FTSgof (version 1.0.0)

fSACF_test: Test based on fSACF

Description

This function performs a hypothesis test using a test statistic computed from functional spherical autocorrelation coefficients of a FTS.

Usage

fSACF_test(
  f_data,
  H = 10,
  alpha = 0.05,
  pplot = FALSE,
  suppress_raw_output = FALSE,
  suppress_print_output = FALSE
)

Value

If suppress_raw_output = FALSE, a list containing the test statistic, the \((1-\alpha)\) quantile of the limiting distribution, and the p-value computed from the specified hypothesis test. Also prints output containing a short description of the test, the p-value, and additional information about the test if suppress_print_output = FALSE.

Arguments

f_data

A \(J \times N\) matrix of functional time series data, where \(J\) represents the number of discrete points in a grid and \(N\) represents the sample size.

H

A positive integer specifying the maximum lag for which test statistic is computed.

alpha

A numeric value between 0 and 1 indicating the significance level for the test.

pplot

A Boolean value. If TRUE, the function will produce a plot of p-values of the test as a function of maximum lag \(H\), ranging from \(H=1\) to \(H=20\), which may increase the computation time.

suppress_raw_output

A Boolean value. If TRUE, the function will not return the list containing the p-value, quantile, and statistic.

suppress_print_output

A Boolean value. If TRUE, the function will not print any output to the console.

Details

The test statistic is the sum of the squared \(L^2\)-norm of the sample spherical autocorrelation coefficients: $$ S_{N,H} = N \sum_{h=1}^H \|\tilde{\rho}_{h}\|^2, $$ where \(\tilde\rho_h=\frac{1}{N}\sum_{i=1}^{N-h} \langle \frac{X_i - \tilde{\mu}}{\|X_i - \tilde{\mu}\|}, \frac{X_{i+h} - \tilde{\mu}}{\|X_{i+h} - \tilde{\mu}\|} \rangle\), and \(\tilde{\mu}\) is the estimated spatial median of the series. This test assesses the cumulative significance of lagged spherical autocorrelation coefficients, up to a user-selected maximum lag \(H\). A higher value of \(S_{N,H}\) suggests a potential departure of the observed series from white noise process. The approximated null distribution of this statistic is developed under the strong white noise assumptions.

References

[1] Yeh CK, Rice G, Dubin JA (2023). “Functional spherical autocorrelation: A robust estimate of the autocorrelation of a functional time series.” Electronic Journal of Statistics, 17, 650–687.

Examples

# \donttest{
data(Spanish_elec)
fACF_test(Spanish_elec, H = 20)
# }