Package 'fdaACF' reference manual

Title:	Autocorrelation Function for Functional Time Series
Description:	Quantify the serial correlation across lags of a given functional time series using the autocorrelation function and a partial autocorrelation function for functional time series proposed in Mestre et al. (2021) <doi:10.1016/j.csda.2020.107108>. The autocorrelation functions are based on the L2 norm of the lagged covariance operators of the series. Functions are available for estimating the distribution of the autocorrelation functions under the assumption of strong functional white noise.
Authors:	Guillermo Mestre Marcos [aut, cre], José Portela González [aut], Gregory Rice [aut], Antonio Muñoz San Roque [ctb], Estrella Alonso Pérez [ctb]
Maintainer:	Guillermo Mestre Marcos <[email protected]>
License:	GPL (>= 2)
Version:	1.0.0
Built:	2024-11-01 03:26:23 UTC
Source:	https://github.com/gmestrem/fdaacf

Daily electricity price profiles from the Day-Ahead Spanish Electricity Market

Description

A dataset containing the hourly electricity prices for Spain in the Day-Ahead Market (MIBEL)

Usage

elec_prices
elec_prices

Format

A data frame with 365 rows and 24 variables:

H1: Electricity price for hour 1
H2: Electricity price for hour 2
H3: Electricity price for hour 3
H4: Electricity price for hour 4
H5: Electricity price for hour 5
H6: Electricity price for hour 6
H7: Electricity price for hour 7
H8: Electricity price for hour 8
H9: Electricity price for hour 9
H10: Electricity price for hour 10
H11: Electricity price for hour 11
H12: Electricity price for hour 12
H13: Electricity price for hour 13
H14: Electricity price for hour 14
H15: Electricity price for hour 15
H16: Electricity price for hour 16
H17: Electricity price for hour 17
H18: Electricity price for hour 18
H19: Electricity price for hour 19
H20: Electricity price for hour 20
H21: Electricity price for hour 21
H22: Electricity price for hour 22
H23: Electricity price for hour 23
H24: Electricity price for hour 24

Source

https://www.esios.ree.es/es/analisis/600

Estimate distribution of the fACF under the iid. hypothesis using Imhof's method

Description

Estimate the distribution of the autocorrelation function under the hypothesis of strong functional white noise. This function uses Imhof's method to estimate the distribution.

Usage

estimate_iid_distr_Imhof(Y, v, autocovSurface, matindex, figure = FALSE, ...)
estimate_iid_distr_Imhof(Y, v, autocovSurface, matindex, figure = FALSE, ...)

Arguments

`Y`	Matrix containing the discretized values of the functional time series. The dimension of the matrix is $(n x m)$ , where $n$ is the number of curves and $m$ is the number of points observed in each curve.
`v`	Discretization points of the curves, by default `seq(from = 0, to = 1, length.out = 100)`.
`autocovSurface`	An $(m x m)$ matrix with the discretized values of the autocovariance operator $\hat{C}_{0}$ , obtained by calling the function `obtain_autocovariance`. The value $m$ indicates the number of points observed in each curve.
`matindex`	A vector containing the L2 norm of the autocovariance function. It can be obtained by calling function `obtain_suface_L2_norm`.
`figure`	Logical. If `TRUE`, plots the estimated distribution.
`...`	Further arguments passed to the `plot` function.

Value

Return a list with:

ex: Knots where the distribution has been estimated
ef: Discretized values of the estimated distribution.

Examples

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
autocovSurface <- obtain_autocovariance(Y,nlags)
matindex <- obtain_suface_L2_norm (v,autocovSurface)
# Remove lag 0
matindex <- matindex[-1]
Imhof_dist <- estimate_iid_distr_Imhof(Y,v,autocovSurface,matindex)
plot(Imhof_dist$ex,Imhof_dist$ef,type = "l",main = "ecdf obtained by Imhof's method")
grid()


# Example 2

N <- 400
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
autocovSurface <- obtain_autocovariance(Y,nlags)
matindex <- obtain_suface_L2_norm (v,autocovSurface)
# Remove lag 0
matindex <- matindex[-1]
Imhof_dist <- estimate_iid_distr_Imhof(Y,v,autocovSurface,matindex)
plot(Imhof_dist$ex,Imhof_dist$ef,type = "l",main = "ecdf obtained by Imhof's method")
grid()

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
autocovSurface <- obtain_autocovariance(Y,nlags)
matindex <- obtain_suface_L2_norm (v,autocovSurface)
# Remove lag 0
matindex <- matindex[-1]
Imhof_dist <- estimate_iid_distr_Imhof(Y,v,autocovSurface,matindex)
plot(Imhof_dist$ex,Imhof_dist$ef,type = "l",main = "ecdf obtained by Imhof's method")
grid()


# Example 2

N <- 400
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
autocovSurface <- obtain_autocovariance(Y,nlags)
matindex <- obtain_suface_L2_norm (v,autocovSurface)
# Remove lag 0
matindex <- matindex[-1]
Imhof_dist <- estimate_iid_distr_Imhof(Y,v,autocovSurface,matindex)
plot(Imhof_dist$ex,Imhof_dist$ef,type = "l",main = "ecdf obtained by Imhof's method")
grid()

Estimate distribution of the fACF under the iid. hypothesis using MC method

Description

Estimate the distribution of the autocorrelation function under the hypothesis of strong functional white noise. This function uses Montecarlo's method to estimate the distribution.

Usage

estimate_iid_distr_MC(Y, v, autocovSurface, matindex, nsimul = 10000,
  figure = FALSE, ...)
estimate_iid_distr_MC(Y, v, autocovSurface, matindex, nsimul = 10000,
  figure = FALSE, ...)

