The Extraction of Elliptical Rules from the Trained Radial Basis Function Neural Network

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1 do: /v The Extracton of Ellptcal Rules from the Traned Radal Bass Functon Neural Network Andrey Bondarenko 1, Arkady Borsov, 1- Rga Techncal Unversty Abstract The paper descrbes an algorthm for approxmaton of traned radal bass functon neural network (RBFNN) classfcaton boundary wth the help of ellptc rules. These rules can later be translated nto IF-THEN form f requred. We provde expermental results of the algorthm for a two-dmensonal case. Currently, neural networks are not wdely used and spread due to dffcultes wth the nterpretaton of classfcaton decson beng made. The formalzed representaton of decson process s requred n many msson crtcal areas, such as medcne, nuclear energy, fnance and others. Keywords radal bass functon networks, knowledge acquston, optmzaton I. INTRODUCTION There are many problems to whch artfcal neural networks are stll a preferable soluton. They can be traned on data wth outlers and are naturally chosen for multple classfcaton problems. However, the way, n whch classfcaton s made, s a black-box algorthm for the user. Due to specfc requrements of dfferent ndustres for clear and formal decson process descrpton for valdaton and knowledge gatherng purposes, knowledge extracton from the traned neural network s a topcal problem. In the current paper, we propose the algorthm for extracton of ellptcal rules from the traned artfcal radal bass functon neural network. Radal bass functon neural networks are local n ther nature,.e., each neuron s responsble for classfcaton decson n a specfc regon based on defned neuron radus and locaton parameters (and weght). Thus, t s possble to defne an optmzaton problem that wll allow us to cover a classfcaton regon wth ellpsods, hence, gvng us a formalzed vew of how classfcaton s made. The structure of the paper s as follows: Secton provdes the background nformaton on a neural network, Secton 3 poses an optmzaton problem and algorthm, Secton 4 shows expermental results, and Secton 5 draws conclusons. II. RBF NEURAL NETWORK WITH TUNABLE NODES Radal bass functon artfcal neural network [1,] can be represented as follows: N (1) (x)= a p( x c ) where a s -th neuron weght, c s -th neuron centre, N s the number of neurons. The norm s usually taken to be Eucldean dstance and the bass functon p s taken to be Gaussan: p( x c )= exp β x c () There are multple strateges for tranng such a network. Network can have neurons wth fxed rad, as t smplfes a learnng procedure. On the one hand, neurons wth dfferent rad reduce the number of neurons requred to get necessary classfcaton accuracy [3]. On the other hand, neurons wth varyng rad requre a specalzed learnng approach. We have used the method descrbed n [3]. The proposed method allows us to buld RBF networks wth a small number of neurons and at the same tme preserve hgh accuracy. A. Optmzaton Problem III. RULE EXTRACTION The extracton of ellptcal rules from the traned RBF neural network can be treated as an optmzaton problem of fndng ellpsods of maxmum volume nscrbed nto the space area(-s) defned by RFBNN decson boundary. It s possble to choose other maxmzaton crtera; nstead of volume t can be the number of ponts covered by a newly shaped ellpsod. Let us denote an ellpsod as: ε = Bu+d u. (3) a unt ball under affne transformaton. As descrbed n [5], we say that B s the n-length vector contanng postve elements, so ellpsod volume s proportonal to detb. We can wrte next optmzaton problem: maxlog (det B )) s.t. RBNN ε.e., to fnd the ellpsod of maxmum volume fully contaned wthn RBFNN defned decson boundary. Apart from that we have explct constrant that the ellpsod should have non-negatve elements n ts radus vector B. The descrbed problem allows us to fnd the frst ellpsod nscrbed nto the RBFNN decson boundary. We should note that due to RBFNN nature, constrants of our problem are nonconvex; thus, the found ellpsod can be a local soluton. In (4) 161

