True Change Points Research Articles

Finding True Change Point When a CUSUM Control Chart is Used

Finding True Change Point When a CUSUM Control Chart is Used

Estimating non-simultaneous changes in the mean of vectors

A sequence of independent vectors with correlated components is considered. It is supposed that there is one change point in the mean of each component and changes need not occur simultaneously. The asymptotic distribution of the change point estimators is studied. If the true change points are well separated, the explicit asymptotic distribution of the change point estimators is presented. In the case the true change points coincide, it is shown that the limit distribution of properly standardized change points estimates exists. It depends not only on the underlying time series dependence structure, but also on the ratio of the sizes of the changes. The asymptotic distribution function is not known, but due to the invariance principle it can be obtained by simulations.

THE SCREENING AND RANKING ALGORITHM FOR CHANGE-POINTS DETECTION IN MULTIPLE SAMPLES.

The chromosome copy number variation (CNV) is the deviation of genomic regions from their normal copy number states, which may associate with many human diseases. Current genetic studies usually collect hundreds to thousands of samples to study the association between CNV and diseases. CNVs can be called by detecting the change-points in mean for sequences of array-based intensity measurements. Although multiple samples are of interest, the majority of the available CNV calling methods are single sample based. Only a few multiple sample methods have been proposed using scan statistics that are computationally intensive and designed toward either common or rare change-points detection. In this paper, we propose a novel multiple sample method by adaptively combining the scan statistic of the screening and ranking algorithm (SaRa), which is computationally efficient and is able to detect both common and rare change-points. We prove that asymptotically this method can find the true change-points with almost certainty and show in theory that multiple sample methods are superior to single sample methods when shared change-points are of interest. Additionally, we report extensive simulation studies to examine the performance of our proposed method. Finally, using our proposed method as well as two competing approaches, we attempt to detect CNVs in the data from the Primary Open-Angle Glaucoma Genes and Environment study, and conclude that our method is faster and requires less information while our ability to detect the CNVs is comparable or better.

A Bayesian model for multiple change point to extremes, with application to environmental and financial data

ABSTRACTAbrupt changes often occur for environmental and financial time series. Most often, these changes are due to human intervention. Change point analysis is a statistical tool used to analyze sudden changes in observations along the time series. In this paper, we propose a Bayesian model for extreme values for environmental and economic datasets that present a typical change point behavior. The model proposed in this paper addresses the situation in which more than one change point can occur in a time series. By analyzing maxima, the distribution of each regime is a generalized extreme value distribution. In this model, the change points are unknown and considered parameters to be estimated. Simulations of extremes with two change points showed that the proposed algorithm can recover the true values of the parameters, in addition to detecting the true change points in different configurations. Also, the number of change points was a problem to be considered, and the Bayesian estimation can correctly identify the correct number of change points for each application. Environmental and financial data were analyzed and results showed the importance of considering the change point in the data and revealed that this change of regime brought about an increase in the return levels, increasing the number of floods in cities around the rivers. Stock market levels showed the necessity of a model with three different regimes.

Bayesian Method for Finding True Change Point when a Control Chart is used

: The process personnel always seek the opportunity to improve the processes. One of the essential steps for process improvement is to quickly find the starting time or the change point of a process disturbance. To do this, after a control chart triggers an out-of-control signal, an order of points in time (known as a plan) should be identified such that if the process examined sequentially at them, the true change point is detected as soon as possible. A typical method is to start the examination of the process from the signal time of the control chart and proceed to neighbouring points. In this paper, we establish a Bayesian method to solve this problem, i.e. to find a plan for examining the process sequentially such that it minimizes the Bayes risk among all other possible plans. At last, our proposed Bayes method is applied to a normal process, and compared to a typical method which is usually used to find the true change point through a series of simulations.

Multiple Change Point Analysis: Fast Implementation and Strong Consistency

One of the main challenges in identifying structural changes in stochastic processes is to carry out analysis for time series with dependency structure in a computationally tractable way. Another challenge is that the number of true change points is usually unknown, requiring a suitable model selection criterion to arrive at informative conclusions. To address the first challenge, we model the data generating process as a segment-wise autoregression, which is composed of several segments (time epochs), each of which modeled by an autoregressive model. We propose a multi-window method that is both effective and efficient for discovering the structural changes. The proposed approach was motivated by transforming a segment-wise autoregression into a multivariate time series that is asymptotically segment-wise independent and identically distributed. To address the second challenge, we derive theoretical guarantees for (almost surely) selecting the true number of change points of segment-wise independent multivariate time series. Specifically, under mild assumptions, we show that a Bayesian Information Criterion (BIC)-like criterion gives a strongly consistent selection of the optimal number of change points, while an Akaike Information Criterion (AIC)-like criterion cannot. Finally, we demonstrate the theory and strength of the proposed algorithms by experiments on both synthetic and real-world data, including the Eastern US temperature data and the El Nino data from 1854 to 2015. The experiment leads to some interesting discoveries about temporal variability of the summer-time temperature over the Eastern US, and about the most dominant factor of ocean influence on climate.

