Bayesian P-splines and advanced computing in R for a changepoint analysis on spatio-temporal point processes

ABSTRACT

This work presents advanced computational aspects of a new method for changepoint detection on spatio-temporal point process data. We summarize the methodology, based on building a Bayesian hierarchical model for the data and declaring prior conjectures on the number and positions of the changepoints, and show how to take decisions regarding the acceptance of potential changepoints. The focus of this work is about choosing an approach that detects the correct changepoint and delivers smooth reliable estimates in a feasible computational time; we propose Bayesian P-splines as a suitable tool for managing spatial variation, both under a computational and a model fitting performance perspective. The main computational challenges are outlined and a solution involving parallel computing in R is proposed and tested on a simulation study. An application is also presented on a data set of seismic events in Italy over the last 20 years.     

Change-points in stochastic ordering

This article studies the problem of testing and locating changepoints in stochas¬tic ordering. We propose a sequential process to detect the changepoints from two multinomial distributions. We also obtain the maximum likelihood estimators of two multinomial probability vectors under the assumption that the cumulative distribu¬tions have a changepoint. Asymptotically unbiased Akaike's information criterion is used to estimate the changepoints of two discrete probability distributions. Finally. we demonstrate our procedure by studying a data set pertaining to average daily insulin dose from the Boston Collaborative Drug Surveillance Program and locate the changepoints in stochastic ordering.     

Robust test for means when population variances are unequal

We consider the problem of testing the equality of two population means when the population variances are not necessarily equal. We propose a Welch-type statistic, say T^* _c, based on Tiku!s ‘1967, 1980’ modified maximum likelihood estimators, and show that this statistic is robust to symmetric and moderately skew distributions. We investigate the power properties of the statistic T^* _c; T^* _c clearly seems to be more powerful than Yuen's ‘1974’ Welch-type robust statistic based on the trimmed sample means and the matching sample variances. We show that the analogous statistics based on the ‘adaptive’ robust estimators give misleading Type I errors. We generalize the results to testing linear contrasts among k population means     

Maximum Test versus Adaptive Tests for the Two-Sample Location Problem

For the non-parametric two-sample location problem, adaptive tests based on a selector statistic are compared with a maximum and a sum test, respectively. When the class of all continuous distributions is not restricted, the sum test is not a robust test, i.e. it does not have a relatively high power across the different possible distributions. However, according to our simulation results, the adaptive tests as well as the maximum test are robust. For a small sample size, the maximum test is preferable, whereas for a large sample size the comparison between the adaptive tests and the maximum test does not show a clear winner. Consequently, one may argue in favour of the maximum test since it is a useful test for all sample sizes. Furthermore, it does not need a selector and the specification of which test is to be performed for which values of the selector. When the family of possible distributions is restricted, the maximin efficiency robust test may be a further robust alternative. However, for the family of t distributions this test is not as powerful as the corresponding maximum test.     

Bayesian binary segmentation procedure for detecting streakiness in sports

Summary.  When an individual player or team enjoys periods of good form, and when these occur, is a widely observed phenomenon typically called 'streakiness'. It is interesting to assess which team is a streaky team, or who is a streaky player in sports. Such competitors might have a large number of successes during some periods and few or no successes during other periods. Thus, their success rate is not constant over time. We provide a Bayesian binary segmentation procedure for locating changepoints and the associated success rates simultaneously for these competitors. The procedure is based on a series of nested hypothesis tests each using the Bayes factor or the Bayesian information criterion. At each stage, we only need to compare a model with one changepoint with a model based on a constant success rate. Thus, the method circumvents the computational complexity that we would normally face in problems with an unknown number of changepoints. We apply the procedure to data corresponding to sports teams and players from basketball, golf and baseball.     

A randomized Baumgartner statistic for multivariate two-sample testing hypothesis

In this paper, multivariate two-sample testing problems were examined based on the Jure?ková–Kalina's ranks of distances. The multivariate two-sample rank test based on the modified Baumgartner statistic for the two-sided alternative was proposed. The proposed statistic was a randomized statistic. Simulations were used to investigate the power of the suggested statistic for various population distributions.     

