Myriam Tami

M Clausel

E Devijver

E Gaussier

J-M Aubert

M Chebre

Decision tree for uncertainty measures

Keywords: Apprentissage machine, Méthodes d'ensemble, Arbres de régression, Mesures incertaines, Variables hétérogènes Machine learning, Ensemble methods, Regression trees, Uncertainty measures, Heterogeneous variables

come

Introduction

When each quantitative observation of a dataset or a part of them has been measured with an uncertainty described by a known probability distribution and a known variance parameter value (but unknown mean parameter), the induced learning set is more informative than a classical learning set used by ensemble methods. To deal with this supplementary information, this work proposes to extend the ensemble methods to this kind of learning set. According to the literature [START_REF] Genuer | Arbres CART et Forêts aléatoires, Importance et sélection de variables[END_REF][START_REF] Gey | Bornes de risque, détection de ruptures, boosting: trois thèmes statistiques autour de CART en régression[END_REF][START_REF] Louppe | Understanding random forests: From theory to practice[END_REF], several algorithms for the construction of a tree are available, but they are all based on the minimization of a risk function and can be summarized into two tasks: a split rule and a prediction rule. For that purpose, each data instance is supposed to belong to a subset of the input variables space. However, in our context of uncertainty, each data instance is associated with a latent instance data. Thus, the construction process of a tree has to take into account the probability that the underlying latent instance belongs to each other disjoint regions. This is made by the knowledge available in the learning set, meaning, probability distributions and their variance parameter values. The induced model and notations are introduced in Section 2. To illustrate the contribution of this work, Section 3 presents the risk function, the prediction rule and the split rule for the CART algorithm and then, a new tree formula, which takes into account the uncertainty, is proposed in Section 4. The risk function formula modified is presented, and the induced split and prediction rules are demonstrated. To conclude, we present a discussion on a set of perspectives of this work in Section 5.

Model and notations

Let (X, Y ) be a concatenation of variables taking values in X × R, where X = (X 1 , • • • , X p ), X = R p is the input space2 , and R is the output space. We assume we have access to a set (x i , y i ) 1≤i≤n of size n of independent past cases of (X, Y ) taken from a universe Ω and a set of probability distributions with their respective variance parameter values. The learning set L n is then defined by

L n = (x 1 i , • • • , x p i , y i ) 1≤i≤n , (P j , σ j ) 1≤j≤p , P Y , σ Y (1)
where P j is the probability distribution associated to the input variable X j for j ∈ {1, • • • , p} and σ 2 j is its associated variance (the corresponding mean is unknown), and similarly P Y is the probability distribution associated to the output variable Y i and σ 2 Y is its associated variance (and the corresponding mean is unknown).

Each observation is independent and identically distributed from the other. Then, probability distributions defined by (P j ) 1≤j≤p , P Y and (σ 2 j ) 1≤j≤p , σ 2 Y do not depend on the index i of the observations and are fixed through the learning set L n .

In the uncertainty measure case, we assume that, for every i ∈ {1, • • • , n}, each observed value x j i is coming from a random variable X j , for j ∈ {1, • • • , p}, which corresponds to a latent variable U j with a measurement error j : formally, each observation (input and output) is defined such as, for

1 ≤ j ≤ p,    Y = f (X) + , X j = U j + j Y = U Y + Y , (2)
where j ∼ P j σ 2 j , Y i ∼ P Y σ 2 Y
, given X is a zero mean measurement error with a variance parameter σ 2 and f is the unknown regression function. The joint probability distribution of (X, Y ) is unknown, then our goal is to learn the function f : X → R from L n . This supervised learning task can be realized by the construction of a regression tree T , providing an estimator f and then a prediction ŷ = f (x) for a new observation x. The estimator f is constructed by minimizing the empirical quadratic risk:

F n (f, f , L m ) = 1 n K k=1 {1≤i≤m:(x i ,y i )∈R k } (y i -f (x i )) 2 . (3)
In this work, we propose to extend the popular CART construction (restricted to the regression case) to the uncertainty measures case. To tackle this issue, we take into account the supplementary knowledge given by ((P j , σ j ) 1≤j≤p ) in the learning set. The knowledge given by (P Y , σ Y ) is not taken into account in this project, but is part of our perspectives.

Regression decision tree: classical case

In this section, the main steps to construct a regression decision tree without uncertainty measures are described. The learning set in this case is defined by

L * n = (x 1 i , • • • , x p i , y i ) 1≤i≤n .
Binary tree is considered. The set X is associated to the root. Then, a node t represents a subset of input observations, which belong to a certain region, the corresponding learning set is denoted by L * n|t and similarly the corresponding estimator is denoted f|t . The construction of the tree is done by induction: we subdivide every node t (or the corresponding region) into two nodes t L and t R (or two subregions) with respect to a split which needs to be defined. In the end, we get a partition of the input space X into K regions (R k ) 1≤k≤K , and then, for 1 ≤ k ≤ K, a prediction γ k is assigned to each region R k . In this work, the regions (R k ) 1≤k≤K are generated by splits (hyper-planes from X ) with split zones (boundaries of R k ) parallel to the axes formed by the input variables: we focus on hyperrectangles. Thus, a tree is defined by its parameters Θ = {(R k , γ k ) 1≤k≤K }, and we estimate the function f by f (x) = T (x; Θ) :

= K k=1 γ k 1 {x∈R k } .
Parameters are estimated by minimizing the quadratic risk (3). It consists of a minimization problem over two multivariate variables, and we decompose this optimization problem into two steps: the split rule and the predictive rule described in the next subsections. Those steps are iterated until a stopping rule (to be defined). It leads to construct a maximal tree, which should be pruned.

