Change-point Problem Research Articles

Empirical Likelihood Ratio Test for the Epidemic Change Model

Change point problem has been studied extensively since 1950s due to its broad applications in many elds such as nance, biology and so on. As a special case of the multiple change point problem, the epidemic change point problem has received a lot of attention especially in medical studies. In this paper, a nonparametric method based on the empirical likelihood is proposed to detect the epidemic changes of the mean after unknown change points. Under some mild conditions, the asymptotic null distribution of the empirical likelihood ratio test statistic is proved to be the extreme distribution. The consistency of the test is also proved. Simulations indicate that the test behaves comparable to the other available tests while it enjoys less constraint on the data distribution. The method is applied to the Standford heart transplant data and detects the change points successfully.

Detecting Change in the Number of Modes for Circular Data

In this paper, we investigate change-point problems for the number of modes in circular data. We use a mixture of two circular normal distributions to model the observed data. Tests for detecting the presence of change-point are derived using the generalized likelihood-ratio method. Asymptotic distributions of the test statistics are obtained. Results of some simulation studies are also presented.

Limit Distribution of the φ-Divergence Based Change Point Estimator

The assumption of stationarity is too restrictive especially for long time series. This paper studies the change point problem through a change point estimator based on the φ-divergence which provides a rich set of distance like measures between pairs of distributions. The change point problem is considered in the following sub-fields: the problem of divergence estimation, testing for the homogeneity between two samples as well as estimating the time of change. The asymptotic distribution of the change point estimator is estimated by the limiting distribution of a stochastic process within given bounds through asymptotic theory surrounding the likelihood theory. The distribution is found to converge to that of a standardized Brownian bridge process.

Discrete Time-Space Stochastic Mathematical Modelling for Quantitative Description of Software Imperfect Fault-Debugging with Change-Point

Statistics and stochastic-process theories, along with the mathematical modelling and the respective empirical evidence support, describe the software fault-debugging phenomenon. In software-reliability engineering literature, stochastic mathemat-ical models based on the non-homogeneous Poisson process (NHPP) are employed to measure and boost reliability too. Since reliability evolves on account of the running of computer test-run, NHPP type of discrete time-space models, or difference-equation, is superior to their continuous time-space counterparts. The majority of these models assume either a constant, monotonically increasing, or decreasing fault-debugging rate under an imperfect fault-debugging environment. However, in the most debugging scenario, a sudden change may occur to the fault-debugging rate due to an addition to, deletion from, or modification of the source code. Thus, the fault-debugging rate may not always be smooth and is subject to change at some point in time called change-point. Significantly few studies have addressed the problem of change-point in discrete-time modelling approach. The paper examines the combined effects of change-point and imperfect fault-debugging with the learning process on software-reliability growth phenomena based on the NHPP type of discrete time-space modelling approach. The performance of the proposed modelling approach is compared with other existing approaches on an actual software-reliability dataset cited in literature. The findings reveal that incorporating the effect of change-point in software-reliability growth modelling enhances the accuracy of software-reliability assessment because the stochastic character-istics of the software fault-debugging phenomenon alter at the change-point.

Optimal sequential tests for detection of changes under finite measure space for finite sequences of networks

This paper considers the change-point problem for finite sequences of networks. To avoid the difficulty of computing the normalization coefficient in the models such as Exponential Random Graphical Model (ERGM) and Markov networks, we construct a finite measure space with measure ratio statistics. A new performance measure of detection delay is proposed to detect the changes in distribution of the network data. And under the performance measure we defined, an optimal sequential test is presented. The good performance of the optimal sequential test is illustrated numerically on ERGM and Erdős–Rényi network sequences.

A Bayesian Theory of Change Detection in Statistically Periodic Random Processes

A new class of stochastic processes called independent and periodically identically distributed (i.p.i.d.) processes is defined to capture periodically varying statistical behavior. A novel Bayesian theory is developed for detecting a change in the distribution of an i.p.i.d. process. It is shown that the Bayesian change point problem can be expressed as an optimal control problem of a Markov decision process (MDP) with periodic transition and cost structures. An optimal control theory is developed for periodic MDPs for discounted and undiscounted total cost criteria. A fixed-point equation is obtained that is satisfied by the optimal cost function. It is shown that a nonstationary but periodic policy is optimal. A value iteration algorithm is obtained to compute the optimal cost function. The results from the MDP theory are then applied to detect a change in the distribution of an i.p.i.d. process. It is shown that while a stopping rule based on a periodic sequence of thresholds is exactly optimal, a single-threshold policy is asymptotically optimal, as the probability of false alarm goes to zero. Numerical results are provided to demonstrate that the asymptotically optimal policy is not strictly optimal.

