By Willi-Hans Steeb
Offers all of the thoughts and techniques utilized in nonlinear dynamics. Discusses Chaos, Fractals, mobile Automata, Neural Networks, Genetic Algorithms, Fuzzy common sense with C++, Java, Symbolic C++ and decrease courses. Softcover.
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Extra info for The nonlinear workbook: chaos, fractals, neural networks, genetic algorithms, fuzzy logic
Xn ) − E [F(X1 , . . , Xn )] ≤ Lσ 2ξ , where P(ξ ≥ t) ≤ e−t for t > 0. When f : R → R is L-Lipschitz, the Cauchy–Schwartz inequality gives L n 1 n 1 n L f (Xi ) − ∑ f (Yi ) ≤ ∑ |Xi −Yi | ≤ √ ∑ n i=1 n i=1 n i=1 n n ∑ (Xi −Yi )2 , i=1 so the function F(X1 , . . , Xn ) = n−1 ∑ni=1 f (Xi ) is (n−1/2 L)-Lipschitz. 6). Concentration inequalities are central tools for the non asymptotic analysis of estimators, and we will meet them in every major proof of this book. 6 at the end of this chapter. 1 About This Book Statistics and Data Analysis Data science is an ever-expanding field.
Such a point of view is definitely more suited to modern data than the classical point of view. Yet, it is sensitive to the choice of the function f . For example, asymptotic results for f (n) = n2 and f (n) = eαn can be very different. If p = 1000 and n = 33, are you in the setting f (n) = n2 or f (n) = en/5 ? An alternative point of view is to treat n and p as they are and provide a non asymptotic analysis of the estimators, which is valid for any value of n and p. Such an analysis avoids the above caveat of the asymptotic analysis.
D. ,p α log(p) = 1 − 1 − P |Z1 | ≥ = 1 − exp − p α log(p) 2 p1−α/2 +O απ (log p)1/2 p1−α/2 (log p)3/2 . 4 23 Principal Component Analysis The Principal Component Analysis (PCA) is tightly linked to the Singular Value Decomposition (SVD). We refer to Appendix C for a reminder on the SVD. For any data points X (1) , . . 0 where ProjV is the orthogonal projection matrix onto V . Let us denote by X = ∑rk=1 σk uk vTk a SVD of the n × p matrix (1) T (X ) X = ... 0 X[,1] (X (n) )T V2 in dimension p = 3.
The nonlinear workbook: chaos, fractals, neural networks, genetic algorithms, fuzzy logic by Willi-Hans Steeb