| Title: | Inference for Multiple Change-Points in Linear Models |
|---|---|
| Description: | Implementation of Narrowest Significance Pursuit, a general and flexible methodology for automatically detecting localised regions in data sequences which each must contain a change-point (understood as an abrupt change in the parameters of an underlying linear model), at a prescribed global significance level. Narrowest Significance Pursuit works with a wide range of distributional assumptions on the errors, and yields exact desired finite-sample coverage probabilities, regardless of the form or number of the covariates. For details, see P. Fryzlewicz (2021) <https://stats.lse.ac.uk/fryzlewicz/nsp/nsp.pdf>. |
| Authors: | Piotr Fryzlewicz [aut, cre]
|
| Maintainer: | Piotr Fryzlewicz <[email protected]> |
| License: | GPL (>= 3) |
| Version: | 1.0.0 |
| Built: | 2024-11-05 06:16:11 UTC |
| Source: | CRAN |
This function simulates the multiscale sup-norm adjusted for the form of the covariates, as described in Section 5.3 of the paper. This is done for i.i.d. N(0,1) innovations.
cov_dep_multi_norm(x, N = 1000)cov_dep_multi_norm(x, N = 1000)
x |
The design matrix with the regressors (covariates) as columns. |
N |
Desired number of simulated values of the norm. |
The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear models", preprint.
Sample of size N containing the simulated norms.
Piotr Fryzlewicz, [email protected]
cov_dep_multi_norm_poly, sim_max_holder
set.seed(1) g <- c(rep(0, 100), rep(2, 100)) x.g <- g + stats::rnorm(200) mscale.norm.200 <- cov_dep_multi_norm(matrix(1, 200, 1), 100) nsp_poly(x.g, 100, thresh.val = stats::quantile(mscale.norm.200, .95))set.seed(1) g <- c(rep(0, 100), rep(2, 100)) x.g <- g + stats::rnorm(200) mscale.norm.200 <- cov_dep_multi_norm(matrix(1, 200, 1), 100) nsp_poly(x.g, 100, thresh.val = stats::quantile(mscale.norm.200, .95))
This function simulates the multiscale sup-norm adjusted for the form of the covariates, as described in Section 5.3
of the paper, for piecewise-polynomial models of degree deg. This is done for i.i.d. N(0,1) innovations.
cov_dep_multi_norm_poly(n, deg, N = 10000)cov_dep_multi_norm_poly(n, deg, N = 10000)
n |
The data length (for which the multiscale norm is to be simulated) |
deg |
The degree of the polynomial model (0 for the piecewise-constant model; 1 for piecewise-linearity, etc.). |
N |
Desired number of simulated values of the norm. |
The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear models", preprint.
Sample of size N containing the simulated norms.
Piotr Fryzlewicz, [email protected]
cov_dep_multi_norm, sim_max_holder
set.seed(1) g <- c(rep(0, 100), rep(2, 100)) x.g <- g + stats::rnorm(200) mscale.norm.200 <- cov_dep_multi_norm_poly(200, 0, 100) nsp_poly(x.g, 100, thresh.val = stats::quantile(mscale.norm.200, .95))set.seed(1) g <- c(rep(0, 100), rep(2, 100)) x.g <- g + stats::rnorm(200) mscale.norm.200 <- cov_dep_multi_norm_poly(200, 0, 100) nsp_poly(x.g, 100, thresh.val = stats::quantile(mscale.norm.200, .95))
This function produces a change-point prominence plot based on the NSP object provided. The heights of the bars are arranged in non-decreasing order and correspond directly to the lengths of the NSP intervals of significance. Each bar is labelled as s-e where s (e) is the start (end) of the corresponding NSP interval of significance, respectively. The change-points corresponding to the narrower intervals can be seen as more prominent.
cpt_importance(nsp.obj)cpt_importance(nsp.obj)
nsp.obj |
Object returned by one of the |
The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear models", preprint.
The function does not return a value.
