come

Epilogue: Of Theories, Typology and empirical data

Bettina Migge University College Dublin, Ireland

This book examines the place of creoles from a typological perspective using modern phylogenetic modeling tools. Exploring the similarities and differences that exist among creoles and between creoles and their input languages, the authors aim to generate new insights into persistent and at times hotly debated topics such as creole genesis and the relationships among creoles and between creoles and other languages, most specifically their input languages. The volume casts a very wide net. It investigates creoles from the Atlantic, Asian, Pacific and Africa region and also considers creoles associated with a range of so-called lexifiers (Arabic, Dutch, English, French and Iberian), common superstrates and some of the substrate inputs. But diversity does not stop there either. The authors also examine data from a range of linguistic levels, including phonological, morphological, syntactic and semantic phenomena.

The first four chapters set the stage for the research reported on in this volume. Chapter 1 introduces readers to the topics to be investigated in this volume via a broad style overview of what the research team consider to be the most important issues in research on creoles. For instance, the chapter sketches some positions on the linguistic processes and agents of creole genesis, opposing views on what constitutes a creole and whether they are a typologically distinct group of languages, and presents some of the proposals as to the reasons for their alleged uniqueness. The second chapter provides a very insightful introduction to current phylogenetic work, including discussion of its origin, development and methods. Chapter three 'scores' creoles from different regions and with different European and substrate input languages using a range of linguistic features (phonology, morphology, constituent ordering, lexicon) while chapter four critically examines prior typological research on creole languages. Chapters five to thirteen report on the different types of phylogenetic analyses carried out by the research team. The relative homogeneity of African input languages to Atlantic creoles and their impact on creoles is examined in chapter five while chapters six, eight, nine and ten examine the relationships among creoles that were influenced by French, Dutch, and Iberian languages, respectively and between them and their main European input language. Chapter seven investigates the similarities and differences between Juba Arabic, one of the Arabic contact varieties, and its input languages. Finally, Chapters twelve and thirteen explore the representation of lexical concepts across a broad range of creoles (Chapter twelve) and among creoles for whom English was an important input language (Chapter thirteen).

Based on the results of the different scoring exercises, the authors arrive at a number of conclusions about the typology of creoles: (a) "there are no linguistic properties that are unique to creole languages" (b) creole grammars are not inherently simpler than those of other languages (c) however, there are broad generalizations that cover most or all creoles (d) the broad similarities between creoles are due to processes of simplification that took place prior to their genesis (e) most creoles with the same European inputs cluster together but they do not form a homogeneous group; sets of subclusters can be identified (f) creoles do not cluster with their substrate inputs The team's use of phylogenetic research methods can be taken to provide an empirically grounded insight into longstanding hunches and issues invoked by people working on creoles and those from related fields. Two things that I keep hearing from linguists working on other languages, including institutions that fund research on language documentation, for instance, is the either openly worded or implicitly presented assumption that all creoles are alike and are naturally simpler or unmarked linguistic entities. 1 The analyses presented in this volume visually represented by the numerous phylogenetic trees clearly belie these received notions, showing that there is, in fact, quite a bit of variety among the languages referred to as creoles. Equally interesting is the fact that this diversity is not simply due to differences in lexifers, but also occurs among creoles that were influenced by the same European language. Leaving aside for a moment the empirical difficulties inherent in defining notions such as simplicity and complexity, chapter three in particular highlights the fact that creoles are not inherently lacking in complexity but, like all languages, show different degrees of complexity across different areas of grammar. While I don't have hopes that the rest of the linguistic community will, on foot of this book, magically deconstruct their long-standing views about contact languages that continue to be deeply rooted in the colonial linguistic enterprise (e.g. [START_REF] Mühleisen | Creole Discourse. Exploring Prestige Formation and Change across Caribbean English-lexicon Creoles[END_REF], it definitely presents another important piece of evidence towards confirming that creoles are natural languages.

Reading the different contributions I was also struck by a number of issues. For reasons of space, I will limit my discussion to only a few that seem most pertinent to me right now. My main caveat concerns the choice of features. While the team made a concerted effort to draw on a wide range of linguistic features, I cannot stop but feeling uneasy when looking at the various phylogenetic trees and the wide-ranging conclusions that are drawn based on them. For once, while the authors identified various problems with previous, in many cases more localized, typological approaches, I am not convinced that the analyses in this volume resolve the main issue that plagues all typological studies: representativeness. The selected features, though enriched by features used for broad cross-linguistic comparisons (e.g. WALS),2 remain abstract and in many ways closely overlap in type and kind with those used in previous attempts. As a result, the authors are not able to report any ground breaking new results as such, but are mostly confirming or adding some (minor?) detail to previous typological endeavors based however on newer data sets for the creoles (e.g. APiCs database) and using enhanced modeling methods.

Second, and more fundamentally problematic are the assumptions underlying this enterprise. I find it surprising or actually disconcerting that after decades of research on a wide range of aspects relating to language it is apparently still considered perfectly normal to assume that the essence of a language can be captured based on a narrowly defined set of structural features. While it is true, and regrettably so in my opinion, that the bulk of linguistic research continues to be preoccupied with a narrow set of phonological, morpho-syntactic, semantic and lexical etc phenomena mostly derived from a decontextualized set of sentences or word lists obtain using behaviorist data collection methods (e.g. elicitation), I am not sure that it is simultaneously warranted to assume that these features are in any way central to defining languages and central to defining relationships between languages. This kind of reasoning seems to be a highly problematic and indicative of a lack of critical reflection among linguists about their endeavors, including its historical roots and developments. If we are interested in getting new insights into the relationships between languages, should we not first deconstruct common assumptions about what constitutes a language and consequently our own endeavors (linguistic research), 3 second consider what is central or indicative about human communication systems and how we can study it and third invest more time and effort in the reconstruction of the contact settings (and the histories of speakers of languages)?4 I submit that this is particularly important when dealing with non-European languages such as creoles that have had to suffer from centuries of willful and accidental distortion by mostly Euro-American (trained and untrained) linguists. My feeling is that uncritical feature crutching is perpetuating and maybe also infusing new life into received notions about language simply by using more sophisticated tools to do it. Just to clarify: I have no problem with feature crutching, but you have to critically assess what you are crunching and to what end. Why do pragmatic and sociolinguistic aspects of language never feature in typological work?

Every time I engage with speakers of Maroon languages in Suriname, French Guiana or beyond in both mundane and more formal settings in urban and rural contexts through oral communication or emedia, I am always struck by the vast amount of meaning making that takes place that has to date remained firmly below the surface of most of our research activities. In fact, research into these aspects of language which require more than just passing knowledge of a few decontextualized features continues to be overtly and covertly discouraged because it is unsettling as it requires engaging with and developing new kinds of research modalities and much closer and democratic collaboration with the speakers of the languages that we study. But I believe that embarking on this unknown adventure will fundamentally expand our current thinking about creole languages and languages in general and will help us to rise above impoverished feature matrixes and dubious word lists. In order to get there, however, we need to critically examine Chomskyan notions of (linguistic) competence that continue to dominate much of what's going on in the field of linguistics and transition towards Hymesian notions of communicative competence and beyond.

Chapters twelve and thirteen make an attempt at breaking away from the stranglehold of typical structural linguistic concerns by focusing on semantics or what the authors call patterns of lexicalizations. Instead of using the Swadish list, the authors are basing their work on current lexico-semantic approaches. However, oddly enough, the comparisons focus for the most part on the nature of the etymological shapes of words and their most basic meaning rather than on a full analysis of the actual meanings and functions of the words [START_REF] Huttar | Gbe and other West African sources of Suriname creole semantic structures: Implications for creole genesis[END_REF]. So again, in order to enable broad comparisons, the languages have been reduced to simplified objects. Can we not imagine a more sophisticated way of doing comparisons that will not require erasure of potentially important information?

A final point about features relates to the notion of stable linguistic features invoked in Chapter five. Here a selection of creoles and African languages (and European languages) are compared "to test the claim that the African languages involved in the colonial settings, and hence in the creation of various creole languages, were typologically homogeneous." However, instead of selecting a range of features, curiously, the authors base their comparison on what they describe as features that are "relatively stable over time" because they "were most likely inherited rather than borrowed, these are assumed to represent well-suited markers of distant genealogical relationships". If creoles arose from creative and in many ways idiosyncratic situated interaction between different sets of linguistic practices employed by human agents to fill their communicative needs, as most people working on creole genesis nowadays believe, then why would either stable or unstable linguistic features be privileged over other features in creole genesis? It seems to me that both features have an equal chance of being propagated in this kind of a setting and thus the assumption that the presence or absence of certain kinds of linguistic features is indicative of a certain kind of relationship between creoles and their input languages is problematic.

There are also other kinds of reductionisms. For instance, in some places languages are depicted as a living organisms and sociolinguistic processes such as language contact are likened to biological processes ("lateral gene transfer"). Are we regressing to the view that languages are bounded objects and that signs of variation, change and 'outside' influence are a sign of degeneration? Even if the book uses analytical tools derived from biology, there is no need for such unwarranted metaphoric extensions. Languages are very different from cells and bodies and any attempt at reheating overcooked parallels between them are not only running the risk of misrepresenting language related matters but are also propagating views from the colonial and nationalist era. Another kind of reductionism relates to the presentation of research on creoles such as creole genesis, for instance. A rather complex and multifaceted area of investigation is reduced to a few somewhat eccentric theoretical approaches (superstratist versus relexification accounts; abrupt versus life-cycle accounts of creole formation) in order to generate a few succinct hypotheses that can serve as a basis for comparative exercises. This seems rather crude especially because there is little reference to existing critiques about these approaches so it is not quite clear what the research in this volume actually contributes to the discussion. And on the other hand, because there are few references to more nuanced approaches, it represents research in this area as somewhat simplistic.