Arguments

`Y`	Matrix containing the discretized values of the functional time series. The dimension of the matrix is $(n x m)$ , where $n$ is the number of curves and $m$ is the number of points observed in each curve.
`v`	Discretization points of the curves, by default `seq(from = 0, to = 1, length.out = 100)`.
`autocovSurface`	An $(m x m)$ matrix with the discretized values of the autocovariance operator $\hat{C}_{0}$ , obtained by calling the function `obtain_autocovariance`. The value $m$ indicates the number of points observed in each curve.
`matindex`	A vector containing the L2 norm of the autocovariance function. It can be obtained by calling function `obtain_suface_L2_norm`.
`nsimul`	Positive integer indicating the number of MC simulations that will be used to estimate the distribution of the statistic. Increasing the number of simulations will improve the estimation, but it will increase the computational time. By default, `nsimul = 10000`.
`figure`	Logical. If `TRUE`, plots the estimated distribution.
`...`	Further arguments passed to the `plot` function.

Value

Return a list with:

ex: Knots where the distribution has been estimated
ef: Discretized values of the estimated distribution.
Reig: Raw values of the i.i.d. statistic for each MC simulation.

Examples

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
autocovSurface <- obtain_autocovariance(Y,nlags)
matindex <- obtain_suface_L2_norm (v,autocovSurface)
# Remove lag 0
matindex <- matindex[-1]
MC_dist <- estimate_iid_distr_MC(Y,v,autocovSurface,matindex)
plot(MC_dist$ex,MC_dist$ef,type = "l",main = "ecdf obtained by MC simulation")
grid()


# Example 2

N <- 400
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 20
autocovSurface <- obtain_autocovariance(Y,nlags)
matindex <- obtain_suface_L2_norm (v,autocovSurface)
# Remove lag 0
matindex <- matindex[-1]
MC_dist <- estimate_iid_distr_MC(Y,v,autocovSurface,matindex)
plot(MC_dist$ex,MC_dist$ef,type = "l",main = "ecdf obtained by MC simulation")
grid()

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
autocovSurface <- obtain_autocovariance(Y,nlags)
matindex <- obtain_suface_L2_norm (v,autocovSurface)
# Remove lag 0
matindex <- matindex[-1]
MC_dist <- estimate_iid_distr_MC(Y,v,autocovSurface,matindex)
plot(MC_dist$ex,MC_dist$ef,type = "l",main = "ecdf obtained by MC simulation")
grid()


# Example 2

N <- 400
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 20
autocovSurface <- obtain_autocovariance(Y,nlags)
matindex <- obtain_suface_L2_norm (v,autocovSurface)
# Remove lag 0
matindex <- matindex[-1]
MC_dist <- estimate_iid_distr_MC(Y,v,autocovSurface,matindex)
plot(MC_dist$ex,MC_dist$ef,type = "l",main = "ecdf obtained by MC simulation")
grid()

fdaACF: Autocorrelation function for Functional Time Series

Description

The fdaACF package provides diagnostic and analysis tools to quantify the serial autocorrelation across lags of a given functional time series in order to improve the identification and diagnosis of functional ARIMA models. The autocorrelation function is based on the L2 norm of the lagged covariance operators of the series. Several real-world datasets are included to illustrate the application of these techniques.

Fit an ARH(p) to a given functional time series

Description

Fit an $ARH(p)$ model to a given functional time series. The fitted model is based on the model proposed in (Aue et al, 2015), first decomposing the original functional observations into a vector time series of n_harm FPCA scores, and then fitting a vector autoregressive model of order $p$ ( $VAR(p)$ ) to the time series of the scores. Once fitted, the Karhunen-Loève expansion is used to re-transform the fitted values into functional observations.

Usage

fit_ARHp_FPCA(y, v, p, n_harm, show_varprop = T)
fit_ARHp_FPCA(y, v, p, n_harm, show_varprop = T)

Arguments

`y`	Matrix containing the discretized values of the functional time series. The dimension of the matrix is $(n x m)$ , where $n$ is the number of curves and $m$ is the number of points observed in each curve.
`v`	Numeric vector that contains the discretization points of the curves.
`p`	Numeric value specifying the order of the functional autoregressive model to be fitted.
`n_harm`	Numeric value specifying the number of functional principal components to be used when fitting the $ARH(p)$ model.
`show_varprop`	Logical. If `show_varprop = TRUE`, a plot of the proportion of variance explained by the first `n_harm` functional principal components will be shown. By default `show_varprop = TRUE`.

References

Aue, A., Norinho, D. D., Hormann, S. (2015). On the Prediction of Stationary Functional Time Series Journal of the American Statistical Association, 110, 378–392. https://doi.org/10.1080/01621459.2014.909317

Examples

# Example 1

# Simulate an ARH(1) process
N <- 250
dv <- 20
v <- seq(from = 0, to = 1, length.out = 20)

phi <- 1.3 * ((v) %*% t(v))

persp(v,v,phi, 
      ticktype = "detailed", 
      main = "Integral operator")

set.seed(3)
white_noise <-  simulate_iid_brownian_bridge(N, v = v)

y <- matrix(nrow = N, ncol = dv)
y[1,] <- white_noise[1,]
for(jj in 2:N){
    y[jj,] <- white_noise[jj,];
    
    y[jj,]=y[jj,]+integral_operator(operator_kernel = phi,
                                    v = v,
                                    curve = y[jj-1,])
}

# Fit an ARH(1) model
mod <- fit_ARHp_FPCA(y = y,
                     v = v,
                     p = 1,
                     n_harm = 5)

# Plot results
plot(v, y[50,], type = "l", lty = 1, ylab = "")
lines(v, mod$y_est[50,], col = "red")
legend("bottomleft", legend = c("real","est"),
       lty = 1, col = c(1,2))

