# Principal Component Analysis And Randomness Tests For Big Data Analysis

## byMieko Tanaka

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This book presents the novel approach of analyzing large-sized rectangular-shaped numerical data (so-called big data). The essence of this approach is to grasp the "meaning" of the data instantly, without getting into the details of individual data. Unlike conventional approaches of principal component analysis, randomness tests, and visualization methods, the authors' approach has the benefits of universality and simplicity of data analysis, regardless of data types, structures, or specific field of science.

First, mathematical preparation is described. The RMT-PCA and the RMT-test utilize the cross-correlation matrix of time series,C=XXT, whereXrepresents a rectangular matrix ofNrows andLcolumns andXTrepresents the transverse matrix ofX. BecauseCis symmetric, namely,C=CT, it can be converted to a diagonal matrix of eigenvalues by a similarity transformation-1=SCSTusing an orthogonal matrixS. WhenNis significantly large, the histogram of the eigenvalue distribution can be compared to the theoretical formula derived in the context of the random matrix theory (RMT, in abbreviation).

Then the RMT-PCA applied to high-frequency stock prices in Japanese and American markets is dealt with. This approach proves its effectiveness in extracting "trendy" business sectors of the financial market over the prescribed time scale. In this case,Xconsists ofNstock- prices of lengthL, and the correlation matrixCis anNbyNsquare matrix, whose element at thei-th row andj-th column is the inner product of the price time series of the lengthLof thei-th stock and thej-th stock of the equal lengthL.

Next, the RMT-test is applied to measure randomness of various random number generators, including algorithmically generated random numbers and physically generated random numbers.

The book concludes by demonstrating two application of the RMT-test: (1) a comparison of hash functions, and (2) stock prediction by means of randomness.

Mieko Tanaka Professor Graduate School of Engineering, Tottori Universitymieko@ike.tottori-u.ac.jp4-101 Koyama-cho Minami, Tottori 680-8550, Japan
Title:Principal Component Analysis And Randomness Tests For Big Data AnalysisFormat:HardcoverPublished:June 13, 2019Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:4431559043

ISBN - 13:9784431559047

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