Single Change Point Research Articles

A composite statistical method for the detection of multiple undocumented abrupt changes in the mean value within a time series

AbstractThe time series of measurements of hydro‐meteorological variables often suffer from imperfections such as missing data, outliers and discontinuities in the mean values. The discontinuity in the mean can be the effect of: instrumental offsets and of their corrections, of changes in the monitoring station or in the surrounding environment. If the discontinuities can be identified with a reasonable precision, a correction of the erroneous data can be made. Several authors have put their great effort into developing techniques to identify non‐climatic inhomogeneities; the resulting statistical methods are especially effective when the series contains a single change point, while their performances decline when the series contains multiple change points or inhomogeneous segments (a portion of the series bounded by two complementary shifts). These limitations also affect the standard normal homogeneity test (SNHT), one of the most effective and widely applied tests. We present a composite method of homogeneity testing, standard normal homogenization composite method (SNHCM), including the SNHT as one component, which improves the SNHT performances with multiple change points and inhomogeneous segments. A number of comparisons among the new method, the SNHT and a powerful optimal segmentation method (OSM‐CM), are illustrated in the paper. SNHCM demonstrates their performances in change‐point detection similar to, or better than, the SNHT and very close to the OSM‐CM. The SNHCM is effective in recognizing complex patterns of discontinuities, especially inhomogeneous segments, which represent a severe problem for SNHT; on the contrary, SNHT performs slightly better only when the series contains a single change point, but the difference between the two methods is negligible. Compared to the OSM‐CM, SNHCM provides very similar performances, with some favourable features deriving from the fact that it is computationally lighter, simpler to implement, can easily handle very long series and is based on statistical hypothesis tests with a well‐defined and adjustable significance level. Copyright © 2010 Royal Meteorological Society

Off-Line Detection of Multiple Change Points by the Filtered Derivative with p-Value Method

This article deals with off-line detection of change points, for time series of independent observations, when the number of change points is unknown. We propose a sequential analysis method with linear time and memory complexity. Our method is based, on a filtered derivative method that detects the right change points as well as false ones. We improve the filtered derivative method by adding a second step in which we compute the p-values associated to every single potential change point. Then, we eliminate false alarms; that is, the change points that have p-values smaller than a given critical level. Next, we apply our method and penalized least squares criterion procedure to detect change points on simulated data sets and then we compare them. Eventually, we apply the filtered derivative with p-value method to the segmentation of heartbeat time series, and the detection of change points in the average daily volume of financial time series.

Asymptotic properties of maximum likelihood estimators in models with multiple change points

Models with multiple change points are used in many fields; however, the theoretical properties of maximum likelihood estimators of such models have received relatively little attention. The goal of this paper is to establish the asymptotic properties of maximum likelihood estimators of the parameters of a multiple change-point model for a general class of models in which the form of the distribution can change from segment to segment and in which, possibly, there are parameters that are common to all segments. Consistency of the maximum likelihood estimators of the change points is established and the rate of convergence is determined; the asymptotic distribution of the maximum likelihood estimators of the parameters of the within-segment distributions is also derived. Since the approach used in single change-point models is not easily extended to multiple change-point models, these results require the introduction of those tools for analyzing the likelihood function in a multiple change-point model.

Intercomparison of homogenization techniques for precipitation data continued: Comparison of two recent Bayesian change point models

In this paper, two new Bayesian change point techniques are described and compared to eight other techniques presented in previous work to detect inhomogeneities in climatic series. An inhomogeneity can be defined as a change point (a time point in a series such that the observations have a different distribution before and after this time) in the data series induced from changes in measurement conditions at a given station. It is important to be able to detect and correct an inhomogeneity, as it can interfere with the real climate change signal. The first technique is a Bayesian method of multiple change point detection in a multiple linear regression. The second one allows the detection of a single change point in a multiple linear regression. These two techniques have never been used for homogenization purposes. The ability of the two techniques to discriminate homogeneous and inhomogeneous series was evaluated using simulated data series. Various sets of synthetic series (homogeneous, with a single shift, and with multiple shifts) representing the typical total annual precipitation observed in the southern and central parts of the province of Quebec, Canada, and nearby areas were generated for the purpose of this study. The two techniques gave small false detection rates on the homogeneous series. Furthermore, the two techniques proved to be efficient for the detection of a single shift in a series. For the series with multiple shifts, the Bayesian method of multiple change point detection performed better. An application to a real data set is also provided and validated with the available metadata.