# Example 1

# Simulate an ARH(1) process
N <- 250
dv <- 20
v <- seq(from = 0, to = 1, length.out = 20)

phi <- 1.3 * ((v) %*% t(v))

persp(v,v,phi, 
      ticktype = "detailed", 
      main = "Integral operator")

set.seed(3)
white_noise <-  simulate_iid_brownian_bridge(N, v = v)

y <- matrix(nrow = N, ncol = dv)
y[1,] <- white_noise[1,]
for(jj in 2:N){
    y[jj,] <- white_noise[jj,];
    
    y[jj,]=y[jj,]+integral_operator(operator_kernel = phi,
                                    v = v,
                                    curve = y[jj-1,])
}

# Fit an ARH(1) model
mod <- fit_ARHp_FPCA(y = y,
                     v = v,
                     p = 1,
                     n_harm = 5)

# Plot results
plot(v, y[50,], type = "l", lty = 1, ylab = "")
lines(v, mod$y_est[50,], col = "red")
legend("bottomleft", legend = c("real","est"),
       lty = 1, col = c(1,2))

Obtain the auto- and partial autocorrelation functions for a given FTS

Description

Estimate both the autocorrelation and partial autocorrelation function for a given functional time series and its distribution under the hypothesis of strong functional white noise. Both correlograms are plotted to ease the identification of the dependence structure of the functional time series.

Usage

FTS_identification(Y, v, nlags, n_harm = NULL, ci = 0.95,
  estimation = "MC", figure = TRUE, ...)
FTS_identification(Y, v, nlags, n_harm = NULL, ci = 0.95,
  estimation = "MC", figure = TRUE, ...)

Arguments

`Y`	Matrix containing the discretized values of the functional time series. The dimension of the matrix is $(n x m)$ , where $n$ is the number of curves and $m$ is the number of points observed in each curve.
`v`	Discretization points of the curves.
`nlags`	Number of lagged covariance operators of the functional time series that will be used to estimate the autocorrelation functions.
`n_harm`	Number of principal components that will be used to fit the ARH(p) models. If this value is not supplied, `n_harm` will be selected as the number of principal components that explain more than 95 % of the variance of the original data. By default, `n_harm = NULL`.
`ci`	A value between 0 and 1 that indicates the confidence interval for the i.i.d. bounds of the partial autocorrelation function. By default `ci = 0.95`.
`estimation`	Character specifying the method to be used when estimating the distribution under the hypothesis of functional white noise. Accepted values are: "MC": Monte-Carlo estimation. "Imhof": Estimation using Imhof's method. By default, `estimation = "MC"`.
`figure`	Logical. If `TRUE`, plots the estimated partial autocorrelation function with the specified i.i.d. bound.
`...`	Further arguments passed to the `plot_FACF` function.

Value

Return a list with:

Blueline: The upper prediction bound for the i.i.d. distribution.
rho_FACF: Autocorrelation coefficients for each lag of the functional time series.
rho_FPACF: Partial autocorrelation coefficients for each lag of the functional time series.

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

Examples

# Example 1 (Toy example)

N <- 50
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
set.seed(15)
Y <- simulate_iid_brownian_bridge(N, v, sig)
FTS_identification(Y,v,3)


# Example 2

data(elec_prices)
v <- seq(from = 1, to = 24)
nlags <- 30
FTS_identification(Y = as.matrix(elec_prices), 
v = v,
nlags = nlags,
ci = 0.95,
figure = TRUE)

# Example 1 (Toy example)

N <- 50
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
set.seed(15)
Y <- simulate_iid_brownian_bridge(N, v, sig)
FTS_identification(Y,v,3)


# Example 2

data(elec_prices)
v <- seq(from = 1, to = 24)
nlags <- 30
FTS_identification(Y = as.matrix(elec_prices), 
v = v,
nlags = nlags,
ci = 0.95,
figure = TRUE)

Integral transformation of a curve using an integral operator

Description

Compute the integral transform of the curve $Y_i$ with respect to a given integral operator $\Psi$ . The transformation is given by

$\Psi(Y_{i})(v) = \int \psi(u,v)Y_{i}(u)du$

Usage

integral_operator(operator_kernel, curve, v)
integral_operator(operator_kernel, curve, v)

Arguments

`operator_kernel`	Matrix with the values of the kernel surface of the integral operator. The dimension of the matrix is $(g x m)$ , where $g$ is the number of discretization points of the input curve and $m$ is the number of discretization points of the output curve.
`curve`	Vector containing the discretized values of a functional observation. The dimension of the matrix is $(1 x m)$ , where $m$ is the number of points observed in the curve.
`v`	Numerical vector specifying the discretization points of the curves.

Value

Returns a matrix the same size as curve with the transformed values.

Examples

# Example 1

v <- seq(from = 0, to = 1, length.out = 20)
set.seed(10)
curve <- sin(v) + rnorm(length(v))
operator_kernel <- 0.6*(v %*% t(v))
hat_curve <- integral_operator(operator_kernel,curve,v)

# Example 1

v <- seq(from = 0, to = 1, length.out = 20)
set.seed(10)
curve <- sin(v) + rnorm(length(v))
operator_kernel <- 0.6*(v %*% t(v))
hat_curve <- integral_operator(operator_kernel,curve,v)

Obtain a fd object from a matrix

Description

This function returns a fd object obtained from the discretized functional observations contained in mat_obj.

It is assumed that the functional observations contained in mat_obj are real observations, hence a poligonal base will be used to obtain the functional object.

Usage

mat2fd(mat_obj, range_val = c(0, 1), argvals = NULL)
mat2fd(mat_obj, range_val = c(0, 1), argvals = NULL)

Arguments

`mat_obj`	A matrix that contains the discretized functional observations.
`range_val`	A numeric vector of length 2 that contains the range of the observed functional data. By default `range_val = c(0,1)`.
`argvals`	Optinal argument that contains a strictly increasing vector of argument values at which line segments join to form a polygonal line. If `argvals = NULL`, it is assumed a equidistant discretization vector. By default `argvals = NULL`.