Piotr Fryzlewicz, [email protected]
draw_rects, draw_rects_advanced
set.seed(1) f <- c(rep(0, 100), 1:100, rep(101, 100)) x.f <- f + 15 * stats::rnorm(300) x.f.n <- nsp_poly(x.f, 100, "sim", deg=1) cpt_importance(x.f.n)set.seed(1) f <- c(rep(0, 100), 1:100, rep(101, 100)) x.f <- f + 15 * stats::rnorm(300) x.f.n <- nsp_poly(x.f, 100, "sim", deg=1) cpt_importance(x.f.n)
This function draws intervals of significance returned by one of the nsp* functions on the current plot. It shows them as shaded
rectangular areas (hence the name of the function).
draw_rects(nsp.obj, yrange, density = 10, col = "red", x.axis.start = 1)draw_rects(nsp.obj, yrange, density = 10, col = "red", x.axis.start = 1)
nsp.obj |
Object returned by one of the |
yrange |
Vector of length two specifying the (lower, upper) vertical limit of the rectangles. |
density |
Density of the shading. |
col |
Colour of the shading. |
x.axis.start |
Time index the x axis starts from. The NSP intervals of significance get shifted by |
The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear models", preprint.
The function does not return a value.
Piotr Fryzlewicz, [email protected]
set.seed(1) h <- c(rep(0, 150), 1:150) x.h <- h + stats::rnorm(300) * 50 x.h.n <- nsp_poly(x.h, 1000, "sim", deg=1) draw_rects(x.h.n, c(-100, 100))set.seed(1) h <- c(rep(0, 150), 1:150) x.h <- h + stats::rnorm(300) * 50 x.h.n <- nsp_poly(x.h, 1000, "sim", deg=1) draw_rects(x.h.n, c(-100, 100))
This function plots the intervals of significance returned by one of the nsp* functions, at appropriate places along the graph of data.
It shows them as shaded rectangular areas (hence the name of the function) "attached" to the graph of the data. Note: the data sequence y
needs to have been plotted beforehand.
draw_rects_advanced( y, nsp.obj, half.height = NULL, show.middles = TRUE, col.middles = "blue", lwd = 3, density = 10, col.rects = "red", x.axis.start = 1 )draw_rects_advanced( y, nsp.obj, half.height = NULL, show.middles = TRUE, col.middles = "blue", lwd = 3, density = 10, col.rects = "red", x.axis.start = 1 )
y |
The data. |
nsp.obj |
Object returned by one of the |
half.height |
Half-height of each rectangle; if |
show.middles |
Whether to display lines corresponding to the midpoints of the rectanlges (rough change-point location estimates). |
col.middles |
Colour of the midpoint lines. |
lwd |
Line width for the midpoint lines. |
density |
Density of the shading. |
col.rects |
Colour of the shading. |
x.axis.start |
Time index the x axis starts from. The NSP intervals of significance get shifted by |
The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear models", preprint.
The function does not return a value.
Piotr Fryzlewicz, [email protected]
set.seed(1) f <- c(rep(0, 100), 1:100, rep(101, 100)) x.f <- f + 15 * stats::rnorm(300) x.f.n <- nsp_poly(x.f, 100, "sim", deg=1) stats::ts.plot(x.f) draw_rects_advanced(x.f, x.f.n, density = 3)set.seed(1) f <- c(rep(0, 100), 1:100, rep(101, 100)) x.f <- f + 15 * stats::rnorm(300) x.f.n <- nsp_poly(x.f, 100, "sim", deg=1) stats::ts.plot(x.f) draw_rects_advanced(x.f, x.f.n, density = 3)
This function runs the bare-bones Narrowest Significance Pursuit (NSP) algorithm on data sequence y and design matrix x
to obtain localised regions (intervals) of the domain in which the parameters of the linear regression model y_t = beta(t) x_t + z_t significantly
depart from constancy (e.g. by containing change-points). For any interval considered by the algorithm,
significance is achieved if the multiscale supremum-type
deviation measure (see Details for the literature reference) exceeds lambda. This function is
mainly to be used by the higher-level functions nsp_poly, nsp_poly_ar and nsp_tvreg
(which estimate a suitable lambda so that a given global significance level is guaranteed), and human users may prefer to use those functions
instead; however, nsp can also be run directly, if desired.
The function works best when the errors z_t in the linear regression formulation y_t = beta(t) x_t + z_t are independent and
identically distributed Gaussians.
nsp(y, x, M, lambda, overlap = FALSE, buffer = 0)nsp(y, x, M, lambda, overlap = FALSE, buffer = 0)
y |
A vector containing the data sequence being the response in the linear model y_t = beta(t) x_t + z_t. |
x |
The design matrix in the regression model above, with the regressors as columns. |
M |
The minimum number of intervals considered at each recursive stage, unless the number of all intervals is smaller, in which case all intervals are used. |
lambda |
The threshold parameter for measuring the significance of non-constancy (of the linear regression parameters), for use with the multiscale supremum-type deviation measure described in the paper. |
overlap |
If |
buffer |
A non-negative integer specifying how many observations to leave out immediately to the left and to the right of a detected interval of significance before recursively continuing the search for the next interval. |
The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear models", preprint.
