Contents




1.   An    introduction to modelling and learning algorithms . . . .      1
     1.1    Introduction to modelling                                     1
     1.2    Modelling, control and learning algorithms                    7
     1.3    The learning problem                                          9
     1.4    Book philosophy and contents overview                        13
            1.4.1 Book overview                                          13
            1.4.2 A historical perspective of adaptive modelling
                   and control                                           21

2.   Basic concepts of data-based modelling                              25
     2.1 Introduction                                                    25
     2.2 State-space models versus input-output models                   26
         2.2.1 Conversion of state-space models
                to input-output models                                   26
         2.2.2 Conversion of input-output models
                to state-space models                                    28
     2.3 Nonlinear modelling by basis function expansion                 29
     2.4 Model parameter estimation                                      31
     2.5 Model quality                      ,                            33
         2.5.1 The bias-variance dilemma                                 33
         2.5.2 Bias-variance balance by model structure regularisation   34
     2.6 Reproducing kernels and regularisation networks                 39
     2.7 Model selection methods                                         42
         2.7.1 Model selection criteria                                  43
         2.7.2 Model selection criteria sensitivity                      44
         2.7.3 Correlation tests                                         45
     2.8 An example: time series modelling                               49

3.   Learning laws for linear-in-the-parameters networks                 53
     3.1 Introduction to learning                                        53
     3.2 Error or performance surfaces                                   55
     3.3 Batch learning laws                                             58
         3.3.1 General learning laws                                     58
         3.3.2 Gradient descent algorithms                               59
XIV     Contents
      3.4 Instantaneous learning laws                              61
          3.4.1 Least mean squares learning                        61
          3.4.2 Normalised least mean squares learning             62
          3.4.3 NLMS weight convergence                            63
          3.4.4 Recursive least squares estimation                 67
      3.5 Gradient noise and normalised condition numbers          68
      3.6 Adaptive learning rates                                  70
4.    Fuzzy and neurofuzzy modelling                               71
      4.1 Introduction to fuzzy and neurofuzzy systems             71
      4.2 Fuzzy systems                                            74
          4.2.1 Fuzzy sets                                         75
          4.2.2 Fuzzy operators                                    83
          4.2.3 Fuzzy relation surfaces                            87
          4.2.4 Inferencing                                        88
          4.2.5 Fuzzification and denazification                   89
      4.3 Functional mapping and neurofuzzy models                 91
      4.4 Takagi-Sugeno local neurofuzzy model                     95
      4.5 Neurofuzzy modelling examples                            97
          4.5.1 Thermistor modelling                               97
          4.5.2 Time series modelling                              99
5.    Parsimonious neurofuzzy modelling                            103
      5.1 Iterative construction modelling                         103
      5.2 Additive neurofuzzy modelling algorithms                 106
      5.3 Adaptive spline modelling algorithm (ASMOD)              107
          5.3.1 ASMOD refinements                                  107
          5.3.2 Illustrative examples of ASMOD                     Ill
      5.4 Extended additive neurofuzzy models                      119
          5.4.1 Weight identification                              122
          5.4.2 Extended additive model structure identification   124
      5.5 Hierarchical neurofuzzy models                           125
      5.6 Regularised neurofuzzy models                            129
          5.6.1 Bayesian regularisation                            129
          5.6.2 Error bars                                         132
          5.6.3 Priors for neurofuzzy models                       133
          5.6.4 Local regularised neurofuzzy models                136
      5.7 Complexity reduction through orthogonal least squares    143
      5.8 A-optimality neurofuzzy model construction (NeuDec)      144
6.    Local neurofuzzy modelling                                   153
      6.1 Introduction                                             153
      6.2 Local orthogonal partitioning algorithms                 157
          6.2.1 k-d Trees .. . :                                   157
          6.2.2 Quad-trees                                         161
                                                            Contents      XV

     6.3   Operating point dependent neurofuzzy models                    164
     6.4   State space representations of operating point dependent
           neurofuzzy models                                              168
     6.5   Mixture of experts modelling                                   173
     6.6   Multi-input-Multi-output (MIMO) modelling
           via input variable selection                                   187
           6.6.1 MIMO NARX neurofuzzy model decomposition                 187
           6.6.2 Feedforward Gram-Schmidt OLS procedure
                  for linear systems                                      191
           6.6.3 Input variable selection via the modified Gram-Schmidt
                  OLS for piecewise linear submodels                      193

7.   Delaunay input space partitioning modelling                         201
     7.1 Introduction                                                    201
     7.2 Delaunay triangulation of the input space                       202
     7.3 Delaunay input space partitioning for locally linear models . . 204
     7.4 The Bezier-Bernstein modelling network                          209
         7.4.1 Neurofuzzy modelling using Bezier-Bernstein
                function for univariate term fi(xi) and bivariate
                term fiujl(xh,xh)                                        209
         7.4.2 The complete Bezier-Bernstein model construction
                algorithm                                                219
         7.4.3 Numerical examples                                        220

8.   Neurofuzzy linearisation modelling
     for nonlinear state estimation                                       225
     8.1 Introduction to linearisation modelling                          225
     8.2 Neurofuzzy local linearisation and the MASMOD algorithm..        228
     8.3 A hybrid learning scheme combining MASMOD
          and EM algorithms for neurofuzzy local linearisation            236
     8.4 Neurofuzzy feedback linearisation (NFFL)                         239
     8.5 Formulation of neurofuzzy state estimators                       245
     8.6 An example of nonlinear trajectory estimation                    249

9.   Multisensor data fusion using Kalman filters based
     on neurofuzzy linearisation                                          255
     9.1 Introduction                                                     255
     9.2 Measurement fusion                                               258
         9.2.1 Outputs augmented fusion (OAF)                             259
         9.2.2 Optimal weighting measurement fusion (OWMF)                259
         9.2.3 On functional equivalence of OAF and OWMF                  260
         9.2.4 On the decentralised architecture                          262
     9.3 State-vector fusion                                              263
         9.3.1 State-vector assimilation fusion (SVAF)                    263
         9.3.2 Track-to-track fusion (TTF)                                264
XVI     Contents
          9.3.3 On the decentralised architecture                           265
      9.4  Hierarchical multisensor data fusion -
          trade-off between centralised and decentralised Architectures .   266
      9.5 Simulation examples                                               267
          9.5.1 On functional equivalence of two measurement fusion
                  methods                                                   267
          9.5.2 On hierarchical multisensor data fusion                     271
10. Support vector neurofuzzy models                                        281
    10.1 Introduction                                                       281
    10.2 Support vector machines                                            282
         10.2.1 Loss functions                                              284
         10.2.2 Feature space and kernel functions                          284
    10.3 Support vector regression                                          286
    10.4 Support vector neurofuzzy networks                                 289
    10.5 SUPANOVA                                                           297
    10.6 A comparison among neural network models                           303
    10.7 Conclusions                                                        304
References                                                                  307
Index                                                                       319
PPN: 099283697
Titel: Adaptive modelling, estimation and fusion from data : a neurofuzzy approach / Chris Harris;
Xia Hong; Qiang Gan. - Berlin : Springer, 2002
ISBN: 3-540-42686-8
Bibliographischer Datensatz im SWB-Verbund