fACF: Functional Autocorrelation Function (fACF) Plot

In FTSgof: White Noise and Goodness-of-Fit Tests for Functional Time Series

Functional Autocorrelation Function (fACF) Plot

Description

This function provides a graphical summary of the fACF of a functional time series (FTS) across different time lags h = 1:H. It also plots the 100 (1-\alpha)\% confidence bounds, developed under both weak white noise (WWN) and strong white noise (SWN) assumptions for all lags h = 1:H.

Usage

fACF(f_data, H = 20, alpha = 0.05, wwn_bound = FALSE, M = NULL)

Arguments

f_data	A J \times N matrix of functional time series data, where J is the number of discrete points in a grid and N is the sample size.
H	A positive integer value. The maximum lag for which to compute the coefficients and confidence bounds.
alpha	A numeric value between 0 and 1 specifying the significance level to be used for the confidence bounds.
wwn_bound	A Boolean value allowing the user to turn on the WWN bound. FALSE by default. Speeds down computation when TRUE.
M	A positive integer value. The number of Monte-Carlo simulations used to compute the confidence bounds under the WWN assumption. If M = NULL, M = \text{floor}((\max(150 - N, 0) + \max(100 - J, 0) + (J / \sqrt{2}))), ensuring that the number of Monte Carlo simulations is adequate based on the dataset size.

Details

This function computes and plots functional autocorrelation coefficients at lag h, for h \in 1:H. Given functional observations, X_1,\ldots, X_N, the sample autocovariance kernel at lag h can be computed by

\hat{\gamma}_{N,h}(t,s)=\frac{1}{N}\sum_{i=1}^{N-h} (X_i(t)-\bar{X}_N(t))(X_{i+h}(s)-\bar{X}_N(s)),\ \ \ \ 0 \le h < N,

where \bar{X}_N(t) = \frac{1}{N} \sum_{i=1}^N X_i(t). Then, the fACF at lag h is defined by measuring the magnitude (L^2-norm) of the lagged autocovariance kernel \hat\gamma_{N,h}:

\hat\rho_h =\frac{\|\hat{\gamma}_{N,h}\|}{\int \hat{\gamma}_{N,0}(t,t)dt}, \ \ \ \ \|\hat{\gamma}_{N,h}\|^2=\iint \hat{\gamma}_{N,h}^2(t,s) dtds.

This function plots estimated asymptotic 100 (1-\alpha)\% confidence bounds under the WWN assumption. Additionally, it computes similar (constant) bounds under the SWN assumption.

Value

Plot of the estimated functional autocorrelation coefficients for lags h \in 1:H with the WWN 100 (1-\alpha)\% upper confidence bound for each lag, as well as the constant SWN 100 (1-\alpha)\% upper confidence bound.

References

[1] Kokoszka P., Rice G., Shang H.L. (2017). Inference for the autocovariance of a functional time series under conditional heteroscedasticity. Journal of Multivariate Analysis, 162, 32-50.

[2] Mestre G., Portela J., Rice G., Roque A. M. S., Alonso E. (2021). Functional time series model identification and diagnosis by means of auto-and partial autocorrelation analysis. Computational Statistics & Data Analysis, 155, 107108.

Examples

data(Spanish_elec) # Daily Spanish electricity price profiles
fACF(Spanish_elec)
fACF(Spanish_elec, H=10, wwn_bound=TRUE)

FTSgof documentation built on Oct. 4, 2024, 1:06 a.m.