Change-point detection based on adjusted shape context cost method

Change-point problems which originated from the field of quality control have become an important area of research. Although existing methods have been successful in detecting change-points, most of them require the underlying data to follow a specific distribution. Heuristically speaking, those methods only perform well when the data set is hypothesised to follow a normal distribution. In this paper, instead of traditional statistical inference, we propose a new algorithm from a shape perspective, which provides a more robust approach to addressing change-point problems. Our new algorithm will define a novel statistic based on shape context, a rich local shape descriptor, to replace the CUSUM test statistic considered by traditional methods. In addition, some areas which do not have change-points are abandoned through segmentation and screening, reducing computational complexity and increasing available storage. At the same time, we introduce the idea of peak recognition, which increases the robustness and effectiveness of the detection. The experimental results demonstrate that the proposed algorithm significantly outperforms some other methods with regard to accuracy and efficiency, especially when a longer time series is under study. We include analyses of two real-world data sets which demonstrate the practical effectiveness of this algorithm.

Time-varying NoVaS Versus GARCH: Point Prediction, Volatility Estimation and Prediction Intervals

Abstract The NoVaS methodology for prediction of stationary financial returns is reviewed, and the applicability of the NoVaS transformation for volatility estimation is illustrated using realized volatility as a proxy. The realm of applicability of the NoVaS methodology is then extended to non-stationary data (involving local stationarity and/or structural breaks) for one-step ahead point prediction of squared returns. In addition, a NoVaS-based algorithm is proposed for the construction of bootstrap prediction intervals for one-step ahead squared returns for both stationary and non-stationary data. It is shown that the “Time-varying” NoVaS is robust against possible nonstationarities in the data; this is true in terms of locally (but not globally) financial returns but also in change point problems where the NoVaS methodology adapts fast to the new regime that occurs after an unknown/undetected change point. Extensive empirical work shows that the NoVaS methodology generally outperforms the GARCH benchmark for (i) point prediction of squared returns, (ii) interval prediction of squared returns, and (iii) volatility estimation. With regard to target (i), earlier work had shown little advantage of using a nonzero α in the NoVaS transformation. However, in terms or targets (ii) and (iii), it appears that using the Generalized version of NoVaS—either Simple or Exponential—can be quite beneficial and well-worth the associated computational cost.

Testing Relevant Hypotheses in Functional Time Series via Self-Normalization

SummaryWe develop methodology for testing relevant hypotheses about functional time series in a tuning-free way. Instead of testing for exact equality, e.g. for the equality of two mean functions from two independent time series, we propose to test the null hypothesis of no relevant deviation. In the two-sample problem this means that an L2-distance between the two mean functions is smaller than a prespecified threshold. For such hypotheses self-normalization, which was introduced in 2010 by Shao, and Shao and Zhang and is commonly used to avoid the estimation of nuisance parameters, is not directly applicable. We develop new self-normalized procedures for testing relevant hypotheses in the one-sample, two-sample and change point problem and investigate their asymptotic properties. Finite sample properties of the tests proposed are illustrated by means of a simulation study and data examples. Our main focus is on functional time series, but extensions to other settings are also briefly discussed.

Multivariate analysis of variance and change points estimation for high‐dimensional longitudinal data

AbstractThis article considers the problem of testing temporal homogeneity of p‐dimensional population mean vectors from repeated measurements on n subjects over T times. To cope with the challenges brought about by high‐dimensional longitudinal data, we propose methodology that takes into account not only the “large p, large T, and small n” situation but also the complex temporospatial dependence. We consider both the multivariate analysis of variance problem and the change point problem. The asymptotic distributions of the proposed test statistics are established under mild conditions. In the change point setting, when the null hypothesis of temporal homogeneity is rejected, we further propose a binary segmentation method and show that it is consistent with a rate that explicitly depends on p,T, and n. Simulation studies and an application to fMRI data are provided to demonstrate the performance and applicability of the proposed methods.

