J Casillas, O Cord6n, F Herrera, Magdalena (Eds.) Interpretability Issues in Fuzzy Modeling Springer-Verlag Berlin Heidelberg GmbH Studies in Fuzziness and Soft Computing, Volume 128 http://www.springer.de/cgi-bin/search_book.pl?series=2941 Editor-in-chief Prof Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul Newelska 01-447 Warsaw Poland E-mail: kacprzyk@ibspan.waw.pl Further volumes of this series can be found on our homepage Vol 118 M Wygralak Cardinalities of Fuzzy Sets 2003 ISBN 3-540-00337-1 Vol 109 R.J Duro J Santos and M Grana (Eds.) Biologically Inspired Robot Behavior Engineering 2003 ISBN 3-7908-1513-6 Vol 119 Karmeshu (Ed.) Entropy Measures Maximum Entropy Principle and Emerging Applications 2003 ISBN 3-540-00242-1 Vol 110 E Fink 112 Y Jin Advanced Fuzzy Systems Design and Applications 2003 ISBN 3-7908-1523-3 Vol 120 H.M Cartwright L.M Sztandera (Eds.) 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Interpretability Issues in Fuzzy Modeling " Springer Dr Jorge Casillas E-mail: casillas@decsai.ugr.es Dr Oscar Cord6n E-mail: ocordon@decsai.ugr.es Dr Francisco Herrera E-mail: herrera@decsai.ugr.es Dpto Ciencias de la Computaci6n e Inteligencia Artificial Escuela Tecnica Superior de Ingenieria Informatica Universidad de Granada E - 18071 Granada Spain ISBN 978-3-642-05702-1 Dr Luis Magdalena E-mail: llayos@mat.upm.es Dpto Matematicas Aplicadas a las Tecnologias de la Informacion Escuela Tecnica Superior de Ingenieros de Telecomunicaci6n Universidad Politecnica de Madrid E - 28040 Madrid Spain ISBN 978-3-540-37057-4 (eBook) DOl 10.1007/978-3-540-37057-4 Library of Congress Cataloging-in-Publication-Data applied for A catalog record for this book is available from the Library of Congress Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliographie; detailed bibliographic data is available in the internet at This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitations, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Violations are liable for prosecution under the German Copyright Law http://www.springer.de © Springer-Verlag Berlin Heidelberg 2003 Originally published by Springer-Verlag Berlin Heidelberg New York in 2003 Softcover reprint of the hardcover 1st edition 2003 The use of general descriptive names, registered names trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: camera-ready by editors Cover design: E Kirchner, Springer-Verlag, Heidelberg 6213020/M - Printed on acid free paper Foreword When I accepted the editors' invitation to write this foreword, I assumed that it would have been an easy task At that time I did not realize the monumental effort that went into the organization and compilation of these chapters, the depth of each contribution, and the thoroughness with which the book's theme had been covered A foreword usually tries to impress upon the reader the importance of the book's main topic, placing the work within a comparative framework, and identifying the new trends or ideas that are pushing the state-of-the-art While doing this, one also tries to relate the book's main theme to some personal experience that will help the reader understand the usefulness and applicability of the various contributions I will my best to achieve at least some of these lofty goals The need for trading off interpretability and accuracy is intrinsic to the use of fuzzy systems Before the advent of soft computing, and in particular of fuzzy logic, accuracy was the main concern of model builders, since interpretability was practically a lost cause In a recent article in which I reviewed hybrid Soft Computing (SC) systems and compared them with more traditional approaches [1], I remarked that the main reason for the popularity of soft computing was the synergy derived from its components In fact, SC's main characteristic is its intrinsic capability to create hybrid systems that are based on the integration of constituent technologies This integration provides complementary reasoning and searching methods that allow us to combine domain knowledge and empirical data to develop flexible computing tools and solve complex problems Soft Computing provides a different paradigm in terms of representation and methodologies, which facilitates these integration attempts For instance, in classical control theory the problem of developing models is usually decomposed into system identification (or system structure) and parameter estimation The former determines the order of the differential equations, while the latter determines its coefficients In these traditional approaches, the main goal is the construction of accurate models, within the assumptions