2 most cases, t wll be nsuffcent to represent RBFNN wth a sngle ellpsod; thus, we need to search for addtonal ellpsods. For ths reason, we need to apply an teratve approach. In contrast to the recursve volume subdvson appled n [6, 7], we have chosen another strategy. Although the recursve subdvson s a feasble approach as t does not requre the objectve functon modfcaton, t generates a large number of ellpsods. Instead of space subdvson for searchable regons on each subsequent teraton, we just look for the nscrbed ellpsod wth maxmum volume not covered by the found ellpsods: max (ε vol Ε vol ) P s.t. RBNN ε here ε vol denotes the volume of the found ellpsod and Ε vol s the volume of already exstng ellpsods. P s the penalty term. Modfcaton n terms of volume calculaton ntroduced large plateau areas wth the objectve functon equal to zero. To allow faster convergence of optmzaton procedure, penalty term P calculates a mnmal dstance between the canddate ellpsod centre and the border of a set formed by ntersecton of all prevously found ellpsods. Thus, the 'further' the centre of canddate ellpsod contaned wthn other found ellpsods the larger penalty term P wll be. In ths way we can ensure that on each teraton faster convergence wll allow us to fnd a new ellpsod, whch wll cover most of the area not yet covered. Of course, there s no guarantee that the found ellpsod wll be a global soluton. B. Algorthm In ths secton we wll descrbe the algorthm for ellptcal rule extracton from traned RBFNN. We should menton that the extracted ellpsods wll be orented parallel to coordnate axes. On the frst teraton we need to nscrbe the ellpsod of maxmum volume nto the RBFNN decson boundary. The objectve functon s computed as follows: (5) log(prod(b))) In general, the algorthm lsted below has the followng nputs: Data tranng nput vectors, C, R, w parameters depctng RBFNN, from whch knowledge n the form of ellptcal rules wll be extracted. Output of the algorthm s Ellpse set, whch contans ellpsods. In the algorthm descrpton crbf s the handle for constrant functon RBF, whch accepts C, R and w, whch are neuron centres, rad and weghts, respectvely. ub, lb and x0 are upper bound, lower bound and ntal startng pont for optmzaton. objvol s the objectve functon, whch accepts the ellpsod vector contanng ellpsod rad and ellpsod centre vector. crbfmod and objvolmod are constrants and objectve functon modfed versons. U ponts covered by RBFNN, but not covered by the suppled set of ellpsods. n s a number of ponts (from U) that are not covered by a newly found ellpsod. If n s 0, then the algorthm adds a newly found ellpsod and returns the result. IN: ( maxellpsodscount, Data, C, R, w ) OUT: ( Ellpsods ) crbf constrantrbf(x, C, R, w); objvol objvolume(x); e1 = solve(crbf, objvol, ub, lb, x0); Ellpsods = e1; = ; whle < maxellpsodscount ++; U = uncoveredponts(data, Ellpsods, C, R, w); crbfmod constrantrbfmodfed(x, C, R, w); objvolmod objvolumemodfed(x); e = solve(crbfmod, objvolmod, ub, lb, x0); n = sze(uncoveredponts(u, e, C, R, w), 1); Ellpsods = Ellpsods + e; f (n == 0) break; end end TABLE I RBFNN ACCURACY, EXTRACTED ELLIPSOID ACCURACY AND NUMBER # of Neurons n RBFNN RBNN Tran RBNN Test Ellpsod Tran Ellpsod Test max Ellpsod number mean mn neurons neurons neurons neurons

3 IV. EXPERIMENTS We have created the algorthm supportng two-dmensonal nput vectors. Furthermore, t has not been our fnal goal to verfy best possble classfcaton accuracy (whch drectly correlates to RBFNN accuracy), but to show that the extracted rules are approxmatng RBFNN as close as possble. We have conducted experments on synthetc two-dmensonal Rpley dataset, whch can be found n [8]. We have mplemented RBFNN constructon algorthm descrbed n [3] to construct several neural networks contanng a varable number of neurons. Furthermore, we have observed only closed RBFNN defned classfcaton boundares, whch may be seen n fgures. Lookng at the algorthm, one can notce maxellpsodscount varable. We have ntalzed t wth a number of neurons n the subject RBFNN, the only excepton s a network wth 9 neurons, for whch the maxmum number of ellpsods to be extracted has been set to 7. Fg. 3. Decson boundary of RBFNN wth 6 neurons and 5 extracted ellpsods Fg. 1. Decson boundary of RBFNN wth neurons and extracted ellpsods Fg. 4. Decson boundary of RBFNN wth 7 neurons and 4 extracted ellpsods As already mentoned, you may notce that the algorthm has not been executed n open (unbounded areas,.e., the decson boundary les outsde lower and upper bounds) decson areas, and RBFNN decson boundares consst of several separate space volumes (lke two neurons formng two separate postve decson areas). Decson boundares and extracted ellpses can be observed n Fg Expermental results can be observed n Table Ш. One can notce that testng accuraces are hgher than tranng ones due to nature of tranng/testng data beng used. Fg.. Decson boundary of RBFNN wth 6 neurons and 4 extracted ellpsods Another pont to menton s algorthms used n volume ntersecton and ellpsod contanment wth RBFNN boundary calculatons. To check whether an ellpsod fully les wthn RBFNN decson boundary, we have created a set of ponts on ts surface and checked each of ponts to belong to a requred set. 163