Predictive rule

For 1 ≤ k ≤ K, a constant γ k has to be assigned to each region R k such that the predictive rule for a new observation x becomes:

x ∈ R k ⇒ f (x) = γ k .
Each constant is computed by minimizing the empirical quadratic risk defined in (3): for all 1 ≤ k ≤ K, γk =argmin

γ k ∈R F n (f, γ k 1 {x i ∈R k } , L * n ) = 1 | {i : x i ∈ R k } | i:x i ∈R k y i . (4)

Split rule

To define the regions, we have to define the variable with respect to which we are splitting, and the splitting value. To select the best variable to split, we compute the best split for every variable, and then, among all those splits, we define the variable which minimizes the quadratic risk. As the construction is done by induction, we fix a node t which corresponds to a region R, and we split it into two disjoint child nodes t L and t R corresponding to the regions R ∩ {X j < ŝj t } and R ∩ {X j ≥ ŝj t }. Fix the variable index 1 ≤ j ≤ p. The split ŝj t is chosen to minimize the impurity in both t L and t R :

ŝj t = argmin {(t L ,t R ):t=t L ∪t R } F n (f, f|t L , L * n|t L ) + F n (f, f|t R , L * n|t R ) (5)
To speed up the computation, we are looking for splits in the set of observed values: we are testing every possible values in the finite set S j = {(x j i ) 1≤i≤n }. Then, we select the index j minimizing the quadratic risk, where the split has been defined beforehand.

Regression decision tree with uncertainty measures

In this section, uncertainty measures are taken into account through the learning set defined in (1) to construct a tree, in order to be as close as possible to the expected tree that would be constructed on the unknown set {((U j i ) 1≤j≤p , U Y i ) 1≤i≤n }. Some intuition to understand the differences between the construction of the two trees is first given. For an observation x i belonging to a region R k , the underlying latent observation of the variable U i can be in a different region R k . Indeed, the measurement error can be large enough to modify the regionbelonging. Thus, given the value x i , the construction of the estimator f has to take into account the probability of

U i ∈ R k for each region k ∈ {1, • • • , K}. We are then interested in P(U i ∈ R k |X i = x i ).
In this work, we assume that the variables are independent, then this probability is decomposed as follows:

P i,k := P(U i ∈ R k |X i = x i ) = p j=1 P(U j i ∈ R j k |X j i = x j i ).
This probability can be computed in some specific cases, e.g. when P j is normal and X is normally distributed. In general, assuming a parametric family, the distribution of X j can be estimated from the observations; one could then obtain the distribution of U j |X j using Fourier transforms (an explicit formula may be obtained when the marginal and conditional distributions are related). We plan to rely on numerical approximations to compute the above probabilities in the general case. We assume in the following that we can compute those probabilities. P denotes the corresponding matrix of size n × K, with coefficients P i,k . Thus, to estimate the function f , we propose a new tree formula f (x) = T (x; Θ) :

= K k=1 γ k P(U ∈ R k |X = x) defined by its parameters Θ = {(R k , γ k ) 1≤k≤K }.

Predictive rule

For 1 ≤ k ≤ K, constant γ k has to be assigned to each region R k such that the predictive rule for a new observation x becomes:

X = x ⇒ f (x) = K k=1 γ k P(U ∈ R k |X = x).
Each constant is computed by minimizing the empirical quadratic risk defined in (3): for all 1 ≤ k ≤ K, γk =argmin

γ k ∈R {F n (f, γ k P(U i ∈ R k |X i = x i ), L * n )} = P T P -1 P T y, (6)

Split Rule

In the case of the learning set (1) and the model (2), the better split ŝj t for a node t minimizes the impurity in both child nodes t L and t R :

ŝj t = argmin {(t L ,t R ):t=t L ∪t R } F n (f, f|t L , L n|t L ) + F n (f, f|t R , L n|t R ) (7)
= argmin

{(t L ,t R ):t=t L ∪t R }   1 n K k=1 {i:x i ∈t L } y i -P i (P T P ) -1 P T • y 2 + 1 n K k=1 {i:x i ∈t R } y i -P i (P T P ) -1 P T • y 2  
Then, it minimizes the residual error in each child nodes t L and t R .

Discussion

From this proposed construction process, we are planning to move to the random forest [START_REF] Breiman | Random forests[END_REF][START_REF] Louppe | Understanding random forests: From theory to practice[END_REF] or to Gradient Boosted Trees (GBT) [START_REF] Ridgeway | Generalized Boosted Models: A guide to the gbm package[END_REF][START_REF] Freund | A short introduction to boosting[END_REF] to improve the performances. Further, we plan to extend Quantile Regression Forests [START_REF] Meinshausen | Quantile regression forests[END_REF] to manage the uncertainty measures from the output data and to take into account the information (P Y , σ Y ) available in the learning set (1). In the end, this approach will be extended to quantile regression GBT [START_REF] Fenske | Identifying risk factors for severe childhood malnutrition by boosting additive quantile regression[END_REF][START_REF] Kriegler | Boosting the quantile distribution: A cost-sensitive statistical learning procedure[END_REF][START_REF] Kriegler | Small area estimation of the homeless in Los Angeles: An application of cost-sensitive stochastic gradient boosting[END_REF] to benefit from both aspects. To this end, we plan to define a pruning step to be available to select the better constructed tree among the sub-trees available via the maximal tree. Thus, the obtained set of classifiers will be better and their weighted vote will be better again. We plan also to extend this work to the classification trees construction.