A list with the following components:
intervals |
A data frame containing the estimated intervals of significance: |
threshold.used |
The threshold |
Piotr Fryzlewicz, [email protected]
nsp_poly, nsp_poly_ar, nsp_tvreg, nsp_selfnorm, nsp_poly_selfnorm
set.seed(1) f <- c(1:100, 100:1, 1:100) y <- f + stats::rnorm(300) * 15 x <- matrix(0, 300, 2) x[,1] <- 1 x[,2] <- seq(from = 0, to = 1, length = 300) nsp(y, x, 100, 15 * thresh_kab(300, .1))set.seed(1) f <- c(1:100, 100:1, 1:100) y <- f + stats::rnorm(300) * 15 x <- matrix(0, 300, 2) x[,1] <- 1 x[,2] <- seq(from = 0, to = 1, length = 300) nsp(y, x, 100, 15 * thresh_kab(300, .1))
This function runs the Narrowest Significance Pursuit (NSP) algorithm on a data sequence y believed to follow the model
y_t = f_t + z_t, where f_t is a piecewise polynomial of degree deg, and z_t is noise. It returns localised regions (intervals) of the
domain, such that each interval must contain a change-point in the parameters of the polynomial f_t
at the global significance level alpha.
For any interval considered by the algorithm,
significant departure from parameter constancy is achieved if the multiscale supremum-type
deviation measure (see Details for the literature reference) exceeds a threshold, which is either provided as input
or determined from the data (as a function of alpha). The function works best when the errors z_t are independent and
identically distributed Gaussians.
nsp_poly( y, M = 1000, thresh.type = "univ", thresh.val = NULL, sigma = NULL, alpha = 0.1, deg = 0, overlap = FALSE )nsp_poly( y, M = 1000, thresh.type = "univ", thresh.val = NULL, sigma = NULL, alpha = 0.1, deg = 0, overlap = FALSE )
y |
A vector containing the data sequence. |
M |
The minimum number of intervals considered at each recursive stage, unless the number of all intervals is smaller, in which case all intervals are used. |
thresh.type |
|
thresh.val |
Numerical value of the significance threshold (lambda in the paper); or |
sigma |
The standard deviation of the errors z_t; if |
alpha |
Desired maximum probability of obtaining an interval that does not contain a change-point (the significance threshold will be determined as a function of this parameter). |
deg |
The degree of the polynomial pieces in f_t (0 for the piecewise-constant model; 1 for piecewise-linearity, etc.). |
overlap |
If |
The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear
models", preprint. For how to determine the "univ" threshold, see Kabluchko, Z. (2007) "Extreme-value analysis of standardized Gaussian increments".
Unpublished.
A list with the following components:
intervals |
A data frame containing the estimated intervals of significance: |
threshold.used |
The threshold value. |
Piotr Fryzlewicz, [email protected]
nsp, nsp_poly_ar, nsp_tvreg, nsp_selfnorm, nsp_poly_selfnorm
set.seed(1) f <- c(1:100, 100:1, 1:100) y <- f + stats::rnorm(300) * 15 nsp_poly(y, 100, deg = 1)set.seed(1) f <- c(1:100, 100:1, 1:100) y <- f + stats::rnorm(300) * 15 nsp_poly(y, 100, deg = 1)
This function runs the Narrowest Significance Pursuit (NSP) algorithm on a data sequence y believed to follow the model
Phi(B)y_t = f_t + z_t, where f_t is a piecewise polynomial of degree deg, Phi(B) is a characteristic polynomial of autoregression of order
ord with unknown coefficients, and z_t is noise. The function returns localised regions (intervals) of the domain, such that each interval
must contain a change-point in the parameters of the polynomial f_t, or in the autoregressive parameters,
at the global significance level alpha.
