Parameters

• n: The number of curves to generate for each group. By default, this is set to 50 per group, resulting in a total of 100 curves.
• p: The number of grid points at which the curves are evaluated over the interval \([0, 1]\). The default is 30 grid points.
• i_sim: An integer between 1 and 8 that specifies which model to use for generating the curves in the second group. The first group of curves is generated in the same way for all values of i_sim.

Function breakdown

The first group of functions is generated by a Gaussian process \[X_1(t)=E_1(t)+e(t),\] where \(E_1(t)=30t^{ \frac{3}{2}}(1-t)\) is the mean function and \(e(t)\) is a centered Gaussian process with covariance matrix \[ Cov(e(t_i),e(t_j))=0.3 \exp\left(-\frac{\lvert t_i-t_j \rvert}{0.3}\right).\]

The second group of functions \(X_i\) are obtained from the first one by perturbing the generation process. Different values of i_sim generate different curves.

For i_sim values of 1, 2, and 3, the curves in the second group exhibit changes in the mean while the covariance matrix remains unchanged. The changes in the mean increase in magnitude as i_sim increases.

• i_sim = 1: \(X_i(t)=X_1(t)+0.5.\)

• i_sim = 2: \(X_i(t)=X_1(t)+0.75.\)

• i_sim = 3: \(X_i(t)=X_1(t)+1.\)

For i_sim 4 and 5, the curves in the second group are obtained by multiplying the covariance matrix by a constant.

• i_sim = 4: \(X_i(t)=E_1(t)+2 \ e(t).\)

• i_sim = 5: \(X_i(t)=E_1(t)+0.25 \ e(t).\)

For i_sim = 6, the curves in the second group are obtained by adding to \(E_1(t)\) a centered Gaussian process \(h(t)\) whose covariance matrix is given by \[ Cov(h(t_i),h(t_j))=0.5 \exp (-\frac{\lvert t_i-t_j\rvert}{0.2})\].

• i_sim = 6: \(X_i(t)=E_1(t)+ \ h(t).\)

For i_sim 7 and 8, the curves in the second group are obtained by a different mean function \(E_2(t)=30t{(1-t)}^2\).

• i_sim = 7: \(X_8(t)=E_2(t)+ h(t).\)
• i_sim = 8: \(X_9(t)=E_2(t)+ e(t).\)

Output

The function returns a data matrix of size \(2n \times p\), where the first \(n\) rows contain the curves from the first group, and the next \(n\) rows contain the curves from the second group, generated according to the selected value of i_sim.

Usage

To simulate the curves, we can follow these steps:

n <- 5
curves <- sim_model_ex1(n = n, i_sim = 1)
dim(curves)
#> [1] 10 30

This code generated 10 curves over the interval [0,1] with 30 grid points, using i_sim=1

Also, we can plot them. To do so, we are going to load external packages. This is just an example of how to plot the curves, but the end user can do it the way they want.

library(tidyr)
library(dplyr)
library(ggplot2)
library(ggsci)
library(cowplot)

plot_curves <- function(curves, title = "", subtitle = "") {
  data <- as.data.frame(curves) |>
    mutate(curve_id = row_number())
  n <- dim(curves)[1] / 2

  data_long <- data |>
    pivot_longer(
      cols = -curve_id,
      names_to = "point",
      values_to = "value"
    ) %>%
    mutate(
      point = as.numeric(sub("V", "", point)),
      group = if_else(curve_id %in% 1:n, 1, 2)
    )

  ggplot(data_long, aes(x = point, y = value, group = curve_id, color = as.factor(group))) +
    geom_line() +
    labs(
      title    = title,
      subtitle = subtitle,
      color    = "Group",
      x        = "",
      y        = ""
    ) +
    scale_color_d3() +
    theme_half_open() +
    theme(legend.position = "none")
}

plot_curves(curves, title = "Type 1 DGPs", subtitle = "i_sim = 1")

The case for i_sim = 7 has the following graph representation:

Clear differences can be seen between both plots. We leave it up to the end user to experiment as they see fit with the simulations.

Parameters

• n: The number of curves to generate for each group. By default, this is set to 50 per group, resulting in a total of 100 curves.
• p: The number of grid points at which the curves are evaluated over the interval \([0, 1]\). The default is 150 grid points.
• i_sim: An integer between 1 and 4. 1 and 2 for one-dimensional functional data and 3 and 4 for the multidimensional case.