Institute of Engineering Univ. Grenoble Alpes

In order to avoid heavy formulas, we present the reduced case X = R p in the sequel, but, with no loss of generality, we can extend this work to X = R p-q × Q q where Q is a finite set of values. Thus, categorical input variables can be managed by this approach. Then, we can also extend this case to some quantitative input observations without uncertainty. For this purpose, one just has to manage these input both categorical and without uncertainty observations classically such as in CART.

Acknowledgments

The authors acknowledge A. HAUDIBERT-AYET TOTAL CTG and S. JANAQI for usefull discussions and Total, that partially funded this research, as well as the Grenoble Alpes Data Institute, supported by the French National Research Agency under the 305 "Investissements d'avenir" program (ANR-15-IDEX-02), and the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01).


Marianna Marino
email: marianna.marino@epfl.ch

Pierpaolo Parrotta

Dario Pozzoli
email: dpozzoli@asb.dk.

Tomaso Duso

Dominique Foray

Marc Gruber

Stéphane Lhuillery

Mariola Pytlikova

Julio Raffo

Does Labor Diversity Promote Entrepreneurship?

Keywords: JEL Classification: C26, J24, L26 Labor diversity, entrepreneurship, transitions to self-employment

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Despite a proliferating literature documenting a variety of aspects of how labor diversity may affect economic outcomes (Alesina and La Ferrara, 2005; Hong and Page, 2001; Lazear, 1999), the relationship between entrepreneurial activities and exposure to heterogeneous workforces is substantially left unexplored. To the best of our knowledge, Audretsch et al. (2010) is the only study analyzing this link at regional level and focusing mainly on the ethnic dimension.

Our aim is to fill such a gap. This study is inspired by the knowledge spillover theory of entrepreneurship (Audretsch and Keilbach, 2007) and the Jack-of-All-Trades theory (Lazear, 2004). The former suggests that the entrepreneurial activity tends to be greater in contexts where investments in knowledge and human capital are high or there is a relatively large amount of under-exploited knowledge useful for commercialization of new ideas. The latter concludes that the accumulation of a balanced skill-mix across different fields of expertise stimulates entrepreneurship as entrepreneurs must be sufficiently well versed in a variety of fields to manage different people and tasks.

Combining the conclusions of both theories, we assess whether a diversified workforce facilitates mechanisms of knowledge transfer (and sharing) that may ultimately stimulate entrepreneurial behavior of employees. The interaction with individuals presenting heterogeneous cultural backgrounds, skills, perspectives and attitudes to problem solving may promote the entrepreneurial behavior of employees, by favoring the accumulation of a balanced skill-mix across different competencies. However, workforce heterogeneity may also hinder these knowledge transfers by creating communication barriers (Lazear, 1999), reducing cooperative behavior and preventing reciprocal learning process. Specifically, we evaluate whether and to what extent the level of diversity characterizing the workforce cultural background, education and demographics stimulates an employee to move to a self-employment status and eventually to establish a new firm. This latter aspect of the entrepreneurial behavior has received attention from scholars as new born firms typically outperform older and larger companies in terms of employment formation and innovative potential (Audretsch et. al, 2004).

The structure of the remainder of this paper is as follows: data, estimation strategy, results and conclusions.

Data

We retrieve demographic information on each employee from the Danish "Integrated Database for Labor Market Research" for the period 1980-2002. Merging this information with data on patent applications ascribed to Danish firms at the European Patent Office and a detailed firm-level database (Generel Firmastatistik), we can distinguish patenting and exporting firms respectively for the period 1996-2002. We use data on patent applications to control for the departure firm innovativeness and to build up an external knowledge indicator based on geographical distance between firms. [START_REF] Alesina | Ethnic Diversity and Economic Performance[END_REF] This indicator accounts for closeness to industrial clusters or to innovative firms that might encourage entrepreneurial activities and lower the fixed costs associated with the start of a new business.

We analyze potential transitions to self-employment only for Danish employees in order to work on a more homogeneous sample and to exclude a potential bias due to forms of segregation eventually experienced by immigrants, as self-employment may represent a strategy to escape discrimination in the labor market. We construct a sample of individuals at risk of entering self-employment between 1996 and 2002 by drawing a random sample of employees that never move to self-employment, and combining it with a sample containing all first transitions to self-employment. [START_REF] Andersen | Commuting Areas in Denmark[END_REF] Thus, the final sample consists of 2.5 million individuals and 23 thousands departure firms over 7 years. Transitions to self-emploment cover about 1.2% of the full sample, whereas just a 0.22% is associated with the formation of new firms. Descriptive statistics in Table 1 show that transitions to self-employment are more likely to come from more ethnically and educationally (less demographically) diversi?ed workforce, providing a prima facie evidence of the phenomenon under analysis.