For any interval considered by the algorithm,
significant departure from parameter constancy is achieved if the multiscale
deviation measure (see Details for the literature reference) exceeds a threshold, which is either provided as input
or determined from the data (as a function of alpha). The function works best when the errors z_t are independent and
identically distributed Gaussians.
nsp_poly_ar( y, ord = 1, M = 1000, thresh.type = "univ", thresh.val = NULL, sigma = NULL, alpha = 0.1, deg = 0, power = 1/2, min.size = 20, overlap = FALSE, buffer = ord )nsp_poly_ar( y, ord = 1, M = 1000, thresh.type = "univ", thresh.val = NULL, sigma = NULL, alpha = 0.1, deg = 0, power = 1/2, min.size = 20, overlap = FALSE, buffer = ord )
y |
A vector containing the data sequence. |
ord |
The assumed order of the autoregression. |
M |
The minimum number of intervals considered at each recursive stage, unless the number of all intervals is smaller, in which case all intervals are used. |
thresh.type |
|
thresh.val |
Numerical value of the significance threshold (lambda in the paper); or |
sigma |
The standard deviation of the errors z_t; if |
alpha |
Desired maximum probability of obtaining an interval that does not contain a change-point (the significance threshold will be determined as a function of this parameter). |
deg |
The degree of the polynomial pieces in f_t (0 for the piecewise-constant model; 1 for piecewise-linearity, etc.). |
power |
A parameter for the MOLS estimator of sigma; the span of the moving window in the MOLS estimator is |
min.size |
(See immediately above.) |
overlap |
If |
buffer |
A non-negative integer specifying how many observations to leave out immediately to the left and to the right of a detected interval of significance before recursively continuing the search for the next interval. |
The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear
models", preprint. For how to determine the "univ" threshold, see Kabluchko, Z. (2007) "Extreme-value analysis of standardized Gaussian increments".
Unpublished.
A list with the following components:
intervals |
A data frame containing the estimated intervals of significance: |
threshold.used |
The threshold value. |
Piotr Fryzlewicz, [email protected]
nsp, nsp_poly, nsp_tvreg, nsp_selfnorm, nsp_poly_selfnorm
set.seed(1) g <- c(rep(0, 100), rep(10, 100), rep(0, 100)) nsp_poly_ar(stats::filter(g + 2 * stats::rnorm(300), .5, "recursive"), thresh.type="sim")set.seed(1) g <- c(rep(0, 100), rep(10, 100), rep(0, 100)) nsp_poly_ar(stats::filter(g + 2 * stats::rnorm(300), .5, "recursive"), thresh.type="sim")
This function runs the Narrowest Significance Pursuit (NSP) algorithm on a data sequence y believed to follow the model
y_t = f_t + z_t, where f_t is a piecewise polynomial of degree deg, and z_t is noise. It returns localised regions (intervals) of the
domain, such that each interval must contain a change-point in the parameters of the polynomial f_t
at the global significance level alpha.
For any interval considered by the algorithm,
significant departure from parameter constancy is achieved if the multiscale
deviation measure (see Details for the literature reference) exceeds a threshold, which is either provided as input
or determined from the data (as a function of alpha). The function assumes independence, symmetry and finite variance of the
errors z_t, but little else; in particular they do not need to have a constant variance across t.
nsp_poly_selfnorm( y, M = 1000, thresh.val = NULL, power = 1/2, min.size = 20, alpha = 0.1, deg = 0, eps = 0.03, c = exp(1 + 2 * eps), overlap = FALSE )nsp_poly_selfnorm( y, M = 1000, thresh.val = NULL, power = 1/2, min.size = 20, alpha = 0.1, deg = 0, eps = 0.03, c = exp(1 + 2 * eps), overlap = FALSE )
y |
A vector containing the data sequence. |
M |
The minimum number of intervals considered at each recursive stage, unless the number of all intervals is smaller, in which case all intervals are used. |
thresh.val |
Numerical value of the significance threshold (lambda in the paper); or |
power |
A parameter for the (rough) estimator of the global sum of squares of z_t; the span of the moving window in that estimator is
|
min.size |
(See immediately above.) |
alpha |
Desired maximum probability of obtaining an interval that does not contain a change-point (the significance threshold will be determined as a function of this parameter). |
deg |
The degree of the polynomial pieces in f_t (0 for the piecewise-constant model; 1 for piecewise-linearity, etc.). |
eps |
Parameter of the self-normalisation statistic as described in the paper; use default if unsure how to set. |
c |
Parameter of the self-normalisation statistic as described in the paper; use default if unsure how to set. |
overlap |
If |
The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear models", preprint.
