| Title: | Piecewise Smooth Regression by Bootstrapped Binary Segmentation |
|---|---|
| Description: | Provides methods for piecewise smooth regression. A piecewise smooth signal is estimated by applying a bootstrapped test recursively (binary segmentation approach). Each bootstrapped test decides whether the underlying signal is smooth on the currently considered subsegment or contains at least one further change-point. |
| Authors: | McDaid Kate [aut], Pein Florian [aut, cre] |
| Maintainer: | Pein Florian <[email protected]> |
| License: | GPL-3 |
| Version: | 1.0-1 |
| Built: | 2024-11-14 03:59:20 UTC |
| Source: | https://github.com/cran/BinSegBstrap |
Provides methods for piecewise smooth regression. The main function BinSegBstrap estimates a piecewise smooth signal by applying a bootstrapped test recursively (binary segmentation approach). A single bootstrapped test for the hypothesis that the underlying signal is smooth versus the alternative that the underlying signal contains at least one change-point can be performed by the function BstrapTest. A single change-point is estimated by the function estimateSingleCp. More details can be found in the vignette. Parts of this work were inspired by Gijbels and Goderniaux (2004).
This work results from a summer research project at the University of Cambridge in 2019. Kate McDaid was supported by a bursary from the summer research programme of the Centre of Mathematics at the University of Cambridge. Florian Pein's position is funded by the EPSRC programme grant 'StatScale: Statistical Scalability for Streaming Data'.
Gijbels, I., Goderniaux, A-C. (2004) Bootstrap test for change-points in nonparametric regression. Journal of Nonparametric Statistics 16(3-4), 591–611.
BinSegBstrap, BstrapTest, estimateSingleCp
n <- 200 signal <- sin(2 * pi * 1:n / n) signal[51:100] <- signal[51:100] + 5 signal[151:200] <- signal[151:200] + 5 y <- rnorm(n) + signal est <- BinSegBstrap(y = y) plot(y) lines(signal) lines(est$est, col = "red") n <- 100 signal <- sin(2 * pi * 1:n / n) signal[51:100] <- signal[51:100] + 5 y <- rnorm(n) + signal test <- BstrapTest(y = y) est <- estimateSingleCp(y = y) plot(y) lines(signal) lines(est$est, col = "red")n <- 200 signal <- sin(2 * pi * 1:n / n) signal[51:100] <- signal[51:100] + 5 signal[151:200] <- signal[151:200] + 5 y <- rnorm(n) + signal est <- BinSegBstrap(y = y) plot(y) lines(signal) lines(est$est, col = "red") n <- 100 signal <- sin(2 * pi * 1:n / n) signal[51:100] <- signal[51:100] + 5 y <- rnorm(n) + signal test <- BstrapTest(y = y) est <- estimateSingleCp(y = y) plot(y) lines(signal) lines(est$est, col = "red")
A piecewise smooth signal is estimated by applying BstrapTest recursively (binary segmentation approach). The final estimator is estimated by kernel smoothing on each segment separately; a joint bandwidth is selected by crossvalidation. More details can be found in the vignette.
BinSegBstrap(y, bandwidth, nbandwidth = 30L, B = 500L, alpha = 0.05, kernel = c("epanechnikov", "gaussian", "rectangular", "triangular", "biweight", "silverman"))BinSegBstrap(y, bandwidth, nbandwidth = 30L, B = 500L, alpha = 0.05, kernel = c("epanechnikov", "gaussian", "rectangular", "triangular", "biweight", "silverman"))
y |
a numeric vector containing the data points |
bandwidth |
the bandwidth, i.e. a numeric with values between |
nbandwidth |
a single integer giving the number of bandwidths (see above) if |
B |
a single integer giving the number of bootstrap samples |
alpha |
a probability, i.e. a single numeric between 0 and 1, giving the significance level of the test |
kernel |
the kernel function, i.e. either a string or a function that takes a single numeric vector and returns the values of the kernel at those locations |
a list with the following components:
- est: the estimated signal
- cps: the estimated change-point locations
- bandwidth: the selected bandwidth
n <- 200 signal <- sin(2 * pi * 1:n / n) signal[51:100] <- signal[51:100] + 5 signal[151:200] <- signal[151:200] + 5 y <- rnorm(n) + signal # default bandwidth and kernel est <- BinSegBstrap(y = y) plot(y) lines(signal) lines(est$est, col = "red") # fixed bandwidth est <- BinSegBstrap(y = y, bandwidth = 0.1) # user specified kernel kernel <- function(x) 1 - abs(x) # triangular kernel est <- BinSegBstrap(y = y, kernel = kernel)n <- 200 signal <- sin(2 * pi * 1:n / n) signal[51:100] <- signal[51:100] + 5 signal[151:200] <- signal[151:200] + 5 y <- rnorm(n) + signal # default bandwidth and kernel est <- BinSegBstrap(y = y) plot(y) lines(signal) lines(est$est, col = "red") # fixed bandwidth est <- BinSegBstrap(y = y, bandwidth = 0.1) # user specified kernel kernel <- function(x) 1 - abs(x) # triangular kernel est <- BinSegBstrap(y = y, kernel = kernel)
Tests whether the underlying signal is smooth or contains at least one change-point. The smooth alternative is estimated by a (crossvalidated) kernel smoother. The single change-point alternative is estimated by estimateSingleCp. Its estimated jump size is used as a test statistic and the critical value is obtained by bootstrapping. More details can be found in the vignette.