Estimation strategy

To investigate the effect of labor diversity on individual's propensity to become self-employed, we implement a standard linear probability model:

y it = γ c Div_c it + γ s Div_s it + γ d Div_d it + x it β + v it y it indicates
whether employee i becomes self-employed at time t; the first three terms at the right-hand side are diversity in cultural background, education and demographics, respectively. Our diversity measures are computed at the firm level and based on the Herfindahl index. Diversity in cultural background is computed by using the main language spoken in employees' country of origin. [START_REF] Audretsch | Cultural diversity and entrepreneurship: a regional analysis for Germany[END_REF] The education-related diversity is based on by the employees' highest achieved educational level while demographic diversity is represented by their age and gender. [START_REF] Audretsch | The Theory of Knowledge Spillover Entrepreneurship[END_REF] The vector x it includes an extensive set of departure firm (firm size; dummies for 3-digit industry, foreign ownership, multi establishment, patenting and exporting activity; shares of [START_REF] Alesina | Ethnic Diversity and Economic Performance[END_REF] The detailed construction of the this indicator is described in Parrotta et al. (2010). [START_REF] Andersen | Commuting Areas in Denmark[END_REF] We make sure that a transition is not preceded by another one since 1980. [START_REF] Audretsch | Cultural diversity and entrepreneurship: a regional analysis for Germany[END_REF] We would like to thank Mariola Pytlikova for the provision of the linguistic classification used in this paper. [START_REF] Audretsch | The Theory of Knowledge Spillover Entrepreneurship[END_REF] The detailed construction of the indexes is described in Parrotta et al. (2010).

3 males, managers, middle-managers, highly educated workers, differently aged and foreign employees; the cited knowledge spillover indicator) and individual characteristics (work experience, departure firm tenure, dummies for gender, education, job position at the departure firm, being a parent, and having a parent with entrepreneurial experience).

As employees may self-select among workplaces with different degrees of labor diversity to improve their entrepreneurial chances, we implement an instrumental variable (IV) strategy à la Card (2001). Specifically, this IV strategy is based on the historical levels5 of workforce diversity in ethnic, education and demographic characteristics at the commuting area where the firm is located. [START_REF] Card | Immigrant Inflows, Native Outflows and the Local Labor Market Impacts of Higher Immigration[END_REF] The commuting area level presents a suitable supply driven instrument for workplace level diversity because commuting areas in Denmark (except for the area around Copenhagen) are relatively small and therefore firms very likely recruit workers from a given local supply of labor, which is characterized by a certain degree of heterogeneity. This argument is further reinforced by the role of networks in the employment process (Munshi, 2003) and rather low residential mobility in Denmark (Deding et al. 2009).

Finally, using only the sample of individuals moving to self-employment we implement the same linear probability model and identification strategy to evaluate to what extent labor diversity is associated with firm formation.

Results

Table 2 reports our main results. It emerges that the educational diversity favors transitions from employment to self-employment, whereas diversity in demographics hinders such transitions. Both OLS and IV show qualitatively similar effects. [START_REF] Deding | Spatial Mobility and Commuting: the Case of Two-Earner Households[END_REF] Looking at the IV with all controls, we find that a standard deviation increase in the educational (demographic) diversity leads to a 0.07 (0.20) standard deviation increase (decrease) in an individual's propensity to become self-employed. The parameter on the ethnic diversity is positive but insignificant in our favorite specification.

Given the transition to self-employment, we find that the probability to establish a new firm is positively associated with the educational diversity but negatively with the demographic one. Specifically, a standard deviation increase in the educational (demographic) diversity is now associated with 0.15 (0.14) standard deviation increase (decrease) in a self-employed propensity to start a new business.

Robustness checks, related to transitions to self-employment, are reported in Table 3. These findings confirm the role of educational and demographic diversity, which are not affected significantly by the exclusion of fore 1950 might have experienced a transition before 1980, the first observed year) , big (multi-establishment) companies that typically attract talented workers. Interestingly, we find that ethnic heterogeneity promotes entrepreneurship in key industries like financial and business services.

Conclusions

We find evidence that both diversity in cultural backgrounds and education favors transitions from employment to self-employment. Conversely, these transitions are lowered by higher degrees of demographic heterogeneity. Further, given the self-employment status, educational diversity seems to foster firm formation.

Our findings support the hypothesis that exposure to higher degrees of cultural and educational heterogeneity facilitates knowledge transfer (and sharing), favoring the exploitation of new ideas. Age and gender differences seem instead to be associated with communication barriers, hindering then the transfer of valuable knowledge among employees. Table 2: The effects of labor diversity on the transition to entrepreneurship.

( Table 3: The effects of labor diversity on the transition to self-employment, robustness checks.

(1) Notes: The dependent variable is the probability to become self-employed. Beta coefficients are reported. Significance levels: ***1%, **5%, *10%.

( 10 )

 10 Estimates by Industry "Copenhagen" county is excluded Born before 1950 excluded Mono-establishment firmsManufacturing Construction Ws and retail trade Transport Financial and business activities Public and personal

Table 1 :

 1 Descriptive statistics

	Sample

  In Columns 1-6, the dependent variable is the probability to become self-employed. In Columns 7 and 8, the dependent variable is the probability to form a new firm conditional on self-employment. Beta coefficients are reported. Significance levels: ***1%, **5%, *10%.

