Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

With the aim of understanding the mathematical structure of the fluctuation-dissipation theorem in non-equilibrium statistical physics and then constructing a mathematical principle in the modeling problem for time series analysis, we have developed the theory of KM2O-Langevin equations for discrete time stochastic processes. In this paper, as a new method for model analysis in the theory of KM2O-Langevin equations, we show that block frames provide a natural mathematical language for dealing with minimum norm expansions of multi-dimensional stochastic processes which do not necessarily satisfy stationarity and non-degeneracy conditions. (C) 2007 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jmaa.2007.12.015

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Language：Japanese  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Berlin  

An automatic detection and a precise picking of the arrival times of seismic waves using digital seismograms are important for earthquake early detection systems. Here we suggest a new method for detecting and picking P- and S-wave signals automatically. Compared to methods currently in use, our method requires fewer assumption with properties of the data time series. We divide a record into intervals of equal lengths and check the "local and weak stationarity" of each interval using the theory of the KM2O-Langevin equations. The intervals are stationary when these include only background noise, but the stationarity breaks abruptly when a seismic signal arrives and the intervals include both the background noise and the P-wave. This break of stationarity makes us possible to detect P-wave arrival. We expand the method for picking of S-waves. We applied our method to earthquake data from Hi-net Japan, and 90% of P-wave auto-picks were found to be within 0.1 s of the corresponding manual picks, and 70% of S-wave picks were within 0.1 s of the manual picks. This means that our method is accurate enough to use as a part of the seismic early detection systems. Copyright © The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS)
The Seismological Society of Japan
The Volcanological Society of Japan
The Geodetic Society of Japan
The Japanese Society for Planetary Sciences
TERRAPUB.

DOI： 10.1186/BF03352719

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TERRA SCIENTIFIC PUBL CO  

An automatic detection and a precise picking of the arrival times of seismic waves using digital seismograms are important for earthquake early detection systems. Here we suggest a new method for detecting and picking P and S-wave signals automatically. Compared to methods currently in use, our method requires fewer assumption with properties of the data time series. We divide a record into intervals of equal lengths and check the "local and weak stationarity" of each interval using the theory of the KM2O-Langevin equations. The intervals are stationary when these include only background noise, but the stationarity breaks abruptly when a seismic signal arrives and the intervals include both the background noise and the P-wave. This break of stationarity makes us possible to detect P-wave arrival. We expand the method for picking of S-waves. We applied our method to earthquake data from Hi-net Japan, and 90% of P-wave auto-picks were found to be within 0.1 s of the corresponding manual picks, and 70% of S-wave picks were within 0. 1 s of the manual picks. This means that our method is accurate enough to use as a part of the seismic early detection system.

DOI： 10.1186/BF03352719

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：BLACKWELL PUBLISHING  

Since the 1970s, deep low-frequency earthquakes (DLF) with depths ranging 20-40 km have been observed just beneath the Japan Island Arc. Almost all of these earthquakes are recognized up to now have had magnitudes less than 2.5, so that we have little information concerning IDLE Employing the theory of KM2O-Langevin equations, we develop a new method to represent the characteristics of the coda parts of DLF, and propose a new concept of 'average dissipation spectrum'.
The new averaging algorithm for the KM2O-Langevin matrix function was applied in the analysis of DLF (M: 1.0), which occurred in Akita prefecture on 2001 July 11, and we succeeded in separating the characteristics of the source vibration system and the source excitation process into the averaged dissipation term and the fluctuation term, respectively. The gaps between the arrival times of the fluctuation term's peaks at three stations near the epicentre are slightly different than the gaps between the S-wave arrival times. Assuming a homogenous crust structure with an S-wave velocity of 4.3 km s(-1) and assuming the depth of the second source to be the same as that of the hypocentre, the second source lies about 1.5 km, N 56 degrees E of the hypocentre. We estimate the common characteristics of this DLF successfully by using the 'average dissipation spectrum', which is made up of typical frequencies, theta(k), attenuation factors, Q(k) and amplitude factors, A(k). The common elements of (theta(k) similar to 1.5, Q(k) similar to -0.3) an (theta(k) similar to 3.25, Q(k) similar to -0.45) among all stations indicate the characteristics of the source dynamics of the Akita DLF.
The major parts of the coda waves of DLF satisfy the stationary property, and the causality values for the linear and odd-degree non-linear transformations are relatively higher than those for the even-degree non-linear transformations. These characteristics are quite different from the characteristics of tectonic earthquakes. This quantitative property is common among all DLF.

