Rank-based multiple change-point detection

Abstract

A nonparametric procedure is proposed to estimate multiple change-points of location changes in a univariate data sequence by using ranks instead of the raw data. While existing rank-based multiple change-point detection methods are mostly based on sequential tests, we treat it as a model selection problem. We derive the corresponding Schwarz’s information criterion for rank-statistics, theoretically prove the consistency of the change-point estimator and use a pruned dynamic programing algorithm to achieve the change-point estimator. Simulation studies show our method’s robustness, effectiveness and efficiency in detecting mean-changes. We also apply the method to a gene dataset as an illustration.     

A Cox-type regression model with change-points in the covariates

We consider a Cox-type regression model with change-points in the covariates. A change-point specifies the unknown threshold at which the influence of a covariate shifts smoothly, i.e., the regression parameter may change over the range of a covariate and the underlying regression function is continuous but not differentiable. The model can be used to describe change-points in different covariates but also to model more than one change-point in a single covariate. Estimates of the change-points and of the regression parameters are derived and their properties are investigated. It is shown that not only the estimates of the regression parameters are [Formula: see text] -consistent but also the estimates of the change-points in contrast to the conjecture of other authors. Asymptotic normality is shown by using results developed for M-estimators. At the end of this paper we apply our model to an actuarial dataset, the PBC dataset of Fleming and Harrington (Counting processes and survival analysis, 1991) and to a dataset of electric motors.     

Testing the Existence of Change-Point in NHPP Software Reliability Models

In the software testing process, the nature of the failure data is affected by many factors, such as the testing environment, testing strategy, and resource allocation. These factors are unlikely to all be kept stable during the entire process of software testing. As a result, the statistical structure of the failure data is likely to experience major changes. Recently, some useful non homogeneous Poisson process (NHPP) models with change-point are proposed. However, in many realistic situations, whether a change-point exists is unknown. Furthermore, some real data seem to have two or more change-points. In this article we propose test statistics to test the existence of change-point(s). The experimental results of real data show that our tests perform well.     

Computational approximation of the likelihood ratio for testing the existence of change-points in a heteroscedastic series

In this work, we present a computational method to approximate the occurrence of the change-points in a temporal series consisting of independent and normally distributed observations, with equal mean and two possible variance values. This type of temporal series occurs in the investigation of electric signals associated to rhythmic activity patterns of nerves and muscles of animals, in which the change-points represent the actual moments when the electrical activity passes from a phase of silence to one of activity, or vice versa. We confront the hypothesis that there is no change-point in the temporal series, against the alternative hypothesis that there exists at least one change-point, employing the corresponding likelihood ratio as the test statistic; a computational implementation of the technique of quadratic penalization is employed in order to approximate the quotient of the logarithmic likelihood associated to the set of hypotheses. When the null hypothesis is rejected, the method provides estimations of the localization of the change-points in the temporal series. Moreover, the method proposed in this work employs a posteriori processing in order to avoid the generation of relatively short periods of silence or activity. The method is applied to the determination of change-points in both experimental and synthetic data sets; in either case, the results of our computations are more than satisfactory.     

Statistical inference for multiple change-point models

In this article, we propose a new technique for constructing confidence intervals for the mean of a noisy sequence with multiple change-points. We use the weighted bootstrap to generalize the bootstrap aggregating or bagging estimator. A standard deviation formula for the bagging estimator is introduced, based on which smoothed confidence intervals are constructed. To further improve the performance of the smoothed interval for weak signals, we suggest a strategy of adaptively choosing between the percentile intervals and the smoothed intervals. A new intensity plot is proposed to visualize the pattern of the change-points. We also propose a new change-point estimator based on the intensity plot, which has superior performance in comparison with the state-of-the-art segmentation methods. The finite sample performance of the confidence intervals and the change-point estimator are evaluated through Monte Carlo studies and illustrated with a real data example.     

Truncating Estimation for the Mean Change-Point in Heavy-Tailed Dependent Observations

ABSTRACT

This article considers the problem of a mean change-point in heavy-tailed dependent observations. A method of change-point estimation by truncating initial process is proposed, which can weaken the affection of outliers. In the infinite variance case, we obtained a generalization Hájek-Rényi type inequality. Consistency and the rate of convergence for the estimated change-point are also established. The results of a simulation study support validity of our method.     