4 Fg. 5. Decson boundary of RBFNN wth 7 neurons and 4 extracted ellpsods Fg. 8. Decson boundary of RBFNN wth 9 neurons and 7 extracted ellpsods Fg. 6. Decson boundary of RBFNN wth 7 neurons and 5 extracted ellpsods Fg. 9. Decson boundary of RBFNN wth 9 neurons and 7 extracted ellpsods Overall computaton tme was partally an ssue for us, because the process of fndng very frst ellpsod turned out to be a rather quck operaton whle the process of searchng for subsequent ellpsods was more CPU ntensve task due to the number of computatons needed to calculate the modfed objectve functon. Although one can use any of the global optmzaton approaches, we beleve GA s not a good opton here, whle the search nvolvng local solvers wth dfferent startng ponts has shown tself to be the best n terms of computaton tme. Overall accuracy of extracted rules ellpses n 3 out of 4 cases les wthn 1%, whch s a good result takng nto account the number of rules beng extracted. Fg. 7. Decson boundary of RBFNN wth 7 neurons and 5 extracted ellpsods We beleve t s possble to lower executon tmes by ntroducton of heurstcs for ntal pont selecton. In our algorthm, we have used an ellpse fully resdng nsde RBFNN decson boundary wthout applyng addtonal constrants. 16

5 V. CONCLUSIONS We have nvestgated the possblty of ellptcal rule extracton from radal bass functon neural networks. We have developed the algorthm successfully appled to twodmensonal nput data and radal bass functon neural network traned on the data. The observed results ndcate that the proposed algorthm can be successfully appled to low dmensonal problems; thus, further research drectons are related to the way of dealng wth ths problem. Apart from that, feasble research drecton s RBF structure explotng to speed up constrant calculaton, along wth that the algorthm s not tested on RBFNN decson boundary, whch covers open sets and several solated space regons. Here by sayng an open set we mean that classfcaton boundary partally les behnd upper and lower boundares. Ths can be solved by extendng boundares further away or even elmnatng them; however, the effect needs to be tested, and t s clear that t depends on optmzaton algorthms used. As mentoned n Secton 3, one can utlze recursve space subdvson for subsequent searches; however, t can mply some lmtatons to the found ellpsods. Overall number of found ellpsods along wth demonstrated accuracy proves the proposed approach to be feasble, especally for low radal bass functon neural networks wth low-to-medum szed nput layer. REFERENCES [1] J. Moody and C.J.Darken, Fast Learnng n networks of locally tuned processng unts, Neural Computaton, 1, pp.81-94, [] T.Poggo and F.Gros, Networks for approxmaton and learnng, Proc. IEEE 78(9), pp , [3] S. Chen, X. Hong, B.L. Luk and C.J. Harrs, Constructon of tunable radal bass functon networks usng orthogonal forward selecton, IEEE Trans. Systems, Man, and Cybernetcs, Part B, Vol.39, No., pp , 009. [4] A. Bondarenko, V. Jumutc, Extracton of Interpretable Rules from Pecewse-Lnear Approxmaton of a Nonlnear Classfer Usng Clusterng-Based Decomposton, Cambrdge, AIKED 011. [5] H.Nunez, C. Angulo and A.Catala, Rule extracton from support vector machnes. Proceedngs of the 10 th European Symposum on Artfcal Neural Networks, pp , Bruges, Belgum, Aprl4-6, 00. [6] S. Boyd, L. Vandenberghe, Convex Optmzaton. Cambrdge Unversty Press, 004. [7] A.Frank and A.Asuncon, UCI Machne Learnng Repostory, [ School of Informaton and Computer Scence, Irvne, Unversty of Calforna. [Accessed Sept. 9, 01] Andrey Bondarenko receved the B.Sc. and Ms.Sc. degrees n 004 and 006 from the Unversty of Latva and Transport and Communcaton Insttute n Rga, Latva. Snce 010 he has been studyng at Rga Techncal Unversty to obtan a doctoral degree n Computer Scence. Currently major feld of study s: extracton of concse rules from