Simulation of Multivariate Linear Model Data
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 –
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
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) and multi-response discussed in rimal et.al. (2018) is the basis for these functions. The function require following arguments which are also parameters for the simulation.
integer for number of training samples. For example:
n: Number of training samples (n)
n = 1000simulates 1000 training observations.
integer for number of predictor variables.
p: Number of predictor variables (p)
p = 150gives data with 150 predictor variables.
integer for the number of predictor variables that are relevant for the
response. For example:
q: Number of relevant predictors (q)
q = 15results 15 predictors out of
prelevant for the response.
A vector of position index of relevant principal components of
x. For instance,
relpos: Position of relevant components
relpos = c(1, 2, 3, 5)will give data with 4 relevant components at position 1, 2, 3 and 5.
decimal value between 0 and 1 specifying the coefficient of
determination. Input of
R2: Coefficient of determination
R2 = 0.8gives data with 0.8 coefficient of determination.
A numeric value greater
than 0. It is a factor controlling exponential decay of eigenvalues of
predictor variables. For
gamma: Decay factor for exponential decay of
eigenvalues of predictor variables
ppredictors, the eigenvalues are computed as
i=1, 2, …, pso that, higher the value of
gammasteeper will be the decay of eigenvalues. Since steeper eigenvalues corresponds to high multicollinearity in data,
gammaalso controls the multicollinearity present in the simulated data.
eta: Decay factor for exponential decay of
eigenvalues of response variables
gamma, it is a factor controlling exponential decay of eigenvalues of response variables. For
mresponses, the eigenvalues are computed as
j=1, 2, …, mso that, higher the value of
etasteeper will be the decay of eigenvalues.
An integer specifying the number of
response variables to simulate. This is only applicable in Multivariate
m:Number of Response Variables (Only applicable
for Univariate Simulation)
The true dimension of response matrix
containing the information can be smaller than the dimension that is
defined by all the response variables. Lets suppose for an example that
only two response components actually contains information that the
predictor (or a subset of predictor) can explain. However, In the
simulation data user might want 5 response variables that contains the
same inforamtion contained in two latent component of the respone
ypos:Position of response components indices to
yposparameters if specified as
list(c(1, 3), c(2, 4, 5))will mix up the inforamtion in response component 1 with uninformative component 3 so that the response variable 1 and 3 contains the same information that was contained in response component 1. A similar description can be made for the second response component which is mixed with non-informative response components 4 and 5.
A vector of two
numbers specifying the correlation between the the two response variable
in bivariate simulation. The first number is for the correlation without
the knownledge of predictors while the second number is for the
correlation given the predictors. These values should be between the
range of a correlation, i.e. -1 to 1.
rho:Correlation between two response variables
(Only applicable for Bivariate Simulation
The package is now available in CRAN and can be installed using
A more recent stable version can be download from GitHub as,
# 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 )
sim_obj is a object with class
simrel and constitue of a list
of simulated linear model data along with other relevant properties.
plot function to overview the situation,
plot_simrel(sim_obj, which = c(1, 2, 4), layout = matrix(c(1, 1, 2, 3), 2, 2, byrow = TRUE))
The wrapper function
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,
plot_simrel(simrel2_obj, which = c(1, 3, 4), 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 wi. 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
plot_simrel(simrel_m_obj, which = 1:4, layout = matrix(1:4, 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.
- Sæbø, S., Almøy, T., & Helland, I. S. (2015). simrel—A versatile tool for linear model data simulation based on the concept of a relevant subspace and relevant predictors. Chemometrics and Intelligent Laboratory Systems, 146, 128-135.
- Rimal, R., Almøy, T., & Sæbø, S. (2018). A tool for simulating multi-response linear model data. Chemometrics and Intelligent Laboratory Systems, 176, 1-10.