I wanted to generate random variables from a multivariate t distribution in R. i am using the mvtnorm package which has the command rmvt for generating random variables from the multivariate t-distribution. Now my question is about the syntax of the function and being able to manipulate it to do what I want. The function requires the following
rmvt(n, sigma = diag(2), df = 1, delta = rep(0, nrow(sigma)),
type = c("shifted", "Kshirsagar"), ...)
where sigma is a correlation matrix. Now what I am having trouble with is how to sample from a multivariate t-distribution with mean m and covariance matrix S. Is the following the appropriate syntax?
rmvt(1,S,df=n) + m
or
rmvt(1,R,df=n)*sigma + m
where my covariance matrix can be decomposed as S = sigma*R (i.e., R is my correlation matrix). I am getting different results when I run the two lines of code so that is partially where my confusion stems from.
Have a look at the help file for rmvt. There is says that sigma is the scale (not correlation) matrix and that the correlation matrix, which is only defined for df>2 is given by sigma * df/(df-2). Therefore is you have a pre-specified covariance matrix S then you should set
sigma=S*(D-2)/D
where D is the degrees of freedom. To generate n samples from the multivariate t-distribution with mean m and covariance matrix S you can either add the mean outside the call to rmvt, as you indicated:
rmvt(n, sigma=S*(D-2)/D, df=D) + m
or by using the mu argument:
rmvt(n, mu=m, sigma=S*(D-2)/D, df=D)
Edit: For whatever reason, rmvt is not loading properly on my machine so I have to type this first to have the function loaded properly:
rmvt <- bfp:::rmvt
Related
I'm wondering if there exist pre-programed R functions that can calculate geometrically weighted edgewise shared partners (GWESP, per Hunter (2007)) for a given adjacency matrix with fixed decay parameter (alpha) and then return the estimated values also in matrix form?
I've looked at xergm and igraph packages but could not find one, they only estimate GWESP during model (network models) fitting but that's not what I want to do here. I only need a function to calculate GWESP for a given adjacency matrix and return the estimated GWESP value (with fixed decay parameter) for each dyad of the adjacency matrix in matrix form as well.
For example
# for a given adjacency matrix (adjm)
adjm <- matrix(sample(0:1, 100, replace=TRUE, prob=c(0.6,0.4)), nc=10)
# apply some functions to calculate GWESP for each dyad of adjm (and fix alpha at some value) and return a matrix of the same dimension filled with estimated GWESP values
somefunction(adjm, alpha = somevalue)
I am have some difficulty understanding some steps in a procedure. They take coordinate data, find the covariance matrix, apply PCA, then extract the standard deviation from the square root of each eigenvalue in short. I am trying to re-produce this process, but I am stuck on the steps.
The Steps Taken
The data set consists of one matrix, R, that contains coordiante paris, (x(i),y(i)) with i=1,...,N for N is the total number of instances recorded. We applied PCA to the covariance matrix of the R input data set, and the following variables were obtained:
a) the principal components of the new coordinate system, the eigenvectors u and v, and
b) the eigenvalues (λ1 and λ2) corresponding to the total variability explained by each principal component.
With these variables, a graphical representation was created for each item. Two orthogonal segments were centred on the mean of the coordinate data. The segments’ directions were driven by the eigenvectors of the PCA, and the length of each segment was defined as one standard deviation (σ1 and σ2) around the mean, which was calculated by extracting the square root of each eigenvalue, λ1 and λ2.
My Steps
#reproducable data
set.seed(1)
x<-rnorm(10,50,4)
y<-rnorm(10,50,7)
# Note my data is not perfectly distirbuted in this fashion
df<-data.frame(x,y) # this is my R matrix
covar.df<-cov(df,use="all.obs",method='pearson') # this is my covariance matrix
pca.results<-prcomp(covar.df) # this applies PCA to the covariance matrix
pca.results$sdev # these are the standard deviations of the principal components
# which is what I believe I am looking for.
This is where I am stuck because I am not sure if I am trying to get the sdev output form prcomp() or if I should scale my data first. They are all on the same scale, so I do not see the issue with it.
My second question is how do I extract the standard deviation in the x and y direciton?
You don't apply prcomp to the covariance matrix, you do it on the data itself.
result= prcomp(df)
If by scaling you mean normalize or standardize, that happens before you do prcomp(). For more information on the procedure see this link that is introductory to the procedure: pca on R. That can walk you through the basics. To get the sdev use the the summary on the result object
summary(result)
result$sdev
You don't apply prcomp to the covariance matrix. scale=T bases the PCA on the correlation matrix and F on the covariance matrix
df.cor = prcomp(df, scale=TRUE)
df.cov = prcomp(df, scale=FALSE)
I read this introduction to sprandn and tried to create a sparse matrix obeying [-1,1] uniform distribution.
using SparseArrays
using Distributions
sprandn(100,100,0.3,Uniform(-1,1))
But it failed. I apologize for not having pasted the error log. Here is an image of what the error says in MethodError.
So how can I generate a [-1,1] uniform distribution sparse matrix?
sprandn is for sampling from a standard normal. However, there is a method of sprand that you can use:
sprand(m::Integer, n::Integer, density::AbstractFloat, rfn::Function)
The last argument is a function used internally for sampling the non-zero values, and you can use it like this:
D = Uniform(-1.0, 1.0)
rf(n) = rand(D, n)
sprand(100, 100, 0.3, rf)
If you want to specify the used RNG, this needs to be passed into rf as another argument in first position.
I have a pdf $f(x)=4x^3$ of a random variable $X$ in which I need to simulate a draw from the distribution.
My solution consists of finding the cdf from the pdf (1st issue):
> pdf <- function(x){4*x^3}
> cdf <- integrate(pdf,lower=0,upper=x)
Error in integrate(pdf, lower = 0, upper = x) : object 'x' not found
Once I get the cdf $U$, I will set $X=F^-1(U)$. I notice that the pdf follows a Beta distribution with $\alpha=4$ and $\beta=1$.
Is it best to find the $F^-1$ via a inverse beta function? Is there a quick way to find the inverse of a beta function in R?
Since you have identified your pdf as beta, just use rbeta to sample.
s1 <- rbeta(5000,4,1)
In the case where the distribution is non-standard and you cannot solve analytically, you can use rejection sampling. Let's pretend we don't know your pdf is beta and we don't know how to integrate/inverse.
pdf <- function(x) 4*x^3 # on [0,1]
First we draw from our proposal distribution
p <- runif(50000)
Calculate the density values under our pdf
dp <- pdf(p)
And randomly accept/reject in proportion
s2 <- p[runif(50000) < dp/max(dp)]
You should find the distributions of s1 and s2 comparable, using histograms or, preferably, a qqplot.
I would like to calculate a covariance matrix where the matrix components are obtained by an exponential moving average (i.e. S(t)=a*Y(t)+(1-a)*S(t-1)).
It seems to me that mewma should do that but when I call the function for a 60,400 matrix I get a 'subsrcript out of bounds error'.
I believe it comes from the line:
ucl <- h4[m1[l], m2[a[2] - 1]]
Can the mewma function be called for a matrix with a dim of more than 9 ?
Otherwise is there an alternative package?
I have followed:
https://stats.stackexchange.com/questions/19328/how-to-compute-an-exponentially-weighted-covariance-matrix-function-in-r