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Detect changepoints in functional data

Source: R/FKWC.R
FKWC.Rd

More specifically, fkwc() uses the functional Kruskal-Wallis tests for covariance changepoint algorithm (FKWC) to detect changes in the covariance operator.

Usage

fkwc(data, depth = c("RPD", "FM", "LTR", "FMd", "RPDd"), k = 0.25)

Arguments

data

Functional data in matrix or data.frame form, where each row is an observation/function and the columns are the grid.

depth

Depth function of choice.

k

Penalty constant passed to pruned exact linear time algorithm.

Value

A list consisting of:

  • $changepoints : Indices of the changepoints detected; will return integer(0) if no changepoints are detected.

  • $method : A string "FKWC"

Note

The options for the depth argument are as follows:

  • RPD: Random projection depth, which generally performs best

  • FM: Frainman-Muniz depth

  • LTR: \(L^2\)-root depth, most suitable for detecting changes in the norm

  • FMd: Frainman-Muniz depth of the data and its first order derivative

  • RPDd: Random projection depth of the data and its first order derivative

    The depth arguments that incorporate the first order derivative (which is approximated using fda.usc::fdata.deriv) result in a more robust detection of changes in the covariance structure (Ramsay and Chenouri, 2025).

The penalty is of the form $$3.74 + k\sqrt{n}$$ where \(n\) is the number of observations. In the case that there is potentially correlated observations, the parameter could be set to \(k=1\). More information could be found in the reference.

References

Killick, R., P. Fearnhead, and I. A. Eckley. “Optimal Detection of Changepoints With a Linear Computational Cost.” Journal of the American Statistical Association 107, no. 500 (2012): 1590–98. https://doi.org/10.1080/01621459.2012.737745.

Ramsay, K., & Chenouri, S. (2025). Robust changepoint detection in the variability of multivariate functional data. Journal of Nonparametric Statistics. https://doi.org/10.1080/10485252.2025.2503891

Examples


set.seed(2)
# Generating 80 observations, with a changepoint (in our case a change in
# kernel) at observation 40
n  <- 80
k0 <- 40
T  <- 30
t  <- seq(0, 1, length.out = T)


# Both kernels K1 and K2 are Gaussian (or squared exponential) kernels but
# with different lengthscale values, and thus we hope to detect it.
K_se <- function(s, t, ell) exp(- ( (s - t)^2 ) / (2 * ell^2))
K1   <- outer(t, t, function(a,b) K_se(a,b, ell = 0.20))
K2   <- outer(t, t, function(a,b) K_se(a,b, ell = 0.07))

L1 <- chol(K1 + 1e-8 * diag(T))
L2 <- chol(K2 + 1e-8 * diag(T))

Z1 <- matrix(rnorm(k0 * T),      k0,      T)
Z2 <- matrix(rnorm((n-k0) * T),  n - k0,  T)

# We finally have an 80 x 30 matrix where the rows are the observations and
# the columns are the grid points.
X  <- rbind(Z1 %*% t(L1), Z2 %*% t(L2))

fkwc(X)
#> $changepoints
#> [1] 19 40
#> 
#> $method
#> [1] "FKWC"
#>