Relationship between Inhaled Corticosteroid Therapy and Rate of Lung Function Decline in Children with Cystic Fibrosis

To assess the relationship between inhaled corticosteroids (ICS) use and lung function decline in children with cystic fibrosis (CF) using the Epidemiologic Study of Cystic Fibrosis, an observational study of patients with CF in North America. We analyzed data from 2978 patients 6 to 17 years old enrolled in ESCF between 1994 to 2004. We estimated the rate of decline in forced expiratory volume in 1 second (FEV(1)) before and after starting ICS therapy with a piecewise linear continuous single change point model, adjusting for potentially confounding covariates. Before initiation of ICS, mean FEV(1) decline was -1.52% predicted/year (95% CI: -1.96 to -1.08% predicted/year). After initiation of ICS therapy, mean FEV(1) decline was -0.44% predicted/year (95% CI: -0.85 to -0.03% predicted/year), which was a significant change (P = .002). ICS use was associated with decreased height for age Z scores and increased insulin/oral hypoglycemic use. In this retrospective analysis of prospectively collected data, ICS therapy in patients with CF was associated with a significant reduction in the rate of FEV(1) decline, decreased linear growth, and increased insulin/oral hypoglycemic use.

Algorithm for Detection of Change Point in On-Line Monitoring Data

The detection of abrupt changes in observed/estimated physical processes is an issue of concern in many applications. An algorithm for single change point detection in Gaussian white noise processes is presented in this paper. The algorithm progressively includes and analyzes the observed data points, and hence is suitable for detection of change point in on-line monitoring data. Two example problems are presented, wherein Monte Carlo simulation technique is used, to illustrate the proposed algorithm. The performance of the algorithm with respect to changes in shift amplitude, intensity of the signal, and rate of sampling are studied, and the observed trends indicate that the proposed algorithm possesses the desirable properties of change detection algorithms.

A Monte Carlo approach for change‐point detection in the Cox proportional hazards model

Detecting a time lag of treatment effect or identifying change points in a hazard function is of great interest and importance in survival analysis. The testing procedures hereto are primarily based on analytical approximations for the asymptotic null distribution of either the likelihood ratio test or the score test. In the presence of random censoring and/or covariates, however, the justification for the limiting distribution often requires some technical assumptions and conditions that are difficult to verify in practice. Moreover, a satisfactory asymptotic theory for testing the existence of multiple change points in hazard function has not emerged. In this paper, we consider maximal score tests for detecting change point(s) in the Cox proportional hazards model with censored data. We propose to use a simple Monte Carlo approach for assessing the statistical significance of tests. The proposed approach is applicable for testing a single change point in the Cox model with covariates and sample stratifications over various types of candidate regions, including discrete time-point sets or disjoint intervals. We also show that the proposed test statistics and the Monte Carlo procedure are well applicable under situations with multiple change points. Simulation studies and an analysis of a real data from a randomized cancer trial are conducted to demonstrate the finite-sample performance of the proposed approach.

