Raju Rimal, Trygve Almøy & Solve Sæbø

Simrel r-package is a versatile tool for simulation of multivariate linear model data. The package consist of four core functions – unisimrel, bisimrel, multisimrel and simrel for simulation. It also has two more functions – one for plotting covariance and rotation matrices and another for plotting different properties of simulated data. As the name suggests, unisimrel function is used for simulating univariate linear model data, bisimrel simulates bivariate linear model data where user can specify the correlation between two responses with and without given X. In addition, this function allows users to get responses (y) having common relevant components.

An extension of bisimrel and unisimrel is multisimrel, by which user can simulate multivariate linear model data with multiple responses. In this simulation, each response must have exclusive set of predictors and relevant predictors components. Following examples will give a clear picture of these functions. The forth simulation function simrel wraps around these function and calls them according to what type of data a user is simulating. Following section discusses about the arguments required for each of these simulation function.

A tool for simulating linear model data with single response discussed in sæbø et.al. (2015) is the basis for this function. The function require following arguments which are also parameters for the simulation.

**`gamma`:** Decay factor for exponential decay of eigenvalues of predictor variables

A numeric value greater than 0. It is a factor controlling exponential decay of eigenvalues of predictor variables. For `p` predictors, the eigenvalues are computed as,

[\lambda_i = e^{-\gamma(i - 1)}, i = 1, 2, \ldots, p]

So that, higher the value of gamma steeper will be the decay of eigenvalues. Since steeper eigenvalues corresponds to high multicollinearity in data, gamma also controls the multicollinearity present in the simulated data.

Following parameters (arguments) are used in these function,

Install the package from GitHub,

# install.pacakges("devtools")
devtools::install_github("simulatr/simrel")
devtools::install_bitbucket("simulatr/simrel")

Simulate a univariate linear model data with 100 training samples and 500 test samples having 10 predictors (X) where only 8 of them are relevant for the variation in the response vector. The population model should explain 80% of the variation present in the response. In addition, only 1st and 3rd principal components of X should be relevant for y and the eigenvalues of X decreases exponentially by a factor of 0.7.

library(simrel)
sim_obj <-
  simrel(
    n      = 100,         # 100 training samples
    p      = 10,          # 10 predictor variables
    q      = 8,           # only 8 of them are relevant
    R2     = 0.8,         # 80% of variation is explained by the model
    relpos = c(1, 3),     # First and third principal components are relevant
    gamma  = 0.7,         # decay factor of eigenvalue of X is 7
    ntest  = 500,         # 500 Test observations
    type   = "univariate" # Univariate linear model data simulation
)

Here sim_obj is a object with class simrel and constitue of a list of simulated linear model data along with other relevant properties. Lets use plot function to overview the situation,

ggsimrelplot(sim_obj, layout = matrix(c(1, 1, 2, 3), 2, 2, byrow = TRUE))

The wrapper function simrel uses bisimrel for simulating bivariate linear model data. Lets consider a situation to simulate data from bivariate distribution with 100 training and 500 test samples. The response vectors y₁ and y₂ have correlation of 0.8 without given X and 0.6 with given X. Among 10 total predictor variables, 5 are relevant for y₁ and 5 are relevant for y₂. However 3 of them are relevant for both of them. Let the predictors explain 80% and 70% of total variation present in population of y₁ and y₂ respectively. In addition, let 1, 2 and 3 components are relevant for y₁ and 3 and 4 components are relevant for y₂. In this case, the third component is relevant for both responses. Let the decay factor of eigenvalues of X be 0.8.

simrel2_obj <-
  simrel(
    n      = 100,                       # 100 training samples
    p      = 10,                        # 10 predictor variables
    q      = c(5, 5, 3),                # relevant variables for y1 and y2
    relpos = list(c(1, 2, 3), c(3, 4)), # relevant components for y1 and y2
    R2     = c(0.8, 0.7),               # Coefficient of variation for y1 and y2
    rho    = c(0.8, 0.6),               # correlation between y1 and y2 with and without given X
    gamma  = 0.8,                       # decay factor of eigenvalues of X
    ntest  = 500,                       # 500 test samples
    type   = "bivariate"
  )

Lets look at the plot,

ggsimrelplot(simrel2_obj, layout = matrix(c(1, 1, 2, 3), 2, 2, byrow = TRUE))

Multivariate simulation uses multisimrel function and can simulate multiple responses. Lets simulate 100 training samples and 500 test samples. The simulated data has 5 responses and 15 predictors. These 5 responses spans 5 latent space out of which only 3 are related to the predictors. Lets denote them by w_i. Let 5, 4 and 4 predictors are relevant for response components w₁, w₁ and w₁ respectively. The position of relevant predictor components for w₁ be 1, 2, 3; for w₂ be 4 and 5. Similarly, predictor components 6 and 8 are relevant for w₃.

Since we need 5 response variables, we mix-up these 3 informative response components with 2 remaining uninformative components so that all simulated response contains information that X are related. Lets combine w₁ with w₄ and w₃ with w₅. So that the predictors that are relevant for response components w₁ will be relevant for response y₁ and y₃ and so on.

In addition to these latent space requirements, let X explains 80% variation present in w₁, 50% in w₂ and 70% in w₃. The eigenvalues of X reduces by the factor of 0.8.

simrel_m_obj <-
simrel(
    n      = 100,                                # 100 training samples
    p      = 15,                                 # 15 predictor variables
    q      = c(5, 4, 4),                         # relevant variables for w1, w2 and w3
    relpos = list(c(1, 2, 3), c(4, 5), c(6, 8)), # relevant components for w1, w2 and y3
    R2     = c(0.8, 0.5, 0.7),                   # Coefficient of variation for w1, w2 and y3
    ypos   = list(c(1, 4), c(2), c(3, 5)),       # combining response components together
    m      = 5,                                  # Number of response
    gamma  = 0.8,                                # decay factor of eigenvalues of X
    ntest  = 500,                                # 500 test samples
    type   = "multivariate"                      # multivariate simulation
  )

Lets look at the simrel plot;

ggsimrelplot(simrel_m_obj, layout = matrix(c(1, 1, 2, 3), 2, 2, byrow = TRUE))

To make the process easier to use, we have created a shiny gadget as an rstudio addons. If you are using Rstuio, you can access this app from Tools > Addins > simulatr. But you can also access this app using simrel::app_simulatr(). This will open the app in a browser from where you can choose all your parameter, see the true population parametrs you will get from the simulation. When the app is closed, it will give an command output on your R console.