My final point relates to the interpretation of phylogenetic trees. On the one side, these visual representations of relationships between languages are great because they allow us to grasp a great amount of complexity pretty much instantaneously. However, isn't this also part of the problem? When you look at the trees in more detail, it is in fact not quite clear how to interpret them reliably because the multitude of intersecting lines is actually quite hard to separate out. The more I look at the trees in the book, the less clear I am about how to derive significance from these webs of lines.

In summary, while I commend the authors on the huge amount of work that they put into the analyses that figure in this volume, I am uncomfortable with the uncritical glossing over a number of pertinent issues and the lack of willingness to push beyond the narrowly defined limits of current typological work.

And thus, due to their alleged lack of unique features, are of lesser interest to structural, documentation and typological research.

SeeKouwenberg (2010a&b) for a discussion of the usefulness of WALS features(Dryer et al 2013) in typological research on creoles.

The interested reader might start with reading some recent publications about what has come to be known as colonial linguistics (e.g.[START_REF] Errington | Linguistics in a colonial world: A story of Language, Meaning and Power[END_REF][START_REF] Makoni | Disinventing and reconstituting languages[END_REF] and to consider recent discussions in the area of language documentation (e.g.[START_REF] Childs | Beyond the Ancestral Code: Towards a Model for Sociolinguistic Language Documentation[END_REF].

In this regard, it is equally important to take on boardKouwenberg's (2010a&b) remarks on whether it makes sense to compare creoles to existing typological databases that are unsuitable for dealing with contact languages.


des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Further Studies in the Lesser-Known Varieties of English. By Jeffrey R. Williams, Edgar W. Schneider, Peter Trudgill & Daniel Schreier (eds.). (Studies in Language Series). Cambridge: Cambridge University Press. 2015. Pp. xvi, 345. Hardback $105. Although much of linguistic research is preoccupied with widely used and known languages and varieties that function as important social capital, there has always also been interest in so-called non-dominant languages. Early work, carried out by missionaries and those employed by imperialist governments often followed an assimilationist agenda, aiming to spread colonial worldviews and life-styles and/or to prove the inadequacy of other languages [START_REF] Errington | Linguistics in a Colonial World: A Story of Language, Meaning and Power[END_REF]. Modern linguistic research on non-dominant languages is predominantly less overtly concerned with larger societal questions and focuses instead on discipline-internal goals such as the discovery of new facts about human language, the testing of theoretical claims about language (Dixon 2008), and the gathering and preservation of linguistic data for posterity [START_REF] Himmelmann | Reproduction and preservation of linguistic knowledge: Linguistics' Response to language endangerment[END_REF].

Varieties of English do not typically figure in language documentation efforts, but research on them follows the broad goals of descriptive linguistics. The editors of the present volume, for instance, justify their focus on so-called lesser-known varieties by arguing that they might provide new "insights into larger questions in linguistics and sociolinguistics" (p. 1). Instead of discussing the precise nature of these insights, the editors tend towards questions of definition though. They propose eight characteristics to define the term lesser-known varieties of English and to justify juxtaposing descriptions of thirteen varieties of English from three broad regions -Europe, Americas, Asia and Pacific -in a single volume. Some criteria refer to broadly linguistic matters (linguistic distinctiveness, emergence from contact), others to sociolinguistic issues (important local means of communication, association with a stable speaker community, speakers as minorities, identity function, endangered status), and yet others are historical in nature (emergence from settler communities, adoption by emerging communities with substantial British inputs).

I first summarize the main socio-historical, sociolinguistic and linguistic characteristics of the thirteen varieties of English. I then assess whether the varieties in the volume constitute a distinctive category of varieties, as suggested by the editors, before commenting on a few final aspects of the volume. I follow the authors' naming conventions for referring to varieties.

The varieties involved originate from different varieties of English. Palauan, and Malta and Gibraltar English are emerging from American and British English respectively but in all three cases an English-speaking settler community was not at the center of these relatively recent developments. American Indian English developed and spread in schools set up for American Indian children while Caribbean varieties developed from European Englishes -English, Irish and Scottish -spoken by indentured laborers and freed slaves, but also received important inputs from American English varieties (e.g. St Eustatius, Gustavia, Saban Englishes). Gullah West (also locally called Seminole and Mascogo) developed from different Caribbean Englishes and Creoles on North American plantations. Finally, Anglo-Paraguayan English originates with a community of Australian English speakers, Traveller English from Irish and English varieties of English.

Migge Bettina. 2016. Review

With respect to their sociolinguistic situation, we find out that all varieties exist in contexts of contact though involving different levels of intensity. On Saba, St Eustatius and Palmerston Island contact with Dutch and Cook Island Māori respectively happens through the education system. In Malta, Gibraltar and Palau there is intense contact with the widely used L1s (Maltese, Yanito, Palauan) and other languages, while Bequia, American Indian, Gullah West, Palmerston Island, Traveller and Pasifika Englishes are mainly in contact with other Englishes. English plays an important role in all communities, but the details differ. Many varieties are the main vernacular of an ethnically or regionally defined community (e.g. Travellers, American Indians, Bequians, Sabans, Gullah West, St Eustatians, Gustavians, Palmerston Islanders) or of closely related communities (Pacific islanders). Some are spoken by national though not always recognized minorities (Traveller, Gustavia, Anglo-Paraguayan, American Indian, Pasifika Englishes) while others 'belong' to entire island communities (Palmerston Island, Bequia, Saba, St Eustatius, Gibraltar, Malta, Palau). Gibraltar, Malta, and Palau English are only now vernacularizing while others are currently acquired as first languages and used as community languages. Speaker numbers also vary. In some cases they are tiny (Palmerston Island English) while in others they are comparatively substantial (e.g. Malta, Gibraltar, Palau). Only some of the varieties are endangered (e.g. Gustavia English), one no longer has any speakers (Anglo-Paraguayan Englishes) and some are under pressure from other varieties of English (Pasifika Englishes, Irish Traveller English, American Indian Englishes). Finally, the varieties also perform different identity functions. Pasifika and American Indian Englishes index pan-community identities, Caribbean Englishes village-based identities that originate in settlement histories, and Palmerston Island English is linked to a clan founded by an Englishman. In Gibraltar, Malta and Palau, it is the hybrid patterns of use that instill a sense of identity.

The detailed linguistic descriptions are based on corpora of sound recordings which, however, vary in size and makeup. Some consist of a mere one hundred minutes (e.g. Anglo-Paraguyan) while others are substantial publicly available corpora (e.g. Maltese English). Some include mostly synchronic interview data and others also include situated language use and have a diachronic component. Despite this diversity, data sets are broadly comparable. The linguistic descriptions focus on distinctive traits, including variable linguistic features, and generally cover phonological, morpho-syntactic, syntactic and lexical properties. Some authors also mention pragmatic features such as discourse markers (Maltese, Traveller, American Indian, Pasifika Englishes) and/or usage-based issues namely code alternation practices (Gibraltar, Malta, Palauan Englishes).

As in other varieties of English, a number of common vocalic features are attested. Vowel mergers commonly affect one or more vowels -FLEECE-KIT, FOOT-GOOSE, LOT-THOUGHT and TRAP-STRUT -but are only socially stratified in some varieties as in Gibraltar where they are characteristic of older people's speech. Monothongization of the FACE and GOAT diphthongs and to a lesser extent those in NEAR and SQUARE (Palauan, Bequia English) occurs widely. Apart from slightly different realizations of vowel nuclei (e.g. PRICE in Saban and Gustavia English), some varieties do not at all (Maltese Englishes) or only variably (Bequia English) reduce vowels in unstressed syllables. In others (Bequia Englishes), long vowels (CLOTH, PALM) are shortened and centralized.

With respect to consonants, rhoticity varies across and within varieties due to social (Maltese English) or phonological factors (St Eustatius, Bequia, Pasifika and Palauan English). Only Gibraltar English appears to be fully non-rhothic and /r/ is also realized as a retroflex (Bequia English) or a trill (Palmerston Island English). THstopping is common but in some varieties there is variation between stop and interdental fricative realizations (St. Eustatius, Gustavia English) or between stops or alveolar fricatives (Palmerston Island, Pasifika English). Alveolar /l/ only is found in Gibraltar, Bequia, American Indian English while variation with velar /l/ is present in others (Palauan English). Other common processes include the merging of /v/ and /w/ (St Eustatius, Gustavia, Palmerston Island Englishes), yod-dropping (Traveller English), palatalization (Gustavia English), and the avoidance of consonant clusters (Palmerston Island English). American Indian English is probably unique in replacing oral stops with the glottal stop. Several articles also mention unique suprasegmental features such as variation in stress placement (Palmerston Island, Palauan, Traveller, Maltese Englishes), differences in pitch patterns (American Indian, Traveller English), and the adoption of a syllable-timed rhythm (Palauan, American Indian, Gibraltar English).