BstrapTest(y, bandwidth, nbandwidth = 30L, B = 500L, alpha = 0.05, kernel = c("epanechnikov", "gaussian", "rectangular", "triangular", "biweight", "silverman"))BstrapTest(y, bandwidth, nbandwidth = 30L, B = 500L, alpha = 0.05, kernel = c("epanechnikov", "gaussian", "rectangular", "triangular", "biweight", "silverman"))
y |
a numeric vector containing the data points |
bandwidth |
the bandwidth, i.e. a numeric with values between |
nbandwidth |
a single integer giving the number of bandwidths (see above) if |
B |
a single integer giving the number of bootstrap samples |
alpha |
a probability, i.e. a single numeric between 0 and 1, giving the significance level of the test |
kernel |
the kernel function, i.e. either a string or a function that takes a single numeric vector and returns the values of the kernel at those locations |
a list with the following components:
- piecewiseSignal: the estimated signal with a single change-point
- cp: the estimated change-point location
- size: the estimated jump size
- bandwidth: the selected bandwidth for the piecewise signal
- bandwidthSmooth: the selected bandwidth for the smooth signal
- smoothSignal: the estimated smooth signal
- critVal: the by bootstrapping obtained critical value
- pValue: the p-Value of the test
- outcome: a boolean saying whether the test rejects the hypothesis of a smooth signal
n <- 100 signal <- sin(2 * pi * 1:n / n) signal[51:100] <- signal[51:100] + 5 y <- rnorm(n) + signal # default bandwidth and kernel test <- BstrapTest(y = y) if (test$outcome) { # null hypothesis of a smooth signal is rejected estimatedSignal <- test$piecewiseSignal } else { # null hypothesis of a smooth signal is accepted estimatedSignal <- test$smoothSignal } plot(y) lines(signal) lines(estimatedSignal, col = "red") # fixed bandwidth test <- BstrapTest(y = y, bandwidth = 0.1) # user specified kernel kernel <- function(x) 1 - abs(x) # triangular kernel test <- BstrapTest(y = y, kernel = kernel)n <- 100 signal <- sin(2 * pi * 1:n / n) signal[51:100] <- signal[51:100] + 5 y <- rnorm(n) + signal # default bandwidth and kernel test <- BstrapTest(y = y) if (test$outcome) { # null hypothesis of a smooth signal is rejected estimatedSignal <- test$piecewiseSignal } else { # null hypothesis of a smooth signal is accepted estimatedSignal <- test$smoothSignal } plot(y) lines(signal) lines(estimatedSignal, col = "red") # fixed bandwidth test <- BstrapTest(y = y, bandwidth = 0.1) # user specified kernel kernel <- function(x) 1 - abs(x) # triangular kernel test <- BstrapTest(y = y, kernel = kernel)
Estimates a single change-point in an otherwise smooth function. The change-point location is estimated as the maximum of the differences of left and right sided running means. The estimate left and right of the change-point are obtained by kernel smoothers. Windows of the running mean and kernel bandwidth are chosen by crossvalidation. More details can be found in the vignette.
estimateSingleCp(y, bandwidth, nbandwidth = 30L, kernel = c("epanechnikov", "gaussian", "rectangular", "triangular", "biweight", "silverman"))estimateSingleCp(y, bandwidth, nbandwidth = 30L, kernel = c("epanechnikov", "gaussian", "rectangular", "triangular", "biweight", "silverman"))
y |
a numeric vector containing the data points |
bandwidth |
the bandwidth, i.e. a numeric with values between |
nbandwidth |
a single integer giving the number of bandwidths (see above) if |
kernel |
the kernel function, i.e. either a string or a function that takes a single numeric vector and returns the values of the kernel at those locations |
a list with the following components:
- est: the estimated function with a single change-point
- cp: the estimated change-point location
- size: the estimated jump size
- bandwidth: the selected bandwidth
n <- 100 signal <- sin(2 * pi * 1:n / n) signal[51:100] <- signal[51:100] + 5 y <- rnorm(n) + signal # default bandwidth and kernel est <- estimateSingleCp(y = y) plot(y) lines(signal) lines(est$est, col = "red") # fixed bandwidth est <- estimateSingleCp(y = y, bandwidth = 0.1) # user specified kernel kernel <- function(x) 1 - abs(x) # triangular kernel est <- estimateSingleCp(y = y, kernel = kernel)n <- 100 signal <- sin(2 * pi * 1:n / n) signal[51:100] <- signal[51:100] + 5 y <- rnorm(n) + signal # default bandwidth and kernel est <- estimateSingleCp(y = y) plot(y) lines(signal) lines(est$est, col = "red") # fixed bandwidth est <- estimateSingleCp(y = y, bandwidth = 0.1) # user specified kernel kernel <- function(x) 1 - abs(x) # triangular kernel est <- estimateSingleCp(y = y, kernel = kernel)