Value

A fd object obtained from the functional observations in mat_obj.

Examples

# Example 1
N <- 100
dv <- 30
v <- seq(from = 0, to = 1, length.out = dv)
set.seed(150) # For replication
mat_func_obs <- matrix(rnorm(N*dv), 
                       nrow = N, 
                       ncol = dv)
fd_func_obs <- mat2fd(mat_obj = mat_func_obs,
                      range_val = range(v),
                      argvals = v)
plot(fd_func_obs)
# Example 1
N <- 100
dv <- 30
v <- seq(from = 0, to = 1, length.out = dv)
set.seed(150) # For replication
mat_func_obs <- matrix(rnorm(N*dv), 
                       nrow = N, 
                       ncol = dv)
fd_func_obs <- mat2fd(mat_obj = mat_func_obs,
                      range_val = range(v),
                      argvals = v)
plot(fd_func_obs)

Estimate the autocorrelation function of the series

Description

Obtain the empirical autocorrelation function for lags $= 0,...,$ nlags of the functional time series. Given $Y_{1},...,Y_{T}$ a functional time series, the sample autocovariance functions $\hat{C}_{h}(u,v)$ are given by:

$\hat{C}_{h}(u,v) = \frac{1}{T} \sum_{i=1}^{T-h}(Y_{i}(u) - \overline{Y}_{T}(u))(Y_{i+h}(v) - \overline{Y}_{T}(v))$

where $\overline{Y}_{T}(u) = \frac{1}{T} \sum_{i = 1}^{T} Y_{i}(t)$ denotes the sample mean function. By normalizing these functions using the normalizing factor $\int\hat{C}_{0}(u,u)du$ , the range of the autocovariance functions becomes $(0,1)$ ; thus defining the autocorrelation functions of the series

Usage

obtain_autocorrelation(Y, v = seq(from = 0, to = 1, length.out =
  ncol(Y)), nlags)
obtain_autocorrelation(Y, v = seq(from = 0, to = 1, length.out =
  ncol(Y)), nlags)

Arguments

`Y`	Matrix containing the discretized values of the functional time series. The dimension of the matrix is $(n x m)$ , where $n$ is the number of curves and $m$ is the number of points observed in each curve.
`v`	Discretization points of the curves, by default `seq(from = 0, to = 1, length.out = 100)`.
`nlags`	Number of lagged covariance operators of the functional time series that will be used to estimate the autocorrelation function.

Value

Return a list with the lagged autocorrelation functions estimated from the data. Each function is given by a $(m x m)$ matrix, where $m$ is the number of points observed in each curve.

Examples

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
lagged_autocor <- obtain_autocorrelation(Y = bbridge,
                                        nlags = nlags)
image(x = v, y = v, z = lagged_autocor$Lag0)


# Example 2
require(fields)
N <- 500
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 4
lagged_autocov <- obtain_autocovariance(Y = bbridge,
                                        nlags = nlags)
lagged_autocor <- obtain_autocorrelation(Y = bbridge,
                                         v = v,
                                         nlags = nlags)

opar <- par(no.readonly = TRUE)
par(mfrow = c(1,2))
z_lims <- range(lagged_autocov$Lag1)
colors <- heat.colors(12)
image.plot(x = v, 
           y = v,
           z = lagged_autocov$Lag1,
           legend.width = 2,
           zlim = z_lims,
           col = colors,
           xlab = "u",
           ylab = "v",
           main = "Autocovariance")
z_lims <- range(lagged_autocor$Lag1)
image.plot(x = v, 
           y = v,
           z = lagged_autocor$Lag1,
           legend.width = 2,
           zlim = z_lims,
           col = colors,
           xlab = "u",
           ylab = "v",
           main = "Autocorrelation")
par(opar)

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
lagged_autocor <- obtain_autocorrelation(Y = bbridge,
                                        nlags = nlags)
image(x = v, y = v, z = lagged_autocor$Lag0)


# Example 2
require(fields)
N <- 500
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 4
lagged_autocov <- obtain_autocovariance(Y = bbridge,
                                        nlags = nlags)
lagged_autocor <- obtain_autocorrelation(Y = bbridge,
                                         v = v,
                                         nlags = nlags)

opar <- par(no.readonly = TRUE)
par(mfrow = c(1,2))
z_lims <- range(lagged_autocov$Lag1)
colors <- heat.colors(12)
image.plot(x = v, 
           y = v,
           z = lagged_autocov$Lag1,
           legend.width = 2,
           zlim = z_lims,
           col = colors,
           xlab = "u",
           ylab = "v",
           main = "Autocovariance")
z_lims <- range(lagged_autocor$Lag1)
image.plot(x = v, 
           y = v,
           z = lagged_autocor$Lag1,
           legend.width = 2,
           zlim = z_lims,
           col = colors,
           xlab = "u",
           ylab = "v",
           main = "Autocorrelation")
par(opar)

Estimate eigenvalues of the autocovariance function

Description

Estimate the eigenvalues of the sample autocovariance function $\hat{C}_{0}$ . This functions returns the eigenvalues which are greater than the value epsilon.

Usage

obtain_autocov_eigenvalues(v, Y, epsilon = 1e-04)
obtain_autocov_eigenvalues(v, Y, epsilon = 1e-04)

Arguments

`v`	Discretization points of the curves, by default `seq(from = 0, to = 1, length.out = 100)`.
`Y`	Matrix containing the discretized values of the functional time series. The dimension of the matrix is $(n x m)$ , where $n$ is the number of curves and $m$ is the number of points observed in each curve.
`epsilon`	Value used to determine how many eigenvalues will be returned. The eigenvalues $\lambda_{j}/\lambda_{1} > \code{epsilon}$ will be returned. By default `epsilon = 0.0001`.

