The objective of our study is to develop a theoretical framework for conditional random fields (CRFs) which consist on non-stationary stochastic processes, using probability density functions of Fourier amplitudes and phases in the frequency domain. The problem area of CRFs in this study is limited to the estimation of stochastic processes conditioned by realized values of time series at one site.
To represent the properties of non-stationary processes, we will introduce group delay time spectra, which are gradient of phase spectra with respect to frequency. Using the style of likelihood method, the conditional probability density functions of Fourier phases are updated by information of group delay time. Then, a method to generate numerically the conditional random fields containing non-stationary processes is developed and it is verified through the numerical examples that the method can give reasonable results.
The theories and methodologies regarding random fields with the observed data as conditions, which we call ‘conditional random fields (CRFs),’ have been investigated by many researchers in the last decade. While the problem area for CRFs covers various situations, we have limited the target for our study to the estimation of stochastic processes conditioned by realized values of time series.
There are two main approaches to this problem: one is a direct approach from the time-space domain [1], [2], [3], [4], [5], [6] and [7], and the other is from the frequency domain [8], [9], [10], [11], [12] and [13]. Most of the former techniques are based on the krigging and its extensions. In principle, they can deal with not only stationary and homogeneous problems but also the non-stationary and non-homogeneous problems. For many actual problems, however, we often face the difficulty of making mathematical models of stochastic properties for non-stationary or non-homogeneous stochastic problems.
On the other hand, the latter approach is based on the Fourier coefficients at each frequency. While this approach makes it easy to understand physical meanings of time-varying phenomena, this method cannot deal with the non-stationary problems, unfortunately.
This study will provide a simple method under the theoretical framework for stationary problems in order to represent the CRFs which consist of non-stationary stochastic processes. The authors have analytical derived the PDFs of Fourier amplitudes and phases for stationary processes conditioned by observed time series [12]. On this basis, we derive analytically the solution for a simple problem that includes two non-stationary processes: that is, one is a process obtained deterministically and the other will be estimated. The presented method, furthermore, can be easily extended to the problem on multiple processes [14].
Generally speaking, the characteristic of non-stationary stochastic processes appear in Fourier phase spectra. To handle the phase spectra, we will thus introduce the group delay time (tgr), which is defined by the gradient of phase spectra. We will combine tgr with the conditional PDFs of Fourier phases using the style of likelihood method, and then present the method to simulate the non-stationary processes conditioned by observed time series.
A method to generate numerically the conditional random fields containing non-stationary time series was developed. This method was straightforward extension of one of the stationary processes, and formulated using the properties of Fourier phases, where we introduced distribution of group delay time that corresponds to the envelope of time history.
Numerical simulation of non-stationary conditioned time series was performed for the simple case. On this bases, it was demonstrated that the method developed herein gives reasonable results.
