非線性時間序列分析

PrefaceVii
AcknowledgementsiX
Part1BasictopicsI
Chapter1Introduction:Whynonlinearmethods?3
Chapter2Lineartoolsandgeneralconsiderations13
2.1Stationarityandsampling13
2.2Testingforstationarity15
2.3Linearcorrelationsandthepowerspectrum18
2.3.1Stationarityandthelow-frequencycomponentinthepowerspectrum22
2.4Linearfilters23
2.5Linearpredictions25
Chapter3Phasespacemethods29
3.1Determinism:Uniquenessinphasespace29
3.2Delayreconstruction33
3.3Findingagoodembedding34
3.4Visualinspectionofdata37
3.5Poincaresurfaceofsection37
Chapter4Determinismandpredictability42
4.1Sourcesofpredictability43
4.2Simplenonlinearpredictionalgorithm44
4.3Verificationofsuccessfulprediction46
4.4Probingstationaritywithnonlinearpredictions49
4.5Simplenonlinearnoisereduction51
Chapter5Instability:Lyapunovexponents58
5.1Sensitivedependenceoninitialconditions58
5.2Exponentialdivergence59
5.3Measuringthemaximalexponentfromdata62
Chapter6Self-similarity:Dimensions69
6.1Attractorgeometryandfractals69
6.2Correlationdimension70
6.3Correlationsumfromatimeseries72
6.4Interpretationandpitfalls75
6.5Temporalcorrelations,nonstationarity,andspacetimeseparationplots81
6.6Practicalconsiderations84
6.7Ausefulapplication:Determinationofthenoiselevel86
Chapter7Usingnonlinearmethodswhendeterminismisweak91
7.1Testingfornonlinearitywithsurrogatedata93
7.1.1Thenullhypothesis95
7.1.2Howtomakesurrogatedatasets96
7.1.3Whichstatisticstouse99
7.1.4Whatcangowrong102
7.1.5Whatwehavelearned103
7.2Nonlinearstatisticsforsystemdiscrimination104
7.3Extractingqualitativeinformationfromatimeseries108
Chapter8Selectednonlinearphenomena112
8.1Coexistenceofattractors112
8.2Transients113
8.3Intermittency114
8.4Structuralstability118
8.5Bifurcations119
8.6Quasi-periodicity121
Part2Advancedtopics123
Chapter9Advancedembeddingmethods125
9.1Embeddingtheorems125
9.1.1Whitney'sembeddingtheorem126
9.1.2Takens'sdelayembeddingtheorem127
9.2Thetimelag130
9.3Filtereddelayembeddings134
9.3.1Derivativecoordinates134
9.3.2Principalcomponentanalysis135
9.4Fluctuatingtimeintervals139
9.5Multichannelmeasurements141
9.5.1Equivalentvariablesatdifferentpositions141
9.5.2Variableswithdifferentphysicalmeanings142
9.5.3Distributedsystems143
9.6Embeddingofinterspikeintervals145
Chapter10Chaoticdataandnoise150
10.1Measurementnoiseanddynamicalnoise150
10.2Effectsofnoise151
10.3Nonlinearnoisereduction154
10.3.1Noisereductionbygradientdescent155
10.3.2Localprojectivenoisereduction156
10.3.3Implementationoflocallyprojectivenoisereduction159
10.3.4Howmuchnoiseistakenout?163
10.3.5Consistencytests167
10.4Anapplication:FoetalECGextraction168
Chapter11Moreaboutinvariantqnantities172
11.1Ergodicityandstrangeattractors173
11.2LyapunovexponentsII174
11.2.1ThespectrumofLyapunovexponentsandinvariantmanifolds174
11.2.2Flowsversusmaps176
11.2.3Tangentspacemethod177
11.2.4Spuriousexponents178
11.2.5Almosttwo-dimensionalflows184
11.3DimensionsII184
11.3.1Generaliseddimensions,multifractals186
11.3.2Informationdimensionfromatimeseries188
11.4Entropies189
11.4.1Chaosandtheflowofinformation189
11.4.2Entropiesofastaticdistribution191
11.4.3TheKolmogorov-Sinaientropy193
11.4.4Entropiesfromtimeseriesdata194
11.5Howthingsarerelated198
11.5.1Pesin'sidentity198
11.5.2Kaplan-Yorkeconjecture199
Chapter12Modellingandforecasting202
12.1Stochasticmodels204
12.1.1Linearfilters204
12.1.2Nonlinearfilters206
12.1.3Markermodels207
12.2Deterministicdynamics207
12.3Localmethodsinphasespace208
12.3.1Almostmodelfreemethods209
12.3.2Locallinearfits209
12.4Globalnonlinearmodels211
12.4.1Polynomials211
12.4.2Radialbasisfunctions212
12.4.3Neuralnetworks213
12.4.4Whattodoinpractice214
12.5Improvedcostfunctions215
12.5.1Overfittingandmodelcosts216
12.5.2Theerrors-in-variablesproblem217
12.6Modelverification219
Chapter13Chaoscontrol223
13.1Unstableperiodicorbitsandtheirinvariantmanifolds224
13.1.1Locatingperiodicorbits225
13.1.2Stable/unstablemanifoldsfromdata229
13.2OGY-controlandderivates231
13.3VariantsofOGY-control234
13.4Delayedfeedback235
13.5Chaossuppressionwithoutfeedback235
13.6Tracking236
13.7Relatedaspects237
Chapter14Otherselectedtopics239
14.1Highdimensionalchaos239
14.1.1Analysisofhigherdimensionalsignals241
14.1.2Spatiallyextendedsystems245
14.2Analysisofspatiotemporalpatterns247
14.3Multiscaleorself-similarsignals,wavelets249
14.3.1Dynamicaloriginofmultiscalesignals250
14.3.2Scalinglaws252
14.3.3Waveletanalysis254
AppendixAEfficientneighboursearching257
AppendixBProgramlistings262
AppendixCDescriptionoftheexperimentaldatasets278
References288
Index300