Positions Of Change-points Research Articles

Exploring the ability of the Pettitt method for detecting change point by Monte Carlo simulation

The Pettitt method, which is a rank-based test method, has been widely used to detect change point in the mean value of observed series. Traditionally the rank-based test has been assumed to be distribution-free and not sensitive to outliers and skewed distributions. However, there has no evidence provided to prove this assumption. Based on the work of Yue and Wang (Stoch Environ Res Risk Assess 16:307–323, 2002), this study defines the success rate of detecting the given change point as the ability of the Pettitt method, and investigates the ability in various circumstances by means of Monte Carlo simulation. Experiment results demonstrate that, the ability of the Pettitt method depends on not only the pre-assigned significance level, but also various properties of the sample data, including the sample size, the magnitude of a shift and the change point position. Besides, the distribution type and the distribution parameters such as the coefficient of variation, the coefficient of skewness and the shape parameter also seriously influence the ability. As expected, it is easier for the method to detect the change point when the sample size is larger, or the magnitude of a change point is bigger, or the variation of the sample data is smaller. And the highest ability is obtained when the change point occurs at the middle position of the series. These simulation results would provide users an extensive and detailed understanding about the use of the Pettitt method for the detection of change point.

Exploring the latent segmentation space for the assessment of multiple change-point models

This paper addresses the retrospective or off-line multiple change-point detection problem. Multiple change-point models are here viewed as latent structure models and the focus is on inference concerning the latent segmentation space. Methods for exploring the space of possible segmentations of a sequence for a fixed number of change points may be divided into two categories: (i) enumeration of segmentations, (ii) summary of the possible segmentations in change-point or segment profiles. Concerning the first category, a dynamic programming algorithm for computing the top $$N$$ N most probable segmentations is derived. Concerning the second category, a forward-backward dynamic programming algorithm and a smoothing-type forward-backward algorithm for computing two types of change-point and segment profiles are derived. The proposed methods are mainly useful for exploring the segmentation space for successive numbers of change points and provide a set of assessment tools for multiple change-point models that can be applied both in a non-Bayesian and a Bayesian framework. We show using examples that the proposed methods may help to compare alternative multiple change-point models (e.g. Gaussian model with piecewise constant variances or global variance), predict supplementary change points, highlight overestimation of the number of change points and summarize the uncertainty concerning the position of change points.

소량자료를 위한 베이지안 다중 변환점 모형

다중 변환점(multiple change-point) 추론에 있어 소량자료에 관한 연구는 많지 않다. 본 논문에서는 소량자료의 다중 변환점 추정을 위해 베이지안 비중심(noncentral) t 분포 변환점 모형을 제안하고, 제안된 모형 추론을 위해 메트로폴리스-해스팅스를 포함한 깁스 샘플링(Metropolis-Hastings-Within-Gibbs sampling) 알고리즘을 이용하였다. 모의실험 및 태풍 발생 수의 실증 분석결과는 제안된 모형과 알고리즘의 우수성을 보여 준다.

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.

Non-Parametric Change-Point Method for Differential Gene Expression Detection

BackgroundWe proposed a non-parametric method, named Non-Parametric Change Point Statistic (NPCPS for short), by using a single equation for detecting differential gene expression (DGE) in microarray data. NPCPS is based on the change point theory to provide effective DGE detecting ability.MethodologyNPCPS used the data distribution of the normal samples as input, and detects DGE in the cancer samples by locating the change point of gene expression profile. An estimate of the change point position generated by NPCPS enables the identification of the samples containing DGE. Monte Carlo simulation and ROC study were applied to examine the detecting accuracy of NPCPS, and the experiment on real microarray data of breast cancer was carried out to compare NPCPS with other methods.ConclusionsSimulation study indicated that NPCPS was more effective for detecting DGE in cancer subset compared with five parametric methods and one non-parametric method. When there were more than 8 cancer samples containing DGE, the type I error of NPCPS was below 0.01. Experiment results showed both good accuracy and reliability of NPCPS. Out of the 30 top genes ranked by using NPCPS, 16 genes were reported as relevant to cancer. Correlations between the detecting result of NPCPS and the compared methods were less than 0.05, while between the other methods the values were from 0.20 to 0.84. This indicates that NPCPS is working on different features and thus provides DGE identification from a distinct perspective comparing with the other mean or median based methods.

Bayesian off-line detection of multiple change-points corrupted by multiplicative noise: application to SAR image edge detection

This paper addresses the problem of Bayesian off-line change-point detection in synthetic aperture radar images. The minimum mean square error and maximum a posteriori estimators of the changepoint positions are studied. Both estimators cannot be implemented because of optimization or integration problems. A practical implementation using Markov chain Monte Carlo methods is proposed. This implementation requires a priori knowledge of the so-called hyperparameters. A hyperparameter estimation procedure is proposed that alleviates the requirement of knowing the values of the hyperparameters. Simulation results on synthetic signals and synthetic aperture radar images are presented.

General fuzzy piecewise regression analysis with automatic change-point detection

Yu et al. (Fuzzy Sets and Systems 105 (1999) 429) performed general piecewise necessity regression analysis based on linear programming (LP) to obtain the necessity area. Their method is the same as that according to data distribution, even if the data are irregular, practitioners must specify the number and the positions of change-points. However, as the sample size increases, the number of change-points increases and the piecewise linear interval model also becomes complex. Therefore, this work devises general fuzzy piecewise regression analysis with automatic change-point detection to simultaneously obtain the fuzzy regression model and the positions of change-points. Fuzzy piecewise possibility and necessity regression models are employed when the function behaves differently in different parts of the range of crisp input variables. As stated, the above problem can be formulated as a mixed-integer programming problem. The proposed fuzzy piecewise regression method has three advantages: (a) Previously specifying the number of change-points, then the positions of change-points and the fuzzy piecewise regression model are obtained simultaneously. (b) It is more robust than conventional fuzzy regression. The conventional regression is sensitive to outliers. In contrast, utilizing piecewise concept, the proposed method can deal with outliers by automatically segmenting the data. (c) By employing the mixed integer programming, the solution is the global optimal rather than local optimal solution. For illustrating more detail, two numerical examples are shown in this paper. By using the proposed method, the fuzzy piecewise regression model with detecting change-points can be derived simultaneously.