	0.00001	(0.00001)	909.18; 866.56; 943.73	yes	yes	yes	yes	30534	0.030
	0.00001	(0.00001)		yes	yes	yes	yes	30534	0.042
	**	(0.00000)	845.22; 638.51; 595.47 426.80; 329.39; 307.50 652.67; 476.61; 442.26	yes yes yes	no yes yes	yes yes yes	no no yes	2553005 2536983 2536983	0.005 0.012 0.012
		(0.00000)		yes	yes	yes	yes	2536983	0.012
				yes	yes	yes	no	2536983	0.011
				yes	no	yes	no	2553005	0.005
			F-stat on excluded instruments	Industry, county and year dummies	Other individual's characteristics	Other departure firm's characteristics	Other parents' characteristics	Observations	R2	Notes:

The prediction of a commuting area diversity is computed by using its early 90s demographic composition and the current population stocks.

In total 104 commuting areas are identified(Andersen A. K., 2000).

The values of F-test always reject the hypothesis that our instruments are weak.


Cristian Ghiaus
email: cristian.ghiaus@usa.net

Fuzzy modeling and controlling of a fan-coil

Keywords: artificial intelligence, fuzzy logic, nonlinear control, thermal comfort

Fuzzy models and controllers are represented by if-then rules and thus can provide a user-friendly and understandable representation of combined quantitative and qualitative description. The qualitative rules result in a quantitative non-linear controller well suited to nonlinear systems controlling thermal comfort with vague defined objectives as fan-coils used in solar energy systems. The fuzzy models are achieved through identification by varying the inputs so that the fan-coil reaches the states encountered in normal operation. These models may be used for control or for simulation. On its domain of validity, the fuzzy controller behaves better then a classical controller. In fact, the main advantage of fuzzy controller is the easiness of introducing operating modes and poor defined objectives specific for solar energy systems such as freeze and overtemperature protection, or draining.

Introduction

Solar energy heating systems convert the solar radiant energy and use air or, more commonly, liquid to transfer the resulted heat to the building. The objectives of solar energy systems are temperature, humidity, air movement, and air purity control. Besides clear objectives (e.g. room air temperature value), other vaguely defined operating rules and objectives characterize the control of solar energy systems, such as heat collection, heat rejection, power outage, freeze protection, and auxiliary heating [START_REF] Ashrae | Heating, Ventilating, and Air-Conditioning Systems and Equipment[END_REF].

Room air temperature is controlled by varying the supply air temperature and flow.

Supply air temperature control presents interest in evaluating the comfort and in control system design: models for evaluating the thermal comfort consider the supply air temperature as input [START_REF] Ghiaus | Evaluating the indoor temperature field using a given air velocity distribution[END_REF], and cascade control of room temperature has the control of supply air temperature as inner loop. Hence the interest of controlling supply air temperature and also having the possibility of including operating rules, specific for the solar energy systems, in the overall algorithm.

Fuzzy controllers are represented by vague if-then rules (such as if the temperature is small then slightly open the control valve) and thus can provide an easy and understandable knowledge representation. The qualitative rules in a fuzzy controller result in a quantitative non-linear controller. As there is no unique relation between a qualitative expression and a quantitative value, a loss of information results in the translation. However, it is an advantage that complex control strategies available in the form of expert knowledge can be implemented in automatic controllers. Generally, the computer implemented controllers with time-invariant parameters may be considered as static functions. The measurements are performed at certain sampling time and the dynamic behavior is achieved by considering time differences of inputs and outputs. The linear controller yields a control (hyper)plane. Fuzzy controllers are in most cases non-linear static functions. They provide a user-friendly method to implement non-linear functions and "interpolative mechanisms, not only in small, but also in very large and complex problems." [START_REF] Dubois | Partial truth is not uncertainty -fuzzy logic versus possibilistic logic[END_REF] The comfort control systems require the combination of comfort parameter control with operating and scheduling rules, in order to achieve poor defined objectives, which are well represented in fuzzy logic.

Fuzzy controllers are usually claimed to be more robust and adequate for nonlinear control. Nevertheless, there is no demonstration of the robustness of fuzzy controllers. A fuzzy algorithm is in fact a multivariable nonlinear static function; if a system controlled with such a function is robust or not depends on the rules that define the function. When the variation of parameters is known (even partially), the fuzzy control algorithm may be designed to be less sensitive to parameter variation. The other claim, that the fuzzy control algorithms are better suited for nonlinear control is true only for that type of systems in which the nonlinearities depend on the error (i.e. the difference between the output and set point) and its derivates. But in many cases, the process nonliniarities depend on other variables, which should be measured. Consequently, using a fuzzy controller requires knowledge of the process, even if it is not expressed in a clear mathematical form [START_REF] Ghiaus | Contributions to Intelligent control of HVAC Systems[END_REF].

An important problem in fuzzy control is the stability. As fuzzy controllers are nonlinear functions, the stability is difficult to be analyzed [START_REF] Driankov | An introduction to fuzzy control[END_REF], but, practically, the stability analysis is replaced by prototype tests [START_REF] Mamdani | Twenty years of fuzzy control: experienced gained and lessons learnt[END_REF].

Applications of classical fuzzy controllers (as introduced by Zadeh and using Mamdani type inference algorithm) in heating, ventilating and air conditioning systems is reported in both research papers and industrial applications. Tobi and Hanfusa [START_REF] Driankov | An introduction to fuzzy control[END_REF] describe an experimental test of fuzzy control of an air conditioning system. The controller has two inputs (the temperature and the humidity of the air-conditioned space) and three outputs (the valves of the heating and cooling coils, and of the humidifier) and uses 22 rules.

