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| Mandic D.P., Chambers J.A. — Recurrent neural networks for prediction: learning algorithms, architectures and stability |
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| Предметный указатель |
A posteriori, algorithms 113 138
A posteriori, computational complexity 138
A posteriori, error 135
A posteriori, techniques 241
Activation functions, continual adaptation 202
Activation functions, definition 239
Activation functions, desirable properties 51
Activation functions, examples 53
Activation functions, properties to map to the complex plane 61
Activation functions, steepness 200
Activation functions, temperature 200
Activation functions, why nonlinear 50
Activation potential 36 54
ADALINE 2
Adaptability 9
Adaptation gain, behaviour of steepest descent 16
Adaptive algorithms, convergence criteria 163
Adaptive algorithms, momentum 211
Adaptive algorithms, performance criteria 161
Adaptive learning 21
Adaptive resonance theory 2
Adaptive systems, configurations 10
Adaptive systems, generic structure 9
Analytic continuation 61
Asymptotic convergence rate 121
Asymptotic stability 116
Attractors 115
Autonomous systems 263
Autoregressive (AR) models, coefficients 37
Autoregressive integrated moving, average (ARIMA) model 171
Autoregressive moving average (ARMA) models, filter structure 37
Averaging methods 163
Backpropagation 18
Backpropagation, normalised algorithm 153
Backpropagation, through time 209
Backpropagation, through time dynamical equivalence 210
Batch learning 20
Bias/variance dilemma 112
Bilinear model 93
Black box modelling 73
Blind equalizer 12
Block-based estimators 15
Bounded Input Bounded Output (BIBO) stability 38 118
Brower’s Fixed Point Theorem 247
Canonical state-space representation 44
Cauchy Riemann equations 228
Channel equalisation 11
Clamping functions 240
Classes of learning algorithm 18
Cognitive science 1
Complex numbers, elementary transformations 227
Connectionist Models 1
Connectionist models, dates 2
Constructive learning 21
Contraction mapping theorem 245
Contractivity of relaxation 252
Curse of dimensionality 22
Data-reusing 114 135 142
Data-reusing, stabilising features 137 145
David Hilbert 47
Delay space embedding 41
Deseasonalising data 265
Deterministic learning 21
Deterministic versus stochastic (DVS) plots 172 175
Directed algorithms 111
Domain of attraction 116
Dynamic multilayer perceptron (DMLP) 79
Efficiency index 135
Electrocardiogram (ECG) 181
Embedded memory 111
Embedding dimension 74 76 174 178
Equation error 104
Equation error, adaptation 107
Equilibrium point 264
Error criterion 101
Error function 20
Exogeneous inputs 40
Exponentially stable 264
Extended Kalman filter 109
Extended recursive least squares algorithm 217 236
Feedforward network, definition 239
Fixed point, iteration 143
Fixed point, theory 117 245
Forgetting behaviour 110
Forgetting mechanism 101
Frobenius matrix 251
Function definitions, conformal 227
Function definitions, entire 227
Function definitions, meromorphic 227
Gamma memory 42
Gaussian variables, fourth order standard factorisation 167
Gear shifting 21
Global asymptotic stability (GAS) 116 118 251 264
Gradient-based learning 12
Grey box modelling 73
Hammerstein model 77
Heart rate variability 181
Heaviside function 55
hessian 15 52
Holomorphic function 227
Hyperbolic attractor 110
Incremental learning 21
Independence assumptions 163
Independent Identically Distributed (IID) 38
Induced local field 36
Infinite Impulse Response (IIR), equation error adaptive filter 107
Input transformations 23
Kalman Filter (KF) algorithm 14
Kolmogorov function 223
Kolmogorov — Sprecher Theorem 224
Kolmogorov’s theorem 6 93 223
Kolmogorov’s theorem, universal approximation theorem 47 48
Learning rate 13
Learning rate adaptation 200
Learning rate, continual adaptation 202
Learning rate, selection 202
Least Mean Square (LMS) algorithm 14 18
Least Mean Square (LMS) algorithm, data-reusing form 136
Linear filters 37
Linear prediction, foundations 31
Linear regression 14
Liouvitle Theorem 61 228
Lipschitz function 224 246 263
Logistic function 36 53
Logistic function, a contraction 118
Logistic function, approximation 58
Logistic function, fixed points of biased form 127
Lorenz equations 174 195
Lyapunov stability 116 143 162
Lyapunov stability, indirect method 162
Mandelbrot and Julia sets 61
Markov model, first order 164
Massive parallelism 6
Misalignment 168 169
Mobius transformation 47 228
Mobius transformation, fixed points 67
Model reference adaptive system (MRAS) 106
Modular group 229
Modular group, transfer function between neurons 66
Modular neural networks, dynamic equivalence 215
Modular neural networks, static equivalence 214
NARMA with eXogeneous inputs (NARMAX) model, compact representation 71
NARMA with eXogeneous inputs (NARMAX) model, validity 95
Nearest neighbours 175
Nesting 130
Neural dynamics 115
Neural network, bias term 50
Neural network, free parameters 199
| Neural network, general architectures for prediction and system identification 99
Neural network, growing and pruning 21
Neural network, hybrid 84
Neural network, in complex plane 60
