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2024 | Buch

Cumulative Sum Processes Kapitel lesen Erstes Kapitel lesen

Change Point Analysis for Time Series

verfasst von: Lajos Horváth, Gregory Rice

Verlag: Springer Nature Switzerland

Buchreihe : Springer Series in Statistics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This volume provides a comprehensive survey that covers various modern methods used for detecting and estimating change points in time series and their models. The book primarily focuses on asymptotic theory and practical applications of change point analysis. The methods discussed in the book go beyond the traditional change point methods for univariate and multivariate series. It also explores techniques for handling heteroscedastic series, high-dimensional series, and functional data. While the primary emphasis is on retrospective change point analysis, the book also presents sequential "on-line" methods for detecting change points in real-time scenarios. Each chapter in the book includes multiple data examples that illustrate the practical application of the developed results. These examples cover diverse fields such as economics, finance, environmental studies, and health data analysis. To reinforce the understanding of the material, each chapter concludes with several exercises.Additionally, the book provides a discussion of background literature, allowing readers to explore further resources for in-depth knowledge on specific topics. Overall, "Change Point Analysis for Time Series" offers a broad and informative overview of modern methods in change point analysis, making it a valuable resource for researchers, practitioners, and students interested in analyzing and modeling time series data.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Cumulative Sum Processes
Abstract
In this chapter we introduce the basic change point in the mean problem for scalar observations. We see that the most logical and straightforward approaches to detect such a change point lead to the consideration of weighted functionals of the cumulative sum (CUSUM) processes computed from the observed data. As such, we begin by developing a comprehensive asymptotic theory for CUSUM processes under conditions that allow for serial dependence in the observations. This includes a careful analysis of how weights applied to the CUSUM process affect the limiting distribution of its functionals, and extensions to multivariate observations.
Lajos Horváth, Gregory Rice
Chapter 2. Change Point Analysis of the Mean
Abstract
We have seen that under the no change in the mean null hypothesis \(H_0\), and assuming the observations satisfy a functional version of the central limit theorem (Assumptions 1.​1.​1 and 1.​2.​2), that the asymptotic distribution of many functionals of the CUSUM process may be computed. Since the CUSUM process arises as the objective function in maximally selecting two sample test statistics to test \(H_0\) versus \(H_A\), it stands to reason that, in the presence of change points in the series, the functionals of the CUSUM process that we have considered should be consistent in the sense that they diverge in probability to positive infinity as the sample size grows. One goal of this chapter is to carefully quantify the asymptotic behaviour of the CUSUM process in the presence of change points.
Lajos Horváth, Gregory Rice
Chapter 3. Variance Estimation, Change Points in Variance, and Heteroscedasticity
Abstract
A crucial step in approximating the distribution of the CUSUM statistics introduced in Chaps. 1 and 2 is estimating the variance parameter \(\sigma ^2\) describing the limiting variance of the partial sum of the observations under Assumption 1.​1.​1.
Lajos Horváth, Gregory Rice
Chapter 4. Regression Models
Abstract
When two or more variables are observed concurrently, it is often of interest to evaluate whether their relationship appears to stay constant over time, or if instead it appears to change. A simple framework to address questions of this type is when the variables are related to each other through a parametric regression model. In this case a change in the relationship may be characterized by a change in the model parameters. This chapter is devoted to the development of asymptotic methods to perform change point analysis in the context of regression models.
Lajos Horváth, Gregory Rice
Chapter 5. Parameter Changes in Time Series Models
Abstract
We develop in this chapter the asymptotic theory surrounding change point methods for many popular time series models. Although up to this point we have generally taken into consideration potential serial dependence in the observations under study, in this chapter we are concerned with detecting change points in the parameters for models specifically designed to capture the serial dependence structure of a time series.
Lajos Horváth, Gregory Rice
Chapter 6. Sequential Monitoring
Abstract
Up to this point we have been concerned with what is usually referred to as “retrospective” or “off-line” change point detection and estimation, in which the goal is to conduct change point analysis retrospectively on an observed series. In this chapter, we shift our focus to sequential or “online” change point detection methods. These aim to detect a change point in the data generating process, relative to a stable training or historical sample, as quickly as possible as we continue to obtain data sequentially. We begin by developing the framework of such sequential detection procedures in the context of a simple mean change in Sect. 6.1, which we then extend to linear and time series models in Sects. 6.2 and 6.3. A key consideration throughout is the distribution of the stopping time, which is the amount of time required in order to detect a change point in the data generating process. The asymptotic distribution of stopping times are investigated in Sect. 6.4.
Lajos Horváth, Gregory Rice
Chapter 7. High-Dimensional and Panel Data
Abstract
We have considered in several instances, see e.g. Sects. 1.​3, 5.​5, and 5.​6, performing change point analysis with multivariate time series. In this chapter we change our notation slightly to denote such multivariate time series data as \(X_{i,t}, \quad t\in \{1,\ldots ,T\}, \;i \in \{1,\ldots ,N\},\) where we think of t and T as denoting “time”, and N denotes the dimension or number of “cross-sectional units” that we observe. For example, such data might comprise real valued observations of N financial or economic time series over T time units.
Lajos Horváth, Gregory Rice
Chapter 8. Functional Data
Abstract
Functional data analysis concerns methods to analyse data that are naturally viewed as taking values in infinite dimensional function spaces. Examples include data that can be imagined as curves or surfaces. A general object of this type is termed a functional data object. When functional data are observed sequentially over time, they are referred to as functional time series. Usually the data that are available in this setting are discrete measurements of such objects from which the full functional data objects must be reconstructed or estimated using curve fitting techniques. In some cases the functional data objects are fully observable on the domain on which they are defined, for instance when they represent probability densities or other summary functions.
Lajos Horváth, Gregory Rice
Backmatter
Metadaten
Titel
Change Point Analysis for Time Series
verfasst von
Lajos Horváth
Gregory Rice
Copyright-Jahr
2024
Electronic ISBN
978-3-031-51609-2
Print ISBN
978-3-031-51608-5
DOI
https://doi.org/10.1007/978-3-031-51609-2

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