Estimation of heterogeneous panels with structural breaks

This paper extends Pesaran’s (2006) work on common correlated effects (CCE) estimators for large heterogeneous panels with a general multifactor error structure by allowing for unknown common structural breaks. Structural breaks due to new policy implementation or major technological shocks, are more likely to occur over a longer time span. Consequently, ignoring structural breaks may lead to inconsistent estimation and invalid inference. We propose a general framework that includes heterogeneous panel data models and structural break models as special cases. The least squares method proposed by Bai (1997a, 2010) is applied to estimate the common change points, and the consistency of the estimated change points is established. We find that the CCE estimator have the same asymptotic distribution as if the true change points were known. Additionally, Monte Carlo simulations are used to verify the main results of this paper.

Identifying the time of a step change in AR(1) auto-correlated simple linear profiles

Assuming a first-order auto-regressive model for the auto-correlation structure between observations, in this paper, a transformation method is first employed to eliminate the effect of auto-correlation. Then, a maximum likelihood estimator (MLE) of a step change in the parameters of the transformed model is derived and three separate EWMA control charts are used to monitor the parameters of the profile. The performance of the proposed change-point estimator is next compared to the one of the built-in change-point estimator of EWMA control chart through some simulation experiments. The results show that the proposed MLE of the change point accurately estimates the true change point and outperforms the built-in estimator of EWMA chart for almost all shift values and auto-correlation coefficients, while the built-in estimator of EWMA chart, in general, underestimates the true change point.

Multiple Change-Points Estimation in Linear Regression Models via Sparse Group Lasso

We consider linear regression problems for which the underlying model undergoes multiple changes. Our goal is to estimate the number and locations of change-points that segment available data into different regions, and further produce sparse and interpretable models for each region. To address challenges of the existing approaches and to produce interpretable models, we propose a sparse group Lasso based approach for linear regression problems with change-points. Under certain mild assumptions and a properly chosen regularization term, we prove that the solution of the proposed approach is asymptotically consistent. In particular, we show that the estimation error of linear coefficients diminishes, and the locations of the estimated change-points are close to those of true change-points. We further propose a method to choose the regularization term so that the results mentioned above hold. In addition, we show that the complexity of the proposed algorithm is much smaller than those of existing approaches. Numerical examples are provided to validate the analytical results.

Change Detection Using Wavelets in Solution Monitoring Data for Nuclear Safeguards

Wavelet analysis is known to be a good option for change detection in many contexts. Detecting changes in solution volumes that are measured with both additive and relative error is an important aspect of safeguards for facilities that process special nuclear material. This paper qualitatively compares wavelet-based change detection to a lag-one differencing option using realistic simulated solution volume data for which the true change points are known. We then show quantitatively that Haar wavelet-based change detection is effective for finding the approximate location of each change point, and that a simple piecewise linear optimization step is effective to refine the initial wavelet-based change point estimate.

Change-Point Detection for High-Dimensional Time Series With Missing Data

This paper describes a novel approach to change-point detection when the observed high-dimensional data may have missing elements. The performance of classical methods for change-point detection typically scales poorly with the dimensionality of the data, so that a large number of observations are collected after the true change-point before it can be reliably detected. Furthermore, missing components in the observed data handicap conventional approaches. The proposed method addresses these challenges by modeling the dynamic distribution underlying the data as lying close to a time-varying low-dimensional submanifold embedded within the ambient observation space. Specifically, streaming data is used to track a submanifold approximation, measure deviations from this approximation, and calculate a series of statistics of the deviations for detecting when the underlying manifold has changed in a sharp or unexpected manner. The approach described in this paper leverages several recent results in the field of high-dimensional data analysis, including subspace tracking with missing data, multiscale analysis techniques for point clouds, online optimization, and change-point detection performance analysis. Simulations and experiments highlight the robustness and efficacy of the proposed approach in detecting an abrupt change in an otherwise slowly varying low-dimensional manifold.

A Bayesian Approach to Change Point Estimation in Multivariate SPC

A Bayesian procedure is developed to estimate the time of a change in the process mean vector for a multivariate process, given that an out-of-control signal was raised on a multivariate control chart. In addition, we can infer simultaneously which variable(s) had a change in mean value, when the change occurred, and the value of the changed mean. All three problems (inferring change point time, variables that shifted, and new values for the shifted variables) are addressed in a single statistical model. Markov chain Monte Carlo (MCMC) methods, through the software WinBUGS, are used to estimate parameters of the change point models. To identify the mean shift in a process with more than two variables, we propose a branch-and-bound search algorithm so that MCMC can be carried out with a predictable computing time in each search step. A simulation study shows that the Bayesian approach has similar performance compared to the maximum likelihood estimation (MLE) in terms of identifying the true change point location when a noninformative prior is assumed; however, it can perform better when proper prior knowledge is incorporated into the estimation procedure. The Bayesian approach provides full posterior distributions for the model and change point, which can contain information that is not available in a likelihood analysis.