Neural network, invertibility 67
Neural network, modularity 26 199 214
Neural network, multilayer feedforward 41
Neural network, nesting 27
Neural network, node structure 2
Neural network, ontogenic 21
Neural network, properties 1
Neural network, radial basis function 60
Neural network, redundancy 113
Neural network, specifications 2
Neural network, spline 56
Neural network, time-delay 42
Neural network, topology 240
Neural network, universal approximators 49 54
Neural network, wavelet 57
Neural network, with locally distributed dynamics (LDNN) 79
Neuron, biological perspective 32
Neuron, definition 3
Neuron, structure 32 36
Noise cancellation 10
Non-recursive algorithm 25
Nonlinear Autoregressive (NAR) model 40
Nonlinear Autoregressive Moving, Average (NARMA) model 39
Nonlinear Autoregressive Moving, recurrent perceptron 97
Nonlinear Finite Impulse Response (FIR) filter, learning algorithm 18
Nonlinear Finite Impulse Response (FIR) filter, normalised gradient descent, optimal step size 153
Nonlinear Finite Impulse Response (FIR) filter, weight update 201
Nonlinear gradient descent 151
Nonlinear parallel model 103
Nonlinearity detection 171 173
Nonparametric modelling 72
Normalised LMS algorithm, learning rate 150
o notation 221
Objective function 20
Ontogenic functions 241
Orthogonal condition 34
Output Error 104
Output error, adaptive infinite impulse response (IIR) filter 105
Output error, learning algorithm 108
Parametric modelling 72
Pattern learning 26
Perceptron 2
Phase space 174
Piecewisedinear model 36
Pipelining 131
Polynomial equations 48
Polynomial time 221
Prediction, basic building blocks 35
Prediction, conditional mean 39 88
Prediction, configuration 11
Prediction, difficulties 5
Prediction, history 4
Prediction, principle 33
Prediction, reasons for using neural networks 5
Preservation of contractivity/expansivity 218
Principal component analysis 23
Proximity functions 54
Pseudolinear regression algorithm 105
Quasi-Newton learning algorithm 15
Rate of convergence 121
Real time recurrent learning (RTRL) 92 108 231
Real time recurrent learning (RTRL), a posteriori form 141
Real time recurrent learning (RTRL), normalised form 159
Real time recurrent learning (RTRL), teacher forcing 234
Real time recurrent learning (RTRL), weight update for static and dynamic equivalence 209
Recurrent backpropagation 109 209
Recurrent backpropagation, static and dynamic equivalence 211
Recurrent neural filter, a posteriori form 140
Recurrent neural filter, fully connected 98
Recurrent neural filter, stability bound for adaptive algorithm 166
Recurrent neural networks (RNNs), activation feedback 81
Recurrent neural networks (RNNs), dynamic behaviour 69
Recurrent neural networks (RNNs), dynamic equivalence 205 207
Recurrent neural networks (RNNs), Elman 82
Recurrent neural networks (RNNs), fully connected, relaxation 133
Recurrent neural networks (RNNs), fully connected, structure 231
Recurrent neural networks (RNNs), Jordan 83
Recurrent neural networks (RNNs), local or global feedback 43
Recurrent neural networks (RNNs), locally recurrent -globally feedforward 82
Recurrent neural networks (RNNs), nesting 130
Recurrent neural networks (RNNs), output feedback 81
Recurrent neural networks (RNNs), pipelined (PRNN) 85 132 204 234
Recurrent neural networks (RNNs), rate of convergence of relaxation 127
Recurrent neural networks (RNNs), relaxation 129
Recurrent neural networks (RNNs), RTRL optimal learning rate 159
Recurrent neural networks (RNNs), static equivalence 205 206
Recurrent neural networks (RNNs), universal approximators 49
Recurrent neural networks (RNNs), Williams — Zipser 83
Recurrent perceptron, GAS relaxation 125
Recursive algorithm 25
Recursive Least-Squares (RLS) algorithm 14
Referent network 205
Riccati equation 15
Robust stability 116
Sandwich structure 86
Santa Fe Institute 6
Saturated-modulus function 57
Seasonal ARIMA model 266
Seasonal behaviour 172
Semiparametric modelling 72
Sensitivities 108
Sequential estimators 15
Set, closure 224
Set, compact 225
Set, dense subset 224
Sigmoid packet 56
Sign-preserving 162
Spin glass 2
Spline, cubic 225
Staircase function 55
Standardisation 23
Stochastic learning 21
Stochastic matrix 253
Stone — Weierstrass theorem 62
Supervised learning 25
Supervised learning, definition 239
Surrogate dataset 173
System identification 10
System linearity 263
Takens’ theorem 44 71 96
Tanh activation function, contraction mapping 124
Teacher forced adaptation 108
Threshold nonlinearity 36
Training set construction 24
Turing machine 22
Unidirected algorithms 111
Uniform approximation 51
Uniform asymptotic stability 264
Uniform stability 264
Unsupervised learning 25
Unsupervised learning, definition 239
Uryson model 77
Vanishing gradient 109 166
Vector and matrix differentiation rules 221
Volterra series, expansion 71
Weierstrass theorem 6 92 224
White box modelling 73
Wiener filter 17
Wiener model 77
Wiener model, represented by NARMA model 80
Wiener — Hammerstein model 78
Wold decomposition 39
Yule — Walker equations 34
Zero-memory nonlinearities 31
Zero-memory nonlinearities, examples 35
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