In relation to morpho-syntax, chapters highlight features in various domains. In the noun phrase, features documented include the variable omission of definite articles (Maltese, Traveller English, American Indian English), the absence of non-redundant plural marking (Traveller, American Indian, Bequia, Saban, Seminole, Palauen Englishes) and/or the use of analytic plural makers (e.g. dem) (Bequia, St Eustatius, Seminole). For pronouns, there is variation in the use of reflexive pronouns (Traveller, Bequia English), an absence of gender distinctions in 3 rd person singular pronouns (American Indian English, Gullah West), and a higher incidence of null subjects (American Indian, Bequia, Palauan Englishes). For the verb phrase, the chapters deal with morphological issues such as the absence of subject-verb agreement, variation in the realization of the copula (Gullah West, Bequia, Saban, St Eustatius, Gustavia English) and marking of temporal distinctions and negation. Varieties use non-standard negation markers such as variants of ain't (Bequia, Saban English) or nah (Seminole). There is cross and intra-variety variation in the interpretation of the unmarked verb form: it may have present and past time (Bequia, Saban, St Eustian, Gullah West) or also future time reference (Palmerston Island English). Past time is also conveyed by special verb forms or by preverbal bin (Seminole, Bequia English) and future by preverbal will or going. Habitual aspect shows the greatest amount of variation being expressed by be and -s (Saban English), do be (Traveller English) and doz, deh, ah and forms used in standardized varieties (Bequia, St Eustatius Englishes). Finally, flexibility in word order is attested in American Indian Englishes.

The overview discussion suggests that the thirteen varieties do not constitute a distinct category of varieties as regards their sociolinguistic status and history as they emerged and currently exist in very different social conditions. Linguistically, there appears to be less diversity. Unless this results from a theoretical bias, this would be evidence in favor of their distinctiveness, if the same phenomena did not also figure prominently in discussions of other varieties of English. There is some indication of subtle differences in the distribution of shared variable features but since most contributions do not systematically explore patterns of variation using quantitative or qualitative sociolinguistic methods, or refer to related studies that do this (e.g. the chapter on Bequia), this level of distinctiveness remains mostly hidden. Finally, there is the issue of origins of features, which does receive attention in the article on Traveller English. Other chapters occasionally invoke different mechanisms of change (substrate or adstrate influence, retention, language-internal changes) or highlight special social conditions such as relative isolation -Gibraltar, Palmerston Island English -and settlement histories -Bequia, Saba English -but do not discuss the linguistic and sociohistorical linkages in-depth.

The chapters also vary in other respects. Some articles are around forty pages long (Malta, Palau Englishes) while others are less than half that length (Bequia, Ango-Paraguyan, Pasifika Englishes). In some, primary importance is given to linguistic aspects (Bequia Englishes), whereas others include substantial sociohistorical sections with a less detailed linguistic section (Anglo-Paraguyan Englishes). Many of the chapters privilege sociohistorical over sociolinguistic matters in line with the current diachronic focus of research on varieties of English. Moreover, patterns of variation are noted, but especially their social correlates receive little systematic attention in most contributions. Given that many of the locations referred to in the volume are little known, it is unfortunate that only three of the chapters (Malta, Bequia, and Saba English) include illustrative maps. Last but not least, a discussion of theoretical issues in research on varieties of English and an assessment of the conclusions that can be derived from the contributions in the introduction or a concluding chapter would have much enhanced the coherence of the volume. As it stands, the primary contribution of the volume seems to be in the area of expanding our empirical base rather than fundamentally tackling wider theoretical or typological questions.

Lesser-known varieties of English may not constitute a unique category of varieties of English, but that does not make them less interesting or worthy of study. The thirteen chapters provide a wealth of insights into the spread and ways of how English is appropriated around the world. They prove wrong the commonly held assumption that linguistic heterogeneity is a property of large urban populations and provide an excellent starting point for sociolinguistic, contact linguistic and linguistic anthropological research projects that will unearth more detailed insights into the dynamics of language use outside of the urban western context.


Julien Vandaele

Simon Duquennoy

Nathalie Mitton

Demo: Large-scale Sensing, Mobility, and Monitoring with the FIT IoT-LAB Testbed

FIT IoT-LAB is a testbed distributed over 7 different sites, 4 hardware platforms, and a total of 2446 nodes. The testbed enables large-scale protocol testing on a wide range of topologies. It offers features such as live interaction with remote nodes, per-node packet sniffers, online energy monitoring, programmable mobility, and per-node public IPv6 addressing, to name a few. In this Demo, we show a webcam view of the testbed with remotely controlled lighting (ceiling LEDs and a mobile robot carrying a torch). A tight grid of 256 sensors will be used to collect light information. We display live updates of the resulting heatmap, live energy profiles and other performance metrics.

Introduction

Testbeds enable realistic experimentation for low-power wireless networking research. There exists a number of open testbeds, such as Indriya [START_REF] Doddavenkatappa | Indriya: A Low-Cost, 3D Wireless Sensor Network Testbed[END_REF] and FlockLab [START_REF] Lim | Flocklab: A testbed for distributed, synchronized tracing and profiling of wireless embedded systems[END_REF], already widely used in our research community. However, no existing solution enables yet large-scale Internet of Things testing (hundreds of nodes) in a variety of environments.

This abstract presents FIT IoT-LAB 1 , a large-scale collection of testbeds distributed over 7 sites. All sites can be accessed via a common interface, enabling the evaluation of all network stack layers protocols on a variety of physical topologies with complementary properties. IoT-LAB offers unique features such as per-node packet sniffers and programmable robots. The demonstrator shows mobile robots, remote lighting control, wireless data collection, and live monitoring of power draw, in a single integrated setup.

Design Overview

Key Features

When performing experiments, IoT-LAB provides the user with the following features:

• Socket I/O access to individual nodes: can be used to collect logs, send commands, or connect a node to the Internet via a public IPv6, e.g., as a RPL border router; • Mobility: certain nodes are mounted on robots. Robots follow pre-recorded tracks selected by the user; • Fine-grained monitoring: users can access power samples, RSSI samples and packet sniffed at runtime; • OS support: the platform offers mature support for Contiki, RIOT, OpenWSN and FreeRTOS, on the different hardware platforms available.

Deployments

FIT IoT-LAB offers a large variety of deployments, in an attempt to cover as many different application scenarios as possible. The deployments vary in scale (up to 866 nodes spanning a building), density (up to 342 nodes in a room), physical topology (building or single room, on ceiling, walls, or under the floor), and ability (static or mobile). In total, there are 7 different sites in France, for a total of 2446 nodes. Fig. 2 sums up the available deployments.

Future Evolutions

FIT IoT-LAB is constantly evolving, and the following new features are currently planned:

• Long-range communication: deploy a testbed of IEEE 802.15.4g compliant nodes; • General-purpose I/O pins: support for fine-grained control and monitoring via I/O pins; • Custom nodes: offer new reference hardware platforms and the ability for users to plug their own hardware to a testbed (requires physical access); • Robots: extend robot deployments (up to 40 nodes) and capabilities with (1) mobility models and (2) usercontrolled mobility; • Benchmarking: support automated benchmarking of low-power communication protocols [START_REF] Duquennoy | Poster Abstract: A Benchmark for Low-power Wireless Networking[END_REF].

Demonstration Description

To illustrate the capabilities of IoT-LAB, we will run an experiment on the Lille testbed, with live monitoring of luminosity and performance metrics. The nodes perform lowpower data collection using Orchestra [START_REF] Duquennoy | Orchestra: Robust Mesh Networks Through Autonomously Scheduled TSCH[END_REF], an autonomous scheduler for RPL+6TiSCH networks. We use the Lille testbed's remote light controller to vary luminosity at runtime. A Web application retrieves light sensor data from the sink node and displays a real-time updated heatmap of the room (see Fig. 3). Finally, a live video stream will provide visual feedback from the testbed room. Alongside, we will also use some robots carrying a torch to showcase mobility and increase lighting dynamics. This demonstration makes use of many features, as detailed next: Large Scale We demonstrate the ability to experiment at large scale with over 200 M3 nodes involved. Mobility By adding a few robots, we will discuss mobility models and possibilities. Live Monitoring We exploit energy monitoring as well as robot localization features. OS Support For low-power data collection, we run one of our readily-available solutions: Contiki with RPL+6TiSCH. Serial Link Aggregation With the serial aggregator tool, we gain remote access to all nodes' socket I/O and collect relevant network performance metrics. Command Line Tools We showcase the command line tools that interact with the RESTful API to submit/stop experiments, update node firmware, draw consumption graph and robot trajectory.

1Figure 1 .

 1 Figure 1. IoT-LAB node components

  IoT-LAB is a constellation of testbeds with well-defined open interfaces (RESTful API, Web portal, CLI). A central manager enables to create, administer, and gather data from experiments on the different deployments. Each deployment offers its own set of IoT-LAB nodes, physically deployed in different buildings and rooms. An IoT-LAB node, as shown on Fig. 1, is made of three components: (1) the Open Node; (2) the Gateway; and (3) the Control Node, as described below: Open Node The Open Node is the IoT device made available to the user. It can be programmed in an unconstrained manner, i.e., with any compatible firmware. There are different types of Open Nodes, covering different application needs in terms of CPU or energy consumption, based on MSP430, Cortex-M3 and Cortex-A8 MCUs. They are equipped with an IEEE 802.15.4 radio chip at 2.4 GHz or 868 MHz. Some of the nodes are mounted on robots, for mobility. Some sites are even envisioned to offer fleets of tens of programmable robots. Gateway The Gateway connects the constrained device to the infrastructure, and takes care of uploading firmwares and downloading monitoring information. It enables live interaction through sockets mapped to the node's serial interface. Control Node The Control Node offers fine-grained, nonintrusive monitoring capabilities. During the experiment, it monitors the energy consumption of the Open Node and performs live packet sniffing and RSSI recording. Energy monitoring is done at every node at a resolution up to 3.5 kHz.

Figure 2 .

 2 Figure 2. Overview of the IoT-LAB nodes and robots available at the time of writing. The Grenoble picture shows M3 nodes before they were hidden under the floor. The Lille picture shows M3 nodes including two mobile robots.

