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 \(h = 1:H\).
fACF(f_data, H = 20, alpha = 0.05, wwn_bound = FALSE, M = NULL)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.
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.
A positive integer value. The maximum lag for which to compute the coefficients and confidence bounds.
A numeric value between 0 and 1 specifying the significance level to be used for the confidence bounds.
A Boolean value allowing the user to turn on the WWN bound. FALSE by default. Speeds down computation when TRUE.
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.
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.
[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.
# \donttest{
data(Spanish_elec) # Daily Spanish electricity price profiles
fACF(Spanish_elec)
fACF(Spanish_elec, H=10, wwn_bound=TRUE)
# }
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