Possible Generational Effects of Habitat Degradation on Alligator Reproduction

ABSTRACT Population decline of the American alligator (Alligator mississippiensis) was observed in Lake Apopka in central Florida, USA, in the early 1980s. This decline was thought to result from adult mortality and nest failure caused by anthropogenic increases in sediment loads, nutrients, and contaminants. Reproductive impairment also was reported. Extensive restoration of marshes associated with Lake Apopka has been conducted, as well as some limited restoration measures on the lake. Monitoring by the Florida Fish and Wildlife Conservation Commission (FFWCC) has indicated that the adult alligator population began increasing in the early 1990s. We expected that the previously reported high proportion of complete nest failure (θ0) during the 1980s may have decreased. We collected clutches from alligator nests in Lake Apopka from 1983 to 2003 and from 5 reference areas from 1988 to 1991, and we artificially incubated them. We used a Bayesian framework with Gibbs sampler of Markov chain Monte Carlo simulation to analyze θ0. Estimated θ0 was consistently higher in Lake Apopka compared with reference areas, and the difference in θ0 ranged from 0.19 to 0.56. We conducted change point analysis to identify and test the significance of the change point in θ0 in Lake Apopka between 1983 and 2003, indicating the point of reproductive recovery. The estimated Bayes factor strongly supported the single change point hypothesis against the no change point hypothesis. The major downward shift in θ0 probably occurred in the mid‐1990s, approximately a generation after the major population decline in the 1980s. Furthermore, estimated θ0 values after the change point (0.21) were comparable with those of reference areas (0.07–0.31). These results combined with the monitoring by FFWCC seem to suggest that anthropogenic habitat degradation caused reproductive impairment of adult females and decreases in θ0 occurred with the sexual maturity of a new generation of breeding females. Long‐term monitoring is essential to understand population changes due to habitat restoration. Such information can be used as an input in planning and evaluating restoration activities.

Bayesian multivariate linear regression with application to change point models in hydrometeorological variables

Multivariate linear regression is one of the most popular modeling tools in hydrology and climate sciences for explaining the link between key variables. Piecewise linear regression is not always appropriate since the relationship may experiment sudden changes due to climatic, environmental, or anthropogenic perturbations. To address this issue, a practical and general approach to the Bayesian analysis of the multivariate regression model is presented. The approach allows simultaneous single change point detection in a multivariate sample and can account for missing data in the response variables and/or in the explicative variables. It also improves on recently published change point detection methodologies by allowing a more flexible and thus more realistic prior specification for the existence of a change and the date of change as well as for the regression parameters. The estimation of all unknown parameters is achieved by Monte Carlo Markov chain simulations. It is shown that the developed approach is able to reproduce the results of Rasmussen (2001) as well as those of Perreault et al. (2000a, 2000b). Furthermore, two of the examples provided in the paper show that the proposed methodology can readily be applied to some problems that cannot be addressed by any of the above‐mentioned approaches because of limiting model structure and/or restrictive prior assumptions. The first of these examples deals with single change point detection in the multivariate linear relationship between mean basin‐scale precipitation at different periods of the year and the summer–autumn flood peaks of the Broadback River located in northern Quebec, Canada. The second one addresses the problem of missing data estimation with uncertainty assessment in multisite streamflow records with a possible simultaneous shift in mean streamflow values that occurred at an unknown date.

Computation of estimates in segmented regression and a liquidity effect model

Weighted least squares (WLS) estimation in segmented regression with multiple change points is considered. A computationally efficient algorithm for calculating the WLS estimate of a single change point is derived. Then, iterative methods of approximating the global solution of the multiple change-point problem based on estimating change points one-at-a-time are discussed. It is shown that these results can also be applied to a liquidity effect model in finance with multiple change points. The liquidity effect model we consider is a generalization of one proposed by Çetin et al. [2006. Pricing options in an extended Black Scholes economy with illiquidity: theory and empirical evidence. Rev. Financial Stud. 19, 493–529], allowing that the magnitude of liquidity effect depends on the size of a trade. Two data sets are used to illustrate these methods.

A Gibbs Sampling Algorithm for a Changing Regression Model with Pooled Binary Response Data

This article presents a Gibbs Sampling algorithm for a changing regression model with a pooled binary dependent variable. A Gibbs Sampling algorithm for a changing regression model with a continuous dependent variable is extended to binary response data using the chained data augmentation of Tanner (1996). The proposed algorithm is applied to numerical examples with a single change point and multiple change points. The results suggest that this algorithm robustly provides accurate estimates of change points, and that this algorithm also provides sharper estimates of regression parameters when it is applied to a pooled data with a larger cross-section sample size.

Supervised learning for change-point detection

The detection of changes in the distribution of process variables is referred to as the change-point problem. Existing methods focus on detecting a single (or few) change point in a univariate (or low-dimensional) process. We consider the important high-dimensional multivariate case with multiple change points and without an assumed distribution. In this work the problem is transformed into a supervised learning problem with time as the output response and the process variables as inputs. Our focus is to identify the subset of variables that change. This important, practical scenario is analysed through a supervised learner with a variable importance measure that is used to identify the variables that change among hundreds of variables. Simulated cases are discussed in the paper to verify the proposed method. Moreover, the same data sets are compared with a multivariate exponentially weighted moving average control chart and the advantages of the supervised learner are illustrated.

