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Chapter 7. Reconstruction of explicit temporal dependencies......................................... 200
7.1. Parameter estimation .................................................................................................. 201
7.1.1. Methods of estimation..................................................................................................... 202
7.1.2. Comparison of methods .................................................................................................. 205
7.1.3. Optimal estimation .......................................................................................................... 208
7.1.4. Robust estimation............................................................................................................ 209
7.1.5. Conclusions..................................................................................................................... 210
7.2. Approximation............................................................................................................ 210
7.2.1. Two problem settings and terms ..................................................................................... 210
7.2.2. Calculation of parameters ............................................................................................... 212
7.2.3. Model size selection, overlearning, and “Occam’s razor” .............................................. 214
7.2.4. Selection of class of approximating functions ................................................................ 218
7.3. Model validation......................................................................................................... 220
7.3.1. Checking independence of residuals............................................................................... 220
7.3.2. Checking normality of residuals ..................................................................................... 221
7.4. Examples of model applications................................................................................. 223
7.4.1. Forecast ........................................................................................................................... 223
7.4.2. Numerical differentiation................................................................................................225
Chapter 8. Model equations: parameter estimation ......................................................... 229
8.1. Parameter estimates and their accuracy...................................................................... 230
8.1.1. Dynamical noise.............................................................................................................. 230
8.1.2. Measurement noise ......................................................................................................... 232
8.2. Hidden variables......................................................................................................... 235
8.2.1. Methods........................................................................................................................... 235
8.2.2. What benefit is got from successes and failures in modeling?........................................ 238
Chapter 9. Model equations: reconstruction of nonlinear characteristics ..................... 243
9.1. Reconstruction procedures and peculiarities of problem ........................................... 244
9.2. Model structure optimization ..................................................................................... 246
9.3. Reconstruction of characteristics for nonlinear element of electric circuit ................ 248
9.4. Specific choice of model structure ............................................................................. 249
9.4.1. Systems under regular external driving........................................................................... 250
9.4.2. Time-delay systems......................................................................................................... 252
Chapter 10. Reconstruction of equations: “black box” .................................................... 254
10.1. Reconstruction of phase orbit ................................................................................... 255
10.1.1. Takens’ theorems .......................................................................................................... 256
10.1.2. Practical algorithms of reconstruction .......................................................................... 264
10.2. Approximation of multivariable functions ............................................................... 271
10.2.1. Model maps................................................................................................................... 271
10.2.2. Model differential equations ......................................................................................... 279
10.3. Forecast with models of different kinds ................................................................... 280
10.4. Model validation....................................................................................................... 284
Chapter 11. Practical applications of empirical models ................................................... 286
11.1. Segmentation of nonstationary time series............................................................... 286
11.2. Confidential information transmission ..................................................................... 289
11.3. Characterization of coupling between oscillators..................................................... 291
11.4. Other applications..................................................................................................... 294
Conclusion............................................................................................................................. 298
Bibliography ......................................................................................................................... 299
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