Semiparametric method for identifying multiple change-points in financial market

Multiple change-points problem has been discussed recently on the background of financial market. As a financial crisis or big event happened, the government should increase the macro-control ability in order to mitigate property damage. The above issue can be resolved through finding a more accurate model which fits the peculiar financial asset price, and finding a more efficient test. This paper proposes a method of detecting multiple change-points under a semiparametric model. Using empirical likelihood technique to acquire the maximum likelihood estimation of multiple change-points, and testing the estimation by loglikelihood ratio. Furthermore, we present a sequential approach to find the number of change-points. The simulation experiments prove that the proposed multiple change-points estimation is more efficient than the nonparametric one. The diagnosis with application for multiple change-points also illustrates the proposed model well.

A modified information criterion for tuning parameter selection in 1d fused LASSO for inference on multiple change points

ABSTRACTInference about multiple change points has been an interesting topic in the statistics literature. Recently, the high throughput technologies became the most popularly used tools in genomic studies and yielded massive data. In particular, when the concern is searching for heterogenous segments in a massive data set, it becomes an interesting problem in statistical change point analysis. That is, one tries to determine if there are multiple change points that separate the data into different parts. Such data have a ‘sparsity’ feature (within each part, the data points are homogenous), and hence penalized regression, such as the 1d fused LASSO, has been recently used for detecting multiple change points in high throughput data. One of the main challenges for detecting change points is to estimate the number of change points which then becomes the problem of how to select an optimal tuning parameter in the LASSO methods for change point problems. Therefore, in this work, we propose to use a modified Bayesian information criterion to estimate the optimal tuning parameter in the 1d fused LASSO for multiple change points detection. We show theoretically that the proposed JMIC consistently identifies the true number of change points via providing the optimal tuning parameter for 1d fused LASSO. Simulation studies and application to a next-generation sequencing data of a breast cancer tumour cell line illustrated the usefulness of the proposed method.

Detecting possibly frequent change-points: Wild Binary Segmentation 2 and steepest-drop model selection

Many existing procedures for detecting multiple change-points in data sequences fail in frequent-change-point scenarios. This article proposes a new change-point detection methodology designed to work well in both infrequent and frequent change-point settings. It is made up of two ingredients: one is “Wild Binary Segmentation 2” (WBS2), a recursive algorithm for producing what we call a ‘complete’ solution path to the change-point detection problem, i.e. a sequence of estimated nested models containing $$0, \ldots , T-1$$ change-points, where T is the data length. The other ingredient is a new model selection procedure, referred to as “Steepest Drop to Low Levels” (SDLL). The SDLL criterion acts on the WBS2 solution path, and, unlike many existing model selection procedures for change-point problems, it is not penalty-based, and only uses thresholding as a certain discrete secondary check. The resulting WBS2.SDLL procedure, combining both ingredients, is shown to be consistent, and to significantly outperform the competition in the frequent change-point scenarios tested. WBS2.SDLL is fast, easy to code and does not require the choice of a window or span parameter.

Blind Signal Detection in Cellular Bands

In this article, we propose a blind wideband signal detection algorithm to detect and locate occupied channels in cellular bands without a priori knowledge of the band channelization. This is formulated as a change-point detection problem. The proposed algorithm is validated using over-the-air measurements taken at different cellular bands and shown to work in a global system for mobile communication, long-term evolution (LTE), and universal mobile telecommunications system bands.

Detection of Multiple Change-Points in the Scale Parameter of a Gamma Distributed Sequence Based on Reversible Jump MCMC

In this paper, the multiple change-point problem in the scale parameter of a sequence of independent gamma distributed observations is discussed. A reversible jump Markov chain Monte Carlo (RJMCMC) algorithm is developed to compute the posterior probabilities of the number and positions of the multiple change-points. Four types of jumps are designed, and the acceptance probability of each type is given. The simulation studies show that the RJMCMC-based method is efficient in the detection of multiple change-points in the scale parameter in gamma distributed sequence, and performs better than a self-normalization based method. In addition, a real data example about successive rises and falls of Shanghai stock exchange composite index yield is used to illustrate the proposed methodology.

