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Index

adaptivefuzzy systems, 72learning rates, 70modelling history, 21networks based fuzzy inferencesyst ems (ANFIS), 155spline modelling algorithm(ASMOD), 107, 226

addit iveneurofuzzy modellingalgorit hms, 106splines, 40

Akaike informati on crite rion(AIC) , 144algebraic sum operator, 85analysis of var iance (ANOVA),18, 106a posteriori prob abili ty, 32A-optimality

design criterion, 145neurofuzzy mod el construc tion(NeuDec) algorit hm, 143

approximat ion err or, 9autocorre lat ion matrix, 56a -cut , 77

backpropagation algorithm, 214Barycentric coordinates, 204basis functi ons, 14

expa nsion, global, localand radi al, 30

batch learning laws, 58Bayesian regularisati on , 129

Bayes theorem, 32, 129bias-vari ance dilemma, 33bivariate

Bernst ein polynomial , 215de Cast eljau algorit hm, 215

B-splines, 30, 41memb ership function, 79

Bezier- Bernsteinpolynomial models, 18, 41mod elling network , 209model construct ionalgorit hm, 219t riangle patch, 216

canonical state spacerepresent ations, 19centre of grav ity defuzzification , 90compact support sets, 14, 77condit ion numb er , 15, 36, 57corre lat ion test s, 45cross valid ation , 44, 105curse of dim ensionality, 17, 73, 156cyclic mod elling, 7

dataacquisit ion, 2preprocessing, 3

de Caste ljau algorithm, 212decentralised dat a fusion , 256

architecture , 262, 265defuzzification , 90

mean of maxima, 90

320 Index

cent re of gravity, 90defuzzifier , 75Delaunay

input space par ti ti oning, 204partitioning modelling, 201simplex, 203trian gulation, 18, 202

dir ect state est imation method , 169divide and conquer principle, 18

Ein st ein 's principle of simplicity, 17empirical

risk, 5minimisation principle (ERM), 10

loss function , 33err or

bar , 132sur face, 55

est imat ion error, 10expectat ion-maximisation (EM)algorit hm, 226, 237, 237exte nded

addit ive neurofuzzy mod el,119, 120

barycentric coordinates, 217

feature space , 284feedback linearisation , 227

neurofuzzy models, 239par ametric feedback form , 241

feedforward Gram-8chimidt OL8pro cedure, 191final prediction error , 43, 144first ord er infinite splines, 299forward const rained regression(FeR) , 177functional equivalence of outputsaugmente d fusion (OAF) andopt imal weighting measurementfusion (OWMF) , 260fuzzification , 89fuzzifier , 75fuzzy

boxtrees, 161dat a clustering algorit hm, 155implicati on, 85intersection , 83memb ership funct ion , 75modelling, 71operators, 83relati on sur face, 87set , 75support , 77systems, 74un ion , 84variable, 78

Gaussianmembership functions, 82radial basis function (RBF) ,41, 285

generallearning laws, 58recurrent neur al network, 243

generalisati on , 1, 13, 72generalised linear models, 6global basis functi ons, 30gradient

descent algorithms, 59, 237noise, 68

Gr am-8chimidt OL8 algorithm, 193grey box models, 4group method of data handling(GMDH) ,104growing and pruning modelconstruction, 107Gustafson-Kessel algorit hm, 155

Hessian matrix, 59, 123hierarchical

multisensor dat a fusion , 266neurofuzzy model, 125

hinging hyp erpl anes, 30hybrid

hierarchical multi sensor dat afusion architecture, 20

learning scheme, 236, 239te nsor / addit ive splines, 41

hypothesis testing, 6, 43

ill-posed problems, 5, 143"incomplete-data" log likelihood,238

independ ent component analysis(ICA), 17ind ex vector , 146indirect state est imationmethods, 169inference engine, 75inferencing, 88informat ion

filter, 249measures, 43

inst ant aneous learning laws, 54, 61intelligent or adaptive modelling , 7input-output models, 26introduction to modelling, 1inverse pro cedure of the deCasteljau algorithm, 209iterative

neurofuzzy model const ruction, 104(cyclic) , 7

Kalman filter , 20, 169, 248, 255Karush-Kuhn-Tucker (KKT)algorit hm, 288kernel functions , 6, 30, 281, 284k-d trees, 17knot insert ion/ removal, 109knowledge base , 74

lat t ice based associative networks,15learning t heory, 9

laws, 53, 58least mean squares , 61

likelihood function , 32, 129linear smoother , 36

Index 321

linguistic vagueness , 71local

basi s functi ons, 30basis funct ion expa nsion(LBF E) , 230linear model t ree (LOLIMOT)algorithm, 159model,136neurofuzzy modelling, 153regulari sed neur ofuzzy

