Signal identification without signal formulation
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When there are signals and noises, physicists try to identify signals by modeling them, whereas statisticians oppositely try to model noise to identify signals. In this study, we applied the statisticians' concept of signal detection of physics data with small-size samples and high dimensions without modeling the signals. Most of the data in nature, whether noises or signals, are assumed to be generated by dynamical systems; thus, there is essentially no distinction between these generating processes. We propose that the correlation length of a dynamical system and the number of samples are crucial for the practical definition of noise variables among the signal variables generated by such a system. Since variables with short-term correlations reach normal distributions faster as the number of samples decreases, they are regarded to be “noise-like” variables, whereas variables with opposite properties are “signal-like” variables. Normality tests are not effective for data of small-size samples with high dimensions. Therefore, we modeled noises on the basis of the property of a noise variable, that is, the uniformity of the histogram of the probability that a variable is a noise. We devised a method of detecting signal variables from the structural change of the histogram according to the decrease in the number of samples. We applied our method to the data generated by globally coupled map, which can produce time series data with different correlation lengths, and also applied to gene expression data, which are typical static data of small-size samples with high dimensions, and we successfully detected signal variables from them. Moreover, we verified the assumption that the gene expression data also potentially have a dynamical system as their generation model, and found that the assumption is compatible with the results of signal extraction.
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