Figure 3 .

 3 Figure 3. Heatmap web application and real time stream from the Lille site.


Parikshit Dutta
email: parikshit.dutta@duke.edu.

Inverse Identification of Inputs of a Random Utility based Model Using Optimal Control

Keywords: Integrated land use and transportation, random utility based multiregional input-output models, Lyapunov stability, optimal control

The random utility based multiregional input-output (RUBMRIO) model is used by several integrated land use and transportation models (ILUTMs) to forecast commodity flows in a region and for spatial allocation of production activities. It makes use of multiregional input-output models which are based on random utility theory. In this work, we use optimal control techniques to find optimal final demand and transportation costs of a RUBMRIO model, that would lead to a desired level of commodity flow and production costs.

At first, the RUBMRIO model is formulated as a discrete time dynamical system. It is shown, using Lyapunov argument, that the fixed point of the dynamical system is input-to-state stable (ISS). Then, a discrete time optimal control problem is formulated with states as commodity flows and production costs, and the final demand and transportation costs as control inputs. An optimization problem is then solved to obtain control inputs that would lead to desired commodity flows and production costs.

Further, the proposed methodology is applied to a numerical example. It is shown that the optimal control based method can achieve user specified commodity flows and production costs up to an acceptable accuracy level.

Introduction

Owing to constantly increasing awareness about environmental protection, urban sustainability has become a key policy issue over the years [START_REF] Nash | The european commission's sustainable consumption and production and sustainable industrial policy action plan[END_REF]. It is well known fact that integrated land use and transportation models play a key role in the context of urban sustainability [START_REF] Spiekermann | Evaluating urban sustainability using land-use transport interaction models[END_REF][START_REF] Black | Sustainable urban transportation: Performance indicators and some analytical approaches[END_REF]. Integrated land use and transport models (ILUTMs) offer invaluable analysis tools for planners working on transportation and urban projects. Analyzing the behavior of ILUTMs is important for policy makers in order to create sustainable cities for future. Hence, study and analysis of ILUTMs are key towards designing better urban environments [START_REF] Dutta | Sensitivity analysis of a large scale integrated land use and transportation system[END_REF].

A number of ILUTMs, for example MEPLAN and TRANUS, make use of multiregional input-output models combined with random utility theory for allocation of activities and land use [START_REF] Hunt | A Description of the MEPLAN Framework for Land Use and Transport Interaction Modelling[END_REF][START_REF] De La Barra | Integrated land use and transport modelling[END_REF]. Since the input-output model varies spatially, between different geographical regions, they are called random utility based input-output (RUBMRIO) models. Typically they combine traditional input-output model with multinomial logit model for calculation of commodity flows and production prices between different regions [START_REF] Zhao | The random-utility-based multiregional input-output model: Solution existence and uniqueness[END_REF][START_REF] Juri | Extending the random-utility-based multiregional input-output model: Incorporating land-use constraints, domestic demand and network congestion in a model of texas trade[END_REF]. The RUBMRIO model is usually solved iteratively, by applying a set of algebraic equations [START_REF] Hunt | A Description of the MEPLAN Framework for Land Use and Transport Interaction Modelling[END_REF]. The outputs of the RUBMRIO model are commodity flows and production costs. The inputs are the transportation costs between different regions and the final demand values. Traditional use of RUBMRIO model is to predict the steady state commodity flows and production prices for given values of transportation costs, final demand and other input parameters. Particularly Kockelman et al. have used the model extensively to forecast land use and transportation scenarios in Texas and United States [START_REF] Du | Tracking transportation and industrial production across a nation[END_REF][START_REF] Ruiz Juri | Evaluation of the trans-texas corridor proposal: Application and enhancements of the random-utility-based multiregional input-output model 1[END_REF]. They have also suggested variants of the original RUBMRIO model, in order to accommodate various objectives [START_REF] Huang | The introduction of dynamic features in a random-utility-based multiregional input-output model of trade, production, and location choice[END_REF][START_REF] Zhao | The random-utility-based multiregional input-output model: Solution existence and uniqueness[END_REF][START_REF] Juri | Extending the random-utility-based multiregional input-output model: Incorporating land-use constraints, domestic demand and network congestion in a model of texas trade[END_REF]. However, a methodology for inverse solution of the RUBMRIO model, where inputs are designed based on desired output values is missing from the literature.

In this work, we have solved an inverse problem to determine the inputs such that the desired output levels are met. The RUBMRIO model is represented as a nonlinear discrete time dynamical system with the states as the outputs of the RUBMRIO model and the controls as the inputs. An optimization problem is then solved to find the inputs that minimize the L 2 norm between calculated RUBMRIO outputs and the desired output values.

Discrete time dynamical systems have been used to represent several "real world" systems over the years, ranging from queuing theory [START_REF] Kleinrock | Queueing systems[END_REF] to cellular automata [START_REF] Chopard | Cellular automata modeling of physical systems[END_REF]. Several other applications include formal languages, temporal logic, perturbation analysis and computer simulation [START_REF] Ben-Naoum | Methodologies for discrete event dynamic systems: a survey[END_REF][START_REF] Sandefur | Discrete dynamical systems: Theory and applications[END_REF]. A nonlinear discrete dynamical system is represented as a sequence of states given by x(k) ∈ R n , where k = 1, 2, . . . , N , whose evolution equation is given by,

x(k + 1) = f (x(k), u(k)) (1)
where u(k) ∈ R m is the sequence of inputs to the dynamical system and f : R n × R m → R n is a vector of bounded nonlinear functionals. The above system is called input-to-state stable (ISS), if the sequence of states x(k) < ∞ given the input sequence u(k) < ∞ ∀ k = 1, . . . , N [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF][START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF]. It ensures that every state trajectory of the discrete system in eqn.

(1) will be bounded if the control inputs are bounded. In the current work, we have shown via Lyapunov argument [START_REF] Robinson | An introduction to dynamical systems: continuous and discrete[END_REF], that the parameters of the RUBMRIO model can be designed such that it is ISS. Further the control inputs are obtained by solving an optimal control problem.

Optimal control deals with the problem of finding a control law for a given dynamical system such that a certain optimality criterion is achieved. An optimal control problem involves solving an optimization problem, which involves optimizing a cost functional that is a function of state and control variables, subject to the states and controls satisfying the equations of the dynamical system [START_REF] Bryson | Applied optimal control: Optimization, estimation, and control[END_REF]. Optimal control has various applications ranging from astrodynamics, where it is used to optimize the entry trajectory of a satellite entering a planet's atmosphere [START_REF] Singh | Verification of optimality and costate estimation using hilbert space projection[END_REF][START_REF] Dutta | Nonlinear estimation of hypersonic state trajectories in bayesian framework with polynomial chaos[END_REF] to quantum mechanics, where it is used to optimize the path of a quantum mechanical system to its final state [START_REF] Peirce | Optimal control of quantummechanical systems: Existence, numerical approximation, and applications[END_REF]. As far as ILUTMs are concerned, researchers have used Bayesian melding, to find optimal parameters in UrbanSim, an ILUTM [START_REF] Sevcíková | Assessing uncertainty in urban simulations using bayesian melding[END_REF]. As for models based on RUBMRIO theory, optimization techniques like MLE has been used by researchers to estimate parameters of TRANUS [START_REF] Dutta | Calibration of an integrated land use and transportation model using maximum likelihood estimation[END_REF]. However, these work assume that the model involved is static, while estimating parameters. Finding the optimal inputs while taking into account the dynamical nature of RUBMRIO model has largely been ignored.

Supply-driven models, for example the Ghosh model [START_REF] Dietzenbacher | In vindication of the ghosh model: a reinterpretation as a price model[END_REF] exist in literature, which solves for the inputs like demand for a given supply.

Hence these models are based on inverse of the demand-driven philosophy, on which the RUBMRIO model is based. The inverse identification method using optimal control solves for the demand while satisfying the RUBMRIO equations which are demand-driven. To the author's knowledge such a model is missing from the literature.

This paper has two key contributions. First, it is shown that the RUBM-RIO model can be represented as a nonlinear discrete time dynamical system and the fixed point of the resulting dynamical system is ISS. Secondly, an optimal control problem is formulated, to find the inputs of the RUBMRIO model that will optimize a given cost functional. Further, the proposed optimal control methodology is applied to solve for the inputs of a RUBMRIO model that minimizes the L 2 norm of the difference between calculated and desired state values of the outputs.

The rest of the paper is organized as follows: in Section 2 the RUBMRIO model is introduced and further in Section 3, it is shown that the discrete time dynamical system is ISS. In Section 4 the optimal control problem is formulated for various cost functionals. In Section 5 the results of application of the optimal control methodology to test problems is discussed and shown. Finally, in Section 6 conclusions are stated highlighting the future work to be done.

Random Utility Based Multiregional Input-Output Model

Input-output theory was originally proposed by Leontief, focusing on a single region's industry interactions with business expenditure [START_REF] Leontief | Studies in the Structure of the American Economy[END_REF]. The multiregional input-output (MIO) model extends the original input-output theory to include several regions [START_REF] Leontief | Input-output economics[END_REF]. The RUBM-RIO model has since been developed, combining the MIO model to random utility theory [START_REF] De La Barra | Integrated land use and transport modelling[END_REF]. As stated earlier, the RUBMRIO model combines traditional input-output model with multinomial logit model for allocation of activities and land use. The final output is obtained by recursively solving a set of nonlinear algebraic equations. An intense mathematical description of the model can be found in Ref. [START_REF] Zhao | The random-utility-based multiregional input-output model: Solution existence and uniqueness[END_REF].