Multiple change-point detection with a genetic algorithm

A common change-point problem is considered where the population mean of a random variable is suspected of undergoing abrupt changes in course of a time series. It is usual in practice that no information on positions or number of such shifts is available beforehand. Finding the change points, i.e. the positions of the shifts, in such a situation is a delicate statistical problem since any considered sample may actually represent a mixture of two or more populations where values from both sides of a yet unrecognized change point are unconsciously assembled. If this is the case, underlying assumptions of an employed statistical two-sample test are usually violated. Consequently, no definite decision should be based on just one value of the test statistic. Such a value is rather, as a precaution, to be regarded as an only approximate indicator of the quality of a hypothesis about change-point positions. Given these conclusions, it is found imperative to treat the problem of multiple change-point detection as one of global optimization. A cost function is constructed in such a manner that the change-point configuration yielding the global optimum is compliant with statistical-theoretical requirements to the utmost extent. The used advanced optimization tool, a genetic algorithm, is both efficient – as it takes advantage of the information about promising change-point positions encountered in previously investigated trial configurations – and flexible (as it is open to any modification of the change-point configuration at any time). Experiments using numerical simulation confirm adequate performance of the method in an application where a common change-point detection procedure based on Student's two-sample t-test is used to detect an arbitrary number of shifts in the mean of a normally distributed random variable.

A general piecewise necessity regression analysis based on linear programming

In possibilistic regression analysis proposed by Tanaka and Ishibuchi (1992), linear programming (LP) formulation of necessity analysis has no feasible solution under the enormous variation of the given data. This work proposes a general piecewise necessity regression analysis based on LP rather than a non-linear interval model that they recommended to obtain the necessity area of the given data. In addition to maintaining a linear property, the proposed method prevents the necessity analysis from having no feasible solution. The problematic univariate example and a multivariate example with respect to different number of change-points are demonstrated by the general piecewise necessity regression. The proposed method characteristic is that, according to data distribution, practitioners can specify the number and the positions of change-points. The proposed method maintains the linear interval model and the order of necessity regression function does not need to be determined.

Minimax estimation of sharp change points

We define the sharp change point problem as an extension of earlier problems in change point analysis related to nonparametric regression. As particular cases, these include estimation of jump points in smooth curves. More generally, we give a systematic treatment of the correct rate of convergence for estimating the position of a “cusp”of an arbitrary order. We propose a test function for the local regularity of a signal that characterizes such a point as a global maximum. In the sample implementation of our method, from observations of the signal at discrete time positions $i/n, i =1 \ldots,n$, we use a wavelet transformation to approximate the position of the change point in the no-noise case. We study the noise effect, in the worst case scenario over a wide class of functions having a unique irregularity of “order $\alpha$” and propose a sequence of estimators which converge at the rate $n_{-1/(1+2\alpha)}$, as $n$ tends to infinity. Finally we analyze the likelihood ration of the problem and show that this is actually the minimaz rate of convertence. Examples of thresholding empirical wavelet coefficients to estimate the position of sharp change points are also presented.

Analytical approach to changepoint detection in Laplacian noise

The paper presents an analytical method using the Bayesian inference framework for the identification of time-series discontinuities, i.e. changepoints, in impulsive Laplacian noise. Exact expressions for the posterior density of the changepoint positions and the associated Bayesian model evidence are given for DC step changes. The performance of the analytical approach is compared to that predicted by a Gaussian assumption to the noise statistics and Markov chain Monte Carlo methods.

Mobility criteria of geared five-bar linkages

This article proposes simple Grashof type criteria to identify the full rotatability of geared five-bar linkages and provides a systematic approach to study the effects of the five-bar loop and gear constraint in the mobility of a geared five-bar linkage. A fully rotatable geared linkage must have a C-C (crank-crank) type host five-bar linkage. All fully rotatable geared linkages are grouped into A, B, and C types according to the joint rotation space of their host linkages. In type A linkages, a gear constraint may have any gear ratio and phase angle unless a change-point position exists. In type B linkages, the gear ratio may be a unit or other values but the phase angle must be properly chosen to achieve full rotatability. In type C linkages, the two terminal gears must rotate with the same speed in the same direction relative to the center link between them and have a proper phase angle. Any geared linkage which does not belong to one of these three types does not have full rotatability. A table summarizing the required phase angle conditions for a fully rotatable geared linkage with a unit gear ratio is also presented.

Über verzweigungslagen bei sphärischen viergelenkgetrieben

In discussing the laws of motion of a specific linkage with variable parameters, the limiting or exceptional cases where a change-point position becomes possible is frequently of interest. In this paper it is explained how to obtain the ratios of velocities for spherical four-bar linkages with change-point positions. The positions of the relative pole axes 31 and 20 (i.e. motion of the input link with respect to the output link or motion of the coupler with respect to the fixed link) are of special interest. Passing to the limit leads to the corresponding plane four-bar linkages. Two examples, one linkage from the plane and one from spherical kinematics, conclude the paper.