Bayesian single change point detection in a sequence of multivariate normal observations

A Bayesian method is used to see whether there are changes of mean, covariance, or both at an unknown time point in a sequence of independent multivariate normal observations. Noninformative priors are used for all competing models: no-change model, mean change model, covariance change model, and mean and covariance change model. We use several versions of the intrinsic Bayes factor of Berger and Pericchi (Berger, J.O. and Pericchi, L.R., 1996, The intrinsic Bayes factor for model selection and prediction. Journal of the American Statistical Association,\\/ 91, 109–122; Berger, J.O. and Pericchi, L.R., 1998, Accurate and stable Bayesian model selection: the median intrinsic Bayes factor. Sankhya Series B, 60, 1–18.) to detect a change point. We demonstrate our results with some simulated datasets and a real dataset.

Modelling Growth and Decline in Lung Function in Duchenne’s Muscular Dystrophy with an Augmented Linear Mixed Effects Model

Summary Longitudinal modelling of lung function in Duchenne’s muscular dystrophy is complicated by a mixture of both growth and decline in lung function within each subject, an unknown point of separation between these phases and significant heterogeneity between individual trajectories. Linear mixed effects models can be used, assuming a single changepoint for all cases; however, this assumption may be incorrect. The paper describes an extension of linear mixed effects modelling in which random changepoints are integrated into the model as parameters and estimated by using a stochastic EM algorithm. We find that use of this ‘mixture modelling’ approach improves the fit significantly.

Trend test and change-point detection for the annual discharge series of the Yangtze River at the Yichang hydrological station / Test de tendance et détection de rupture appliqués aux séries de débit annuel du fleuve Yangtze à la station hydrologique de Yichang

The trend test and change-point analysis have been carried out on several annual discharge series of the Yangtze River at the Yichang hydrological station, including the discharge series of the annual maximum, annual minimum and annual mean during the period 1882–2001. The results of both the Mann-Kendall and Spearman’s rho trend tests show that, at the 5% significance level, the annual maximum flood series did not have any statistically significant trend, but the annual minimum flow series and the annual mean discharge series exhibited a sign of decreasing trend. For single change-point detection, a Bayesian model was established to study the abrupt change in the mean levels of the time series. Given the observed hydrological data, the Bayesian model can estimate the posterior probability distribution of each change-point location by using the Monte Carlo Markov Chain (MCMC) sampling method. The results of the Bayesian model show that, during the past 120 years, the mean levels of both the annual minimum discharge series and the annual mean discharge series have decreased by 8% and 6% respectively. Further analysis suggested that, for the annual minimum discharge series and the annual mean discharge series, both the trend term and the abrupt change term are very closely related and very hard to distinguish from each other.

Estimation of a change point in a hazard function based on censored data.

The hazard function plays an important role in reliability or survival studies since it describes the instantaneous risk of failure of items at a time point, given that they have not failed before. In some real life applications, abrupt changes in the hazard function are observed due to overhauls, major operations or specific maintenance activities. In such situations it is of interest to detect the location where such a change occurs and estimate the size of the change. In this paper we consider the problem of estimating a single change point in a piecewise constant hazard function when the observed variables are subject to random censoring. We suggest an estimation procedure that is based on certain structural properties and on least squares ideas. A simulation study is carried out to compare the performance of this estimator with two estimators available in the literature: an estimator based on a functional of the Nelson-Aalen estimator and a maximum likelihood estimator. The proposed least squares estimator tums out to be less biased than the other two estimators, but has a larger variance. We illustrate the estimation method on some real data sets.