Input-output models characterize the interaction between market elements, aggregated into economic sectors. The model calculates flow of commodities or services between sectors, giving rise to productions and demand. For a multiregional model, flows can occur between different regions, adding a spatial element. The different regions are called geographical zones. If the region of interest is a city, then examples of economic sectors can be students or industry workers and even employments. Examples of geographical zones may be a neighborhood near an university or near a chemical factory.

The inputs to a multiregional input-output model are seen as the final demand for a sector in a given geographical zone and associated transportation cost that for a sector to be transported between two zones. From these inputs the intermediate demands are calculated. To meet the intermediate demands, production of services and commodities is required. The production costs and the commodity flow for a sector in a zone is essentially seen as the output of the model. If the multiregional input-output structure is determined by a multinomial logit model, we have the random utility based multiregional input-output model.

Economic Definition of the Variables

Let us consider, M ∈ N economic sectors and J ∈ N geographical regions. Let x n ij be the commodity flow of sector n from region i to region j and b n j be the price of producing sector n in region j. Also let Y n j be the final demand of sector n in region j and d n ij be the price of transporting one unit of sector n from region i to region j. Hence n = 1, . . . , M and i, j = 1, . . . , J. As we will see later, x n ij and b n j are the states and Y n j and d n ij are control inputs of the discrete time dynamical system. developed later in the paper.

The other variables that are germane to the RUBMRIO model are the technical coefficients a mn j and the multinomial logit dispersion parameter λ n . a mn j represents the amount of sector m (in dollars) required to produce an dollar worth of sector n in region j and λ n is a dispersion parameter for sector n, which is used in the multinomial logit equation of the RUBMRIO model. Typically both a mn j > 0, ∀ m, n = 1, . . . , M and j = 1, . . . , J, and λ n > 0, ∀ n = 1, . . . , M . An important point to note is that M m=1 a mn j < 1, as the total amount of inputs required to produce a dollar of sector n should be less than a dollar.

Governing Equations

The RUBMRIO model consists of several equations and variables having different economic interpretations. However, for simplicity, the development has been kept discreet and in this work, only the final equations of production cost and commodity flow have been presented. Detailed equations for the RUBMRIO model can be found in Appendix A. Interested readers are directed towards Ref. [START_REF] Zhao | The random-utility-based multiregional input-output model: Solution existence and uniqueness[END_REF] and Ref. [START_REF] Hunt | A Description of the MEPLAN Framework for Land Use and Transport Interaction Modelling[END_REF] for detailed description of the RUBMRIO model.

The evolution equation of the production price b n j and the commodity flow x n ij can be written as,

b n j (k + 1) = M m=1 a mn j       J i=1 exp(-λ m (b m i (k) + d m ij ))(b m i (k) + d m ij ) J r=1 exp(-λ m (b m r (k) + d m rj ))       x n ij (k + 1) = M m=1 a nm j J l=1 x m jl (k) + Y n j       exp(-λ n (b n i (k) + d n ij )) J r=1 exp(-λ n (b n r (k) + d n rj ))       (2)
The above system of equations (eqn. ( 2)) can be seen as a discrete dynamical system with states x n ij and b n j , and d n ij and Y n j as the control inputs. a mn j is assumed to be known and is constant. λ n is a parameter which is set by the user. It has been proven that the set of equation in eqn. (2) converge to a fixed point under certain regularity conditions on the states and the inputs [START_REF] Zhao | The random-utility-based multiregional input-output model: Solution existence and uniqueness[END_REF]). As will be seen later, λ n is key in ensuring that the fixed point solution of the system is ISS.

Input-to-State Stability of the Solution

This section investigates the input-to-state stability (ISS) of the RUBM-RIO dynamical system [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF]. If a dynamical system is ISS, then it is ensured that for bounded control input the state trajectories of that dynamical system will be bounded. But first some basic definitions are presented which will help us show that the RUBMRIO model is ISS.

Basic Definitions

Let us consider the following definitions,

Definition 1 (Class K function). A continuous function γ : R + → R + is said to belong to class K if it is strictly increasing and γ(0) = 0. Definition 2 (Class K ∞ function). A continuous function γ : R + → R + is said to belong to class K ∞ if it is a class K function and lim s→∞ γ(s) = ∞.
Definition 3 (ISS-Lyapunov function). [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF] A continuous function V : R n → R is called an ISS-Lyapunov function if the following holds.

1. There exists class K ∞ functions α 1 , α 2 , such that,

α 1 (|s|) ≤ V (|s|) ≤ α 2 (|s|), ∀s ∈ R n .
2. There exists class K ∞ function α 3 , and a class K function β such that

V (f (s, w)) -V (s) ≤ -α 3 (|s|) + β(|w|), ∀s ∈ R n , w ∈ R m where f (•, •) is the function in eqn. (1).
Lemma 1. [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF]. If the dynamical system in eqn. ( 1) admits a continuous ISS-Lyapunov function then the fixed point of the system is ISS.

Theorem 1 (Mean value theorem for vector-valued functions.). [START_REF] Hall | The mean value theorem for vector valued functions: A simple proof[END_REF]. Given two vectors spaces V 1 and V 2 and a continuous map h, we have

|h(v) -h(v )| < sup δ ||∇h(δ)|| • |(v -v )|, |v| < |δ| < |v |. (3)
where

h : V 1 → V 2 is a vector valued function, mapping vector space V 1 to space V 2 , and δ, v, v ∈ V 1 . h is continuous in [v, v ] and differentiable in (v, v ). | • | is the norm defined in V 1 and || • || is the usual norm, defined for linear maps from V 1 to V 2 . Remark: Let the dimensions of V 1 and V 2 be n v 1 and n v 2 respectively. Then, we can write, ∇h(δ) = {∇h(δ)} nv 2 ×nv 1 . Let ∇h(δ) = ∇h i,j (δ), where i ∈ N[1, n v 2 ] and j ∈ N[1, n v 1 ]. If h(δ) = [h 1 (δ), h 2 (δ), . . . , h nv 2 (δ)] and δ = [δ 1 , δ 2 , . . . , δ nv 1 ] , then ∇h i,j (δ) = ∂h i ∂δ j .

ISS-Lyapunov Stability of the RUBMRIO Model

We will use Lemma 1 to show that the discrete time system in eqn. ( 2) is an ISS system. We first consider a candidate Lyapunov function and then show that it is an ISS-Lyapunov function for the system in eqn. (2). Let the candidate Lyapunov function be given by,

V (|b n j |, |x n ij |) = J j=1 M n=1 (b n j ) 2 + J i=1 J j=1 M n=1 (x n ij ) 2 (4)
Lemma 2. There exists K ∞ functions α 1 and α 2 such that V (•, •) in eqn.

(4) satisfies Property 1 of an ISS-Lyapunov function.

Proof: Let us consider two class K ∞ functions, α 1 (|b n j |, |x n ij |) = C 1 •V (|b n j |, |x n ij |) and α 2 (|b n j |, |x n ij |) = C 2 • V (|b n j |, |x n ij |). where 0 < C 1 < 1 and C 2 > 1 are two constants. Clearly, α 1 (|b n j |, |x n ij |) ≤ V (|b n j |, |x n ij |) ≤ α 2 (|b n j |, |x n ij |),
which shows the Property 1 of ISS-Lyapunov function holds for V (•, •) in eqn.

(4).

In the context of RUBMRIO model, let us represent eqn. (1) as,

b(k + 1) = g 1 (b(k), x(k); d, Y; Λ) (5a) x(k + 1) = g 2 (b(k), x(k); d, Y; Λ), ( 5b
) where b(k), x(k), d, Y and Λ are vectorized forms of b n j (k), x n ij (k), d n ij , Y n j and λ n .
We will now show that Property 2 for an ISS-Lyapunov function holds for the candidate Lyapunov function. For notational simplicity, henceforth we will drop the discrete time index k.

Lemma 3. There exists Λ such that the sequences of states generated by eqn. (5a) and eqn. (5b) are contractions in L 2 .

Proof: We will use the mean value theorem for vector-valued function (Theorem 1) to prove the current lemma. If the origin is a fixed point of the system in eqn. ( 5), then it is sufficient to show that ∃ δ ∈ (0, v) such that eqn. (3) holds and sup δ ∇h(δ) < 1, where v is an arbitrary vector of states and controls.

Let h = [g 1 , g 2 ] , also note that if b, x, d and Y are 0 then h is 0. Let, v = 0 and v = [b, x, d, Y] , eqn. (3) can be written as,

g 1 (b, x; d, Y) g 2 (b, x; d, Y) < sup δ ∂g 1 ∂b ∂g 1 ∂x ∂g 1 ∂d ∂g 1 ∂Y ∂g 2 ∂b ∂g 2 ∂x ∂g 2 ∂d ∂g 2 ∂Y δ • b x d Y , (6)
where v < |δ| < v.

