Piecewise Stationary Process Research Articles

A Composite Likelihood-based Approach for Change-point Detection in Spatio-temporal Processes

This paper develops a unified and computationally efficient method for change-point estimation along the time dimension in a non-stationary spatio-temporal process. By modeling a non-stationary spatio-temporal process as a piecewise stationary spatio-temporal process, we consider simultaneous estimation of the number and locations of change-points, and model parameters in each segment. A composite likelihood-based criterion is developed for change-point and parameter estimation. Under the framework of increasing domain asymptotics, theoretical results including consistency and distribution of the estimators are derived under mild conditions. In contrast to classical results in fixed dimensional time series that the localization error of change-point estimator is , exact recovery of true change-points is possible in the spatio-temporal setting. More surprisingly, the consistency of change-point estimation can be achieved without any penalty term in the criterion function. In addition, we further establish consistency of the change-point estimator under the infill asymptotics framework where the time domain is increasing while the spatial sampling domain is fixed. A computationally efficient pruned dynamic programming algorithm is developed for the challenging criterion optimization problem. Extensive simulation studies and an application to the U.S. precipitation data are provided to demonstrate the effectiveness and practicality of the proposed method.

Algorithm of variance estimation in weighted least squares method

The representation of a time series model as a piecewise-stationary process is provided, wherein it is regarded as a collection of successive stationary intervals. An algorithm has been developed for identifying the domain containing the trend within this model. It is recognized that applying the least squares method directly for trend determination is not commonly employed in statistical analysis and econometric software packages. Typically, the weighted least squares method is utilized to ideally eliminate non-stationarity. The authors presents an algorithm for estimating the weight coefficients for this method through piecewise-stationary modeling. The algorithm has been tested on time series of various natures.

A two-step procedure for time-dependent reliability-based design optimization involving piece-wise stationary Gaussian processes

A two-step procedure for time-dependent reliability-based design optimization involving piece-wise stationary Gaussian processes

The Method of the Main Tone Detection in the Structure of Electromyographic Signals for the Task of Broken Human Communicative Function Compensation

In the work the method of electromyographic signals processing for the task of broken human communicative function compensation is developed. The method allows to detect the signs of main tone in the structure of electromyographic signals, that were recorded from the surface of patients neck near the vocal folds. Using this signs it is possible to identify the mentally spoken vowel and vocalised consonant phonemes and to identify the speech of patients with broken or lost communicative function. The developed method includes two stages, namely: preparatory and basic. The purpose of the preparatory stage is to obtain data on the individual features of the patient's speach, in particular the approximate value of the main tone frequency and the frequency interval of the main tone frequency when patient is trying to utter test sequences of sounds at certain points in time. These data are necessary to enable the use of the basic stage of the method, which involves the processing of electromyographic signals recorded in an arbitrary attempt to pronounce arbitrary sounds, words or phrases by the patient. It is proposed to processing the electromyographic signals by methods of spectral-correlation analysis using the sliding window method if presenting of such biosignals in the form of a piecewise stationary random process to detect the time intervals of the presence of main tone signs. Within each sliding window the estimates of the power spectral density distribution are calculated and averaged over the frequency and power within the predetermined interval of existence of the main tone frequency. The obtained averaged estimates make it possible to set the time intervals of the main tone presence and, accordingly, the subsequent identification of vowel and consonant vocalised phonemes. The experimentally registered EMG signal was processed by the developed method with different values of the sliding window width.

Nonparametric change point detection in multivariate piecewise stationary time series

ABSTRACTDetecting change points in multivariate time series is an important problem with numerous applications. We develop a nonparametric method to detect multiple change points in multivariate piecewise stationary processes when the locations and number of change points are unknown. Based on a test statistic that measures differences in the spectral density matrices through the norm, we sequentially identify points of local maxima in the test statistic and test for the significance of each of them being change points. In addition, the components responsible for the change in the covariance structure at each detected change point are identified. The asymptotic properties of the test for significant change points under the null and alternative hypothesis are derived. We illustrate the better performance of our method in comparison to some of the recent methods through a few simulation examples and discuss applications of our method in seismology and finance.

