Conditional random fields containing non-stationary stochastic processes

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. resent 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.