DOI： 10.1111/j.1635-246X.2006.02838.x

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：BLACKWELL PUBLISHING  

Since the 1970s, deep low-frequency earthquakes (DLF) with depths ranging 20-40 km have been observed just beneath the Japan Island Arc. Almost all of these earthquakes are recognized up to now have had magnitudes less than 2.5, so that we have little information concerning IDLE Employing the theory of KM2O-Langevin equations, we develop a new method to represent the characteristics of the coda parts of DLF, and propose a new concept of 'average dissipation spectrum'.
The new averaging algorithm for the KM2O-Langevin matrix function was applied in the analysis of DLF (M: 1.0), which occurred in Akita prefecture on 2001 July 11, and we succeeded in separating the characteristics of the source vibration system and the source excitation process into the averaged dissipation term and the fluctuation term, respectively. The gaps between the arrival times of the fluctuation term's peaks at three stations near the epicentre are slightly different than the gaps between the S-wave arrival times. Assuming a homogenous crust structure with an S-wave velocity of 4.3 km s(-1) and assuming the depth of the second source to be the same as that of the hypocentre, the second source lies about 1.5 km, N 56 degrees E of the hypocentre. We estimate the common characteristics of this DLF successfully by using the 'average dissipation spectrum', which is made up of typical frequencies, theta(k), attenuation factors, Q(k) and amplitude factors, A(k). The common elements of (theta(k) similar to 1.5, Q(k) similar to -0.3) an (theta(k) similar to 3.25, Q(k) similar to -0.45) among all stations indicate the characteristics of the source dynamics of the Akita DLF.
The major parts of the coda waves of DLF satisfy the stationary property, and the causality values for the linear and odd-degree non-linear transformations are relatively higher than those for the even-degree non-linear transformations. These characteristics are quite different from the characteristics of tectonic earthquakes. This quantitative property is common among all DLF.

DOI： 10.1111/j.1635-246X.2006.02838.x

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATH SOC JAPAN  

In this paper, we shall develop the linear causal analysis for the system consisting of two flows in a real inner product space and give an algorithm for calculating the non-linear filter for a discrete stochastic system which is given by two discrete time stochastic processes, to be called a signal process and an observation process, based upon the theory of KM2O-Langevin equations.

DOI： 10.2969/jmsj/1150287304

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATH SOC JAPAN  

In this paper, we shall develop the linear causal analysis for the system consisting of two flows in a real inner product space and give an algorithm for calculating the non-linear filter for a discrete stochastic system which is given by two discrete time stochastic processes, to be called a signal process and an observation process, based upon the theory of KM2O-Langevin equations.

DOI： 10.2969/jmsj/1150287304

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

In calculations with matrices, block calculations play an important role. To elucidate the essential structures of block decompositions, we shall in this paper introduce a simple system of axioms which guarantees block calculations of rings. The axioms can be interpreted as rules of some kind of information filters. We shall also give another system of axioms in terms of idempotents which is equivalent to the above mentioned system and is similar to the Kolmogorov axioms of probability spaces. (C) 2004 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jalgebra.2004.10.012

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SIAM PUBLICATIONS  

In previous papers, we have investigated degenerate stochastic processes (or more generally, degenerate flows in inner product spaces), which is important in the analysis of time series obtained from deterministic dynamical systems. In this paper, we shall generalize the results in [M. Matsuura, Methodol. Comput. Appl. Probab., 5 (2003), pp. 369 - 387] and derive recursive relations for the minimum norm coefficients of the equations which describe the time evolutions of degenerate nonstationary flows. The obtained results can be considered as recursive relations among the minimum norm least squares solutions of matrix equations.