Multiple change-point detection for non-stationary time series using Wild Binary Segmentation

We propose a new technique for consistent estimation of the number and locations of the change-points in the second-order structure of a time series. The core of the segmentation procedure is the Wild Binary Segmentation method(WBS), a technique which involves a certain randomised mechanism. The advantage of WBS over the standard Binary Segmentation lies in its localisation feature, thanks to which it works in cases where the spacings between change-points are short. In addition, we do not restrict the total number of change-points a time series can have. We also ameliorate the performance of our method by combining the CUSUM statistics obtained at different scales of the wavelet periodogram, our main change-point detection statistic, which allows a rigorous estimation of the local autocovariance of a piecewise-stationary process. We provide a simulation study to examine the performance of our method for different types of scenarios. A proof of consistency is also provided. Our methodology is implemented in the R package wbsts, available from CRAN.

Stochastic Projection-Factoring Method Based on Piecewise Stationary Renewal Processes for Mid- and Long-Term Traffic Flow Modeling and Forecasting

Forecasting traffic over a long period of time is of considerable interest and usefulness, but accurate forecasting is very difficult. Traditional projection and factoring methods for mid- and long-term cumulative traffic forecasting are deterministic and only provide a point prediction without specifying a statistical measure of prediction reliability. This paper constructs a stochastic projection and factoring method by casting long-term traffic volume counts into an integrated and rigorous framework of a more refined structural time series component model with piecewise stationary renewal processes capturing time-of-day, day-of-week, monthly, and yearly variations. By doing so, the new method roots itself in a solid theoretical foundation and generates two advantages. First, it results in a more accurate point prediction of cumulative traffic by taking into account the time-of-day traffic count variation in the modeling of unobservable future long-term traffic flow at temporary count stations or at a site under investigation as a mixture of piecewise stationary renewal processes with different means and variances. Second, it allows an interval prediction to be estimated by incorporating uncertainty into the modeling and forecasting process.

Inference for Multiple Change Points in Time Series via Likelihood Ratio Scan Statistics

Summary We propose a likelihood ratio scan method for estimating multiple change points in piecewise stationary processes. Using scan statistics reduces the computationally infeasible global multiple-change-point estimation problem to a number of single-change-point detection problems in various local windows. The computation can be efficiently performed with order O{npt log (n)}. Consistency for the estimated numbers and locations of the change points are established. Moreover, a procedure is developed for constructing confidence intervals for each of the change points. Simulation experiments and real data analysis are conducted to illustrate the efficiency of the likelihood ratio scan method.

Reliability analysis of offshore structures within a time varying environment

The analysis and design of offshore structures necessitates the consideration of environmental loads. Realistic modeling of the environmental loads is particularly important to ensure reliable performance of these structures. In this paper, structural reliability analysis of offshore structures subjected to a time varying environment is investigated. In this work, an extreme value statistical model for the wave height is adopted as a basis for the performance assessment of a jacket structure. Due to the changing environment, the model parameters are modeled to be time varying. To deal with this issue, two segmentation algorithms are proposed and applied to the observed data in order to derive piecewise stationary processes for a statistical analysis. The investigation includes the extreme value modeling of the wave height in the characterization of the sea load. The implementation of the segmentation algorithms in the original data eventually leads to approximations of the safety quality of the existing structure within different time interval. The computed result is developed to reflect the time varying effects in the failure probability of structures. The results are compared with the traditional extreme values approach in view of the accuracy and information content. The investigation is also extended to a case where the design of the structure ignores the time varying property.

Detection of Multiple Structural Breaks in Multivariate Time Series

We propose a new nonparametric procedure (referred to as MuBreD) for the detection and estimation of multiple structural breaks in the autocovariance function of a multivariate (second-order) piecewise stationary process, which also identifies the components of the series where the breaks occur. MuBreD is based on a comparison of the estimated spectral distribution on different segments of the observed time series and consists of three steps: it starts with a consistent test, which allows us to prove the existence of structural breaks at a controlled Type I error. Second, it estimates sets containing possible break points and finally these sets are reduced to identify the relevant structural breaks and corresponding components which are responsible for the changes in the autocovariance structure. In contrast to all other methods proposed in the literature, our approach does not make any parametric assumptions, is not especially designed for detecting one single change point, and addresses the problem of multiple structural breaks in the autocovariance function directly with no use of the binary segmentation algorithm. We prove that the new procedure detects all components and the corresponding locations where structural breaks occur with probability converging to one as the sample size increases and provide data-driven rules for the selection of all regularization parameters. The results are illustrated by analyzing financial asset returns, and in a simulation study it is demonstrated that MuBreD outperforms the currently available nonparametric methods for detecting breaks in the dependency structure of multivariate time series. Supplementary materials for this article are available online.