Temporal trends in United States dew point temperatures

In this study, hourly data for the 1951–1990 period for 178 stations in the coterminous United States were used to establish temporal trends in dew point temperature. Although the data had been quality controlled previously (Robinson, 1998. Monthly variations of dew point temperatures in the coterminous United States. International Journal of Climatology18: 1539–1556), comparisons of values between nearby stations suggested that instrumental changes, combined with locational changes, may have modified the results by as much as 1°C during the 40-year period. Nevertheless, seasonally averaged results indicated an increase over much of the area, of slightly over 1°C/100 years in spring and autumn, slightly less than this in summer. Winter displayed a drying of over 1°C/100 years. When only the 1961–1990 period was considered, the patterns were similar and trends increased by approximately 1–2°C/100 years, except in autumn, which displayed a slight drying. Analyses for specific stations indicated periods of both increasing and decreasing Td, the change between them varying with observation hour. No single change point was common over a wide area, although January commonly indicated maximum values early in the period in the east and west, and much later in the north-central portion. Rates of increase were generally higher in daytime than at night, especially in summer. Investigation of the inter-decadal differences in dew point, as a function of wind conditions, indicated that changes during calm conditions were commonly similar in magnitude to that of the overall average changes, suggesting an important role for the local hydrologic cycle in driving changes. Other inter-decadal changes could be attributed to the changes in the frequency and moisture content of invading air-streams. This was particularly clear for the changes in north–south flow in the interior. Copyright © 2000 Royal Meteorological Society

Change point analysis of a Gaussian model

In this paper, the testing and estimation of a single change point in means and variances of a sequence of independent Gaussian normal random variables are studied. The Schwarz Information Criterion, SIC, is used to search for the change point. The unbiased version of the SIC for this change point problem is also derived for the finite sample case. Other properties of the SIC test statistic are given as well. Finally, two examples are given at the end of this paper to illustrate the method proposed, and changes are successfully detected.

Outcomes of an educational‐outreach service for community medical practitioners: non‐steroidal anti‐inflammatory drugs

Exploration of longer-term outcomes of an ongoing educational-outreach service for community doctors. Quasi-experimental, with parallel and historical comparisons. Since 1992, a teaching-hospital-based service has been providing advice and information on drugs and therapeutic strategies to community medical practitioners. 210 doctors practising in a particular area of metropolitan Adelaide (79% general practitioners; 21% specialists). Two surgery visits during 1992 focused on better use of prescribed non-steroidal anti-inflammatory drugs (NSAIDs). Subsequent visits on other topical therapeutic issues have occurred regularly. Doctor participation in the service; supply of prescription NSAIDs; hospital admissions for gastrointestinal (GI) effects of NSAID use. 89% of doctors practising within the service area received the first visit on NSAIDs and 86% received the second visit. More than 85% continue to receive the service. Relative to a comparison area, aggregate reductions of 9% and 28%, respectively, were observed in two different measures of NSAID use. During an 11-year observation period, a single change point in the number of hospital admissions for GI disorders occurred in the service area, coinciding with delivery of the NSAID program. In the five years since the visits commenced, a 70% reduction in admissions was observed. No notable changes in hospital admission rates occurred in the comparison area. A continuing education and support service for community medical practitioners which uses principally academic detailing methods in its contact with doctors has contributed to sustained changes in prescribed NSAID use over a five-year period. A focus on risk-minimisation in prescribing of NSAIDs appears to have contributed to reductions in hospitalisations for GI adverse events.

Piecewise convex function estimation: pilot estimators

Given noisy data, function estimation is considered when the unknown function is known a priori to be either convex or concave on each of a small number of regions where the function. When the number of regions is unknown, the model selection problem is to determine the number of convexity change points. For kernel estimates in Gaussian noise, the number of false change points is evaluated as a function of the smoothing parameter. To insure that the number of false convexity change points tends to zero, the smoothing level must be larger than is generically optimal for minimizing the mean integrated square error (MISE). A two-stage estimator is proposed and shown to achieve the optimal rate of convergence of the MISE. In the first stage, the number and location of the change points is estimated using strong smoothing. In the second stage, a constrained smoothing spline fit is performed with the smoothing level chosen to minimize the MISE. The imposed constraint is that a single change point occur in a region about each empirical change point from the first-stage estimate. This constraint is equivalent to the requirement that the third derivative of the second-stage estimate has a single sign in a small neighborhood about each first-stage change point. The change points from the second stage are first-stage change points, but need not be at the identical locations.