loss functions, 11 , 284

Mamdani model, 155maximum

a posteriori (MAP) est imate ,32

likelihood (ML) est imate,33, 54, 236

likelihood (ML) identifi cation, 129max-NLMS , 64mean

of maxima (MOM), 90squared error , 27

measurement fusionmethods, 255, 258minimal capt ure zone, 66, 69minimum description lengt h, 44mixture of experts

algorithm, 18, 227modelling, 173, 204, 236

modelapproximation erro r, 9bias , 34construction algorithm, 143cross validat ion, 44, 105identificat ion , 3parameter est imatio n, 31parsimony, 34quality, 33regressor order, 14selection criteria sensit ivity, 44selection methods, 42tran sparency, 3, 73st ructural regular isat ion , 34

322 Index

valid ity tests, 46varia nce , 34

modi fied ada pt ive spline modelling(MA8MOD) algorit hm, 244mul t i-dim ensional kernels, 286mult ilayer perceptron (MLP) , 55,41mult isensor dat a fusion , 20, 255

neurofuzzyfeedback linea risation, 239Kalm an filtering, 173local function basis expa nsionmodels, 230local linearis ation (NFLL) , 225modelling, 71models, 91network, 94condit ioning, 15, 36, 57

regulari sed model, 129state esti mators , 245systems, 72

Newtonalgorit hm, 122- Raphson algorithm, 59

nonli nearaffine syste ms, 240autoregress ive models(NARX), 28auto regress ive movingaverage models (NARMAX) , 28finit e impulse responsemodels, 28output error (NOE)model, 28regression approximat ion, 288

normal fuzzy set, 77normalised

condition numbers, 68least mean squ ares (NLM8)weight convergence , 63

learning , 62, 166, 170

operati ng point neuro fuzzymodel, 164optimal weight ing measurementfusion (OWMF) , 259ort hogonal least squares (OL8)algorithm, 143, 177over-determined learning, 13Occam 's razor , 15output

augmented fusion (OAF), 259feedb ack linearisation , 19, 241measurement fusion , 20

par ametric-pure-feedback form , 241parsimonious

neurofuzzy modelling, 103par allel modelling algorit hm, 183

partition of unity, 78polynomial

kernels, 41, 286Bezier-Bern stein , 41, 215B-splines, 30, 41, 286

principal component analysis(PCA), 17, 144priors for neuro fuzzy model, 133problem

computational, 282conceptual , 282

quad- trees models, 17

radi albasis functi ons (RBF), 30const ruction, 30

recurrent neur al network , 242recursive least squares est imat ion,67, 166regulari sat ion , 5, 36

Bayesian , 129function , 283networks, 39techniques, 106, 143

regul arised neurofuzzy model, 129reg ular iser coefficient , 35regression mat rix , 27reproducing kernels, 39

Hilb ert spaces (RKHS) , 6, 39, 281rid ge const ruct ion, 30Robbins- Munro stochasticapproximation algorit hm, 70rul e

confidence , 86completeness , 78

sgn-NLMS, 65sigmoidal neural networks, 30simplexes, 18, 203

Delaunay, 203singular value decomposition(SVD) , 143slack varia bles, 287splines , 41, 286

te nsor/ addit ive, 41st agewise const ruction, 110state

dir ect estimation method , 169feedback linearisation , 240indirect state estimationmethods, 169-sp ace models, 26-space representations ofneurofuzzy mod els, 168vector fusion methods, 255, 263vector assimil ati on fusion(SVAF ),263vector fusion , 20

stat ist icallearn ing theory, 10significance metrics, 105

steepes t descent, 60st rict feedback syste m, 241st ructural

risk minimisation , 11,226, 230, 245, 282

regularisat ion , 5

Index 323

support vectors, 288analysis of var iance(SUPANOVA) algorit hm,20, 297, 298

machine, 6, 282models, 281, 286, 289neural networks, 39neurofuzzy network

S-norm, 85

Takagi-Sugeno (T- S) mod elsfuzzy ty pe, 18local neurofuzzy mod el,95, 155, 181, 225

target tracking, 20tensor

multiplication/tensorsubmodel split t ing, 108product , 30, 286product splines , 40

track-t o-track fusion , 264t ransparency, 3, 73T sukam oto model, 155T-norm, 83

under-d etermined learning, 13unimodal fuzzy sets, 76univariate Bezier-Bernst einpolynomials, 209

valid ation /verificati on , 3Vapnik-Cherr onenki s (VC) , 11very fast simulated reannealing(VFSR) algorit hm, 206

wavelet s, 30weighting

fun cti on , 204identificati on , 122

Wiener and Hammerst ein mod els,28