DGPs introduced in the package
ehymet introduces two functions which generate different
Data Generation Processes (DGPs) for functional data in one or multiple
dimension. The first function, sim_model_ex1, generate DGPs
in one dimension as originally described by (Flores, Lillo, and Romo 2018). The second
function, sim_model_ex2, produces DGPs in both one and two
dimensions first introduced by (Martino et al.
2019).
These functions are useful for simulating functional data to test statistical methods or to understand the behavior of different models under controlled conditions. Each of the two functions display different DGPs depending on the given parameters, each of them with different characteristics.
Type 1 DGPs
The first set of DGPs is generated by function
sim_model_ex1, which produces eight different DGPs, each
corresponding to a different value of i_sim. Here’s a
breakdown of how each DGP is generated and how the function works:
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:
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.
Type 2 DGPs
The second set of DGPs is generated by function
sim_model_ex2, which produces four different DGPs, each
corresponding to a different value of i_sim. The first
two DGPs are in one dimension, while the remaining two DGPs are
two-dimensions functional datasets. Here’s a breakdown of how each DGP
is generated and how the function works:
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.
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$deriv
deriv2 <- funspline_result$deriv2The 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.