Pedrycz [START_REF] Pedrycz | Fuzzy Control and Fuzzy Systems[END_REF] gives an example of fuzzy control implementation on an air conditioning system at Mitsubishi Heavy Industry. To develop this system of 25 rules for heating and 25 rules for cooling, three days were necessary to write down the initial rules, a month to determine the membership functions shapes and about three months to finally tune the controller. Altrock et al. [9] claim that a control system based on heuristic rules is efficient for heating systems. Dounis et al. [START_REF] Dounis | Design of a fuzzy set environment comfort system[END_REF] develop and test (on a numerical model of the building) a fuzzy controller having the comfort criteria (Predicted Mean Vote, PMV) and the outdoor temperature as inputs, and heating, cooling and window opening angle as outputs. Fraisse, Virgone and Roux [START_REF] Fraisse | Thermal control of a discontinuously occupied building using a classical and a fuzzy logic approach[END_REF] present comparison between classical and fuzzy controllers for a tertiary building, achieved by simulations with TRNSYS software. Sousa et al. [START_REF] Sousa | Fuzzy predictive control applied to an air-conditioning system[END_REF] present a more sophisticated approach of a predictive Sugeno-fuzzy controller tested on the same experimental stand as the one used to test the model and controller presented in this paper.

Without being exhaustive, the above presentation of fuzzy control applications to heating, ventilating and air conditioning systems reveals the importance of having a model of the system, even if it is not expressed in a mathematical form. The fuzzy model may be achieved through heuristic methods or, as an alternative, by using identification techniques.

Identification of fuzzy models

The techniques for identifying fuzzy models are developed for fuzzy linguistic models, as defined by Zadeh [START_REF] Zadeh | Fuzzy Sets[END_REF], which use Mamdani inference algorithm [START_REF] Mamdani | Application of Fuzzy Algorithm for simple Dynamic Plant[END_REF], or fuzzy linear models, as introduced by Sugeno [START_REF] Sugeno | Industrial applications of fuzzy control[END_REF][START_REF] Jang | Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence[END_REF]. The models are achieved based on measured inputs and outputs; they may be used for simulation or for designing controllers.

Identifying linguistic models and relations of Mamdani type, based on input-output data, is similar to determining a look-up table [START_REF] Pedrycz | An identification algorithm in fuzzy relational systems[END_REF]; consequently, more rules are required for a better approximation of a given function. The training data should be relevant for the input and output spaces otherwise the model reaches states never attended during the training, states for which the output is unpredictable. When the model is used for recursive prediction (i.e. the output is computed based on the output given by the model at previous time steps), the model easily gets in unidentified states where it may be unstable or deadlocked. The first methods for identifying the Sugeno type fuzzy models consider that the membership functions are trapezoidal and their number is given; membership functions vertices are found by minimizing a criterion, usually the error square mean [START_REF] Takagi | Fuzzy identification of systems and its applications to modelling and control[END_REF]. Another approach is based on the fact that the input-output relation for a multi-input single-output system forms a surface in the space defined by the Cartesian product of inputs and outputs. The identifying algorithm approximates this surface with a number of planes a priori specified [START_REF] Yoshinari | Construction of fuzzy models through clustering techniques[END_REF][START_REF] Yager | Unified structure and parameter identification of fuzzy models[END_REF]. The form of membership functions may be based on clustering methods, when the number of clusters is given [START_REF] Ruspini | A New Approach to Clustering[END_REF][START_REF] Gustafson | Fuzzy clustering with a fuzzy covariance matrix[END_REF][START_REF] Bezdek | Pattern Recognition with Fuzzy Objective Function[END_REF]; the number of clusters may be reduced to a minimum, prior to determining the membership functions [START_REF] Kaymak | Compatible Cluster Merging for Fuzzy Modelling[END_REF][START_REF] Chiu | Fuzzy Model Identification Based on Cluster Estimation[END_REF].

The identification algorithm of a Sugeno type fuzzy system considers a set of rules with fuzzy sets in the premise and a linear equation in the consequent:

If x1 is A1 and … xk is Ak then y=p0+p1 x1+…+pm xm (1)
Each rule may be interpreted as a local linear model of the system. Identifying the system described by a set of rules of the type (1) using input and output data implies choosing:

1. x1,…xk the variables in the fuzzy implicants;

2. A1, … Ak the membership functions of premise fuzzy sets;

3. p0,…pm the parameters of the consequent.

The membership functions may be chosen arbitrarily or may be obtained algorithmically, using the fuzzy ISODATA or Fuzzy C-Mean Clustering algorithm [START_REF] Gustafson | Fuzzy clustering with a fuzzy covariance matrix[END_REF][START_REF] Bezdek | Pattern Recognition with Fuzzy Objective Function[END_REF][START_REF] Dumitrescu | Fuzzy training procedures, I[END_REF][START_REF] Dumitrescu | Fuzzy training procedures, II[END_REF]. This

  = =         =   M k ik c i ik ik c i i M f M k k f f 1 1 1 , , 0 1 , , 1 1 0 (2)
The i-th row of the matrix F represents a discrete membership function, while fik shows the degree of membership of element xk to the set i. The second condition shows that every fuzzy set is not empty and does not include all the elements. Noting i the center of fuzzy sets, the index to be minimized is:

(3)
where M is a positive defined matrix [START_REF] Chiu | Fuzzy Model Identification Based on Cluster Estimation[END_REF][START_REF] Dumitrescu | Fuzzy training procedures, I[END_REF]. The index Q is in fact the sum of the dispersion of the elements around the center of the fuzzy sets; by minimizing this index, the partition matrix, F, can be found.