Approximation model of piecewise stationary stochastic process autocorrelation function

In order to deal with the frequently encountered non-stationary random signals in signal processing,they can be divided into sub-stationary random signals,and autocorrelation function can be used to reflect the essential characteristics of sub-stationary signals.The computation of piecewise stationary stochastic process autocorrelation function was discussed.In order to reduce the amount of calculation and errors of the existing function models,a new model to approximate autocorrelation function of piecewise stationary stochastic process was proposed in this paper.The computer simulation shows that the model can effectively approximate autocorrelation function.The computing speed is faster,and the errors are much fewer and smoother.Applying the model to the restoration of blurred digital images,a very good restoration effect can be got.

GNSS dataless signal tracking with a delay semi-open loop and a phase open loop

In this article we propose to process the code and phase delay of a dataless GNSS signal in open and semi-open loops. The aims of this processing are to get an accurate estimate of the phase and code delays by integration of the GNSS signal. We show that in an open loop the code delay evolution is a piecewise stationary process and we propose to model the phase delay as a circular random variable distributed according to a von Mises distribution. In this context the code tracking is realised on the GNSS signal by estimating the abrupt changes in the code discriminator values. We propose a Bayesian modeling of the problem in order to define the change point estimator. The proposed estimator involves in its definition the inaccurate prior information of Doppler and signal to noise density ratio provided by the phase delay tracking loop. Furthermore we propose a circular Bayesian modeling of the observations provided by the phase open loop. From this model, we derive a circular recursive filter for the estimation of phase delay and frequency of the carrier. The proposed tools are assessed using synthetic and real data.

JPEG stream soft-decoding technique based on autoregressive modeling

This paper introduces a new model-based soft decoding technique to restore the widely used joint photographic expert group (JPEG) streams. The image is modeled as a two dimensional (2D) piecewise stationary autoregressive process, and the decoding task is formulated as a constrained optimization problem. All the constraints are given by the quantization intervals which available at the decoder freely. The autoregressive model serves as an important regularization term of the objective function of the optimization, and the model parameters are solved on the decoded image locally using a weighted total least square method. In addition, a novel bilateral dualside weighting scheme is proposed to minimize the influence of the blocking artifact on the accuracy of parameter estimation. Extensive experimental results suggest that the proposed algorithm systematically improves the quality of JPEG images and also outperforms existing JPEG postprocessing algorithms in a wide bit-rate range both in terms of peak signal-to-noise ratio (PSNR) and subjective quality

A New Optimized Method of Excitation Signal for Closed-Loop Identification of Power System Based on Ambient Signals

Excitation signal optimization is an important part in the power system identification based on ambient signals. An adaptive discrete Kalman filter method is proposed to select an optimal signal for closed-loop identification in this paper. This method is carried out through the use of the measurement innovation sequence as piecewise stationary process inside an estimation window. It also overcomes the shortcomings of relying on the correctness of the mathematical and statistical models excessively. The feature of random load changing in power system is fully considered in this method. Then under the energy constraints of input and output signals, this method can be used to solve the excitation signal which satisfies the performance of power system and the noise covariance estimation matrices are acquired. By using this method, the optimal identification model can be obtained. Simulation results show the effective performance of the proposed method. Compared with other methods, the quality of the closed-loop identification model based on ambient signals is improved by using the excitation signal optimal method proposed in this paper.

Bayesian Change-Points Estimation Applied to GPS Signal Tracking

A hierarchical Bayesian model is applied to off-line segmentation of the GPS signal discriminator. The purpose of this work is to estimate the code delay of the receiving GPS CDMA code in order to retime the local receiver code and to estimate the pseudorange satellite receiver. The goal of our approach is to obtain a high-rate accurate positioning in the dynamic navigation case. We show that the behaviour of the coherent discriminator of a GPS pilot channel can be modelized by a piecewise stationary process. In our approach the discriminator behaviour in each stationary segment is approximated by a constant acceleration model, and the code delay at each end of the segments is known. The interest of this approach is that we use the coherent values of the discriminator in each segment to estimate the change instants of the process and to get in this case an accurate estimation of the code delays. In this context, a simultaneous estimation of the change instants is considered. We define the a posteriori distribution which integrates in its expression the signal change instants and the parameters of its statistical model. The proposed model leads after marginalization to a penalized contrast function that we minimize to estimate the discriminator change instants. The interest of the proposed model is that we can integrate in our estimate prior information on the roughly known values of the signal-to-noise ratio and relative speed satellite receiver. The potential of the proposed method is shown on experimentations realized on synthetic and real data for millisecond receiver localization.