Value

A vector containing the $k$ eigenvalues greater than epsilon.

Examples

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
lambda <- obtain_autocov_eigenvalues(v = v, Y = Y)

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
lambda <- obtain_autocov_eigenvalues(v = v, Y = Y)

Estimate the autocovariance function of the series

Description

Obtain the empirical autocovariance function for lags $= 0,...,$ nlags of the functional time series. Given $Y_{1},...,Y_{T}$ a functional time series, the sample autocovariance functions $\hat{C}_{h}(u,v)$ are given by:

$\hat{C}_{h}(u,v) = \frac{1}{T} \sum_{i=1}^{T-h}(Y_{i}(u) - \overline{Y}_{T}(u))(Y_{i+h}(v) - \overline{Y}_{T}(v))$

where $\overline{Y}_{T}(u) = \frac{1}{T} \sum_{i = 1}^{T} Y_{i}(t)$ denotes the sample mean function.

Usage

obtain_autocovariance(Y, nlags)
obtain_autocovariance(Y, nlags)

Arguments

`Y`	Matrix containing the discretized values of the functional time series. The dimension of the matrix is $(n x m)$ , where $n$ is the number of curves and $m$ is the number of points observed in each curve.
`nlags`	Number of lagged covariance operators of the functional time series that will be used to estimate the autocorrelation function.

Value

Return a list with the lagged autocovariance functions estimated from the data. Each function is given by a $(m x m)$ matrix, where $m$ is the number of points observed in each curve.

Examples

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
lagged_autocov <- obtain_autocovariance(Y = bbridge,
                                        nlags = nlags)
image(x = v, y = v, z = lagged_autocov$Lag0)


# Example 2

N <- 500
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 10
lagged_autocov <- obtain_autocovariance(Y = bbridge,
                                        nlags = nlags)
image(x = v, y = v, z = lagged_autocov$Lag0)
image(x = v, y = v, z = lagged_autocov$Lag10)

# Example 3

require(fields)
N <- 500
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 4
lagged_autocov <- obtain_autocovariance(Y = bbridge,
                                        nlags = nlags)
z_lims <- range(lagged_autocov$Lag0)
colors <- heat.colors(12)
opar <- par(no.readonly = TRUE)
par(mfrow = c(1,5))
par(oma=c( 0,0,0,6)) 
for(k in 0:nlags){
   image(x=v,
         y=v,
         z = lagged_autocov[[paste0("Lag",k)]],
         main = paste("Lag",k),
         col = colors,
         xlab = "u",
         ylab = "v")
}
par(oma=c( 0,0,0,2.5)) # reset margin to be much smaller.
image.plot( legend.only=TRUE, legend.width = 2,zlim=z_lims, col = colors)
par(opar)

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
lagged_autocov <- obtain_autocovariance(Y = bbridge,
                                        nlags = nlags)
image(x = v, y = v, z = lagged_autocov$Lag0)


# Example 2

N <- 500
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 10
lagged_autocov <- obtain_autocovariance(Y = bbridge,
                                        nlags = nlags)
image(x = v, y = v, z = lagged_autocov$Lag0)
image(x = v, y = v, z = lagged_autocov$Lag10)

# Example 3

require(fields)
N <- 500
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 4
lagged_autocov <- obtain_autocovariance(Y = bbridge,
                                        nlags = nlags)
z_lims <- range(lagged_autocov$Lag0)
colors <- heat.colors(12)
opar <- par(no.readonly = TRUE)
par(mfrow = c(1,5))
par(oma=c( 0,0,0,6)) 
for(k in 0:nlags){
   image(x=v,
         y=v,
         z = lagged_autocov[[paste0("Lag",k)]],
         main = paste("Lag",k),
         col = colors,
         xlab = "u",
         ylab = "v")
}
par(oma=c( 0,0,0,2.5)) # reset margin to be much smaller.
image.plot( legend.only=TRUE, legend.width = 2,zlim=z_lims, col = colors)
par(opar)

Obtain the autocorrelation function for a given functional time series.

Description

Estimate the lagged autocorrelation function for a given functional time series and its distribution under the hypothesis of strong functional white noise. This graphic tool can be used to identify seasonal patterns in the functional data as well as auto-regressive or moving average terms. i.i.d. bounds are included to test the presence of serial correlation in the data.

Usage

obtain_FACF(Y, v, nlags, ci = 0.95, estimation = "MC", figure = TRUE, ...)
obtain_FACF(Y, v, nlags, ci = 0.95, estimation = "MC", figure = TRUE, ...)

Arguments

`Y`	Matrix containing the discretized values of the functional time series. The dimension of the matrix is $(n x m)$ , where $n$ is the number of curves and $m$ is the number of points observed in each curve.
`v`	Discretization points of the curves, by default `seq(from = 0, to = 1, length.out = 100)`.
`nlags`	Number of lagged covariance operators of the functional time series that will be used to estimate the autocorrelation function.
`ci`	A value between 0 and 1 that indicates the confidence interval for the i.i.d. bounds of the autocorrelation function. By default `ci = 0.95`.
`estimation`	Character specifying the method to be used when estimating the distribution under the hypothesis of functional white noise. Accepted values are: "MC": Monte-Carlo estimation. "Imhof": Estimation using Imhof's method. By default, `estimation = "MC"`.
`figure`	Logical. If `TRUE`, plots the estimated autocorrelation function with the specified i.i.d. bound.
`...`	Further arguments passed to the `plot_FACF` function.

Value

Return a list with:

Blueline: The upper prediction bound for the i.i.d. distribution.
rho: Autocorrelation values for each lag of the functional time series.

References

Examples

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 5)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
obtain_FACF(Y,v,20)


# Example 2

data(elec_prices)
v <- seq(from = 1, to = 24)
nlags <- 30
obtain_FACF(Y = as.matrix(elec_prices), 
v = v,
nlags = nlags,
ci = 0.95,
figure = TRUE)

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 5)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
obtain_FACF(Y,v,20)


# Example 2

data(elec_prices)
v <- seq(from = 1, to = 24)
nlags <- 30
obtain_FACF(Y = as.matrix(elec_prices), 
v = v,
nlags = nlags,
ci = 0.95,
figure = TRUE)

Obtain the partial autocorrelation function for a given FTS.

Description

Estimate the partial autocorrelation function for a given functional time series and its distribution under the hypothesis of strong functional white noise.

Usage

obtain_FPACF(Y, v, nlags, n_harm, ci = 0.95, estimation = "MC",
  figure = TRUE, ...)
obtain_FPACF(Y, v, nlags, n_harm, ci = 0.95, estimation = "MC",
  figure = TRUE, ...)

Arguments

`Y`	Matrix containing the discretized values of the functional time series. The dimension of the matrix is $(n x m)$ , where $n$ is the number of curves and $m$ is the number of points observed in each curve.
`v`	Discretization points of the curves.
`nlags`	Number of lagged covariance operators of the functional time series that will be used to estimate the partial autocorrelation function.
`n_harm`	Number of principal components that will be used to fit the ARH(p) models.
`ci`	A value between 0 and 1 that indicates the confidence interval for the i.i.d. bounds of the partial autocorrelation function. By default `ci = 0.95`.
`estimation`	Character specifying the method to be used when estimating the distribution under the hypothesis of functional white noise. Accepted values are: "MC": Monte-Carlo estimation. "Imhof": Estimation using Imhof's method. By default, `estimation = "MC"`.
`figure`	Logical. If `TRUE`, plots the estimated partial autocorrelation function with the specified i.i.d. bound.
`...`	Further arguments passed to the `plot_FACF` function.

Value

Return a list with:

Blueline: The upper prediction bound for the i.i.d. distribution.
rho: Partial autocorrelation coefficients for each lag of the functional time series.

References

Examples

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 5)
sig <- 2
set.seed(15)
Y <- simulate_iid_brownian_bridge(N, v, sig)
obtain_FPACF(Y,v,10, n_harm = 2)


# Example 2

data(elec_prices)
v <- seq(from = 1, to = 24)
nlags <- 30
obtain_FPACF(Y = as.matrix(elec_prices), 
v = v,
nlags = nlags,
n_harm = 5, 
ci = 0.95,
figure = TRUE)

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 5)
sig <- 2
set.seed(15)
Y <- simulate_iid_brownian_bridge(N, v, sig)
obtain_FPACF(Y,v,10, n_harm = 2)


# Example 2

data(elec_prices)
v <- seq(from = 1, to = 24)
nlags <- 30
obtain_FPACF(Y = as.matrix(elec_prices), 
v = v,
nlags = nlags,
n_harm = 5, 
ci = 0.95,
figure = TRUE)

Obtain L2 norm of the autocovariance functions

Description

Returns the L2 norm of the lagged autocovariance functions $\hat{C}_{h}$ . The L2 norm of these functions is defined as

$\sqrt(\int \int \hat{C}^{2}_{h}(u,v)du dv)$

Usage

obtain_suface_L2_norm(v, autocovSurface)
obtain_suface_L2_norm(v, autocovSurface)

Arguments

`v`	Discretization points of the curves, by default `seq(from = 0, to = 1, length.out = 100)`.
`autocovSurface`	An $(m x m)$ matrix with the discretized values of the autocovariance operator $\hat{C}_{0}$ , obtained by calling the function `obtain_autocovariance`. The value $m$ indicates the number of points observed in each curve.

Value

A vector containing the L2 norm of the lagged autocovariance functions autocovSurface.

Examples

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
autocovSurface <- obtain_autocovariance(Y=Y,nlags = nlags)
norms <- obtain_suface_L2_norm(v = v,autocovSurface = autocovSurface)
plot_autocovariance(fun.autocovariance = autocovSurface,lag = 1)
title(sub = paste0("Lag ",1," - L2 Norm: ",norms[2]))


# Example 2

N <- 400
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 2
autocovSurface <- obtain_autocovariance(Y=Y,nlags = nlags)
norms <- obtain_suface_L2_norm(v = v,autocovSurface = autocovSurface)
opar <- par(no.readonly = TRUE)
par(mfrow = c(1,3))
plot_autocovariance(fun.autocovariance = autocovSurface,lag = 0)
title(sub = paste0("Lag ",0," - L2 Norm: ",norms[1]))
plot_autocovariance(fun.autocovariance = autocovSurface,lag = 1)
title(sub = paste0("Lag ",1," - L2 Norm: ",norms[2]))
plot_autocovariance(fun.autocovariance = autocovSurface,lag = 2)
title(sub = paste0("Lag ",2," - L2 Norm: ",norms[3]))
par(opar)

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
autocovSurface <- obtain_autocovariance(Y=Y,nlags = nlags)
norms <- obtain_suface_L2_norm(v = v,autocovSurface = autocovSurface)
plot_autocovariance(fun.autocovariance = autocovSurface,lag = 1)
title(sub = paste0("Lag ",1," - L2 Norm: ",norms[2]))


# Example 2

N <- 400
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
Y <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 2
autocovSurface <- obtain_autocovariance(Y=Y,nlags = nlags)
norms <- obtain_suface_L2_norm(v = v,autocovSurface = autocovSurface)
opar <- par(no.readonly = TRUE)
par(mfrow = c(1,3))
plot_autocovariance(fun.autocovariance = autocovSurface,lag = 0)
title(sub = paste0("Lag ",0," - L2 Norm: ",norms[1]))
plot_autocovariance(fun.autocovariance = autocovSurface,lag = 1)
title(sub = paste0("Lag ",1," - L2 Norm: ",norms[2]))
plot_autocovariance(fun.autocovariance = autocovSurface,lag = 2)
title(sub = paste0("Lag ",2," - L2 Norm: ",norms[3]))
par(opar)

Generate a 3D plot of the autocovariance surface of a given FTS

Description

Obtain a 3D plot of the autocovariance surfaces of a given functional time series. This visualization is useful to detect any kind of dependency between the discretization points of the series.

Usage

plot_autocovariance(fun.autocovariance, lag = 0, ...)
plot_autocovariance(fun.autocovariance, lag = 0, ...)

Arguments

`fun.autocovariance`	A list obtained by calling the function `obtain_autocovariance`.
`lag`	An integer between 0 and `nlags`, indicating the lagged autocovariance function to be plotted. By default 0.
`...`	Further arguments passed to the `persp` function.

Examples

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
lagged_autocov <- obtain_autocovariance(Y = bbridge,nlags = nlags)
plot_autocovariance(lagged_autocov,1)


# Example 2

N <- 500
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 4
lagged_autocov <- obtain_autocovariance(Y = bbridge,nlags = nlags)
opar <- par(no.readonly = TRUE)
par(mfrow = c(1,5))
for(k in 0:nlags){
   plot_autocovariance(lagged_autocov,k)
}
par(opar)

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
lagged_autocov <- obtain_autocovariance(Y = bbridge,nlags = nlags)
plot_autocovariance(lagged_autocov,1)


# Example 2

N <- 500
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 4
lagged_autocov <- obtain_autocovariance(Y = bbridge,nlags = nlags)
opar <- par(no.readonly = TRUE)
par(mfrow = c(1,5))
for(k in 0:nlags){
   plot_autocovariance(lagged_autocov,k)
}
par(opar)

Plot the autocorrelation function of a given FTS

Description

Plot a visual representation of the autocorrelation function of a given functional time series, including the upper i.i.d. bound.

Usage

plot_FACF(rho, Blueline, ci, ...)
plot_FACF(rho, Blueline, ci, ...)

Arguments

`rho`	Autocorrelation values for each lag of the functional time series obtained by calling the function `obtain_FACF`.
`Blueline`	The upper prediction bound for the i.i.d. distribution obtained by calling the function `obtain_FACF`.
`ci`	Value between 0 and 1 that was used when calling the function `obtain_FACF`. This value is only used to display information in the figure.
`...`	Further arguments passed to the `plot` function.

Examples

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 15
upper_bound <- 0.95
fACF <- obtain_FACF(Y = bbridge,v = v,nlags = nlags,ci=upper_bound,figure = FALSE)
plot_FACF(rho = fACF$rho,Blueline = fACF$Blueline,ci = upper_bound)


# Example 2

N <- 200
v <- seq(from = 0, to = 1, length.out = 30)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 15
upper_bound <- 0.95
fACF <- obtain_FACF(Y = bbridge,v = v,nlags = nlags,ci=upper_bound,figure = FALSE)
plot_FACF(rho = fACF$rho,Blueline = fACF$Blueline,ci = upper_bound)

# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 15
upper_bound <- 0.95
fACF <- obtain_FACF(Y = bbridge,v = v,nlags = nlags,ci=upper_bound,figure = FALSE)
plot_FACF(rho = fACF$rho,Blueline = fACF$Blueline,ci = upper_bound)


# Example 2

N <- 200
v <- seq(from = 0, to = 1, length.out = 30)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 15
upper_bound <- 0.95
fACF <- obtain_FACF(Y = bbridge,v = v,nlags = nlags,ci=upper_bound,figure = FALSE)
plot_FACF(rho = fACF$rho,Blueline = fACF$Blueline,ci = upper_bound)

Obtain the reconstructed curves after PCA

Description

This function reconstructs the functional curves from a score vector using the basis obtained after applying functional PCA. This allows the user to draw estimations from the joint density of the FPCA scores and reconstruct the curves for those new scores.

Usage

reconstruct_fd_from_PCA(pca_struct, scores, centerfns = T)
reconstruct_fd_from_PCA(pca_struct, scores, centerfns = T)

Arguments

`pca_struct`	List obtained after calling function `pca.fd`.
`scores`	Numerical vector that contains the scores of the fPCA decomposition for one functional observation.
`centerfns`	Logical value specifying wheter the FPCA performed used `centerfns = T` or `centerfns = F`. By default `centerfns = T`.

Value

Returns a object of type fd that contains the reconstructed curve.

Examples

# Example 1

# Simulate fd
nobs <- 200
dv <- 10
basis<-fda::create.bspline.basis(rangeval=c(0,1),nbasis=10)
set.seed(5)
C <- matrix(rnorm(nobs*dv), ncol =  dv, nrow = nobs)
fd_sim <- fda::fd(coef=t(C),basis)

# Perform FPCA
pca_struct <- fda::pca.fd(fd_sim,nharm = 6)

# Reconstruct first curve
fd_rec <- reconstruct_fd_from_PCA(pca_struct = pca_struct, scores = pca_struct$scores[1,])
plot(fd_sim[1])
plot(fd_rec, add = TRUE, col = "red")
legend("topright",
       legend = c("Real Curve", "PCA Reconstructed"),
       col = c("black","red"),
       lty = 1)

# Example 2 (Perfect reconstruction)

# Simulate fd
nobs <- 200
dv <- 7
basis<-fda::create.bspline.basis(rangeval=c(0,1),nbasis=dv)
set.seed(5)
C <- matrix(rnorm(nobs*dv), ncol =  dv, nrow = nobs)
fd_sim <- fda::fd(coef=t(C),basis)

