fdaACF

Autocorrelation Function for Functional Time Series

Guillermo Mestre Marcos, guillermo.mestre@comillas.edu

José Portela González, jose.portela@iit.comillas.edu

Gregory Rice, grice@uwaterloo.ca

Antonio Muñoz San Roque, Antonio.Munoz@iit.comillas.edu

Estrella Alonso Pérez, ealonso@icai.comillas.edu

This is the development repository for the fdaACF package. Functions within this package can be used for quantifying the serial correlation across lags of a given functional time series.

The current version of this package can be installed from Github:

install.packages('devtools')
library(devtools)
install_github('GMestreM/fdaACF')

To install the stable version from CRAN, simply run the following from an R console:

install.packages('fdaACF')

Bug reports

Please submit any bug reports (or suggestions) using the issues tab of the GitHub page.

Functions

The main purpose of this package is identifying the underlying structure of a given functional time series. Functional data is expected as an (m x p) matrix, where m is the number of functional observations and p denotes the number of discretization points of each curve.

Start by loading the package and the sample dataset.

library(fdaACF)
data(elec_prices)
matplot(t(elec_prices), type = "l",lty = 1,xlab = "Hours", ylab = "Price (€/MWh)")

The obtain_FACF and obtain_FPACF functions estimates the lagged autocorrelation function and partial autocorrelation function for a given functional time series and its distribution under the hypothesis of strong functional white noise. The visual representation of the lagged ACF and PACF can be used to identify seasonal patterns in the functional data as well as auto-regressive or moving average terms. In addition, i.i.d. bounds are included to test the presence of serial correlation in the data.

# Define the discretization axis
v <- seq(from = 1, to = 24)
nlags <- 30
ci <- 0.95

# Autocorrelation function for functional data
FACF <- obtain_FACF(Y = as.matrix(elec_prices), 
                    v = v,
                    nlags = nlags,
                    ci = ci,
                    figure = TRUE)

# Partial autocorrelation function for functional data
FPACF <- obtain_FPACF(Y = as.matrix(elec_prices), 
                      v = v,
                      nlags = nlags,
                      n_harm = 5, 
                      ci = ci,
                      figure = FALSE)

par(mfrow = c(1,2))
plot_FACF(rho = FACF$rho,
          Blueline = FACF$Blueline,
          ci = ci,
          main = "FACF")
plot_FACF(rho = FPACF$rho,
          Blueline = FPACF$Blueline,
          ci = ci,
          main = "FPACF")
par(mfrow = c(1,1))

The shape of the functional ACF indicates the presence of a daily seasonal effect in the data, as well as the presence of serial correlation.

Function FTS_identification estimates both the FACF and the FPACF of a given functional time series, providing an user-friendly way of quantifying the dependence structure of a functional dataset.

FunAutCorr <- FTS_identification(Y = as.matrix(elec_prices), 
                                 v = v,
                                 nlags = nlags,
                                 ci = ci,
                                 figure = TRUE)

The functional ACF and PACF are based on the L2 norm of the covariance functions of the data, which can be estimated with function obtain_autocovariance and plotted with function plot_autocovariance. In order to test the i.i.d. hypothesis, a functional white noise series will be simulated with function simulate_iid_brownian_bridge.

# Simulate white noise series
N <- 400
v <- seq(from = 0, to = 1, length.out = 24)
sig <- 1
set.seed(3) # For replication
Y <- simulate_iid_brownian_bridge(N, v, sig)
matplot(t(Y), type = "l",lty = 1,xlab = "v", ylab = "", main = "Functional Brownian Bridge")

# Estimate the lagged covariance functions
nlags <- 4
lagged_autocov <- obtain_autocovariance(Y = Y,nlags = nlags)


# Plot lagged covariance functions
par(mfrow = c(1,5))
for(k in 0:nlags){
   plot_autocovariance(lagged_autocov,k)
}
par(mfrow = c(1,1))

As illustrated by the previous image, the L2 norm of these covariance surfaces is close to 0. This can be easily checked by obtaining the ACF function of these data.

# Autocorrelation function for functional data
FACF_iid <- obtain_FACF(Y = Y, 
                        v = v,
                        nlags = 30,
                        ci = ci,
                        figure = TRUE)

As all the autocorrelation values fall below the i.i.d. bound, the i.i.d. hypothesis cannot be rejected.

References

Mestre G., Portela J., Rice G., Muñoz San Roque A., Alonso E. (2021). Functional time series model identification and diagnosis by means of auto- and partial autocorrelation analysis. Computational Statistics & Data Analysis, 155, 107108. https://doi.org/10.1016/j.csda.2020.107108

Mestre, G., Portela, J., Muñoz-San Roque, A., Alonso, E. (2020). Forecasting hourly supply curves in the Italian Day-Ahead electricity market with a double-seasonal SARMAHX model. International Journal of Electrical Power & Energy Systems, 121, 106083. https://doi.org/10.1016/j.ijepes.2020.106083

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. https://doi.org/10.1016/j.jmva.2017.08.004

License

This package is released in the public domain under the General Public License GPL.

Association

Package created in the Institute for Research in Technology (IIT), link to homepage