A list with the following components:
intervals |
A data frame containing the estimated intervals of significance: |
threshold.used |
The threshold value. |
Piotr Fryzlewicz, [email protected]
nsp_poly, nsp_poly, nsp_poly_ar, nsp_tvreg, nsp_selfnorm
set.seed(1) g <- c(rep(0, 100), rep(10, 100), rep(0, 100)) x.g <- g + stats::rnorm(300) * seq(from = 1, to = 4, length = 300) nsp_poly_selfnorm(x.g, 100)set.seed(1) g <- c(rep(0, 100), rep(10, 100), rep(0, 100)) x.g <- g + stats::rnorm(300) * seq(from = 1, to = 4, length = 300) nsp_poly_selfnorm(x.g, 100)
This function runs the self-normalised Narrowest Significance Pursuit (NSP) algorithm on data sequence y and design matrix x
to obtain localised regions (intervals) of the domain in which the parameters of the linear regression model y_t = beta(t) x_t + z_t significantly
depart from constancy (e.g. by containing change-points). For any interval considered by the algorithm,
significant departure from parameter constancy is achieved if the self-normalised multiscale
deviation measure (see Details for the literature reference) exceeds lambda. This function is
used by the higher-level function nsp_poly_selfnorm
(which estimates a suitable lambda so that a given global significance level is guaranteed), and human users may prefer to use that function
if x describe polynomial covariates; however, nsp_selfnorm can also be run directly, if desired.
The function assumes independence, symmetry and finite variance of the errors z_t, but little else; in particular they do not need to have a constant
variance across t.
nsp_selfnorm( y, x, M, lambda, power = 1/2, min.size = 20, eps = 0.03, c = exp(1 + 2 * eps), overlap = FALSE )nsp_selfnorm( y, x, M, lambda, power = 1/2, min.size = 20, eps = 0.03, c = exp(1 + 2 * eps), overlap = FALSE )
y |
A vector containing the data sequence being the response in the linear model y_t = beta(t) x_t + z_t. |
x |
The design matrix in the regression model above, with the regressors as columns. |
M |
The minimum number of intervals considered at each recursive stage, unless the number of all intervals is smaller, in which case all intervals are used. |
lambda |
The threshold parameter for measuring the significance of non-constancy (of the linear regression parameters), for use with the self-normalised multiscale supremum-type deviation measure described in the paper. |
power |
A parameter for the (rough) estimator of the global sum of squares of z_t; the span of the moving window in that estimator is
|
min.size |
(See immediately above.) |
eps |
Parameter of the self-normalisation statistic as described in the paper; use default if unsure how to set. |
c |
Parameter of the self-normalisation statistic as described in the paper; use default if unsure how to set. |
overlap |
If |
The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear models", preprint.
A list with the following components:
intervals |
A data frame containing the estimated intervals of significance: |
threshold.used |
The threshold |
Piotr Fryzlewicz, [email protected]
nsp_poly, nsp_poly, nsp_poly_ar, nsp_tvreg, nsp_poly_selfnorm
set.seed(1) g <- c(rep(0, 100), rep(10, 100), rep(0, 100)) x.g <- g + stats::rnorm(300) * seq(from = 1, to = 4, length = 300) wn003 <- sim_max_holder(100, 500, .03) lambda <- as.numeric(stats::quantile(wn003, .9)) nsp_selfnorm(x.g, matrix(1, 300, 1), 100, lambda)set.seed(1) g <- c(rep(0, 100), rep(10, 100), rep(0, 100)) x.g <- g + stats::rnorm(300) * seq(from = 1, to = 4, length = 300) wn003 <- sim_max_holder(100, 500, .03) lambda <- as.numeric(stats::quantile(wn003, .9)) nsp_selfnorm(x.g, matrix(1, 300, 1), 100, lambda)
This function runs the Narrowest Significance Pursuit (NSP) algorithm on data sequence y and design matrix x
to return localised regions (intervals) of the domain in which the parameters of the linear regression model y_t = beta(t) x_t + z_t significantly
depart from constancy (e.g. by containing change-points), at the global significance level alpha. For any interval considered by the algorithm,
significant departure from parameter constancy is achieved if the multiscale
deviation measure (see Details for the literature reference) exceeds a threshold, which is either provided as input
or determined from the data (as a function of alpha).