Variable dimension via stochastic volatility model using FX rates

In this paper, changepoint analysis is applied to stochastic volatility (SV) models which aim to understand the locations and movements of high frequency FX financial time series. Bayesian inference using the Markov Chain Monte Carlo method is performed using a process called variable dimension for SV parameters. Interesting results are that FX series have locations where one or more positions of the sequence correspond to systemic changes, and overall non-stationarity, in the returns process. Furthermore, we found that the changepoint locations provide an informative estimate for all FX series. Importantly in most cases, the detected changepoints can be identified with economic factors relevant to the country concerned. This helps support the fact that macroeconomics news and the movement in financial price are positively related.     

Exact and efficient Bayesian inference for multiple changepoint problems

We demonstrate how to perform direct simulation from the posterior distribution of a class of multiple changepoint models where the number of changepoints is unknown. The class of models assumes independence between the posterior distribution of the parameters associated with segments of data between successive changepoints. This approach is based on the use of recursions, and is related to work on product partition models. The computational complexity of the approach is quadratic in the number of observations, but an approximate version, which introduces negligible error, and whose computational cost is roughly linear in the number of observations, is also possible. Our approach can be useful, for example within an MCMC algorithm, even when the independence assumptions do not hold. We demonstrate our approach on coal-mining disaster data and on well-log data. Our method can cope with a range of models, and exact simulation from the posterior distribution is possible in a matter of minutes.     

Comparison of two weibull distributions under random censoring

The two-sample problem for comparing Weibull scale parameters is studied for randomly censored data. Three different test statistics are considered and their asymptotic properties are established under a sequence of local alternatives, It is shown that both the test statistic based on the mlefs (maximum likelihood estimators) and the likelihood ratio test are asymptotically optimum. The third statistic based only on the number of failures is not, Asymptotic relative efficiency of this statistic is obtained and its numerical values are computed for uniform and Weibull censoring, Effects of uniform random censoring on the censoring level of the experiment are illus¬trated, A direct proof for the joint asymptotic normality of the mlefs of the shape and the scale parameters is also given     

Robust pairwise multiple comparisons under short-tailed symmetric distributions

In one-way ANOVA, most of the pairwise multiple comparison procedures depend on normality assumption of errors. In practice, errors have non-normal distributions so frequently. Therefore, it is very important to develop robust estimators of location and the associated variance under non-normality. In this paper, we consider the estimation of one-way ANOVA model parameters to make pairwise multiple comparisons under short-tailed symmetric (STS) distribution. The classical least squares method is neither efficient nor robust and maximum likelihood estimation technique is problematic in this situation. Modified maximum likelihood (MML) estimation technique gives the opportunity to estimate model parameters in closed forms under non-normal distributions. Hence, the use of MML estimators in the test statistic is proposed for pairwise multiple comparisons under STS distribution. The efficiency and power comparisons of the test statistic based on sample mean, trimmed mean, wave and MML estimators are given and the robustness of the test obtained using these estimators under plausible alternatives and inlier model are examined. It is demonstrated that the test statistic based on MML estimators is efficient and robust and the corresponding test is more powerful and having smallest Type I error.     

A computationally efficient nonparametric approach for changepoint detection

In this paper we build on an approach proposed by Zou et al. (2014) for nonparametric changepoint detection. This approach defines the best segmentation for a data set as the one which minimises a penalised cost function, with the cost function defined in term of minus a non-parametric log-likelihood for data within each segment. Minimising this cost function is possible using dynamic programming, but their algorithm had a computational cost that is cubic in the length of the data set. To speed up computation, Zou et al. (2014) resorted to a screening procedure which means that the estimated segmentation is no longer guaranteed to be the global minimum of the cost function. We show that the screening procedure adversely affects the accuracy of the changepoint detection method, and show how a faster dynamic programming algorithm, pruned exact linear time (PELT) (Killick et al. 2012), can be used to find the optimal segmentation with a computational cost that can be close to linear in the amount of data. PELT requires a penalty to avoid under/over-fitting the model which can have a detrimental effect on the quality of the detected changepoints. To overcome this issue we use a relatively new method, changepoints over a range of penalties (Haynes et al. 2016), which finds all of the optimal segmentations for multiple penalty values over a continuous range. We apply our method to detect changes in heart-rate during physical activity.     