Knowing the fuzzy sets in the premises of the rules, the parameters of the consequences are found by minimizing the quadratic criterion of the difference between the output of the model and the output of the system. The parameters of the consequences are found from:

P=(X T X) -1 X T Y ( 4
)
where P is the vector of parameters, X is a matrix formed by the fuzzy sets of the antecedents of the rules, and Y is the output vector.

As a simple example, let us consider a nonlinear function defined as follows:

y=0.25x, 0<x  3.3 y=x 2 +0.75, 3.3<x  6.6 y=0.25x+8.25, 6.6<x  10
Using the membership functions A1, A2, and A3 defined in Figure 1 (b), the resulted Sugeno type fuzzy model is:

If x is A1 then y=1.50x+0.82 If x is A2 then y=3.57x-12.86 If x is A3 then y=0.97x
For the same system, the fuzzy membership functions obtained using Fuzzy C-Means

Clustering are shown in Figure 2 (b). In this case, the model is:

If x is A1 then y=0.13x+ 0.34 If x is A2 then y=2.83x -9.39 If x is A3 then y=0.12x+ 9.34
Comparing the behavior of the initial system and of the identified fuzzy model reveals that the model obtained using Fuzzy C-Means Clustering is better than using predefined membership functions due to the non linearities introduced by the membership functions.

The identification method of Sugeno fuzzy models presented by Babuska [START_REF] Babuska | Fuzzy Modeling for Control[END_REF] was used to model a fan-coil; based on this model, a fuzzy control algorithm of the supply air temperature was designed.

Sugeno-type fuzzy model of a fan-coil

Solar energy systems that use liquid in the collector loop have convective radiation heating terminal units that may be of fan-coil type. Fan-coils move air by forced convection through the conditioned space, filter the circulating air, and introduce outside ventilation air in order to control the temperature, humidity, air speed, and air purity [START_REF] Ashrae | Heating, Ventilating, and Air-Conditioning Systems and Equipment[END_REF].

The fan-coil unit capacity must be related to the room thermal load and to the power provided by the solar energy collector by controlling the coil water flow, air bypass, fan speed, or a combination of these. However, air bypass and fan speed are imposed by humidity and air speed requirements. Room temperature is controlled by fan-coil air supply temperature that may vary in a range imposed by thermal comfort criteria.

The experiments needed for modeling were carried out on a commercial type fan-coil.

The fan-coil has two separate coils for heating and cooling, equipped with control valves, and two adjustable dampers, for outdoors and recirculated air. Each damper has three parts, covering one seventh, two sevenths, and four seventh of the damper cross-section.

The model has in the consequent of the rules the prediction of the output at the next time step as a linear function of inputs and output at the current time step:

If y(k) is Ai then y(k+1) = b1u1(k) + b2u2(k) + b3u3(k) + ay(k)

where: the linguistic meanings small, medium and high, respectively.

u1
The model was tested in two situations: control, as a one step predictor, and simulation, as a recursive predictor. The one step predictor test implies that the output at the next time step, y(k+1), is predicted based on measurements at current time step, k; this type of test is relevant for control, since the measured output is known. The recursive predictor test requires only the initial value of the output; then, the output, y(k+1), is computed based on previous value of the output, y(k), resulted from the model output, not from measurement. This test is tougher because once the model gets in states that were unattained during the identification it may stuck or may evolve unpredictably. The results for the one step predictor are very good, the errors being within the measurement error range. The model gives also good results in simulation (recursive predictor), in tests conducted on input sequences different from those used in training (Figure 4).

Fuzzy control algorithm

The good adequacy of the identified model allows us to synthesize the control algorithm for each linear model of the consequent of the fuzzy rules using the well developed techniques of linear dynamic systems. Since the position of the dampers (inlet and recirculated air) is imposed by the indoor air quality requirements, the dampers are not input variables for the controller; they act as disturbances that the controller should reject. As a result, the model of the fan-coil in the consequence of the fuzzy rules is a first order one-input (control valve position) one-output (fan-coil outlet temperature) model. A more sophisticated model may be considered, taking into account, for example, the air temperature before the coil or the indoor and outdoor temperatures. Nevertheless, complex models require more transducers and the use of more complicated design technique [START_REF] Sousa | Fuzzy predictive control applied to an air-conditioning system[END_REF].

The one-input one-output model was obtained by removing the variables corresponding to the dampers from the complete fuzzy model and the improper fuzzy rules (the first and the fifth). This form of the model allows us to have a geometric representation of the model surface and the measured points (Figure 5). For each of the three fuzzy rules, a linear PI control algorithm was designed for the linear model present in the rule consequent, by using the pole allocation method, though any other classical approach may be applied.

The performances of the fuzzy algorithm and of a PID with anti wind-up controller were compared in range of quasi-similar experiments over a month. Using Ziegler-Nichols procedure, the PID controller was tuned by trial and error around the working point. A comparison of the performance of the fuzzy and PID with anti wind-up control systems is given in Figure 6. For the test presented in Figure 6, the fuzzy algorithm achieves the imposed performance of 2.5°C overshot and has a settle time of about 100s (the initial time is 75s). On the other hand, the PID controller has a slightly smaller overshot but the settling time is about 225s. Both controllers reject the perturbation introduced by the outdoor air temperature. (Note that the temperature values used in experiments are higher with about 15°C than in normal operation. In fact, we tried to have differences of temperature that usual in normal operation.)