The function works best when the errors z_t in the linear regression formulation y_t = beta(t) x_t + z_t are independent and
identically distributed Gaussians.
nsp_tvreg( y, x, M = 1000, thresh.val = NULL, sigma = NULL, alpha = 0.1, power = 1/2, min.size = 20, overlap = FALSE )nsp_tvreg( y, x, M = 1000, thresh.val = NULL, sigma = NULL, alpha = 0.1, power = 1/2, min.size = 20, overlap = FALSE )
y |
A vector containing the data sequence being the response in the linear model y_t = beta(t) x_t + z_t. |
x |
The design matrix in the regression model above, with the regressors as columns. |
M |
The minimum number of intervals considered at each recursive stage, unless the number of all intervals is smaller, in which case all intervals are used. |
thresh.val |
Numerical value of the significance threshold (lambda in the paper); or |
sigma |
The standard deviation of the errors z_t; if |
alpha |
Desired maximum probability of obtaining an interval that does not contain a change-point (the significance threshold will be determined as a function of this parameter). |
power |
A parameter for the MOLS estimator of sigma; the span of the moving window in the MOLS estimator is |
min.size |
(See immediately above.) |
overlap |
If |
The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear models", preprint.
A list with the following components:
intervals |
A data frame containing the estimated intervals of significance: |
threshold.used |
The threshold value. |
Piotr Fryzlewicz, [email protected]
nsp, nsp_poly, nsp_poly_ar, nsp_selfnorm, nsp_poly_selfnorm
set.seed(1) f <- c(1:100, 100:1, 1:100) y <- f + stats::rnorm(300) * 15 x <- matrix(0, 300, 2) x[,1] <- 1 x[,2] <- seq(from = 0, to = 1, length = 300) nsp_tvreg(y, x, 100)set.seed(1) f <- c(1:100, 100:1, 1:100) y <- f + stats::rnorm(300) * 15 x <- matrix(0, 300, 2) x[,1] <- 1 x[,2] <- seq(from = 0, to = 1, length = 300) nsp_tvreg(y, x, 100)
This function simulates a sample of size N of values of the Holder-like norm of the Wiener process discretised with step 1/n.
The sample can then be used to find a suitable threshold for use with the self-normalised NSP.
sim_max_holder(n, N, eps, c = exp(1 + 2 * eps))sim_max_holder(n, N, eps, c = exp(1 + 2 * eps))
n |
Number of equispaced sampling points for the Wiener process on |
N |
Desired number of simulated values of the norm. |
eps |
Parameter of the self-normalisation statistic as described in the paper. |
c |
Parameter of the self-normalisation statistic as described in the paper; use default if unsure how to set. |
The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear models", preprint.
Sample of size N containing the simulated norms.
Piotr Fryzlewicz, [email protected]
nsp_selfnorm, nsp_poly_selfnorm, cov_dep_multi_norm, cov_dep_multi_norm_poly
set.seed(1) g <- c(rep(0, 100), rep(10, 100), rep(0, 100)) x.g <- g + stats::rnorm(300) * seq(from = 1, to = 4, length = 300) wn003 <- sim_max_holder(100, 500, .03) lambda <- as.numeric(stats::quantile(wn003, .9)) nsp_poly_selfnorm(x.g, M = 100, thresh.val = lambda)set.seed(1) g <- c(rep(0, 100), rep(10, 100), rep(0, 100)) x.g <- g + stats::rnorm(300) * seq(from = 1, to = 4, length = 300) wn003 <- sim_max_holder(100, 500, .03) lambda <- as.numeric(stats::quantile(wn003, .9)) nsp_poly_selfnorm(x.g, M = 100, thresh.val = lambda)
This function computes the theoretical threshold, corresponding to the given significance level alpha, for the multiscale sup-norm
if the underlying distribution is standard normal.
thresh_kab(n, alpha = 0.1, method = "asymp")thresh_kab(n, alpha = 0.1, method = "asymp")
n |
The sample size. |
alpha |
The significance level. |
method |
"asymp" for the asymptotic method; "bound" for the Bonferroni method. |
For the underlying theory, see Z. Kabluchko (2007) Extreme-value analysis of standardized Gaussian increments. Unpublished.
The desired threshold.
Piotr Fryzlewicz, [email protected]
cov_dep_multi_norm, cov_dep_multi_norm_poly, sim_max_holder
set.seed(1) f <- c(1:100, 100:1, 1:100) y <- f + stats::rnorm(300) * 15 x <- matrix(0, 300, 2) x[,1] <- 1 x[,2] <- seq(from = 0, to = 1, length = 300) nsp(y, x, 100, 15 * thresh_kab(300, .1))set.seed(1) f <- c(1:100, 100:1, 1:100) y <- f + stats::rnorm(300) * 15 x <- matrix(0, 300, 2) x[,1] <- 1 x[,2] <- seq(from = 0, to = 1, length = 300) nsp(y, x, 100, 15 * thresh_kab(300, .1))