmentation of households, so developing tools to help modellers estimate prices and demand functions at the same time, would be much appreciated.

We also think it is crucial to explore more the potential overfitting issues related to the number of shadow prices present in the current state of Tranus. The synthetic scenario generation could be used to test this hypothesis, validating the performance of the predictive capabilities of Tranus when some shadow prices are removed. The shadow prices were initially added to correct for the un-modelled effects in the utilities, making the model fit the observed data perfectly. Determining which shadow prices are really necessary, and at what level of fit

Appendices

Appendix A

Details on Tranus' shadow price iteration scheme

The work from Hyman [START_REF] Hyman | The Calibration of Trip Distribution Models[END_REF]) represent the first attempt to propose a systematic calibration for spatial interaction models. He proposes an iterative scheme and utilises the average trip length S as the calibration indicator. The iterative approach computes in each iteration the new parameters λ as a fraction of the previous iteration Sn and the observed value S * . This example with the details of the computation of S can be found in [START_REF] Batty | Urban Modelling[END_REF]). Equation (A.1) explicits the relationship between the parameter λ n+1 in iteration n + 1, and iteration n.

Hyman, also suggested a linear interpolation procedure to speed up the convergence of the system. Thus utilising the computed values of two previous iterations, as shown in equation Appendix C

Definition of generalised Sobol' indices

Consider the following model:

where y is the output of the model f , x the input vector and d the dimension of the input space. Let (Ω, A, P) be a probability space. The uncertainty on the inputs is modelled by a random vector X = (X 1 , . . . , X d ) whose components are independent. Denote by Y the corresponding output:

Let P X = P X 1 ⊗ . . . ⊗ P X d denote the distribution of X. Suppose that f ∈ L 2 (P X ) and that the covariance matrix of Y , denoted by Σ, is positive definite. Let u be a subset of {1, . . . , d} and denote by ∼ u its complementary. We set X u = (X i ) i∈u and X ∼u = (X i ) i∈{1,...,d}\u .

Recall the following Hoeffding [START_REF] Hoeffding | A class of statistics with asymptotically normal distributions[END_REF]) decomposition of f :

where

By taking the covariance matrix of each side of (C.1), due to orthogonality we get:

Let M be a matrix of dimensions p × p, Equation (C.2) can be projected on a scalar as follows: When u = (v , w ) is a 2-subset of {1, . . . , d}, the influence of the interaction between v and w is quantified by the second-order generalized Sobol' index defined by: S (v ,w ) (f ) -

Classical estimation of S u (f ) The classical estimation procedure for S u (f ) is a generalization of the one used in the univariate case [START_REF] Sobol | Sensitivity estimates for non linear mathematical models[END_REF]). The procedure consists of a Monte-Carlo pick-freeze method. In the pick-freeze method, the Sobol index is viewed as the regression coefficient between the output of the model and its pick-freezed replication. This replication is obtained by holding the value of the variable of interest X u (frozen variable)

and by sampling the other variables X ∼u (picked variables).

We set Y u = f (X u , X ∼u ) where X ∼u is an independent copy of X ∼u . Let N > 0 be an integer and Y 1 , . . . , Y N (resp. Y u 1 , . . . , Y u N ) be N independent copies of Y (resp. Y u ) where:

As in (Janon et al. 2014; Monod, Naud, and Makowski 2006), the following estimator of S u (f ) is constructed: We propose here an extension of the replication method to the multivariate case. With this new approach, first-order and second-order generalized Sobol' indices can be estimated with fewer model evaluations.

Replication procedure for S u (f ) The replication method relies on the construction of two replicated designs of experiments X and X' defined as follows: