Generation and visualization of multidimensional data with R

The ability to generate data with a given correlation is very important for modeling. In R, an extensive set of tools is expected — packages and functions for generating and visualizing data from multidimensional distributions. The basic function for generating multidimensional normally distributed data is mvrnorm() from the MASS package, part R, although the mvtnorm package also offers functions for simulating both multidimensional normal and t-distribution.

The code block below generates 5,000 samples from a two-dimensional normal distribution with an average (0, 0) and Sigma covariance matrix given in the code. The kde2d() function, also from the MASS package, generates a two-dimensional sound density distribution estimate .

 #    # https://stat.ethz.ch/pipermail/r-help/2003-September/038314.html #      library(MASS) #     mu <- c(0,0) #  Sigma <- matrix(c(1, .5, .5, 1), 2) #   

 # > Sigma # [,1] [,2] # [1,] 1.0 0.1 # [2,] 0.1 1.0 

 #    N(mu, Sigma) bivn <- mvrnorm(5000, mu = mu, Sigma = Sigma ) #   MASS head(bivn) #      bivn.kde <- kde2d(bivn[,1], bivn[,2], n = 50) #   MASS 

R offers several ways to visualize the distribution. The next two lines of code impose a contour plot on the heat map, which assigns a color gradient to the point density.

 #     ,     image(bivn.kde) #     contour(bivn.kde, add = TRUE) #     

The graph shows incorrect contours of simulated data. The code below, using the ellipse() function from the ellipse package, generates the classic two-dimensional normal distribution graph found in many textbooks.

 #      library(ellipse) rho <- cor(bivn) y_on_x <- lm(bivn[,2] ~ bivn[,1]) #  Y ~ X x_on_y <- lm(bivn[,1] ~ bivn[,2]) #  X ~ Y plot_legend <- c("99%  ", "95%  ","90%  ", "Y  X ", "X  Y ") plot(bivn, xlab = "X", ylab = "Y", col = "dark blue", main = "     ") lines(ellipse(rho), col="red") # ellipse()   ellipse lines(ellipse(rho, level = .99), col="green") lines(ellipse(rho, level = .90), col="blue") abline(y_on_x) abline(x_on_y, col="brown") legend(3,1,legend=plot_legend,cex = .5, bty = "n") 

The following piece of code generates a couple of three-dimensional surface graphs. The second is the rgl chart, which can be rotated and viewed from different angles directly on the screen.

 #   #    persp(bivn.kde, phi = 45, theta = 30, shade = .1, border = NA) #     #   RGL library(rgl) col2 <- heat.colors(length(bivn.kde$z))[rank(bivn.kde$z)] persp3d(x=bivn.kde, col = col2) 

Now we’ll write some code to get the x, y, and z values ​​from the table coordinates of the core density estimate. They will allow you to build a surface using the new scatterplot3js () function from the htmlwidgets, threejs javascript package. This visualization does not depict the surface at the same level of detail as the rgl plot. Nevertheless, it shows some basic functions of pdf and has a significant advantage - it is easy to integrate into web pages. I suppose graphics in the form of html-widgets will be easier and easier to use.

 # threejs Javascript- library(threejs) #      kde x <- bivn.kde$x; y <- bivn.kde$y; z <- bivn.kde$z #   x,y,z xx <- rep(x,times=length(y)) yy <- rep(y,each=length(x)) zz <- z; dim(zz) <- NULL #    ra <- ceiling(16 * zz/max(zz)) col <- rainbow(16, 2/3) #  3D-  scatterplot3js(x=xx,y=yy,z=zz,size=0.4,color = col[ra],bg="black") 

The code below uses the rtmvt() function from the tmvtnorm package to generate a two-dimensional t distribution. The rgl plot depicts the surface of the sound density distribution estimate in detail.

 #      t- library (tmvtnorm) Sigma <- matrix(c(1, .1, .1, 1), 2) #   X1 <- rtmvt(n=1000, mean=rep(0, 2), sigma = Sigma, df=2) #   tmvtnorm t.kde <- kde2d(X1[,1], X1[,2], n = 50) #   MASS col2 <- heat.colors(length(bivn.kde$z))[rank(bivn.kde$z)] persp3d(x=t.kde, col = col2) 

The real value of multidimensional distribution functions in terms of data science is to simulate data sets with more than two variables. The functions proposed above are suitable for this task, but there are some technical considerations and, of course, visualization will not be available. The code snippet below generates 10 variables from a multidimensional normal distribution with a given covariance matrix. Please note that the genPositiveDefmat() function from the clusterGeneration package was used to generate the covariance matrix. This is because the mvrnorm() function will mvrnorm() error, as it should theoretically happen if the covariance matrix is ​​not positively defined, and selecting a combination of elements of a multidimensional matrix to make it positively defined will require decent luck and computation time.

After generating the matrix, I use the corrplot() function from the corrplot package to derive a beautiful graph of pairwise correlations determined by color and shape. corrplot() scales well with an increase in the number of variables and will produce a decent result for 40-50 variables. (Note that ggcorrplot is now used for ggplot2 graphs.) You can use other options to build pairwise scatterplots, and R offers many alternatives.

 #  library(corrplot) library(clusterGeneration) mu <- rep(0,10) pdMat <- genPositiveDefMat(10,lambdaLow=10) Sigma <- pdMat$Sigma dim(Sigma) mvn <- mvrnorm(5000, mu = mu, Sigma = Sigma ) corrplot(cor(mvn), method="ellipse", tl.pos="n", title=" ") 

Finally, what about other multidimensional distributions other than the normal and t-distributions? R offers several functions, such as rlnorm() from the packages of compositions , which generates random variables from a multidimensional lognormal distribution. They are as easy to use as mvrorm() , but she will have to look for others like her. I think a more fruitful approach if you really need to work with probability distributions is to get acquainted with the copula package.