Bayesian variable selection in Poisson change-point regression analysis

In this article, we develop a Bayesian variable selection method that concerns selection of covariates in the Poisson change-point regression model with both discrete and continuous candidate covariates. Ranging from a null model with no selected covariates to a full model including all covariates, the Bayesian variable selection method searches the entire model space, estimates posterior inclusion probabilities of covariates, and obtains model averaged estimates on coefficients to covariates, while simultaneously estimating a time-varying baseline rate due to change-points. For posterior computation, the Metropolis-Hastings within partially collapsed Gibbs sampler is developed to efficiently fit the Poisson change-point regression model with variable selection. We illustrate the proposed method using simulated and real datasets.     

A Bayesian finite mixture change-point model for assessing the risk of novice teenage drivers

The driving risk during the initial period after licensure for novice teenage drivers is typically the highest but decreases rapidly right after. The change-point of driving risk is a critical parameter for evaluating teenage driving risk, which also varies substantially among drivers. This paper presents latent class recurrent-event change-point models for detecting the change-points. The proposed model is applied to the Naturalist Teenage Driving Study, which continuously recorded the driving data of 42 novice teenage drivers for 18 months using advanced in-vehicle instrumentation. We propose a hierarchical BFMM to estimate the change-points by clusters of drivers with similar risk profiles. The model is based on a non-homogeneous Poisson process with piecewise-constant intensity functions. Latent variables which identify the membership of the subjects are used to detect potential clusters among subjects. Application to the Naturalistic Teenage Driving Study identifies three distinct clusters with change-points at 52.30, 108.99 and 150.20?hours of driving after first licensure, respectively. The overall intensity rate and the pattern of change also differ substantially among clusters. The results of this research provide more insight in teenagers' driving behaviour and will be critical to improve young drivers' safety education and parent management programs, as well as provide crucial reference for the GDL regulations to encourage safer driving.     

Multiple change-point estimation with U-statistics

We consider a multiple change-point problem: a finite sequence of independent random variables consists of segments given by a known number of the so-called change-points such that the underlying distribution differs from segment to segment. The task is to estimate these change-points under no further assumptions on the within-segment distributions. In this completely nonparametric framework the proposed estimator is defined as the maximizing point of weighted multivariate U-statistic processes. Under mild moment conditions we prove almost sure convergence and the rate of convergence.     

Sequential Monitoring Variance Changes in Location Model

In this paper, a moving monitoring procedure is proposed to detect potential variance change of the location model with dependent errors. The procedure is motivated by the problem that the existing square CUSUM test is insensitive to a late variance change of the location model. The asymptotic distribution of the statistics under the null hypothesis and the consistency under the alternative hypothesis are derived. Simulations show that our monitoring procedure compared to the square CUSUM test offers better power and can more quickly detect change. Moreover, the effectiveness of our procedure is illustrated by applying it to two data sets.     

A method for choosing the smoothing parameter in a semi-parametric model for detecting change-points in blood flow

In a smoothing spline model with unknown change-points, the choice of the smoothing parameter strongly influences the estimation of the change-point locations and the function at the change-points. In a tumor biology example, where change-points in blood flow in response to treatment were of interest, choosing the smoothing parameter based on minimizing generalized cross-validation (GCV) gave unsatisfactory estimates of the change-points. We propose a new method, aGCV, that re-weights the residual sum of squares and generalized degrees of freedom terms from GCV. The weight is chosen to maximize the decrease in the generalized degrees of freedom as a function of the weight value, while simultaneously minimizing aGCV as a function of the smoothing parameter and the change-points. Compared with GCV, simulation studies suggest that the aGCV method yields improved estimates of the change-point and the value of the function at the change-point.     

Maximum likelihood estimator in a multi-phase random regression model

We consider a random regression model with several-fold change-points. The results for one change-point are generalized. The maximum likelihood estimator of the parameters is shown to be consistent, and the asymptotic distribution for the estimators of the coefficients is shown to be Gaussian. The estimators of the change-points converge, with n ^?1 rate, to the vector whose components are the left end points of the maximizing interval with respect to each change-point. The likelihood process is asymptotically equivalent to the sum of independent compound Poisson processes.     

A class of goodness-of-fit test for the additive hazards model with case-cohort data

The case-cohort design is commonly used in epidemiological studies due to its cost-effectiveness. The additive hazards model is widely used in survival analysis when the hazards difference is constant. In this article, we propose a class of goodness-of-fit test statistics for the assumption of the additive hazards model with case-cohort data through a class of asymptotically mean-zero multiparameter stochastic processes. We also establish the asymptotic theory of the proposed test statistics and a resampling scheme is adopted to approximate its asymptotic distribution. The performance of the proposed test statistics is evaluated through simulation studies and a real dataset is analyzed to illustrate the proposed method.     

Temporal dynamics of inter-limb coordination in ice climbing revealed through change-point analysis of the geodesic mean of circular data

This study examined the temporal dynamics of the inter-limb angles of skilled and less skilled ice climbers to determine how they explored ice fall properties to adapt their coordination patterns during performance. We observed two circular time series corresponding to the upper- and lower-limbs of seven expert and eight inexperienced ice climbers. We analyzed these data through a multiple change-point analysis of the geodesic (or Fréchet) mean on the circle. Guided by the nature of the geodesic mean obtained by an optimization procedure, we extended the filtered derivative method, known to be computationally very cheap and fast, to circular data. Local estimation of the variability was assessed through the number of change-points computed via the filtered derivatives with p-value method for the time series and integrated squared error (ISE). Results of this change-point analysis did not reveal significant differences of the number of change-points between groups but indicated higher ISE that supported the existence of plateaux for beginners. These results emphasized higher local variability of limb angles for experts than for beginners suggesting greater dependence on the properties of the performance environment and adaptive behaviors in the former. Conversely, the lower local variance of limb angles assessed in beginners may reflect their independence of the environmental constraints, as they focused mainly on controlling body equilibrium.     

A Bayesian multiple structural change regression model with autocorrelated errors

This paper develops a new Bayesian approach to change-point modeling that allows the number of change-points in the observed autocorrelated times series to be unknown. The model we develop assumes that the number of change-points have a truncated Poisson distribution. A genetic algorithm is used to estimate a change-point model, which allows for structural changes with autocorrelated errors. We focus considerable attention on the construction of autocorrelated structure for each regime and for the parameters that characterize each regime. Our techniques are found to work well in the simulation with a few change-points. An empirical analysis is provided involving the annual flow of the Nile River and the monthly total energy production in South Korea to lead good estimates for structural change-points.     

Exact Bayesian inference for off-line change-point detection in tree-structured graphical models

We consider the problem of change-point detection in multivariate time-series. The multivariate distribution of the observations is supposed to follow a graphical model, whose graph and parameters are affected by abrupt changes throughout time. We demonstrate that it is possible to perform exact Bayesian inference whenever one considers a simple class of undirected graphs called spanning trees as possible structures. We are then able to integrate on the graph and segmentation spaces at the same time by combining classical dynamic programming with algebraic results pertaining to spanning trees. In particular, we show that quantities such as posterior distributions for change-points or posterior edge probabilities over time can efficiently be obtained. We illustrate our results on both synthetic and experimental data arising from biology and neuroscience.     

Change-Point Tests for the Error Distribution in Non-parametric Regression

Abstract.  Several testing procedures are proposed that can detect change-points in the error distribution of non-parametric regression models. Different settings are considered where the change-point either occurs at some time point or at some value of the covariate. Fixed as well as random covariates are considered. Weak convergence of the suggested difference of sequential empirical processes based on non-parametrically estimated residuals to a Gaussian process is proved under the null hypothesis of no change-point. In the case of testing for a change in the error distribution that occurs with increasing time in a model with random covariates the test statistic is asymptotically distribution free and the asymptotic quantiles can be used for the test. This special test statistic can also detect a change in the regression function. In all other cases the asymptotic distribution depends on unknown features of the data-generating process and a bootstrap procedure is proposed in these cases. The small sample performances of the proposed tests are investigated by means of a simulation study and the tests are applied to a data example.     

Testing for a change in repeated measures data

Four nonparametric test statistics for the change-point problem with repeated measures data are proposed. In a Monte Carlo simulation study, critical values for the proposed test statistics are simulated and the performances of the proposed tests are compared with the performances of some competitive tests in terms of asymptotic behavior and power. We provide appropriate recommendations for different occurrences of the change-point and illustrate the testing methods using a set of real data.     

Bootstrapping an inhomogeneous point process

In this paper, we focus on resampling non-stationary weakly dependent point processes in two dimensions to make inference on the inhomogeneous K function ( Baddeley et al., 2000). We provide theoretical results that show a consistency result of the bootstrap estimates of the variance as the observation region and resampling blocks increase in size. We present results of a simulation study that examines the performance of nominal 95% confidence intervals for the inhomogeneous K function obtained via our bootstrap procedure. The procedure is also applied to a rainforest dataset.     

Exact posterior distributions and model selection criteria for multiple change-point detection problems

In segmentation problems, inference on change-point position and model selection are two difficult issues due to the discrete nature of change-points. In a Bayesian context, we derive exact, explicit and tractable formulae for the posterior distribution of variables such as the number of change-points or their positions. We also demonstrate that several classical Bayesian model selection criteria can be computed exactly. All these results are based on an efficient strategy to explore the whole segmentation space, which is very large. We illustrate our methodology on both simulated data and a comparative genomic hybridization profile.