The partial derivatives in eqn. ( 6), can be calculated from eqn. (2). Assuming,

P n ij = exp(-λ n (b n i + d n ij )) J l=1 exp(-λ n (b n l + d n lj ))
, it is easy to see that [START_REF] Zhao | The random-utility-based multiregional input-output model: Solution existence and uniqueness[END_REF],

∂g 1 ∂b = ∂g n 1j ∂b n 1 j 1 = ∂ ∂b n 1 j 1            M m=1 a mn j       J i=1 exp(-λ m (b m i + d m ij ))(b m i + d m ij ) J r=1 exp(-λ m (b m r + d m rj ))                  = a n 1 n j P n 1 j 1 j [1 -λ n 1 {b n 1 j 1 + d n 1 j 1 j - J l=1 P n 1 lj (b n 1 l + d n 1 lj )}]. (7)
Similarly, we have,

∂g 1 ∂x = ∂g n 1j ∂x n 1 i 1 j 1 = 0, ( 8a
)
∂g 1 ∂d = ∂g n 1j ∂d n 1 i 1 j 1 = ∂ ∂d n 1 i 1 j 1 M m=1 a mn j J i=1 P m ij (b m i + d m ij ) =        a n 1 n j 1 P n 1 i 1 j 1 [1 -λ n 1 {b n 1 i 1 + d n 1 i 1 j 1 - J l=1 P n 1 lj 1 (b n 1 l + d n 1 lj 1 )}], if, j = j 1 0, otherwise. (8b) ∂g 1 ∂Y = ∂g n 1j ∂Y n 1 j 1 = 0 (8c)
Also,

∂g 2 ∂b = ∂g n 2ij ∂b n 1 j 1 =                  λ n 1 P n 1 j 1 j (P n 1 j 1 j -1) M m=1 a n 1 m j J l=1 x m jl + Y n 1 j , if, i = j 1 , n = n 1 , λ n 1 P n 1 ij P n 1 j 1 j M m=1 a n 1 m j J l=1 x m jl + Y n 1 j , if, n = n 1 , 0, otherwise. ( 9a
)
∂g 2 ∂d = ∂g n 2ij ∂d n 1 i 1 j 1 =                  λ n 1 P n 1 i 1 j 1 (P n 1 i 1 j 1 -1) M m=1 a n 1 m j 1 J l=1 x m j 1 l + Y n 1 j 1 , if, i = i 1 , j = j 1 , n = n 1 λ n 1 P n 1 ij 1 P n 1 i 1 j 1 M m=1 a n 1 m j 1 J l=1 x m j 1 l + Y n 1 j 1 , if, j = j 1 , , n = n 1 , 0, otherwise. ( 9b
)
∂g 2 ∂x = ∂g n 2ij ∂x n 1 i 1 j 1 = a nn 1 i 1 P n ii 1 , if, j = i 1 , 0, otherwise. ( 9c
)
∂g 2 ∂Y = ∂g n 2ij ∂Y n 1 j 1 = P n 1 ij 1 , if, j = j 1 , n = n 1 , 0, otherwise. (9d)
As we need to show that state sequences in eqn. ( 5) are contractions in L 2 , we will assume that the vector spaces V 1 and V 2 are equipped with L 2 norm. Hence the operator norm • for the Jacobian matrix in eqn. ( 6) is the L 2 norm or the spectral norm. But from the equivalence of norms we know that for a matrix A ∈ R m×n , we have A 2 < √ m A ∞ , where subscripts represent appropriate norms. Hence eqn. ( 6) can be rewritten as,

g 1 (b, x; d, Y) g 2 (b, x; d, Y) 2 < √ 4M + 6J sup δ ∂g 1 ∂b ∂g 1 ∂x ∂g 1 ∂d ∂g 1 ∂Y ∂g 2 ∂b ∂g 2 ∂x ∂g 2 ∂d ∂g 2 ∂Y δ ∞ ∇h(δ) ∞ •     b x d Y     2 , (10)
Considering the ∞-norm of the Jacobian matrix in eqn. (10), we have,

||∇h(δ)|| ∞ = max 1≤i≤nv 2 nv 1 j=1 |∇h i,j (δ)| = max n∈N[1,M ] i,j∈N[1,J] ( 1 S n j , 2 S n ij ) (11)
where, | • | represents the absolute value. Let

C m j (k + 1) = M n=1 a mn j X n j (k + 1) + Y m j ,
then we have,

1 S n j = M n 1 =1 J j 1 =1 a n 1 n j P n 1 j 1 j [1 -λ n 1 {b n 1 j 1 + d n 1 j 1 j - J l=1 P n 1 lj (b n 1 l + d n 1 lj )}] + M n 1 =1 J i 1 =1 a n 1 n j P n 1 i 1 j [1 -λ n 1 {b n 1 i 1 + d n 1 i 1 j - J l=1 P n 1 lj (b n 1 l + d n 1 lj )}] , 2 S n ij = λ n P n ij (P n ij -1)C n j + M j 1 =1 j 1 =i λ n P n ij P n j 1 j C n j + λ n P n ij (P n ij -1)C n j + M i 1 =1 i 1 =i λ n P n ij P n i 1 j C n j + M n 1 =1 J j 1 =1 a nn 1 j P n ij + P n ij .
All the terms in the expression of 2 S n ij are positive since 0 < P n ij < 1 and

λ n , C n j , a nn 1 j ≥ 0. Further factorizing 1 S n j and 2 S n ij and letting M n 1 =1 a nn 1 j ≤ A < ∞ , we have, 1 S n j = 2 × M n 1 =1 J j 1 =1 a n 1 n j P n 1 j 1 j [1 -λ n 1 {b n 1 j 1 + d n 1 j 1 j - J l=1 P n 1 lj (b n 1 l + d n 1 lj )}] , ( 12a
) 2 S n ij = 2 3 λ n P n ij C n ij (1 -P n ij ) + P n ij (J M n 1 =1 a nn 1 j + 1) ≤ 2 3 λ n P n ij C n ij (1 -P n ij ) + P n ij (JA + 1). (12b) Let pn 1 j = J l=1 P n 1 ij (b n 1 l + d n 1 lj ), then factorizing eqn. (12a) we have, 1 S n j ≤ 2 M n 1 =1 J j 1 =1 a n 1 n j P n 1 j 1 j [1 -λ n 1 {b n 1 j 1 + d n 1 j 1 j -pn 1 j }] = 2 M n 1 =1 a n 1 n j J j 1 =1 P n 1 j 1 j [1 -λ n 1 {b n 1 j 1 + d n 1 j 1 j -pn 1 j }] ≤ 2 M n 1 =1 J j 1 =1 P n 1 j 1 j [1 -λ n 1 {b n 1 j 1 + d n 1 j 1 j -pn 1 j }] (13)
The above result is obtained using Holder's inequality and properties of norm monotonicity, and further noting that

M n 1 =1 |a n 1 n j | < 1, ∀ n, j (
refer to Section 2.1 for explanation).

To show that the sequence of states in eqn. ( 5) is a contraction we need to show that,

√ 4M + 6J sup δ ∇h(δ) ∞ < 1 ⇒ √ 4M + 6J   sup δ   max n∈N[1,M ] i,j∈N[i,j] ( 1 S n j , 2 S n ij )     < 1 ⇒ √ 4M + 6J   sup δ   max n∈N[1,M ] i,j∈N[i,j] 2 M n 1 =1 J j 1 =1 P n 1 j 1 j [1 -λ n 1 {b n 1 j 1 + d n 1 j 1 j -pn 1 j }] , 2 3 λ n P n ij C n ij (1 -P n ij ) + P n ij (JA + 1) < 1. ( 14
)
Point to note is that the arguments of the max function in the above equation are uniformly continuous ∀ λ n . As λ n → ∞, P n ij → 0, hence it is easy to see that, lim

λ 1 ,λ 2 ,...,λ n →∞ 2 M n 1 =1 J j 1 =1 P n 1 j 1 j [1 -λ n 1 {b n 1 j 1 + d n 1 j 1 j -pn 1 j }] = 0,
and lim

λ 1 ,λ 2 ,...,λ n →∞ 2 3 λ n P n ij C n ij (1 -P n ij ) + P n ij (JA + 1) = 0.
Therefore, we can choose large enough λ n , n = 1, . . . , M , such that both the arguments of the max function in eqn. ( 14) can be made arbitrarily small ∀ δ ∈ (0, v). Hence, we can say that the sets,

Λ S 1 := {λ n 1 : λ n 1 ∈ R, 2 M n 1 =1 J j 1 =1 P n 1 j 1 j [1 -λ n 1 {b n 1 j 1 + d n 1 j 1 j -pn 1 j }] < 1} and Λ S 2 := {λ n : λ n ∈ R, 2 3 λ n P n ij C n ij (1 -P n ij ) + P n ij (JA + 1) < 1}, are nonempty. So clearly, Λ :=    λ n : λ n ∈ R, √ 4M + 6J   sup δ∈(0,v)   max n∈N[1,M ] i,j∈N[i,j] ( 1 S n j , 2 S n ij )     < 1    = ∅ (15) Hence, letting √ 4M + 6J   sup δ∈(0,v)   max n∈N[1,M ] i,j∈N[i,j] ( 1 S n j , 2 S n ij )     < L ≤ 1
Hence we can write eqn. (10) as

g 1 (b, x; d, Y) g 2 (b, x; d, Y) 2 < L • b x d Y 2 . ( 16
)
Lemma 4. There exists a class K ∞ function α 3 , and a class K function β such that V (•, •) in eqn. (4) satisfies Property 2 of an ISS-Lyapunov function.

Proof: From eqn. ( 16) we have,

M n=1 J j=1 (g n 1j ) 2 + M n=1 J i=1 J j=1 (g n 2ij ) 2 < L J j=1 M n=1 (b n j ) 2 + J i=1 J j=1 M n=1 (x n ij ) 2 + J i=1 J j=1 M n=1 (d n ij ) 2 + J j=1 M n=1 (Y n j ) 2 (17)
The left hand side of Property 2 for the RUBMRIO model given by eqn. ( 5), with the candidate ISS-Lyapunov function in eqn. (4), can be written as, 4) is an ISS-Lyapunov function and hence the fixed point of the RUBMRIO model given by eqn. ( 2) is stable.

