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.

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, ...)

Arguments

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)

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

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,
  ...)

Arguments

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()

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, ...)

Arguments

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()

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)

Arguments

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))

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)

Arguments

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)

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)

Arguments

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)

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,
  ...)

Arguments

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

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

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, ...)

Arguments

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

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

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)

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)

Arguments

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)

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)

Arguments

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)

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)

Arguments

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)

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)

Arguments

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)

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, ...)

Arguments

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)

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, ...)

Arguments

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)

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)

Arguments

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)

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)

Arguments

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")

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)

Arguments

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")