Authors: Chitta Ranjan and Somrita Sarkar.
This repository provides implemented codes in R for generation of a full, sparse, and block covariance matrices.
A randomly generated covariance matrix is usually used for simulating random correlated data. The correlation structures are desired to have different characteristics for different problems. The provided functions help generate covariances for most of the problems.
A covariance is a positive definite matrix: it’s symmetric and invertible.
Using SimulateCov(p, corrRange, sdRange) we generate a positive definite covariance matrix.
> covarianceMatrix <- SimulateCov(p = 5,
corrRange = list(min = -0.7, max = 0.9),
sdRange = list(min = 0.5, max = 0.75))
> covarianceMatrix
[,1] [,2] [,3] [,4] [,5]
[1,] 0.42173609 0.231879364 0.05170357 -0.188813524 0.07804691
[2,] 0.23187936 0.418240784 -0.06683895 0.001306713 -0.05796761
[3,] 0.05170357 -0.066838954 0.28883131 -0.096017196 -0.18772132
[4,] -0.18881352 0.001306713 -0.09601720 0.430207248 0.03353824
[5,] 0.07804691 -0.057967609 -0.18772132 0.033538245 0.27670991
> print(PlotMatrix(covarianceMatrix))
The inputs to the function are,
p the number variables (rows/columns in the covariance matrix).
corrRange the range for the correlations between the p variables. The range can be between [-1.0, 1.0].
sdRange the range for the standard deviations of the p variables. Its range can be (0, infinity).
The function SimulateCov works by first randomly generating a correlation matrix and then converts it into a positive definite matrix, which is used at the covariance. The argument sdRange is required for this conversion to yield a unique solution. Without defining the sdRange, there can be several solutions to the correlation-to-covariance conversion. The sdRange becomes the upper and lower limit for the diagonal elements in covarianceMatrix.
We generate a sparse covariance matrix by just adding a sparsity parameter to the SimulateCov function.
> sparseCovariance <- SimulateCov(p = 10,
corrRange = list(min = -0.7, max = 0.9),
sdRange = list(min = 0.5, max = 0.75),
sparsity = 0.1)
> sparseCovariance
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 4.636766e-01 0.000000e+00 0.43998843 0.0000000 0.0000000 -2.553911e-18 0.0000000 0.0000000 -1.012174e-18 0.0000000
[2,] 0.000000e+00 5.432883e-01 0.00000000 0.0000000 0.0000000 -8.846332e-17 0.0000000 0.0000000 -2.804806e-17 0.0000000
[3,] 4.399884e-01 0.000000e+00 0.53617202 0.0000000 0.0000000 -7.158354e-02 0.0000000 0.0000000 -2.043421e-02 0.0000000
[4,] 0.000000e+00 0.000000e+00 0.00000000 0.4818959 0.0000000 0.000000e+00 0.0000000 0.0000000 0.000000e+00 0.0000000
[5,] 0.000000e+00 0.000000e+00 0.00000000 0.0000000 0.3339609 0.000000e+00 0.0000000 0.0000000 0.000000e+00 0.0000000
[6,] -2.553911e-18 -8.846332e-17 -0.07158354 0.0000000 0.0000000 5.193895e-01 0.0000000 0.0000000 2.311240e-16 0.0000000
[7,] 0.000000e+00 0.000000e+00 0.00000000 0.0000000 0.0000000 0.000000e+00 0.2922367 0.0000000 0.000000e+00 0.0000000
[8,] 0.000000e+00 0.000000e+00 0.00000000 0.0000000 0.0000000 0.000000e+00 0.0000000 0.5018156 0.000000e+00 0.0000000
[9,] -1.012174e-18 -2.804806e-17 -0.02043421 0.0000000 0.0000000 2.311240e-16 0.0000000 0.0000000 4.699101e-01 0.0000000
[10,] 0.000000e+00 0.000000e+00 0.00000000 0.0000000 0.0000000 0.000000e+00 0.0000000 0.0000000 0.000000e+00 0.4043639
> print(PlotMatrix(sparseCovariance))
Other than the inputs required for Full Covariance, we have,
sparsity The sparsity level in covariance. Only up to a fraction, equal to the sparsity, of the covariance matrix will be non-zero (excluding the diagonal). The sparsity should be between 0 and 1.
We can generate block covariance using SimulateBlockCovariance(totalVars, maxVarsInABlock, corrRange, sdRange).
> blockCovariance <- SimulateBlockCovariance(totalVars = 10,
maxVarsInABlock = 3,
corrRange = list(min = -0.5, max = 0.9),
sdRange = list(min = 0.5, max = 0.75))
> blockCovariance
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 0.5253886 0.09911930 0.12001999 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.00000000 0.00000000
[2,] 0.0991193 0.55887218 -0.06217979 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.00000000 0.00000000
[3,] 0.1200200 -0.06217979 0.30964417 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.00000000 0.00000000
[4,] 0.0000000 0.00000000 0.00000000 0.2866114 0.1846021 0.1510859 0.0000000 0.0000000 0.00000000 0.00000000
[5,] 0.0000000 0.00000000 0.00000000 0.1846021 0.4078875 0.3638950 0.0000000 0.0000000 0.00000000 0.00000000
[6,] 0.0000000 0.00000000 0.00000000 0.1510859 0.3638950 0.5286350 0.0000000 0.0000000 0.00000000 0.00000000
[7,] 0.0000000 0.00000000 0.00000000 0.0000000 0.0000000 0.0000000 0.3764742 -0.1397405 0.00000000 0.00000000
[8,] 0.0000000 0.00000000 0.00000000 0.0000000 0.0000000 0.0000000 -0.1397405 0.2542494 0.00000000 0.00000000
[9,] 0.0000000 0.00000000 0.00000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.29512556 -0.08383803
[10,] 0.0000000 0.00000000 0.00000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 -0.08383803 0.43995215
> print(PlotMatrix(blockCovariance))
The new inputs in this function are,
totalVars Total number of variables (rows/columns) in the generated block covariance matrix.
maxVarsInABlock The maximum number of variables allowed in one block.
As can be seen in the above sample output, the blocks sizes are no more than 3 (ranging from 1 to 3). Also, the standard deviations (on the diagonal) are between 0.5 and 0.75.
The code is supported by the Science team at ProcessMiner, Inc. If you have any questions/comments, please leave them here in the comments section or email at [email protected].