V (|g n 1j |, |g n 2ij |) -V (|b n j |, |x n ij |) = J j=1 M n=1 [(g n 1j ) 2 -(b n j ) 2 ] + J i=1 J j=1 M n=1 [(g n 2ij ) 2 -(x n ij ) 2 ] < -[1 -L]   J j=1 M n=1 (b n j ) 2 + J i=1 J j=1 M n=1 (x n ij ) 2   α3 (class K∞ function) + L   J i=1 J j=1 M n=1 (d n ij ) 2 + J j=1 M n=1 (Y n j ) 2   β (class K function) = -α 3 (|b n j |, |x n ij |) + β(|d n ij |, |Y n j |), (18
Proof: From Lemma 2 and Lemma 4, we can see that V (•, •) in eqn. ( 4) is an ISS-Lyapunov function for the system in eqn. (2). Now using Lemma 1, it can be said that the fixed point of the discrete dynamical system in eqn.

(2) is ISS.

Discussion

The above analysis in Section 3.2 implies, for the solution of the RUBM-RIO model to be ISS the dispersion parameters should be in the set defined in eqn. ( 15). The economic interpretation of this is, to guarantee input-to-state stability of eqn. ( 2) the commodity flows should be local and concentrated. The majority of the flow to the destination region should originate in the region offering the minimum total cost, defined as p n ij = b n j + d n ij (see [START_REF] Zhao | The random-utility-based multiregional input-output model: Solution existence and uniqueness[END_REF]), Section 3.1.2 for explanation).

For arbitrary values of the dispersion parameter λ n , the purchase probability, defined as P n ij here, is small only when the total cost p n ij is sufficiently large, such that, lim

p n ij →∞ 2 M n 1 =1 J j 1 =1 P n 1 j 1 j [1 -λ n 1 {b n 1 j 1 + d n 1 j 1 j -pn 1 j }] = 0,
and lim

p n ij →∞ 2 3 λ n P n ij C n ij (1 -P n ij ) + P n ij (JA + 1) = 0.
for all values of n = 1, . . . , M and i, j = 1, . . . , J. If the above happens, it is easy to see that the values of 1 S n j and 2 S n ij are sufficiently small such that,

√ 4M + 6J   sup δ∈(0,v)   max n∈N[1,M ] i,j∈N[i,j] ( 1 S n j , 2 S n ij )     < 1
Hence we can say that for sufficiently large value of total cost p n ij the solution of the discrete dynamical system in eqn. (2) is ISS.

However, nothing can be concluded regarding input-to-state stability of the solutions of eqn. (2), if both the dispersion parameter λ n and the total cost p n ij are arbitrary.

Inverse Identification Methodology for RUBMRIO Model Based on Optimal Control

The method of inverse identification of the RUBMRIO model inputs derives its roots from the theory of inverse problems [START_REF] Tarantola | Inverse problem theory: Methods for data fitting and model parameter estimation[END_REF]. In this work, inverse identification done by posing the problem of "calibrating the exogenous inputs" of the RUBMRIO model as an optimization problem. The resulting optimization problem is then solved in optimal control framework. For the sake of brevity the authors have decided to omit discussion on inverse problems. Interested readers may consult [START_REF] Hansen | Discrete inverse problems: insight and algorithms[END_REF] for introductory materials regarding inverse problems. However, in the sections following, we have discussed the motivation behind proposing such a methodology in the context in RUBMRIO models and the significant advantages it provides.

Motivation

The original iterative solution to the RUBMRIO model finds the commodity flows and the production costs, x n ij and b n j respectively in eqn. ( 2), given the exogenous inputs ( which are the final demand and the transportation costs here) specified by the social planner. In the context of inverse identification we will call it as forward model. The solution obtained by solving the forward model is called as the forward solution. However often a desired level of steady state commodity flow or production cost is sought.

Here the social planner's task is to calibrate the RUBMRIO model such that the exogenous inputs that he/ she decides, leads to the desired level of commodity flows and production costs. We will call it as the inverse problem. The exogenous input levels found by solving the inverse problem is called as inverse solution.

To this end, we were interested in only the forward model, where the social planner's job was to provide the exogenous input values. The forward solutions were obtained when the RUBMRIO model reached a steady state solution. Inverse solutions are found by solving an optimization problem where the objective is that the optimal flows and production costs meet the desired level of flows and production costs. The optimal solution is constrained such that it satisfies the original RUBMRIO iterative equations. Thus the proposed inverse identification methodology finds the exogenous inputs that will yield the desired level of commodity flows and production costs when the forward model is solved, up to an acceptable order of accuracy.

However, the proposed inverse identification methodology will yield meaningful solutions, only if solution exists for the optimal control problem. In the present work, we will operate under this caveat, whenever we formulate the optimal control problem for inverse identification of inputs. Interested readers are directed towards Ref. [START_REF] Bryson | Applied optimal control: Optimization, estimation, and control[END_REF] for an in-depth analysis on existence and uniqueness of solutions to optimal control problems.

Advantages of Using an Inverse Identification Methodology

Traditionally, for RUBMRIO model, the social planner chooses the exogenous inputs carefully and checks by solving the forward model, if the outputs of the RUBMRIO model match the one that the social planner expects. This is often a delicate task requiring years of expertise and is also time consuming [START_REF] Dutta | Calibration of an integrated land use and transportation model using maximum likelihood estimation[END_REF]. The goal of inverse identification methodology is to help the social planner choose inputs. Formulating the problem of designing the exogenous inputs as an optimal control problem, eases the task of calibration. Moreover, It is also a robust method as the inputs are solutions of an optimization problem. Once the formulation is complete, one can use readily available optimization solvers to solve the problem [START_REF] Nocedal | Numerical optimization[END_REF].

Similarity to Supply-Driven Models

The technique of inverse identification presented here has some similarities to supply-driven models, like Ghosh's model [START_REF] Ghosh | Input-output approach in an allocation system[END_REF], in essence that it is an inverse identification of a RUBMRIO model which is demanddriven. But it is well known that Ghosh's model becomes implausible for market economies [START_REF] Oosterhaven | On the plausibility of the supply-driven input-output model[END_REF]. The extension to Ghosh's model has been proved to be implausible for centrally planned economies too [START_REF] Oosterhaven | Adding supply-driven consumption makes the ghosh model even more implausible[END_REF].

Hence, in this work we use equations of the RUBMRIO model which come from the original demand-driven input-output model proposed by Leontief [START_REF] Leontief | Studies in the Structure of the American Economy[END_REF]. The solution of proposed inverse identification problem is subject to it satisfying the RUBMRIO model.

Problem Formulation

For the current case, we find the control inputs, i.e. the final demand and the transportation costs given desired values for production costs and commodity flows. We have formulated two optimal control problems depicting two different scenarios, to find the inputs to the RUBMRIO model. However the inverse problem formalism for the RUBMRIO model is not limited to these two cases and can be extended to any number of scenarios.

Scenario 1

Here an optimal control problem is solved to design the control inputs and the parameters such that the steady state production costs bn j and the commodity flows xn ij obtained by solving eqn. ( 2) are close to the desired costs and flows given by bn j and xn ij . Since we have assumed that the states of the RUBMRIO model in eqn. ( 2) are in L 2 , the formulation of the optimal control problem is posed as a L 2 norm minimization problem, which is given by, min

Y m k ,d m rk m∈N[1,M ] r,k∈N[1,J] M n=1 J j=1 ( bn j -bn j ) 2 + M n=1 J i=1 J j=1 (x n ij -xn ij ) 2 , (19a) subject to, bn j = M m=1 a mn j       J i=1 exp(-λ m ( bm i + d m ij ))( bm i + d m ij ) J r=1 exp(-λ m ( bm r + d m rj ))       xn ij = M m=1 a nm j J l=1 xm jl + Y n j       exp(-λ n ( bn i + d n ij )) J r=1 exp(-λ n ( bn r + d n rj ))       (19b)
As stated in Section 2.1, we find the final demands Y m k and the transporta-

tion costs d m rk , m ∈ N[1, M ] and r, k ∈ N[1, J],
such that the desired objective in eqn. (19a) is met, while satisfying the constraints in eqn. (19b).

Scenario 2

In this scenario, the control inputs (final demand and transportation costs) are found such that the commodity flow and the production prices track pre specified trajectories. Let the trajectories of the state variables are given by bn j (k) and xn ij (k) where k = 1, . . . , N is the iteration counter with N being the maximum number of iteration allowed. The optimization problem is then formulated as min

Y m k ,d m rs m∈N[1,M ] r,s∈N[1,J] N k=1 M n=1 J j=1 ( bn j (k) -b n j (k)) 2 + M n=1 J i=1 J j=1 (x n ij (k) -x n ij (k)) 2 , (20a)
subject to,

b n j (k + 1) = M m=1 a mn j       J i=1 exp(-λ m (b m i (k) + d m ij ))(b m i (k) + d m ij ) J r=1 exp(-λ m (b m r (k) + d m rj ))       x n ij (k + 1) = M m=1 a nm j J l=1 x m jl (k) + Y n j       exp(-λ n (b n i (k) + d n ij )) J r=1 exp(-λ n (b n r (k) + d n rj ))       (20b)

Solution Methodology

The problem given in eqn. ( 19) can be solved using general constrained optimization techniques. A nonlinear programming solver can be used to solve the problem. Details of how the solution can be obtained via nonlinear programming has been omitted here and can be found in Ref. [START_REF] Nocedal | Numerical optimization[END_REF].

The second problem in eqn. ( 20) can be solved using discrete time optimal control techniques. Let us represent the discrete dynamical system as in eqn.

