nsp_poly_selfnorm {nsp}

Self-normalised Narrowest Significance Pursuit algorithm for piecewise-polynomial signals

Description

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.

Usage

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
)

Arguments

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 NULL if the threshold is to be determined from the data.

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

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 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_poly, nsp_poly, nsp_poly_ar, nsp_tvreg, nsp_selfnorm

Examples

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)