The goal of the present control algorithm is to regulate the fan-coil outlet temperature and to reject disturbances introduced by damper position and fan speed. Its set-point is imposed by the room temperature controller with which controls in cascade the room temperature. Supply air temperature is also an input to models that predict thermal comfort in air conditioning rooms [START_REF] Ghiaus | Evaluating the indoor temperature field using a given air velocity distribution[END_REF].

The advantage of formulating the control algorithm as fuzzy rules resides not primarily in its performance, but mainly in its ability to accommodate operating rules and smoothly shift the operating point. A fuzzy control algorithm of Sugeno type is an interpolation of local linear algorithms; consequently, the fuzzy controller is at least as good as the corresponding linear algorithm. More than that, expert and/or operating fuzzy rules may be added. The expert rules, derived from reasoning and past experience rather than mathematical models, augment the validity range of the control algorithm. Solar energy system operating rules (such as freeze and over-temperature protection, draining, auxiliary energy to load, etc.) may be formulated as fuzzy rules and included in the control algorithm.

Conclusions

Practically, designing a Sugeno fuzzy controller based on an identified fuzzy model results in better and constant performance over all the operating range and eliminates the retuning procedure required by the classical PID controller for different working points.

The identified model is valid only for the range of inputs and states achieved during the identification; consequently, the control algorithm has the same range of applicability.

Nevertheless, the identification procedure may continue during the normal functioning of the fan-coil, resulting in expanding the initial range of validity. The fuzzy Sugeno system is in fact a nonlinear function obtained by interpolating between linear systems; the interpolation is achieved through weighted mean, the weights being given by the membership functions.

The main advantage of fuzzy controller resides not mainly in performance but in the easiness of understanding and including linguistic scheduling and expert type knowledge;

or, in Lofti Zadeh's words, "in almost every case you can build the same product without fuzzy logic, but fuzzy is faster and cheaper."

  algorithm considers M vectors x1, …,xM seen as elements in an r-dimensional space where the distance ) , ( • • d between two elements is defined. The main notion of this algorithm is the partition matrix, F, with c rows and m columns, having the properties:

  is the position of the control valve; u2 -the position of the indoor air damper; u3 -the position of the outdoor air damper; y -the outlet temperature of the fan-coil;Ai, i=1,..,[START_REF] Driankov | An introduction to fuzzy control[END_REF] -the fuzzy sets defined on the universe of discourse of the output.The control valve position and the fuzzy sets defined on the universe of discourse of the output are given in Figure3(b). The universe of discourse of the output equals the output range, in this case 20 … 70 o C. For a correct identification, the input variables should contain the frequency specter and amplitude range characteristic for the normal working conditions of the system; in other words, the system should reach representative states for the normal working conditions during the identification. The command of the control valve position was chosen as a sum of sine functions with different amplitudes and frequencies, representative for the normal operating conditions of the fan-coil (Figure3(a)). The aim is to cover the output range (30…60°C), mainly in the zones of interest (40… 55°C).Prior to use the experimental data for identification, the input and output were scaled and centered. The model identified for the mean values y=40 o C, u1=70%, u2=37% and u3=65%, using a sample time t=3 s, is:if y(k) is A1 then y(k+1) = -0.1569 u1(k) + 0.1608 u2(k) -0.007851 u3(k) -3.846y(k) if y(k) is A2then y(k+1) = 0.007371 u1(k) -0.00127 u2(k) -0.000466 u3(k) + 0.9867 y(k)if y(k) is A3 then y(k+1) = 0.013834 u1(k) -0.00385 u2(k) -0.001104 u3(k) + 0.9791 y(k) if y(k) is A4 then y(k+1) = 0.01383 u1(k) -0.00134 u2(k) -0.000292 u3(k) + 0.9878 y(k) if y(k) is A5 then y(k+1) = -0.01573 u1(k) + 0.0005 u2(k) + 0,001693 u3(k) + 1.0223 y(k)It is easy to notice that the first and fifth rules are different as compared with the other rules. Analyzing the parameters in rule consequent reveals that these rules are apparently incorrect as their parameters indicate that the opening of the heating control valve results in temperature decreasing (the inverse relation between output y(k+1) and input u1); moreover, the fifth rule defines an unstable system (the sign of y(k) is positive). Referring to Figure3(b), the poor quality of the first and fifth rules is foreseeable since the data used for their identification is irrelevant. The second, third and fourth rules define a simple and intuitive model by assigning to the fuzzy sets A2, A3, and A4 of Figure 3 (b)

Figure 1 Figure 2

 12 Figure 1Identifying a Sugeno fuzzy model with given membership functions in

Figure 3 4 Figure 6

 346 Figure 3 Characteristics of the signals used for identification: (a) time sequence of

Acknowledgments

The experiments described in this paper were conducted in 1995 at the Indoor Climatecontrol and Refrigerating Engineering Laboratory of Delft University of Technology, The Netherlands, sponsored by the Netherlands organization for international cooperation in higher education (NUFFIC). The paper was finalized at University of La Rochelle, France, supported by the Council of Poitou-Charentes Region with a research scholarship. Special thanks are addressed to dr. ir. A.H.C. van Paassen and dr. ing. R. Babuška for their support.