prof. dr. H.G.Dehling prof. dr. F.Takens
contents
1 Introduction. 5
1.1 Time series . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Nonlinear models . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Chaotic dynamical systems . . . . . . . . . . . . . . . . . . 18
1.4 State space reconstruction . . . . . . . . . . . . . . . . . . . 24
1.5 Ergodicity and mixing . . . . . . . . . . . . . . . . . . . . . 27
1.6 U-statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.7 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Review of correlation dimension estimators 37
2.1 Grassberger-Proccacia estimator . . . . . . . . . . . . . . . 38
2.2 Takens estimator . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3 Other estimators . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Consistency of the Takens estimator for the correlation di-
mension 47
3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Weak consistency of U-statistics . . . . . . . . . . . . . . . 50
3.3 Application to the Takens estimator . . . . . . . . . . . . . 55
3.4 Numerical example . . . . . . . . . . . . . . . . . . . . . . . 56
4 Limit theorems for functionals of absolutely regular pro-
cesses 59
4.1 Basic de nitions and coupling lemmas . . . . . . . . . . . . 60
4.2 Correlation and moment inequalities . . . . . . . . . . . . . 66
4.3 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . 73
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4.4 Weak convergence of the empirical process indexed by functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 Limit theorems for U-statistics and U-statistics processes 91
5.1 CLT for U-statistics of functionals of absolutely regular sequences
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Weak convergence of the empirical process of U-statistics
structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3 Bounds on the remainder Rn(t) . . . . . . . . . . . . . . . . 102
5.4 U-process for functionals of absolutely regular sequences . . 112
5.5 U-statistics with unbounded kernel function . . . . . . . . . 117
5.6 Applications to the dimension estimation . . . . . . . . . . 121
6 Estimating the variance of the sample correlation integral127
6.1 Bootstrap for the data from dynamical systems . . . . . . . 129
6.1.1 General background of the bootstrap . . . . . . . . . 129
6.1.2 Consistency of the bootstrap for the functionals of
mixing processes . . . . . . . . . . . . . . . . . . . . 131
6.1.3 Hoe ding decomposition method and bootstrap for
U-statistics . . . . . . . . . . . . . . . . . . . . . . . 138
6.2 Application to a chaotic time series . . . . . . . . . . . . . . 140
6.2.1 Monte-Carlo method . . . . . . . . . . . . . . . . . . 142
6.2.2 Application of the bootstrap . . . . . . . . . . . . . 142
6.2.3 Hoe ding decomposition method . . . . . . . . . . . 146
6.2.4 Application of the bootstrap to fh1(Xi)g . . . . . . 147
6.3 Comparison with a data set of independent observations . . 148
7 Prediction 151
7.1 Kernel autoregression estimation for time series . . . . . . . 153
7.2 Variation of kernel smoothing method . . . . . . . . . . . . 156
7.3 Prediction by Neural Networks . . . . . . . . . . . . . . . . 161
7.3.1 Feedforward Neural Networks . . . . . . . . . . . . . 162
7.3.2 Projection Pursuit Learning . . . . . . . . . . . . . . 164
7.4 Application to real-life time series . . . . . . . . . . . . . . . 167
7.4.1 Fluidized bed time series . . . . . . . . . . . . . . . 167
7.4.2 Gas export time series . . . . . . . . . . . . . . . . . 170
7.4.3 Comparisons . . . . . . . . . . . . . . . . . . . . . . 177 | 