(5), which for the present case are the constraints. In the context of the current work, let

y k = [b(k) x(k)]
y k+1 = h(y k , u) (21)
where h has the same meaning as in Lemma 3. Further, augmenting the objective function φ(y k ) using KKT multipliers Γ T k , we have,

L = φ(y N ) + N -1 k=1 φ(y k ) + Γ k+1 (y k -h(y k , u)) (22)
The optimal u for the above problem is found using Pontryagin's minimum principle [START_REF] Athans | The matrix minimum principle[END_REF] which translates to solving the following system of equations simultaneously,

y k+1 = h(y k , u) (23a)
Γ k = ∂h(y k , u) ∂y k Γ k+1 + ∂φ(y k ) ∂y k (23b) 0 = ∂h(y k , u) ∂u Γ k+1 + ∂φ(y k ) ∂u (23c) 0 = Γ N - ∂φ(y N ) ∂y N dy N (23d)
Details of how eqn. (23a)-( 23d) are obtained can be found in Chapter 2 in Ref. [START_REF] Bryson | Applied optimal control: Optimization, estimation, and control[END_REF].

Application of the Proposed Optimal Control Methodology

In this section we demonstrate the proposed optimal control methodology to find inputs of a RUBMRIO model when applied to an example problem. The example that is used is same as the one in Ref. [START_REF] Zhao | The random-utility-based multiregional input-output model: Solution existence and uniqueness[END_REF]. However, in the present case we consider the inputs, i.e. the transportation costs (d n ij ) and the final demand (Y n j ), to be unknowns. The inputs are found such that the states i.e. the commodity flows (x n ij ) and the production prices (b n j ) satisfy desired optimality criterion. The two scenarios presented in Section 4 are investigated.

The number of economic sectors (M ) and the number of geographical zones (J) in the example problem are 2 and 2, respectively. The technical coefficients (a mn j ) and the dispersion parameter (λ n ) are assumed to be constants. Their values are given in Table 1. We first apply the optimal control problem presented in Section 4.4.1 to the example problem. Here our task is to solve the find control inputs that lead to a desired steady state value. The desired steady state values for each state are given in Table 2. These values are the same as the steady state values obtained in Ref. [START_REF] Zhao | The random-utility-based multiregional input-output model: Solution existence and uniqueness[END_REF]). We will check to see if the optimal control methodology is able to predict the control input values needed to obtain the state state. The optimal control problem given in eqn. ( 19) is first converted into a nonlinear programming (NLP) problem. Then NLP solvers are used to solve the resulting problem. In this work we used trust region based methods to solve the NLP problem. The NLP solver user here was fmincon from MATLAB R [START_REF] Branch | MATLAB: optimization toolbox: user's guide version 1[END_REF].

Table 3, Column 2 shows the actual input values reported in Ref. [START_REF] Zhao | The random-utility-based multiregional input-output model: Solution existence and uniqueness[END_REF] that are used to get the steady state values in Table 2. The percentage errors of the initial guess values supplied to the optimization problem and the final optimized input values, from the actual input values are also presented. The percentage error is calculated as follows,

E i = u i -u a u a × 100, E o = u o -u a u a × 100,
where i and E o are initial and optimized percentage error respectively. u a is the actual control input value and u i and u o are initial and optimized input values, respectively. We can see that there is a large error between initial guesses of inputs supplied to the optimal control problem and the actual values of inputs. However the final optimized values obtained after solving the optimal control problem in eqn. ( 19) are close to the actual values, for all the inputs except for d 2 12 . For a sanity check, the steady state values for the RUBMRIO model are found using the input values obtained after optimization. It is observed that the even with different values of d 1 12 same steady state levels are obtained. Hence, the inverse problem (finding the inputs from states) in this case has non unique solutions. The actual value of d 1 12 is recovered with better initial guess.

Figure 1 shows how the objective function and the normed difference between the control input values obtained from the NLP algorithm and the actual input values, vary with each iteration of the NLP solver.

Application of Scenario 2

For the optimization problem given in eqn. (20), we use the same example as used in the Section 5.1. The maximum number of iterations allowed is fixed to N = 100. The reference trajectories for the state variables are set to the ones followed, when the values of the inputs are same as given in Table 3, Column 2.

Figure 2 shows plot for error between state trajectories before optimization and the reference state trajectory. Figure 3 shows the plot for error between the reference values and the optimized values of commodity flows and the production costs (which are the states), respectively. The optimized . values are the state values obtained after applying the proposed optimal control methodology described in Section 4.4.2.

It is observed that the proposed optimal control methodology is able to reduce error between reference and optimized state trajectory up to an acceptable level of accuracy. Hence it can be said that the optimal control methodology that is formulated in eqn. ( 20) is able to find inputs to the RUBMRIO model which will track a reference trajectory for the states.

Conclusions and Scope for Future Work

This work presents a novel methodology based on optimal control, to obtain inputs to the random utility based multiregional input-output (RUBM-RIO) model given desired values for states. The states of the RUBMRIO model are the commodity flows and the production costs and the inputs are the final demand and the transportation costs.

At first it was shown that the RUBMRIO model can be formulated as a nonlinear discrete time dynamical system. Further it was proven that the resulting dynamical system is input-to-state stable. Then an optimal control methodology was postulated which would find the transportation costs and the final demand of the RUMBRIO model given desired commodity flow values and production costs.

The optimal control methodology was formulated for two different scenarios. Further they were applied to an example problem. For the first scenario, it was found that the proposed optimal control methodology could predict the all, but one of the inputs, upto an acceptable level of accuracy. For the second scenario, it was observed that the optimized control inputs could track a reference trajectory for the states accurately.

In future we are interested in applying the optimal control methodology to a large scale land use and transportation system, modeled based on random utility theory. The land use and transportation model would consists of several zones and sectors. Moreover, a comparison between traditional and the proposed inverse identification method for solving the RUBMRIO model will also be included. Further our goal is to generalize the optimal control methodology such that it can be applied to any land use and transportation model, regardless of the definition of the dynamical system.

)

  where α 3 (|b n j |, |x n ij |) and β(|d n ij |, |Y n j |) are class K ∞ and class K functions as defined above. This shows that the candidate ISS-Lyapunov function in eqn. (4) satisfies Property 2 of an ISS-Lyapunov function. Theorem 2. The function V (•, •) in eqn. (

  be the states of the RUBMRIO model and u = [d, Y] be the control inputs. Then without loss of generality, the optimization problem in eqn. (20) can be then written as,

Figure 1 :

 1 Figure1: Variation of the objective function and the normed difference between calculated and the actual input values. u refers to control input values obtained using the NLP solver at a given iteration. u a is the actual input value (Table3Column 2).

Figure 2 :

 2 Figure 2: Figure showing the error between reference and initial trajectories of the state variables.

Figure 3 :

 3 Figure 3: Figure showing the error between reference and optimized trajetories of the state variables.

Table 1 :

 1 Values of a mn j and λ n .

	Variable Value Zones
	a 11 j	0.2	j = 1, 2
	a 12 j	0.8	j = 1, 2
	a 21 j	0.7	j = 1, 2
	a 22 j	0.1	j = 1, 2
	λ 1	15
	λ 2	0.2
	5.1. Application of Scenario 1

Table 2 :

 2 Desired steady state values for x n ij and b n j .

	Sector 1 Sector 2	xn ij (in units of commodity) x1 11 = 307.072, x1 12 = 0.0 x1 21 = 0.040, x1 22 = 1730.404 b1 bn j (in $s) b1 1 = 18.958 2 = 11.554 x2 11 = 132.386, x2 12 = 49.736 b2 1 = 18.875 x2 21 = 120.777, x2 22 = 1359.615 b2 2 = 11.335

Table 3 :

 3 Comparison of values of the control inputs before and after optimization to the actual input values used in Ref.[START_REF] Zhao | The random-utility-based multiregional input-output model: Solution existence and uniqueness[END_REF]. Y n j has the unit of units of commodity demanded and d n ij has the unit of $s.

	Input	Actual values (u a ) optimization (u i ) error (E i ) optimization (u o ) Values before Percentage Values after	Percentage error (E o )
	Y 1 1 Y 1 2 Y 2 1 Y 2 2	100 200 20 50	302.871 602.401 61.456 150.426	202.871% 207.281% 201.201% 200.851%	100.00 200.001 20.001 50.001	-1.28 × 10 -5 % 7.78 × 10 -4 % 3.27 × 10 -4 % 2.78 × 10 -3 %
	d 1 11 d 1 12 d 1 21 d 1 22 d 2 11 d 2 12 d 2 21 d 2 22	2 10 10 1 2 10 10 1	7.265 32.377 32.878 5.547 8.747 31.967 30.107 5.802	263.264% 223.766% 228.785% 454.739% 337.360% 219.672% 201.071% 480.198%	1.999 10.000 10.000 1.000 2.000 15.000 1.000 0.999	-0.008% 0.0002% 0.0009% 0.0026% 0.0074% 50.00% -0.0004% -0.0075%

Acknowledgments

The author of the manuscript would like to thank Dr. Kara Maria Kockelman for her immense help regarding the principles RUBMRIO model, and its working.

Appendix A. All Equations of the RUBMRIO Model

x n ij (k + 1)

a mn j × c m j (k + 1), (A.1g)

• In eqn. (A.1a) ϑ n ij is the utility of purchasing a dollar worth of sector n from region i to use for production in region j. n ij is a random error term following i.i.d. Gumbel distribution.

• In eqn. (A.1b) υ n ij is called the systematic utility. • In eqn. (A.1c) X m i is the total commodity outflow of sectors m in region i.

• In eqn. (A.1d) C m j is the total consumption (inflow) of sector m to region j.

• In eqn. (A.1f) c n j is the average cost of using sector n for production in region j.