used for the model construction However, the models' interpretability is very limited, given the rigidity of the underlying representation language The equation "model = structure + parameters" 1, followed by the traditional approaches to model building, does not change with the advent of soft computing However, with soft computing we have a much richer repertoire to represent the structure, to tune the parameters, and to iterate this process This repertoire enables us to choose among different trade1 It is understood that the search method used to postulate the structures and find the parameter values is an important and implicit part of the above equation, and needs to be chosen carefully for efficient model construction v offs between the model's interpretability and accuracy For instance, one approach aimed at maintaining the model's transparency might start with knowledge-derived linguistic models, where the domain knowledge is translated into an initial structure and parameters Then the model's accuracy could be improved by using global or local data-driven search methods to tune the structure and/or the parameters An alternative approach aimed at building more accurate models might start with data-driven search methods Then, we could embed domain knowledge into the search operators to control or limit the search space, or to maintain the model's interpretability Postprocessing approaches could also be used to extract more explicit structural information from the models This book provides a comprehensive yet detailed review of all these approaches In the introduction the reader will find a general framework, within which these approaches can be compared, and a description of alternative methods for achieving different balances between models' interpretability and accuracy The book is mainly focused on the achievement of the mentioned tradeoff by improving the interpretability in fuzzy modeling It delves with the use of flexible rule structures to improve legibility, the issues of complexity reduction in linguistic or precise fuzzy models, the interpretability constraints in Takagi-Sugeno-Kang models, the use of measures to assess the interpretability loss, and the applicability of fuzzy rule-based models to interpret black-box models These topics are germane to many applications and resonate with recent issues that I have addressed Therefore, I would like to illustrate the pervasiveness of this book's main theme by relating it to a personal experience By virtue of working in an industrial research center, I am constantly faced with the constraints derived from real-world problems There are situations in which the use of black-box models is not acceptable, due to legal or compliance reasons On the other hand, the same situations require a degree of accuracy that is usually prohibitive for purely transparent models An example of such a situation is the automation of the insurance underwriting process, which consists in evaluating an applicant's medical and personal information to assess his/her potential risk and determine the appropriate rate class corresponding to such risk To address this problem, we need to maintain full accountability of the model decisions, i.e full transparency This legal requirement, imposed by the states insurance commissioners, is necessary since the insurance companies need to notify their customers and explain to them the reasons for issuing policies that are not at the most competitive rates Yet, the model must also be extremely accurate to avoid underestimating the applicants' risk, which would decrease the company's profitability, or overestimating it, which would reduce the company's competitive position in the market We solved this problem by creating several hybrid SC models, some of them transparent, for use in production, and some of them opaque, for use in VI quality assurance The commonalities among these models are the tight integration of knowledge and data, leveraged in their construction, and the loose integration of their outputs, exploited in their off-line use In different parts of this project we strived to achieve different balances between interpretability and accuracy This project exemplifies the pervasiveness of the theme and highlights the timeliness of this book, which fills a void in the technical literature and describes a topic of extreme relevance and applicability Piero P Bonissone General Electric Global Research Center Schenectady, New York, 12308, USA [1] "Hybrid Soft Computing Systems: Industrial and Commercial Applications", P P Bonissone, Y-T Chen, K Goebel and P S Khedkar, Proceedings of the IEEE, pp 