Discrete Denoising With Shifts

We introduce S-DUDE, a new algorithm for denoising DMC-corrupted data. The algorithm, which generalizes the recently introduced DUDE (Discrete Universal DEnoiser) of Weissman et al., aims to compete with a genie that has access, in addition to the noisy data, also to the underlying clean data, and can choose to switch, up to $m$ times, between sliding window denoisers in a way that minimizes the overall loss. When the underlying data form an individual sequence, we show that the S-DUDE performs essentially as well as this genie, provided that $m$ is sub-linear in the size of the data. When the clean data is emitted by a piecewise stationary process, we show that the S-DUDE achieves the optimum distribution-dependent performance, provided that the same sub-linearity condition is imposed on the number of switches. To further substantiate the universal optimality of the S-DUDE, we show that when the number of switches is allowed to grow linearly with the size of the data, \emph{any} (sequence of) scheme(s) fails to compete in the above senses. Using dynamic programming, we derive an efficient implementation of the S-DUDE, which has complexity (time and memory) growing only linearly with the data size and the number of switches $m$. Preliminary experimental results are presented, suggesting that S-DUDE has the capacity to significantly improve on the performance attained by the original DUDE in applications where the nature of the data abruptly changes in time (or space), as is often the case in practice.

Local Spectral Analysis via a Bayesian Mixture of Smoothing Splines

In many practical problems, time series are realizations of nonstationary random processes. These processes can often be modeled as processes with slowly changing dynamics or as piecewise stationary processes. In these cases, various approaches to estimating the time-varying spectral density have been proposed. Our approach in this article is to estimate the log of the Dahlhaus local spectrum using a Bayesian mixture of splines. The basic idea of our approach is to first partition the data into small sections. We then assume that the log spectral density of the evolutionary process in any given partition is a mixture of individual log spectra. We use a mixture of smoothing splines model with time varying mixing weights to estimate the evolutionary log spectrum. The mixture model is fit using Markov chain Monte Carlo techniques that yield estimates of the log spectra of the individual subsections. In addition to an estimate of the local log spectral density, the method yields pointwise credible intervals. We use a reversible jump step to automatically determine the number of different spectral components.

Stochastic modeling of neurobiological time series: Power, coherence, Granger causality, and separation of evoked responses from ongoing activity

In this article we consider the stochastic modeling of neurobiological time series from cognitive experiments. Our starting point is the variable-signal-plus-ongoing-activity model. From this model a differentially variable component analysis strategy is developed from a Bayesian perspective to estimate event-related signals on a single trial basis. After subtracting out the event-related signal from recorded single trial time series, the residual ongoing activity is treated as a piecewise stationary stochastic process and analyzed by an adaptive multivariate autoregressive modeling strategy which yields power, coherence, and Granger causality spectra. Results from applying these methods to local field potential recordings from monkeys performing cognitive tasks are presented.

Joint segmentation of the wind speed and direction

We present in this article a Bayesian estimation method for the joint segmentation of a set of piecewise stationary processes. The estimate we propose is based on the maximization of the posterior distribution of the change instants conditionally to the process parameter estimation. It is defined as a penalized contrast function with a first term related to the fit to the observation and a second term of penalty. The expression of the contrast function is deduced from the log-likelihood of the parametric distribution that models the statistic evolution of processes in the stationary segments. In the case of joint segmentation the penalty term is deduced from the prior law of change instants. It is composed of parameters that guide the number and the position of changes and of parameters that will bring prior information on the joint behavior of processes. This work is applied to the estimation of wind statistics parameters. We use data available from a cup anemometer and a wind vane, supposed to be piecewise stationary. The contrast function is deduced from the circular Von Mises distribution for the wind direction and from the log-normal distribution for the speed. The feasibility and the contribution of our method are shown on synthetic and real data.

Attitude Determination from GNSS Using Adaptive Kalman Filtering

An adaptive Kalman filtering approach is proposed for attitude determination to replace the fixed (conventional) Kalman filtering approach. The filter is used to adaptively reflect system dynamics changes or rapid changes in vehicle trajectory. The estimation procedure is carried out through the use of a measurement residual sequence. The sequence is used as a piece-wise stationary process inside an estimation window. The measurement noise covariance matrix and the system noise matrix are adaptively estimated. An extended Kalman filter approach with iteration of the states within-an-epoch was performed to overcome the non-linearity of the observation equations. A test was performed to evaluate the proposed technique. Different trajectory scenarios are presented to show the difference in performance between the adaptive Kalman filter and the conventional one. Results show that the proposed adoptive filtering approach has a better performance than the conventional filter.