DeCAFS {DeCAFS}

Main DeCAFS algorithm for detecting abrupt changes

Description

This function implements the DeCAFS algorithm to detect abrupt changes in mean of a univariate data stream in the presence of local fluctuations and auto-correlated noise. It detects the changes under a penalised likelihood model where the data, y_1, ..., y_n, is

y_t = \mu_t + \epsilon_t

with \epsilon_t an AR(1) process, and for t = 2, ..., N

\mu_t = \mu_{t-1} + \eta_t + \delta_t

where at time t if we do not have a change then \delta_t = 0 and \eta_t \sim N(0, \sigma_\eta^2); whereas if we have a change then \delta_t \neq 0 and \eta_t=0. DeCAFS estimates the change by minimising a cost equal to twice the negative log-likelihood of this model, with a penalty \beta for adding a change. Note that the default DeCAFS behavior will assume the RWAR model, but fit on edge cases is still possible. For instance, should the user wish for DeCAFS to fit an AR model only with a piecewise constant signal, or similarly a model that just assumes random fluctuations in the signal, this can be specified within the initial parameter estimation, by setting the argument: modelParam = estimateParameters(y, model = "AR"). Similarly, to allow for negative autocorrelation estimation, set modelParam = estimateParameters(Y$y, phiLower = -1).

Usage

DeCAFS(
  data,
  beta = 2 * log(length(data)),
  modelParam = estimateParameters(data, warningMessage = warningMessage),
  penalties = NULL,
  warningMessage = TRUE
)

Arguments

data

A vector of observations y

beta

The l0 penalty. The default one is 2 * log(N) where N is the length of the data.

modelParam

A list of 3 initial model parameters: sdEta, the SD of the drift (random fluctuations) in the signal, sdNu, the SD of the AR(1) noise process, and phi, the autocorrelation parameter of the noise process (so the stationary variance of the AR(1) noise process is sdnu^2 / (1 - phi^2). Defaulted to estimateParameters(data, K = 15), to perform automatically estimation of the three. See estimateParameters() for more details.

penalties

Can be used as an alternative to the model parameters, a list of 3 initial penalties: lambda, the l2-penalty penalising over the lag-1 of the signal, gamma, penalising over the lag-1 of the AR(1) noise process, phi, the autocorrelation parameter. These are related to the modelParam list by list(lambda = 1 / sdEta ^ 2, gamma = 1 / sdNu ^ 2, phi = phi). Only one argument between penalties and modelParam should be specified. Defaulted to NULL.

warningMessage

When TRUE prints a message to warn the user that the automatic parameter estimation is employed. Defaults to TRUE.

Value

Returns an s3 object of class DeCAFSout where:

$changepoints

is the vector of change-point locations,

$signal

is the estimated signal without the auto-correlated noise,

$costFunction

is the optimal cost in form of piecewise quadratics at the end of the sequence,

$estimatedParameters

is a list of parameters estimates (if estimated, otherwise simply the initial modelParam input),

$data

is the sequence of observations.

References

Romano, G., Rigaill, G., Runge, V., Fearnhead, P. (2021). Detecting Abrupt Changes in the Presence of Local Fluctuations and Autocorrelated Noise. Journal of the American Statistical Association. doi:10.1080/01621459.2021.1909598.

Examples

library(ggplot2)
set.seed(42)
Y <- dataRWAR(n = 1e3, phi = .5, sdEta = 1, sdNu = 3, jumpSize = 15, type = "updown", nbSeg = 5)
y <- Y$y
res = DeCAFS(y)
ggplot(data.frame(t = 1:length(y), y), aes(x = t, y = y)) +
  geom_point() +
  geom_vline(xintercept = res$changepoints, color = "red") +
  geom_vline(xintercept = Y$changepoints, col = "blue",  lty = 3)