1641-1667, vol 87, no 9, September 1999 VII Preface System modeling with fuzzy rule-based systems, i.e fuzzy modeling, usually comes with two contradictory requirements in the obtained model: the interpretability, capability to express the behavior of the real system in an understandable way, and the accuracy, capability to faithfully represent the real system Obtaining high degrees of interpretability and accuracy is a contradictory purpose and, in practice, one of the two properties prevails over the other While linguistic fuzzy modeling (mainly developed by linguistic fuzzy systems) is focused on the interpretability, precise fuzzy modeling (mainly developed by Takagi-Sugeno-Kang fuzzy systems) is focused on the accuracy The relatively easy design of fuzzy systems, their attractive advantages, and their emergent proliferation have made fuzzy modeling suffer a deviation from the seminal purpose directed towards exploiting the descriptive power of the concept of a linguistic variable Instead, in the last few years, the prevailing research in fuzzy modeling has focused on increasing the accuracy as much as possible, paying little attention to the interpretability of the final model Nevertheless, a new tendency in the fuzzy modeling scientific community that looks for a good balance between interpretability and accuracy is increasing in importance This searching of the desired trade-off is usually performed from two different perspectives, mainly using mechanisms to improve the interpretability of accurate fuzzy models, or to improve the accuracy of linguistic fuzzy models with a good interpretability From both perspectives, a tendency emerges as one of the most important issues: the interpretability improvements This book aims at present a state-of-the-art on the recent proposals that address it More specifically, the book presents the following structure Section introduces an overview of the different interpretability improvement mechanisms existing in the recent literature Section proposes interpretability improvement tools that consider alternative, more legible fuzzy rule structures Sections and 4, devoted to linguistic and precise fuzzy modeling respectively, are composed of a set of contributions showing how to improve the interpretability by, mainly, decreasing the complexity of the fuzzy models as a consequence of the reduction in the number of rules, variables, linguistic terms, etc Section shows several contributions that improve the interpretability of Takagi-Sugeno-Kang fuzzy systems by imposing constraints to their parameters Section collects a set of contributions that mainly propose new criteria to assess the interpretability loss Finally, Section presents two proposals that allow a translation from black-box models to fuzzy models, thus improving the interpretability of the former ones We believe that this volume presents an up-to-date state of the current IX research that will be useful for non expert readers, whatever their background, to easily get some knowledge about this area of research Besides, it will also support those specialists who wish to discover the latest results as well as the latest trends in research work in fuzzy modeling Finally, we would like to express our most sincere gratitude to SpringerVerlag (Heidelberg, Germany) and in particular to Prof J Kacprzyk, for having given us the opportunity to prepare the text and for having supported and encouraged us throughout its preparation We would also like to acknowledge our gratitude to all those who have contributed to the books by producing the papers that we consider to be of the highest quality We also like to mention the somehow obscure and altruistic, though absolutely essential, task carried out by a group of referees (all the contributions have been reviewed by two of them), who, through their comments, suggestions, and criticisms, have contributed to raising the quality of this edited book Granada and Madrid (Spain) January 2003 x Jorge Casillas, Oscar Cordon, Francisco Herrera, and Luis Magdalena Table of Contents OVERVIEW Interpretability improvements to find the balance interpretabilityaccuracy in fuzzy modeling: an overview J Casillas, O Cordon, F Herrera, L Magdalena IMPROVING THE INTERPRETABILITY WITH FLEXIBLE RULE STRUCTURES Regaining comprehensibility of approximative fuzzy models via the use of linguistic hedges J.G Marin-Blazquez, Q Shen 25 Identifying flexible structured premises for mining concise fuzzy knowledge N Xiong, L Litz 54 COMPLEXITY REDUCTION IN LINGUISTIC FUZZY MOD· ELS A multiobjective genetic learning process for joint feature selection and granularity and contexts