This book presents the novel approach of analyzing large-sized 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, where X represents a rectangular matrix of N rows and L columns and XT represents the transverse matrix of X. The RMT-PCA uses N samples of time series of length L. The RMT-test uses N elements of length L by cutting a single data to N pieces. Because C is symmetric, namely, C = CT, it can be converted to a diagonal matrix of eigenvalues by a similarity transformation SCST using an orthogonal matrix S. When N is 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 is applied to high-frequency stock prices in Japanese and American markets. This approach proves its effectiveness in extracting trendy business sectors of the financial market over the prescribed time scale. In this case, X consists of N stock- prices of length L, and the correlation matrix C is an N by N square matrix, whose element at the i-th row and j-th column is the inner product of the price time series of the length L of the i-th stock and the j-th stock of the equal length L.
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 three applications of the RMT-test: (1) a comparison of hash functions, (2) choice of safe stocks, and (3) prediction of stock index by means of a sudden change of randomness.