Power of the adjusted Q statistic to evaluate heterogeneity in meta-analyses of cluster randomized trials

Because of its simplicity, the Q statistic is frequently used to test the heterogeneity of the estimated intervention effect in meta-analyses of individually randomized trials. However, it is inappropriate to apply it directly to the meta-analyses of cluster randomized trials without taking clustering effects into account. We consider the properties of the adjusted Q statistic for testing heterogeneity in the meta-analyses of cluster randomized trials with binary outcomes. We also derive an analytic expression for the power of this statistic to detect heterogeneity in meta-analyses, which can be useful when planning a meta-analysis. A simulation study is used to assess the performance of the adjusted Q statistic, in terms of its Type I error rate and power. The simulation results are compared to that obtained from the proposed formula. It is found that the adjusted Q statistic has a Type I error rate close to the nominal level of 5%, as compared to the unadjusted Q statistic commonly used to test for heterogeneity in the meta-analyses of individually randomized trials with an inflated Type I error rate. Data from a meta-analysis of four cluster randomized trials are used to illustrate the procedures.     

Approximation of distributions by using the Anderson Darling statistic

ABSTRACT

In practice, it is often not possible to find an appropriate family of distributions which can be used for fitting the sample distribution with high precision. In these cases, it seems to be opportune to search for the best approximation by a family of distributions instead of an exact fit. In this paper, we consider the Anderson–Darling statistic with plugged-in minimum distance estimator for the parameter vector. We prove asymptotic normality of the Anderson–Darling statistic which is used for a test of goodness of approximation. Moreover, we introduce a measure of discrepancy between the sample distribution and the model class.     

Phi-Divergence Statistics for Testing Linear Hypotheses in Logistic Regression Models

In this paper we introduce and study two new families of statistics for the problem of testing linear combinations of the parameters in logistic regression models. These families are based on the phi-divergence measures. One of them includes the classical likelihood ratio statistic and the other the classical Pearson's statistic for this problem. It is interesting to note that the vector of unknown parameters, in the two new families of phi-divergence statistics considered in this paper, is estimated using the minimum phi-divergence estimator instead of the maximum likelihood estimator. Minimum phi-divergence estimators are a natural extension of the maximum likelihood estimator.     

Testing Normality for Linear AR( p ) Models

Abstract

This paper proposes a nonparametric mixed test for normality of linear autoregressive time series. The test is based on the best one-step forecast in mean square with time reverse. The test statistic is the mixture of a goodness of fit statistic and Cramer–Von Mises statistic. Some asymptotic properties are developed for the test. Simulated results have shown that the test is easy to use and has good powers. Three examples of applying the test to real data are also included.     

A Nonparametric Test for the Parallelism of Two First-Order Autoregressive Processes

A nonparametric testing procedure for the parallelism of two first-order autoregressive processes is presented. This paper discuss the Mann–Whitney statistic, its natural competitor two-sample t -test, and the bootstrap method. It studies the asymptotic efficacies of the studentized Mann–Whitney statistic and the t -test statistic with their relative efficiency. Simulation results for comparing the powers of these test statistics are also presented.     

Some results on estimating a change-point using non-parametric type statistics

A Mann-Whitney type statistic is used to estimate a change-point when a change, at an unknown point in a sequence of random variables, has taken place. This estimate is compared, using Monte Carlo techniques, with the normal theory maximum likelihood estimate, when a location change has occurred, for different underlying distributions ranging from the normal to the long tailed “normal over uniform” distribution. The distribution of the Mann-Whitney type estimate remains fairly constant over the various distributions. Two generalisations of the statistic are considered and investigated.     

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.     

Combined sample scores versus two-sample scores under censoring

For testing the equality of K-populations against ordered alternatives a quadratic form based on two-sample scores of (K-l) pairs of adjacent samples has been suggested as an alternative to the statistic proposed by Beslow(1970). The Pitman asymptotic efficiency of the proposed statistic relative to the Breslow's statistic is found to be one or very close to one in all cases considered. The Bahadur asymptotic efficiency has also been studied and found to be at least one for the situations taken up in this paper.