learning in fuzzy rule-based classification systems O Cordon, M.J del Jesus, F Herrera, L Magdalena, P Villar 79 Extracting linguistic fuzzy models from numerical data-AFRELI algorithm J Espinosa, J Vandewalle 100 XI (2) ,I1R }-Jo~ -:iI.L H< -X2 Fig The Min-t-norm of the max in the definition of the Lukasiewicz-t-norm the firing degree is left side of the diagonal of the xl-x2-rectangle and increases then orthogonally to this line In the n-dimensional space such a rule starts with firing degree at a hyperplane and then increases orthogonally to this hyperplane until it reaches the firing degree When we have two different classes on both sides of a hyperplane H, then we just need two rules, one for the first class that starts with at H and increases into the direction of the class that it represents, and the second rule to the same into the other direction In this case at each time we have one rule with a firing degree greater than 0, but when there are several hyperplanes describing the classes, then we have to use several rules for one datum This is to be described in the following We assume that we have only two classes To get the more general case, we just have to consider one class against all the others and then continue with the second class against the remaining and so on o on the Lukasiewicz Classification System with h Hyperplanes We have already seen that the classification based on fuzzy clustering is characterised by hyperplanes We want to construct a classifier that defines exactly the same hyperplanes as class boundaries As a first step we not consider the whole data space but partition it into cuboids of the form [ai, bll x x [an, bnl The cuboids are chosen in such a way that they contain h hyperplanes that intersect in one point and define a convex region Inside this region we have one class, outside this region another class (compare figure 9) The point of intersection should be placed in one corner of the cuboid, but can also be situated outside Figure illustrates the partition of the space for the two- dimensional case We will treat this point more detailed in section 629 Fig An example how to partition the space into handy rectangles In order to be able to illustrate the construction of the rules, we will explain the method for the 3-dimensional case, but the technique easily extends to higher dimensions It is possible to partition the space into several n-dimensional cuboids that have hyperplanes for the separation that not bend If they bend, we have to divide the cuboid again into several smaller ones as illustrated in figure In order to consider the region as a bounded set, we sometimes have to consider the borders of the cuboid as separating hyperplanes 4.1 Basic Principles We have a region that is bounded by the different hyperplanes The interior of the region belongs to one class, while the outer part belongs to another class The most interesting points for the construction of the fuzzy classifier are those where in the n-dimensional space n hyperplanes meet In the 3dimensional space e.g planes meet in lines, and these lines meet in points The very special case that more than n hyperplanes meet in the same point will not be considered here We will depict the basic principles of the construction in the three-dimensional case We have n = hyperplanes that meet in one point P s They mark a section that belongs to one class, while the surroundings belong to another class (see figure 9) To construct the rules we use a straight line lM that starts at P s and is continued inside the section We can draw it e.g through the centre of gravity of the section Figure 10 shows a cut through the section that is orthogonal to this straight line Now we construct two auxiliary planes for each plane These planes mark the points, where the assigned rules adopt the value o The first auxiliary plane Mi includes lM and its image in figure 10 is parallel to Hi, while Bi is situated in the middle between Hi and Mi The rule RBi that starts with firing degree at Bi increases twice as fast as RMi' so that it "overtakes" RMi exactly at Hi 630 Fig Three hyperplanes partition the space into two classes- inside and outside the section that is marked by the planes Fig 10 A cut through the three-dimensional case orthogonally to the straight line 1M The resulting system consists of two rules for each hyperplane One rule RMi gives a firing degree for the class inside the section The other rule RBi has a firing degree lower than that one of RMi inside the section, but at Hi it adopts the same firing degree, and outside the section it is the winning rule itself When we construct these