Use of cumulative sums for detection of changepoints in the rate parameter of a poisson process

This paper studies the problem of multiple changepoints in rate parameter of a Poisson process. We propose a binary segmentation algorithm in conjunction with a cumulative sums statistic for detection of changepoints such that in each step we need only to test the presence of a simple changepoint. We derive the asymptotic distribution of the proposed statistic, prove its consistency and obtain the limiting distribution of the estimate of the changepoint. A Monte Carlo analysis shows the good performance of the proposed procedure, which is illustrated with a real data example

Variance changes detection in multivariate time series

This paper studies the detection of step changes in the variances and in the correlation structure of the components of a vector of time series. Two procedures are considered. The first is based on the likelihood ratio test and the second on cusum statistics. These two procedures are compared in a simulation study and we conclude that the cusum procedure is more powerful. The procedures are illustrated in two examples.

USE OF CUMULATIVE SUMS FOR DETECTION OF CHANGEPOINTS IN THE RATE PARAMETER OF A POISSON PROCESS

This paper studies the problem of multiple changepoints in rate parameter of a Poisson process. We propose a binary segmentation algorithm in conjunction with a cumulative sums statistic for detection of changepoints such that in each step we need only to test the presence of a simple changepoint. We derive the asymptotic distribution of the proposed statistic, prove its consistency and obtain the limiting distribution of the estimate of the changepoint. A Monte Carlo analysis shows the good performance of the proposed procedure, which is illustrated with a real data example.

Multivariate analysis in vector time series

This paper reviews the applications of classical multivariate techniques for discrimination, clustering and dimension reduction for time series data. It is shown that the discrimination problem can be seen as a model selection problem. Some of the results obtained in the time domain are reviewed. Clustering time series requires the definition of an adequate metric between univariate time series and several possible metrics are analyzed. Dimension reduction has been a very active line of research in the time series literature and the dynamic principal components or canonical analysis of Box and Tiao (1977) and the factor model as developed by Peña and Box (1987) and Peña and Poncela (1998) are analyzed. The relation between the nonstationary factor model and the cointegration literature is also reviewed

Model selection criteria and quadratic discrimination in ARMA and SETAR time series models

We show that analyzing model selection in ARMA time series models as a quadratic discrimination problem provides a unifying approach for deriving model selection criteria. Also this approach suggest a different definition of expected likelihood that the one proposed by Akaike. This approach leads to including a correction term in the criteria which does not modify their large sample performance but can produce significant improvement in the performance of the criteria in small samples. Thus we propose a family of criteria which generalizes the commonly used model selection criteria. These ideas can be extended to self exciting autoregressive models (SETAR) and we generalize the proposed approach for these non linear time series models. A Monte-Carlo study shows that this family improves the finite sample performance of criteria such as AIC, corrected AIC and BIC, for ARMA models, and AIC, corrected AIC, BIC and some cross-validation criteria for SETAR models. In particular, for small and medium sample size the frequency of selecting the true model improves for the consistent criteria and the root mean square error of prediction improves for the efficient criteria. These results are obtained for both linear ARMA models and SETAR models in which we assume that the threshold and the parameters are unknown

A note on prediction and interpolation errors in time series

In this note we analyze the relationship between one-step ahead prediction errors and interpolation errors in time series. We obtain an expression of the prediction errors in terms of the interpolation errors and then we show that minimizing the sum of squares of the one step-ahead standardized prediction errors is equivalent to minimizing the sum of squares of standardized interpolation errors

MULTIVARIATE ANALYSIS IN VECTOR TIME SERIES

This paper reviews the applications of classical multivariate techniques for discrimination, clustering and dimension reduction for time series data. It is shown that the discrimination problem can be seen as a model selection problem. Some of the results obtained in the time domain are reviewed. Clustering time series requires the definition of an adequate metric between univariate time series and several possible metrics are analyzed. Dimension reduction has been a very active line of research in the time series literature and the dynamic principal components or canonical analysis of Box and Tiao (1977) and the factor model as developed by Peña and Box (1987) and Peña and Poncela (1998) are analyzed. The relation between the nonstationary factor model and the cointegration literature is also reviewed.

VARIANCE CHANGES DETECTION IN MULTIVARIATE TIME SERIES

This paper studies the detection of step changes in the variances and in the correlation structure of the components of a vector of time series. Two procedures are considered. The first is based on the likelihood ratio test and the second on cusum statistics. These two procedures are compared in a simulation study and we conclude that the cusum procedure is more powerful. The procedures are illustrated in two examples.R

A NOTE ON PREDICTION AND INTERPOLATION ERRORS IN TIME SERIES

In this note we analyze the relationship between one-step ahead prediction errors and interpolation errors in time series. We obtain an expression of the prediction errors in terms of the interpolation errors and then we show that minimizing the sum of squares of the one step-ahead standardized prediction errors is equivalent to minimizing the sum of squares of standardized interpolation errors.

MODEL SELECTION CRITERIA AND QUADRATIC DISCRIMINATION IN ARMA AND SETAR TIME SERIES MODELS

We show that analyzing model selection in ARMA time series models as a quadratic discrimination problem provides a unifying approach for deriving model selection criteria. Also this approach suggest a different definition of expected likelihood that the one proposed by Akaike. This approach leads to including a correction term in the criteria which does not modify their large sample performance but can produce significant improvement in the performance of the criteria in small samples. Thus we propose a family of criteria which generalizes the commonly used model selection criteria. These ideas can be extended to self exciting autoregressive models (SETAR) and we generalize the proposed approach for these non linear time series models. A Monte-Carlo study shows that this family improves the finite sample performance of criteria such as AIC, corrected AIC and BIC, for ARMA models, and AIC, corrected AIC, BIC and some cross-validation criteria for SETAR models. In particular, for small and medium sample size the frequency of selecting the true model improves for the consistent criteria and the root mean square error of prediction improves for the efficient criteria. These results are obtained for both linear ARMA models and SETAR models in which we assume that the threshold and the parameters are unknown.