the traned artfcal neural network. Prevous publcatons: Polytope Classfer: A Symbolc Knowledge Extracton from Pecewse-Lnear Support Vector Machne. LNCS 011, Volume 6881/011, pp Research nterests nclude: machne learnng, computer vson, cognton. E-mal: Arkady Borsov s Professor of Computer Scence at the Faculty of Computer Scence and Informaton Technology, Rga Techncal Unversty. He holds a degree of Doctor of Techncal Scences n Control of Techncal Systems and a habltaton degree n Computer Scence. Hs research nterests nclude fuzzy sets, fuzzy logc, computatonal ntellgence and bonformatcs. He has 0 publcatons n the felds of computer scence and nformaton technology. Contact nformaton: 1 Kalku Street, Rga, LV-1658, phone: , e-mal: Andrejs Bondarenko, Arkādjs Borsovs. Elptsko lēmumu zvlkšana no apmācīta radālo bāzes funkcju neronu tīkla Šobrīd plaš peletot neronu tīklus dažādās nozarēs traucē to peņemto klasfkācjas rsnājumu necaurredzamība. Paskadrojum r nepecešam gan peņemtā rsnājuma valdācja, gan jaunu znāšanu par eejas datos esošajām lkumsakarībām zgūšana. Šajā rakstā r pedāvāts algortms elptsku lkumu zgūšana. Šet elptske lēmum r defnēt kā x^/a^ +y^/b^ <=1, kur a un b r elpses rādus. Domēna eksperts var vegl zanalzēt šādus lkumus, kā arī tos var vegl transformēt uz IF-THEN-ELSE va ctu lkumu/znāšanu formu no apmācīta mākslīgā neronu tīkla ar radālo bāzes funkcju. Tek aplūkots nezlekts optmzācjas uzdevums, kas tek rsnāts ar globālās pārmeklēšanas palīdzību. Pedāvātas algortms r realzēts un peletots RBF neronu tīklam ar eejas slān, kura zmērs r dv. Tā kā pētījuma mērķs bja algortma espēju zpēte, nevs klasfkācjas precztāte, tka zmantota vena sntētska datu kopa, ar kuras palīdzību tka apmācīts RBF tīkls, zmantojot ortogonālās pazīmju zvēles algortmu. Šs algortms ļauj vedot RBF tīklus ar mnmālu neronu skatu ar dažādem rādusem (kas orentēt paralēl koordnātu asīm). Rezultātā egūte neronu tīkl uzrāda augstu klasfkācjas līmen. Rakstā aprakstīt ekspermentu rezultāt un vzualzēt zgūte elptske lkum, kas tka zgūt no tīklem ar dvem, sešem, septņem un devņem neronem. Maksmālas lkumu skats, kas zgūt no tīkla, r septņ. Ekspermentu rezultāt lecna, ka trjos (no četrem) gadījumos elptske lkum uzrādīja klasfkācjas kļūdu, kura salīdznājumā ar RBF tīkla klasfkācjas kļūdu r slktāka par mazāk nekā venu procentu. Pedāvāt arī turpmāke pētījuma vrzen. Андрей Бондаренко, Аркадий Борисов. Извлечение эллиптических правил из обученной искусственной нейронной сети на радиальных базисных функциях На сегодняшний день широкому применению искусственных нейронных сетей в различных отраслях препятствует непрозрачность принимаемого ими классификационного решения. Объяснение требуется как для валидации принятого решения, так и для извлечения новых знаний о взаимосвязях во входных данных. В данной статье предлагается алгоритм извлечения эллиптических правил вида x^/a^ +y^/b^ <=1, где a и b суть радиусы эллипса. Подобные правила могут быть относительно легко проанализированы доменным экспертом, либо трансформированы в IF_THEN_ELSE или другие виды правил/знаний из обученной искусственной нейронной сети на радиальных базисных функциях. Приводится постановка невыпуклой оптимизационной задачи, которая решается путем глобального поиска. Предложенный алгоритм реализован и применен к РБФ нейронной сети с размером входного слоя, равным двум. Поскольку целью исследования было изучение возможностей алгоритма, а не точности классификации, был использован один искусственный набор данных, на котором была обучена РБФ сеть с применением алгоритма ортогонального выбора признаков. Данный алгоритм позволяет строить РБФ сети с минимальным набором нейронов с различными радиусами (ориентированными параллельно осям координат). В результате полученные нейронные сети показывают высокий уровень классификации. В статье приведены результаты экспериментов, а также визуализированы извлеченные эллиптические правила. Проведено извлечение правил из сетей с двумя, шестью, семью и девятью нейронами. Максимальное количество правил, извлеченных из сети, равно семи. Результаты экспериментов показывают минимальную потерю точности в классификации между извлеченными эллиптическими правилами и аппроксимируемой искусственной нейронной РБФ сетью (в 3 из 4 случаев разница менее 1%). Предложены направления для дальнейших исследований. 165

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