two rules for each hyperplane, we get a fuzzy classification system that solves our classification problem correctly The next section describes the calculations for the construction of the rules in the ndimensional case 4.2 Steps for the Construction of the Classification System Let Ps be the point where all the hyperplanes HI, , Hh meet We want to distinguish between the section that is bounded by the hyperplanes and the 631 region outside this section Two hyperplanes meet in one "line" (hyperplane of dimension n - 2) These lines of intersection between two hyperplanes are called lij, i,j E {l, ,n} As we just consider one section marked by the hyperplanes, there are only h lines that are relevant Each hyperplane is only involved in the definition of two lines, those where it meets its neighbour First of all we have to calculate the centre of gravity of the marked section We need the vectors Xi, i = 1, ,h, that are directed from Ps to the point, where Ii meets the border of the rectangle Then the centre of gravity Pg is calculated by Pg := Ps h + h LXi i=1 By drawing a line between Ps and Pg we get a line (a E JR.) that is situated inside the section that we want to describe by the fuzzy classification system Now we have to construct the auxiliary planes The following construction has to be done for each hyperplane Hi separately nHi Fig 11 YH i is a linear combination of (Pg - Ps) and nHi Construction of the Auxiliary Planes First we need the vector belongs to Hi, and that can be written as a linear combination of (Pg - P s ) with nH i being the normal vector of Hi: YH i = a· nH i + (3 (Pg - Ps ), YHi nH i that and (a, (3 E JR.) The principle can be seen in figure 11, that shows a cut through lM and Hi If a > 0, then nHi is pointing into the direction of 1M , otherwise into the other direction 632 Now we construct an orthogonal basis for Hi that includes YH i • Replacing in this basis YH i by (Pg - Ps ), we get a basis for the auxiliary plane Mi We calculate the normal form = nMi x + d Mi of Mi with the normal vector nMi' When choosing a point PHi of Hi, we calculate PHi = nMi PHi + d If PHi> 0, then nMi is pointing into the direction of Hi, otherwise into the other direction The same can be done for a point PMi of Mi For Bi has to be in the middle between Hi and M i , we use the normal vectors nBi and nHi of the two planes The normal vector nB i of Bi is calculated by · n Bi = n 'M - n'H'• I n Mi• I I nH, I if n~i is pointing towards Mi and n'u, towards Hi This can be achieved by using n'u, = sgn(pMi ) • nMi and n~, = sgn(pH.) nHi · As P s has to belong to B i , we can calculate d Bi = -Ps nB i Then Bi is described by nB i x + dBi = O Construction of the Rule R M , Now we have to determine the rules RMi and RBi' that belong to the two planes Mi and B i The firing degrees of the rules have to start with p = at Mi resp Bi As the t-norm is the Lukasiewicz t-norm, we calculate RMi and RBi by n RMi (Xl, , Xn) = LP~~, (Xt) + 1- nand t=l n RBi (Xl, , Xn) = L V~~i (Xt) + 1- t=l First of all we construct the rule RMi that has to start with R M , (X) = at any X E Mi and to increase until it reaches R M , (Pi) = at the corner Pi = (PI, ,Pn) of the cuboid The fuzzy degrees have to be between and 1, therefore all fuzzy degrees have to be in Pi to fulfil 2::;=1 P~~ (Pt) + - n = As we want to have linear fuzzy sets, we choose • (7) with [al; bil x x [an; bnl being the cuboid Then the at, t = 1, , n, are the unknown values of the fuzzy sets and have to stay between and We can denote this by P~~i (Xt) = + a~ (Xt - Pt) (8) with a~ = at if Pt = at and a~ = -at if Pt = bt Now we have to calculate the values a~ Let the hyperplane Mi be described by n Lit' Xt + c = O (9) t=l 633 n The values It are the components of the normal vector nM; of Mi We multiply the equation with ,:= C +£,:;-1 , so that we get t=l ;t Pt t t=l I~ n Xt C C + Lt=l It Pt = (10) with I~ = I· It instead of equation (9) As the firing degree of the rule has to be at the hyperplane M i , this rule has to fulfil the condition n n L>-t~~; (xt)+I-n t=l Therefore we define (11) are equivalent: = n L(1-a~.