open access

https://doi.org/10.11501/3168542

https://hdl.handle.net/2115/51636

A study on multi-dimensional non-linear information analysis and time series analysis

日本語

2000-03-24

Hokkaido University

53p.

In this thesis, we studied the following seven subjects: (a) a study on a non-linear information analysis for multi-dimensional stochastic processes with local time parameter and a construction of a generator with a nest structure for the non-linear information space (b) a study on a non-linear prediction formula to calculate the non-linear predictor for multi-dimensional stochastic processes with local time parameter (c) a study on formulations and characterization theorems of two new concepts of weak causality and non-instantaneous weak causality in cause-result relation based upon the multi-dimensional non-linear information analysis and their characterization (d) a study on a partial non-linear information space which is necessary for data analysis (e) a study on a forinulation of non-linear caufality of finite rank based upon the partial non-linear information space (f) a study on a method of model selection and prediction formula based upon the non-linear causal analysis (g) a study on a usage for applying multi-dimensional non-linear information analysis to data analysis (h) a study on effectiveness of both the multi-dimensional non-linear information analysis and the multi-dimensional non-linear causal analysis The non-linear prediction problem has been first solved by Masani-Wiener for the onedimensional strictly stationary process with global time parameter. They required both the boundedness condition and the non-degenerate conditon for the process. Though the prediction problem was solved theoretically, however, their result lacked for the computable algorithm for calculating the non-linear predictor. Under the same conditions as in Masani-Wiener's work, Okabe-Ootsuka have given a computable algorithm for calculating the non-linear predictor, by using the theory of KM20-Langevin equations for non-degenerate stochastic processes. Moreover, by developing the theory of KM20-Langevin equations for degenerate flows in inner product spaces, Matsuura-Okabe gave a computable algorithm for calculating the non-linear predictor for the one-dimensional stochastic processes under the only exponentially integrable condition weaker than the boundedness condition. On the other hand, various kinds of methods in non-linear time series analysis such as stationary analysis, causal analysis, modeling analysis and prediction analysis have been developed in the framework of the theory of KM20-Langevin equations. However, the non-linearity used in such analyses come from the analysis for one-dimensional stochastic processes. In the course of carrying out the time series analysis for concrete data, we found that it is necessary to construct a theory of information analysis which covers not only the non-linear analysis for one-dimensional stochastic processes, but also the non-linear analysis for multidimensional stochastic processes. This indicates that we need a multi-dimensional theory not as an general extension of one-dimensional theory, but as an essential extension. We shall explain it by giving an example. Let X ＝ (X(n); 0 ≤ n ≤ N) be a one-dimensional stochastic process which we want to investigate and Y ＝ (Y(n); 0 ≤ n ≤ N） be an another one-dimensional stochastic process which seems to affect X in some sense. Let (Ω, β, P) be a probability space on which X and Y are defined. Moreover, we assume that X(n), Y(n) (0 ≤ n ≤ N) are square-integrable. By using the non-linear information analysis for Y, we can examine the causal relation from Y to X. In this case, we calculate PNn0(Y)X(n), where βno(Y) is the minimal σ-algebra with respect to which Y(k) (0 ≤ k ≤ n) are measurable, Nn0(Y) ＝ L2(Ω, βn0(Y), P) and PNn0(Y) is the projection operator from L2(Ω, β, P) onto Nn0(Y). The space Nn0(Y) is insufficient for analyzing the non-linear mutual information between X and Y, because Nn0(Y) uses only the non-linear information of Y. The above discussion indicates the necessity of the non-linear mutual information analysis for X and Y. For that purpose, we need to calculate P-Nn0(X,Y)X(n), where β∼n0(X, Y) is the minimal σ-algebra with respect to which X(m) (0 ≤ m ≤ n - 1), Y(k) (0 ≤ k ≤ n) are measurable, N∼n0(X, Y) ＝ L2(Ω,β∼n0(X, Y), P) and P-Nn0(X,Y) is the projection operator from L2(Ω, β, P) onto N∼n0(X,Y). In this way, we need the multi-dimensional non-linear information analysis for two-dimensional stochastic process t(X, Y). The state of affairs is still more for the case where Y is a multi-dimensional stochastic process. Now the contents of this thesis are stated below. We study the non-linear information analysis for the multi-dimensional stochastic processes with local time parameter ((a)). Also we obtain the non-linear prediction formula, which is based upon the theory of KM2O-Langevin equations for degenerate flows ((b)). We discuss two new causal relations, to be called weak causality and non-instantaneous weak causality, as applications of the multi-dimensiollal non-linear information analysis. Moreover, we give a quantitative characterization for each causal relation ((c)). It is theoretically possible for us to give a complete expression of the non-linear predictor and the non-linear causality by using the infinitely many non-linearity. However, we have to approximate the non-linear predictor and the non-linear causality by using a part of the non-linear information, because we cannot use the entire non-linear information in data analysis. For that purpose, we discuss the partial non-linear information space which bridges between theory and practice ((d)). It is preferable to select the best partial non-linearity according to certain criteria. We propose such a criteria by using the causal analysis. In order to express the relation between the causality and the partial non-linear information space, we define the concept of the non-linear causality of finite rank and discuss the relation between the causality and the partial non-linear information ((e)). In addition, we propose a method of model selection by using the non-linear causality of finite rank and give a formula for calculating the non-linear predictor for each selected model ((f)). The part mentioned above is a theoretical result of this thesis. In the remaining part of this thesis, we first show a procedure of non-linear time series analysis based upon the theorical results ((g)). Finally, as an example of time series analysis for concrete data, we deal with three kinds of data connected with EINino∼ and find that the multi-dimensional information analysis is effective for the causal analysis and the prediction analysis based upon the cauf al analysis ((h)).

2000-03-24

甲第4978号

博士(理学)

博士論文

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