CPAT

Version 0.1.0

CPAT is a package implementing some statistical tests for detecting structural change in a series of data. The tests made publicly available are:

• The CUSUM test, via CUSUM.test()
• The Darling-Erdös test, via DE.test()
• The Hidalgo-Seo test, via HS.test()
• Andrews test, via Andrews.test()
• The Rényi-type test via HR.test()

This package was written to facilitate the simulations performed in a paper by Horváth, Rice and Miller (see the documentation for HR.test() for a citation) and thus is geared to change point tests capable of detecting early/late changes in a sample. That said, it is general purpose.

Installation

CPAT is available on CRAN. You should be able to install CPAT with the command install.packages("CPAT").

You can install CPAT from GitHub via the R command devtools::install_github("ntguardian/CPAT").

It is possible that you have uncompressed the archive containing not only the source code of the package but also the source code of the project, including data files, plots, etc. If that's the case, start an R session while in the base directory of the project then run the command devtools::install(). This should install the project.

Change Point Testing

Change point testing is performed on sequential data (such as time series) to determine whether the data shares a common structure. In particular, let where is a random noise process, with . With the exception of Andrews' test, the tests mentioned above can be used to decide between the hypotheses:

is an integer satisfying . Critically, is not assumed to be known, so the alternative hypothesis states that the change occurs at an unknown location in the sample.

Andrews' test is an exception; his test assumes some information about where the change occured. Suppose for with and known. His test decides between the hypotheses:

In this case, .

Change point testing traces its roots to quality control procedures; if one imagines being some measurement of a part produced by an machine, states that the machine was always calibrated while the alternative hypothesis claimes that the machine became uncalibrated at some unknown porint in time. Another view of a change point test is that it's yet another test to check if a time series is stationary, which is a critical assumption made in most analyses involving time series.

While the above formulation describes a test primarily concerned about the mean of a process, these tests, even in this form, can make statements about structural change other than the mean. For instance, if represents the residuals of a regression model, the test checks whether the parameters of the model are stable over time or not.

All of the tests included in this package are asymptotic tests; the test performs better for large and one should be cautious when using these tests for small . Simulations studies suggest that the Rényi-type test seems to perform best when a change occurs near the ends of a sample, while the CUSUM test seemed to perform best when the change occured mid-sample.

Example

Each function can accept a univariate dataset as input, and will return an htest-class object with the results of the test.

library(CPAT)

set.seed(20180924)
x <- c(rnorm(250), rnorm(50) + 1)

CUSUM.test(x)
#> 
#> 	CUSUM Test for Change in Mean
#> 
#> data:  x
#> A = 1.8875, p-value = 0.001609
#> sample estimates:
#>  t* 
#> 250
DE.test(x)
#> 
#> 	Darling-Erdos Test for Change in Mean
#> 
#> data:  x
#> A = 3.7528, a(log(T)) = 1.8661, b(log(T)) = 3.1872, p-value =
#> 0.04582
#> sample estimates:
#>  t* 
#> 251
HS.test(x)
#> 
#> 	Hidalgo-Seo Test for Change in Mean
#> 
#> data:  x
#> A = 11.783, Correlated Residuals = 1, p-value = 0.00551
#> sample estimates:
#>  t* 
#> 251
HR.test(x)
#> 
#> 	Horvath-Rice Test for Change in Mean
#> 
#> data:  x
#> D = 2.2145, log(T) = 5.7038, p-value = 0.1043
#> sample estimates:
#>  t* 
#> 284
Andrews.test(x, M = 250)
#> 
#> 	Andrews' Test for Structural Change
#> 
#> data:  x
#> S = 52.63, m = 50, p-value = 0.2388

Often these tests return a test statistic, a -value, and the estimated location of the change. This latter quantity is the arg-max of the terms the test statistics maximize. (There is theory supporting this estimator for the CUSUM test, but no theory for the Rényi-type test.)

Andrews.test() also allows for testing for structural change in a linear model directly.

df <- data.frame(x = x, y = 1 + 2 * x + c(rnorm(250), rnorm(50) + 1))
Andrews.test(df, M = 250, formula = y ~ x)
#> 
#> 	Andrews' Test for Structural Change
#> 
#> data:  df
#> S = 6.3494, m = 50, p-value < 2.2e-16

Project Replication

The source of this package does more than make the package available to R. It also includes all code necessary to recreate the simulations performed in the paper by Horvath, Miller and Rice ("A new class of change point test statistics of Rényi type", ????). Some versions of the source may even include the files containing the simulation studies already created.

While the R/ directory contains definitions for functions useful both for implementing these tests and performing our simulations, the exec/ directory contains the scripts (executable from the command line at least on UNIX-like systems) that perform the simulations and the file Makefile in the root directory uses GNU make to manage the structure of the project. Together these allow for simulations to be replicated.

We recommend learning how to use GNU make to fully appreciate the contents of Makefile (the manual, when read according to the authors' recommendations, is a good introduction), but one can get started without fully appreciating make's intricacies. Below I assume GNU make is installed and the base directory of the project is the working directory:

• The command make clean will remove every file associated with the simulations studies; this represents a clean build. Beware, though, that some simulations can take days to complete even on a supercomputer, so don't make clean without careful consideration. The command make mostlyclean will clean up many of the files associated with the project but does not remove files generated from extensive simulations studies.
• make will automatically update all files that are out of date, based on the timestamps of the files in question. All dependencies are tracked in Makefile.
• If the project is clean, running make may not create all simulations since the files that would contain the results of plots don't exist, causing make to decide that they don't need to be updated. The command make init will create dummy versions of some of these files, which will certainly have newer time stamps than their dependencies. (These are three dummy PDF files that aren't even binary files; beware that this will overwrite those files if they already exist.) Running make immediately after make init should rebuild the project.
• Since the simulations take a long time to perform with the full sample sizes, the command make small will do the same as make but with much fewer replications (by default, 20 replications per simulation). This is good for testing to see if a setup works.
• A number of parameters are defined in make variables defined at the beginning of Makefile, and these variables can be overridden in the command line. For example, make LEVEL=0.01 will remake the project but with the tests involved in the power simulations using a significance level of 0.01 rather than 0.05 (this will remake the simulations only if the relevant files, specifically the CSV metadata files, are out of date). make small SMALLREPLICATIONS=50 is like make small but performs 50 replications rather than the default 20. See Makefile for all variables used.

We once used the R package packrat to manage package dependencies but unfortunately this would not work with critical systems for performing the simulations. There's also no tracking of system dependencies. We tracked the packages we needed in the file exec/GetPackages.R, which can be executed from the command line using either the command Rscript exec/GetPackages.R or exec/GetPackages.R directly (depending on whether the file is marked as executable). If executed, R will attempt to download and install all project dependencies we are aware of. As for system dependencies, a sufficiently recent version of R and Rscript are needed, and LaTeX with the tikz package should be available and accessible to the tikzDevice package. Additionally, CPAT itself should be installed; the scripts in exec/ not only use CPAT but use private CPAT functions (accessed via :::). If the source has been uncompressed, running the R command devtools::install() to install the package should be sufficient.

We do not guarantee that once the simulations are run with make the result will be a valid R package (let alone CRAN-compliant), but we have never had issues yet running simulations then using devtools::install() in the base directory of the source of the package to install CPAT.

Planned Future Features

• Currently, Andrews.test() only works for late changes, but Andrews' paper allows for tests for instability in both the beginning and middle of the sample. Andrews.test() should support these types of tests.
• We could include more tests, including those mentioned in the paper by Aue and Horváth (2013) and Horváth and Rice (2014).