(xt-Pt))+I-n = - t=l L a~·(xt-Pt)+1 t=l = 0(11) := I~ With this construction, the equations (10) and =0 -¢:> L~l It Xt -¢:> (Xl, , Xn) +C =0 E M 1, so that the J1-~~ define a rule that starts at the hyperplane Mi with firing degree and inc~eases until it reaches firing degree at the point Pi The result is the same, if we have the the normal vector of Mi pointing into the opposite direction, i.e when we have -nM; instead of nM; The fuzzy sets that we have constructed also take values that not belong to [0,1] By scaling and cutting them in the very end we obtain fuzzy sets that range between and This will be described in section 4.2 Construction of the Rule RBi Now we turn towards the other rule RBi that has to start at the hyperplane Bi and increase faster than RMi until it "overtakes" RMi at the hyperplane Hi For the construction we first consider the fuzzy sets Vi to be linear, i.e we also allow them to adopt values below or above Later we cut them at and so that we get V~~i (x) = max{O, min{l, bt + f3t x}} In figure 12 we illustrate how these rules behave The horizontal line represents the way from Mi to Pi passing the other two auxiliary planes, while the vertical axis shows the firing degree of the rules (RMi and R B ,) 634 lC;etjc.c · 1: f L 2' ·· Bi f L : Mi Fig 12 The rule for the class C2 'overtakes' the rule for C2 We can use the same construction as in the previous section, if we have a point Pi that fulfils the function that Pi has for R M , , i.e that the rule reaches firing degree at Pi- As Bi is situated in the middle between Mi and Hi, and as RBi has to reach the same firing degree at Hi as R M " the rule RBi has to increase twice as fast as R Mi For the calculation of Pi, we have to consider the construction as shown in figure 13 Let Pi lie in the plane that is orthogonal to Mi , Bi and Hi and that includes Pi Let SPi be the point where Hi and the line from Pi to M i , that is orthogonal to M i , meet The rules R M , and RBi are to have the same firing degree on Hi and therefore also in S P, Fig 13 An orthogonal cut through the planes and Pi Now we have to put Mi ) dist(Sp" M i ) dist(~, Pi that way that dist(Pi, Bi) dist(Sp" B i ) 635 with the distance dist(P;, H) = P; nH, by the normal form Pi' nH, + d H = O + dH dist(P"M,) _ dist(Sp, ,M,) - if a hyperplane H is described dist(P; ,B,) dist(Sp, ,B,) As the right side of the last equation is a scalar, we get the normal form of a hyperplane H: that is parallel to B i The firing degree of the rule R B , is increasing orthogonally to Bi until it reaches at HI N ow we can choose the point PI E Pi has to belong to and to be on the line S'p, + 0: nB" 0: E lR When we construct the rule R B , the same way as we have constructed RMi in the previous section starting with firing degree at Bi and increasing until it reaches at PI, then also H: H: is fulfilled Scaling the Rules Now we have the two rules that we needed to describe the classification performed by the hyperplane Hi We have to this for all the hyperplanes HI"'" Hh separately and then combine them by using the maximum as t-conorm Now we have to make sure that the rules R M , and R Mj (resp RBi and R Bj ) not disturb each other when they collide at the lij Therefore the rules for the two hyperplanes that meet at lij must have the same firing degree Anyway the two rules for one hyperplane Hi have the same firing degree RBi (aI, , an) = RMi (aI, , an) for each point (a1, , an) of Hi Now we require the rules for the two hyperplanes Hi and H j to have the same membership degree R Mi (a1, ,an) = R Mj (a1, ,an) for each point (al, , an) of lij We can choose any point for our procedure This can be achieved by scaling the fuzzy sets for the rules The aim is that a rule RMi reaches firing degree tSi in P; and rule RBi reaches tSi in PI instead of tSi has to be in ]0; 1] and maXiE{I, ,h}{tSi} = If we just consider two hyperplanes Hi and H j meeting at lij with RMi Sij:= (aI, , an) R Mj (a1, ,a n ) for any (aI, , an) E lij, then Sij would be the scaling multiplier We determine the scaling multiplier for each h By using the equation Sil := Sij Sjl we calculate the other scaling multipliers, so that we have one for each pair of hyperplanes We determine S := maXi,j{ Sij} = Spq, and then the p tells us 636 the hyperplane Hp that stays the same, while the other Hj , j of P, are to be scaled with Spj < l This means that they are to reach the firing degree Spj in Pi instead of the firing degree As changing the firing degree of a rule in a point results in a complex system of equations, the easiest way to achieve this is to the same construction as we described it in the previous sections for a point Pi (and resp PI) for a new point Pi (resp PI).that is situated more closely to Mi Let SM, be the orthogonal projection of on Mi If T is defined by SM, + T' nM, = Pi, then we choose a point Pi(SM, + Spj' T' nM,) instead of Pi to construct the rules If for the point the construction would result in a rule with the fuzzy sets Ji-~~i (Xt) = 1- at(Xt - Pt), then the rule for P; would result in n n _(t) ( ) ;y ( Xt - Pt Ji-RMi Xt = - at' -::; (t) ) + T' ( 1- ) Spj n Mi with;y := (c + 2:;=1 It(Pt - T (1 - Spj) n