Beta approximation and its applications

ABSTRACTDuring the past 30 years, many statistics have been proposed for tackling change-point problems. However, the null distributions of these statistics are not easy to obtain in both theory and practice. In this paper, we consider to use the beta distribution to approximate a standard distribution by matching their first four moments. Sufficient conditions are derived for the beta approximation to any distribution. The distance is used to evaluate the beta approximation to a distribution. To demonstrate the advantage of this approximation over the Rayleigh approximation, the Gumbel approximation and the empirical saddlepoint approximation in change-point analysis, three applications and one real data example are provided for illustration.

Consistent selection of the number of change-points via sample-splitting

In multiple change-point analysis, one of the major challenges is to estimate the number of change-points. Most existing approaches attempt to minimize a Schwarz information criterion which balances a term quantifying model fit with a penalization term accounting for model complexity that increases with the number of change-points and limits overfitting. However, different penalization terms are required to adapt to different contexts of multiple change-point problems and the optimal penalization magnitude usually varies from the model and error distribution. We propose a data-driven selection criterion that is applicable to most kinds of popular change-point detection methods, including binary segmentation and optimal partitioning algorithms. The key idea is to select the number of change-points that minimizes the squared prediction error, which measures the fit of a specified model for a new sample. We develop a cross-validation estimation scheme based on an order-preserved sample-splitting strategy, and establish its asymptotic selection consistency under some mild conditions. Effectiveness of the proposed selection criterion is demonstrated on a variety of numerical experiments and real-data examples.

Univariate mean change point detection: Penalization, CUSUM and optimality

The problem of univariate mean change point detection and localization based on a sequence of $n$ independent observations with piecewise constant means has been intensively studied for more than half century, and serves as a blueprint for change point problems in more complex settings. We provide a complete characterization of this classical problem in a general framework in which the upper bound $\sigma ^{2}$ on the noise variance, the minimal spacing $\Delta $ between two consecutive change points and the minimal magnitude $\kappa $ of the changes, are allowed to vary with $n$. We first show that consistent localization of the change points is impossible in the low signal-to-noise ratio regime $\frac{\kappa \sqrt{\Delta }}{\sigma }\preceq \sqrt{\log (n)}$. In contrast, when $\frac{\kappa \sqrt{\Delta }}{\sigma }$ diverges with $n$ at the rate of at least $\sqrt{\log (n)}$, we demonstrate that two computationally-efficient change point estimators, one based on the solution to an $\ell _{0}$-penalized least squares problem and the other on the popular wild binary segmentation algorithm, are both consistent and achieve a localization rate of the order $\frac{\sigma ^{2}}{\kappa ^{2}}\log (n)$. We further show that such rate is minimax optimal, up to a $\log (n)$ term.

Mandelbrot's Stochastic Time Series Models

I survey and illustrate the main time series models that Mandelbrot introduced into time series analysis in the 1960s and 1970s. I focus particularly on the members of the additive fractional stable family including Lévy flights and fractional Brownian motion (fBm), noting some of the less well‐known aspects of this family, such as the cases when the self‐similarity exponent H and the Hurst exponent J differ. I briefly discuss the role of multiplicative models in modeling the physics of cascades. I then recount the still little‐known story of Mandelbrot's work on fractional renewal models in the late 1960s, explaining how these differ from their more familiar fBm counterpart and form a “missing link” between fBm and the problem of random change points. I conclude by highlighting the frontier problem of damped fractional models.

Modified information criterion for testing changes in skew normal model

In this paper, we study the change point problem for the skew normal distribution model from the view of model selection problem. The detection procedure based on the modified information criterion (MIC) for change problem is proposed. Such a procedure has advantage in detecting the changes in early and late stage of a data comparing to the one based on the traditional Schwarz information criterion which is well known as Bayesian information criterion (BIC) by considering the complexity of the models. Due to the difficulty in deriving the analytic asymptotic distribution of the test statistic based on the MIC procedure, the bootstrap simulation is provided to obtain the critical values at the different significance levels. Simulations are conducted to illustrate the comparisons of performance between MIC, BIC and likelihood ratio test (LRT). Such an approach is applied on two stock market data sets to indicate the detection procedure.