nsp_tvreg {nsp}

Narrowest Significance Pursuit algorithm with general covariates

Description

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.

Usage

nsp_tvreg(
  y,
  x,
  M = 1000,
  thresh.val = NULL,
  sigma = NULL,
  alpha = 0.1,
  power = 1/2,
  min.size = 20,
  overlap = FALSE
)

Arguments

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 NULL if the threshold is to be determined from the data (see thresh.type).

sigma

The standard deviation of the errors z_t; if NULL then will be estimated from the data via the MOLS estimator described in the paper.

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(n, max(round(n^power), min.size)), where n is the length of y.

min.size

(See immediately above.)

overlap

If FALSE, then on discovering a significant interval, the search continues recursively to the left and to the right of that interval. If TRUE, then the search continues to the left and to the right of the midpoint of that interval.

Details

The NSP algorithm is described in P. Fryzlewicz (2021) "Narrowest Significance Pursuit: inference for multiple change-points in linear models", preprint.

Value

A list with the following components:

intervals

A data frame containing the estimated intervals of significance: starts and ends is where the intervals start and end, respectively; values are the values of the deviation measure on each given interval; midpoints are their midpoints.

threshold.used

The threshold value.

Author(s)

Piotr Fryzlewicz, [e-mail]

See Also

nsp, nsp_poly, nsp_poly_ar, nsp_selfnorm, nsp_poly_selfnorm

Examples

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)