DOI： 10.1137/S089547980343964

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Language：English   Publisher：Research Association of Statistical Sciences  

CiNii Books

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：KLUWER ACADEMIC PUBL  

The theory of KM2O-Langevin equations for stochastic processes (or more generally, flows in inner product spaces) have been developed in view of applications to time series analysis (e.g., Okabe and Nakano, 1991; Okabe, 1999, 2000; Okabe and Matsuura, 2000). In Klimek et al. (2002) and Matsuura and Ckabe (2001, 2003). we have investigated degenerate flows, which is important in the analysis of time series obtained from deterministic dynamical systems. As a continuation, we shall in this paper derive an efficient algorithm by which the minimum norm coefficients of KM2O-Langevin equations are explicitly obtained in degenerate cases. The obtained results have close relations to the calculations of conditional expectations such as nonlinear predictors of stochastic processes (Matsuura and Okabe, 2001). The method has also potential applications to financial mathematics.

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In this paper, the notion of Moore-Penrose biorthogonal systems is generalized. In [Fiedler, Moore-Penrose biorthogonal systems in Euclidean spaces, Lin. Alg. Appl. 362 (2003), pp. 137-143], transformations of generating systems of Euclidean spaces are examined in connection with the Moore-Penrose inverses of their Gram matrices. In this paper, g-inverses are used instead of the Moore-Penrose inverses and, in particular, the details of transformations derived from reflexive g-inverses are studied. Furthermore, the characterization theorem of Moore-Penrose inverses in [Fiedler and Markham, A characterization of the Moore-Penrose inverse, Lin. Alg. Appl. 179 (1993), pp. 129-133] is extended to any reflexive g-inverse.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE INC  

Let A and B be any n x n real symmetric matrices. The following fact is well known: If \I-n - alphaA - betaB\ = \I-n - alphaA\\I-n - betaB\ for any alpha, beta is an element of R, then AB = 0. There exist various proofs. In this paper, we refine Olkin's method [Linear Algebra Appl. 264 (1997) 217]. Furthermore, his determinantal result is generalized. (C) 2003 Elsevier Science Inc. All rights reserved.

DOI： 10.1016/S0024-3795(02)00615-8

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATH SOC JAPAN  

We have developed the theory of KM2O-Langevin equations for stationary and non-degenerate flow in an inner product space. As its generalization and refinement of the results in [14], [15], [16], we shall treat in this paper a general flow in an inner product space without both the stationarity property and the non-degeneracy property. At first, we shall derive the KM2O-Langevin equation describing the time evolution of the flow and prove the fluctuation-dissipation theorem which states that there exists a relation between the fluctuation part and the dissipation part of the above KM2O-Langevin equation. Next we shall prove the characterization theorem of stationarity property, the construction theorem of a flow with any given nonnegative definite matrix function as its two-point covariance matrix function and the extension theorem of a stationary flow without losing stationarity property.

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Language：English   Publishing type：Research paper (scientific journal)  

We give a short proof, based on the geometry of inner product spaces, of the fluctuation-dissipation theorem that asserts applicability of the Whittle-Wiggins-Robinson algorithm in the context of the KM2O-Langevin equations also in degenerate and non-stationary cases. © 2002 by the University of Notre Dame. All rights reserved.

DOI： 10.14492/hokmj/1350911904

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.4099/math1924.27.51

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In this paper, we prove an extension theorem for a given stationary flow from the pointview of the fluctuation-dissipation theorem and apply it to an extension problem for a given positive definite matrix function with Toeplitz condition defined on a finite set. © 2000 by the University of Notre Dame. All rights reserved.

DOI： 10.14492/hokmj/1350912977

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Language：English   Publisher：MATH SOC JAPAN  

We have developed the theory of KM2O-Langevin equations for stationary and non-degenerate flow in an inner product space. As its generalization and refinement of the results in [14], [15], [16], we shall treat in this paper a general flow in an inner product space without both the stationarity property and the non-degeneracy property. At first, we shall derive the KM2O-Langevin equation describing the time evolution of the flow and prove the fluctuation-dissipation theorem which states that there exists a relation between the fluctuation part and the dissipation part of the above KM2O-Langevin equation. Next we shall prove the characterization theorem of stationarity property, the construction theorem of a flow with any given nonnegative definite matrix function as its two-point covariance matrix function and the extension theorem of a stationary flow without losing stationarity property.

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Language：English  

We give a short proof, based on the geometry of inner product spaces, of the fluctuation-dissipation theorem that asserts applicability of the Whittle-Wiggins-Robinson algorithm in the context of the KM2O-Langevin equations also in degenerate and non-stationary cases. © 2002 by the University of Notre Dame. All rights reserved.

DOI： 10.14492/hokmj/1350911904

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