# Perform FPCA
pca_struct <- fda::pca.fd(fd_sim,nharm = dv)

# Reconstruct first curve
fd_rec <- reconstruct_fd_from_PCA(pca_struct = pca_struct, scores = pca_struct$scores[1,])
plot(fd_sim[1])
plot(fd_rec, add = TRUE, col = "red")
legend("topright",
       legend = c("Real Curve", "PCA Reconstructed"),
       col = c("black","red"),
       lty = 1)
# Example 1

# Simulate fd
nobs <- 200
dv <- 10
basis<-fda::create.bspline.basis(rangeval=c(0,1),nbasis=10)
set.seed(5)
C <- matrix(rnorm(nobs*dv), ncol =  dv, nrow = nobs)
fd_sim <- fda::fd(coef=t(C),basis)

# Perform FPCA
pca_struct <- fda::pca.fd(fd_sim,nharm = 6)

# Reconstruct first curve
fd_rec <- reconstruct_fd_from_PCA(pca_struct = pca_struct, scores = pca_struct$scores[1,])
plot(fd_sim[1])
plot(fd_rec, add = TRUE, col = "red")
legend("topright",
       legend = c("Real Curve", "PCA Reconstructed"),
       col = c("black","red"),
       lty = 1)

# Example 2 (Perfect reconstruction)

# Simulate fd
nobs <- 200
dv <- 7
basis<-fda::create.bspline.basis(rangeval=c(0,1),nbasis=dv)
set.seed(5)
C <- matrix(rnorm(nobs*dv), ncol =  dv, nrow = nobs)
fd_sim <- fda::fd(coef=t(C),basis)

# Perform FPCA
pca_struct <- fda::pca.fd(fd_sim,nharm = dv)

# Reconstruct first curve
fd_rec <- reconstruct_fd_from_PCA(pca_struct = pca_struct, scores = pca_struct$scores[1,])
plot(fd_sim[1])
plot(fd_rec, add = TRUE, col = "red")
legend("topright",
       legend = c("Real Curve", "PCA Reconstructed"),
       col = c("black","red"),
       lty = 1)

Simulate a FTS from a brownian bridge process

Description

Generate a functional time series from a Brownian Bridge process. If $W(t)$ is a Wiener process, the Brownian Bridge is defined as $W(t) - tW(1)$ . Each functional observation is discretized in the points indicated in v. The series obtained is i.i.d. and does not exhibit any kind of serial correlation.

Usage

simulate_iid_brownian_bridge(
  N,
  v = seq(from = 0, to = 1, length.out = 100),
  sig = 1
)
simulate_iid_brownian_bridge(
  N,
  v = seq(from = 0, to = 1, length.out = 100),
  sig = 1
)

Arguments

`N`	The number of observations of the simulated data.
`v`	Discretization points of the curves, by default `seq(from = 0, to = 1, length.out = 100)`.
`sig`	Standard deviation of the Brownian Motion process, by default `1`.

Value

Return the simulated functional time series as a matrix.

Examples

N <- 100
v <- seq(from = 0, to = 1, length.out = 20)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
matplot(v,t(bbridge), type = "l", xlab = "v", ylab = "Value")
N <- 100
v <- seq(from = 0, to = 1, length.out = 20)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
matplot(v,t(bbridge), type = "l", xlab = "v", ylab = "Value")

Simulate a FTS from a brownian motion process

Description

Generate a functional time series from a Brownian Motion process. Each functional observation is discretized in the points indicated in v. The series obtained is i.i.d. and does not exhibit any kind of serial correlation

Usage

simulate_iid_brownian_motion(
  N,
  v = seq(from = 0, to = 1, length.out = 100),
  sig = 1
)
simulate_iid_brownian_motion(
  N,
  v = seq(from = 0, to = 1, length.out = 100),
  sig = 1
)

Arguments

`N`	The number of observations of the simulated data.
`v`	Discretization points of the curves, by default `seq(from = 0, to = 1, length.out = 100)`.
`sig`	Standard deviation of the Brownian Motion process, by default `1`.

Value

Return the simulated functional time series as a matrix.

Examples

N <- 100
v <- seq(from = 0, to = 1, length.out = 20)
sig <- 2
bmotion <- simulate_iid_brownian_motion(N, v, sig)
matplot(v,t(bmotion), type = "l", xlab = "v", ylab = "Value")
N <- 100
v <- seq(from = 0, to = 1, length.out = 20)
sig <- 2
bmotion <- simulate_iid_brownian_motion(N, v, sig)
matplot(v,t(bmotion), type = "l", xlab = "v", ylab = "Value")

Package 'fdaACF'

Help Index

Daily electricity price profiles from the Day-Ahead Spanish Electricity Market

Description

Usage

Format

Source

Estimate distribution of the fACF under the iid. hypothesis using Imhof's method

Description

Usage

Arguments

Value

Examples

Estimate distribution of the fACF under the iid. hypothesis using MC method

Description

Usage

Arguments

Value

Examples

fdaACF: Autocorrelation function for Functional Time Series

Description

Fit an ARH(p) to a given functional time series

Description

Usage

Arguments

References

Examples

Obtain the auto- and partial autocorrelation functions for a given FTS

Description

Usage

Arguments

Value

References

Examples

Integral transformation of a curve using an integral operator

Description

Usage

Arguments

Value

Examples

Obtain a fd object from a matrix

Description

Usage

Arguments

Value

Examples

Estimate the autocorrelation function of the series

Description

Usage

Arguments

Value

Examples

Estimate eigenvalues of the autocovariance function

Description

Usage

Arguments

Value

Examples

Estimate the autocovariance function of the series

Description

Usage

Arguments

Value

Examples

Obtain the autocorrelation function for a given functional time series.

Description

Usage

Arguments

Value

References

Examples

Obtain the partial autocorrelation function for a given FTS.

Description

Usage

Arguments

Value

References

Examples

Obtain L2 norm of the autocovariance functions

Description