Changepoint Estimation in a Segmented Linear Regression via Empirical Likelihood

For a segmented regression system with an unknown changepoint over two domains of a predictor, a new empirical likelihood ratio statistic is proposed to test the null hypothesis of no change. Under the null hypothesis of no change, the proposed test statistic is shown empirically to be Gumbel distributed with robust location and scale estimators against various parameter settings and error distributions. A power analysis is conducted to illustrate the performance of the test. Under the alternative hypothesis with a changepoint, the test statistic is utilized to estimate the changepoint between the two domains. A comparison of the frequency distributions between the proposed estimator and two parametric methods indicates that the proposed method is effective in capturing the true changepoint.

Asymptotics of maximum likelihood estimator in a two-phase linear regression model

This paper considers two-phase random design linear regression models with arbitrary error densities and where the regression function has a fixed jump at the true change-point. It obtains the consistency and the limiting distributions of maximum likelihood estimators of the underlying parameters in these models. The left end point of the maximizing interval with respect to the change point, herein called the maximum likelihood estimator r ̂ n of the change-point parameter r, is shown to be n-consistent and the underlying likelihood process, as a process in the standardized change-point parameter, is shown to converge weakly to a compound Poisson process. This process obtains maximum over a bounded interval and n( r ̂ n−r) converges weakly to the left end point of this interval. These results are different from those available in the literature for the case of the two-phase linear regression models when jump sizes tend to zero as n tends to infinity.

Detection and estimation of abrupt changes in the variability of a process

Detection of change-points in normal means is a well-studied problem. The parallel problem of detecting changes in variance has had less attention. The form of the generalized likelihood ratio test statistic has long been known, but its null distribution resisted exact analysis. In this paper, we formulate the change-point problem for a sequence of chi-square random variables. We describe a procedure that is exact for the distribution of the likelihood ratio statistic for all even degrees of freedom, and gives upper and lower bounds for odd (and also for non-integer) degrees of freedom. Both the liberal and conservative bounds for χ 1 2 degrees of freedom are shown through simulation to be reasonably tight. The important problem of testing for change in the normal variance of individual observations corresponds to the χ 1 2 case. The non-null case is also covered, and confidence intervals for the true change point are derived. The methodology is illustrated with an application to quality control in a deep level gold mine. Other applications include ambulatory monitoring of medical data and econometrics.

STRUCTURAL CHANGE IN AR(1) MODELS

This paper investigates the consistency of the least squares estimators and derives their limiting distributions in an AR(1) model with a single structural break of unknown timing. Let β1 and β2 be the preshift and postshift AR parameter, respectively. Three cases are considered: (i) |β1| < 1 and |β2| < 1; (ii) |β1| < 1 and β2 = 1; and (iii) β1 = 1 and |β2| < 1. Cases (ii) and (iii) are of particular interest but are rarely discussed in the literature. Surprising results are that, in both cases, regardless of the location of the change-point estimate, the unit root can always be consistently estimated and the residual sum of squares divided by the sample size converges to a discontinuous function of the change point. In case (iii), [circumflex over beta]2 does not converge to β2 whenever the change-point estimate is lower than the true change point. Further, the limiting distribution of the break-point estimator for shrinking break is asymmetric for case (ii), whereas those for cases (i) and (iii) are symmetric. The appropriate shrinking rate is found to be different in all cases.

A comparison of unconditional and conditional solutions to the maximum likelihood estimation of a change-point

In this paper, we compare the performance of unconditional and conditional solutions to the problem of estimating an unknown change-point by the method of maximum likelihood estimation. We begin by presenting some asymptotic results for the first two moments of the unconditional solution. Then, considering the cases of normal and exponential distributions, we evaluate through simulations the accuracy of large sample distribution theory results for the above two solutions through bias, mean square error, and graphical displays. In both normal and exponential cases, the unconditional solution shows better performance whenever the amount of change is small and when the true parameters are unknown. The differences are particular sharp when the true change-point is towards the tail of the data. When the amount of change is large, both methods perform at similar levels of accuracy. Finally, we illustrate the two solutions through the data on British coal mine explosions.

Two-stage change-point estimators in smooth regression models

We consider a fixed design regression model where the regression function is assumed to be smooth, i.e., Lipschitz continuous, except for a point where it has only one-sided limits and a local discontinuity occurs. We propose a two-step estimator for the location of this change point and study its asymptotic convergence properties. In a first step, initial pilot estimates of the change point and associated asymptotically shrinking intervals which contain the true change point with probability converging to 1 are obtained. In the second step, a weighted mean difference depending on the assumed location of the change point is maximized within these intervals and the maximizing argument is then the final change point estimator. It is shown that this estimator attains the rate O p ( n −1) in the fixed jump case. In the contiguous case, the estimator attains the rate O p ( n −1 Δ n −2), where Δ n is the sequence of jump sizes which in this case is assumed to converge to 0. For the contiguous case an invariance principle is established. A sequence of appropriately scaled deviation processes is shown to converge to a two-sided Brownian motion with triangular drift.

Partial parameter consistency in a misspecified structural change model

This paper discusses the consequences of underspecifying the number of change points in a simple structural change model. It is shown that the change-point estimator converges to one of the true change points, instead of a weighted average of them.