Function breakdown

For the one-dimensional case the first group of functions is generated by \[X_{1}(t)=E_3(t)+ A(t),\] where \(E_3(t)=t(1-t)\) is the mean function, and \[A(t) = \sum_{k=1}^{100} Z_k\sqrt{\rho_k}\theta_k(t),\] with \(\{Z_k, k=1,...,100\}\) being independent standard normal variables, and \(\{ \rho_k,k\geq 1 \}\) a positive real numbers sequence defined as \[\rho_k = \left\{ \begin{array}{lll} \frac{1}{k+1} & if & k \in \{1,2,3\}, \\ \frac{1}{{(k+1)}^2} & if & k \geq 4, \end{array} \right. \] in such a way that the values of \(\rho_k\) are chosen to decrease faster when \(k\geq 4\) in order to have most of the variance explained by the first three principal components. The sequence \(\{\theta_k, k\geq 1\}\) is an orthonormal basis of \(L^2(I)\) defined as \[\theta_k(t) = \left\{ \begin{array}{lllll} I_{[0,1]}(t) & if & k=1, & \\ \sqrt{2}\sin{(k\pi t)}I_{[0,1]}(t) & if & k \geq 2,\\ & & k \ even,\\ \sqrt{2}\cos{((k-1)\pi t)}I_{[0,1]}(t) & if & k \geq 3,\\ & & k \ odd, \end{array} \right. \] where \(I_A(t)\) stands for the indicator function of set \(A\).

The second group of data is generated as follows, depending on the value of i_sim:

• i_sim = 1: \(X_{i}(t)=E_4(t)+ A(t),\) where \(E_4(t)=E_2(t)+\displaystyle \sum_{k=1}^3\sqrt{\rho_k}\theta_k(t).\)

• i_sim = 2: \(X_{i}(t)=E_5(t)+ A(t),\) where \(E_5(t)=E_2(t)+\displaystyle \sum_{k=4}^{100}\sqrt{\rho_k}\theta_k(t).\)

For the two-dimensional case the first group of functions is generated by \[\mathbf{X}_{1}(t)=\mathbf{E}_6(t)+ B(t),\] where \(\mathbf{E}_6(t)= \begin{pmatrix} t(1-t)\\ 4t^2(1-t) \end{pmatrix}\) is the mean function of this process, and \[\mathbf{B}(t) = \sum_{k=1}^{100} \mathbf{Z}_k\sqrt{\rho_k}\theta_k(t),\] where \[\mathbf{E}_6(t)= \begin{pmatrix} t(1-t)\\ 4t^2(1-t) \end{pmatrix}\] is the mean function of this process, \(\{\mathbf{Z}_k, k=1,...,100\}\) are independent bivariate normal random variables, with mean \(\mathbf{\mu = 0}\) and covariance matrix \[\Sigma= \begin{pmatrix} 1 & 0.5\\ 0.5 & 1 \end{pmatrix}.\] The second group of data is generated as follows, depending on the value of i_sim:

• i_sim = 3: \(X_{i}(t)=\mathbf{E}_7(t)+ B(t),\) where \(\mathbf{E}_7(t)=\mathbf{E}_6(t)+\mathbf{1}\displaystyle \sum_{k=1}^3\sqrt{\rho_k}\theta_k(t),\) is the mean function of this process, where \(\mathbf{1}\) represents a vector of 1s.

• i_sim = 4: \(X_{i}(t)=\mathbf{E}_8(t)+ B(t),\) where \(\mathbf{E}_8(t)=\mathbf{E}_6(t)+\mathbf{1}\displaystyle \sum_{k=4}^{100}\sqrt{\rho_k}\theta_k(t).\)

Output

The function returns for the one-dimensional case (i_sim=1,2) a data matrix of size \(2n \times p\). The output for the two-dimensional case (i_sim=3,4)is an array of dimensions \(2n \times p \times 2\).

Usage

This function generates functional curves in one dimension for values of i_sim equal to 1 and 2, and functional datasets in two dimensions for values of i_sim equal to 3 and 4. Here, we would like to point out how to simulate a multivariate functional dataset with i_sim = 3 or i_sim = 4.

n <- 5
curves <- sim_model_ex2(n = n, i_sim = 4)
dim(curves)
#> [1]  10 150   2

As can be seen, now we don’t have a matrix but a 3-dimensional array. We are going to plot the first dimension:

And now the second dimension:

For the sake of visualization, we can also obtain the derivatives of the curves using the funspline function, which is an internal function of the ehymet package. It can be used with ehymet:::.

funspline_result <- ehymet:::funspline(curves)
deriv <- funspline_result$deriv
deriv2 <- funspline_result$deriv2

The plot for first dimension of the derivative is the following one:

And for the second dimensions:

We can observe a clear overlapping between first derivatives curves on both dimensions.