psandovalv — Jan 11, 2013, 4:45 PM
##########################################
# prerequisites:
#
# install the CRAN package "corpcor"
# (version >= 1.4.4)
# this can be done, e.g., by running
# install.packages("corpcor")
# from within R
##########################################
# shrinkage covariance estimator
# compared with usual unbiased sample estimator
##########################################
# Example script (to be run in R) #
##########################################
# small data data set
# with 10 variables (columns)
# and 6 repetitions (rows)
X <- as.matrix(read.table("smalldata.txt")) # tab-delimited data
dim(X) # 6x10 matrix
[1] 6 10
# load "corpcor" R package
library("corpcor")
Warning: package 'corpcor' was built under R version 2.15.1
# estimate 10x10 covariance matrix
s1 <- cov(X) # usual unbiased sample estimate
s2 <- cov.shrink(X) # shrinkage estimate
Estimating optimal shrinkage intensity lambda.var (variance vector): 0.6015
Estimating optimal shrinkage intensity lambda (correlation matrix): 0.7315
# shrinkage intensity variances: 0.6015
# shrinkage intensity correlations: 0.7315
# compare ranks and conditions
rank.condition(s1) # rank=5, condition=Inf
$rank
[1] 5
$condition
[1] Inf
$tol
[1] 9.763e-14
rank.condition(s2) # rank=10, condition=4.5130
$rank
[1] 10
$condition
[1] 4.513
$tol
[1] 3.54e-14
# compare positive definiteness
is.positive.definite(s1) # FALSE
[1] FALSE
is.positive.definite(s2) # TRUE
[1] TRUE
# which estimator can be inverted?
solve(s1) # throws error because s1 is singular
Error: system is computationally singular: reciprocal condition number =
5.57281e-19
solve(s2) # no problem at all (s2 has full rank and is positive definite)
V1 V2 V3 V4 V5 V6 V7
V1 0.184857 0.014059 -0.007324 -0.0132236 -0.022154 -0.018317 -0.022948
V2 0.014059 0.109423 0.001655 -0.0028444 -0.009440 -0.002232 -0.018430
V3 -0.007324 0.001655 0.239947 0.0260179 -0.028809 0.016652 0.003523
V4 -0.013224 -0.002844 0.026018 0.1143938 -0.005208 -0.016895 -0.019651
V5 -0.022154 -0.009440 -0.028809 -0.0052076 0.235981 0.004431 -0.027097
V6 -0.018317 -0.002232 0.016652 -0.0168949 0.004431 0.132830 -0.022672
V7 -0.022948 -0.018430 0.003523 -0.0196507 -0.027097 -0.022672 0.223272
V8 0.022174 -0.013944 0.004888 -0.0007556 -0.015381 0.013334 0.003636
V9 0.009165 0.005039 -0.005042 0.0018101 -0.026167 0.019858 0.014345
V10 0.008012 -0.016680 0.022028 -0.0136779 0.001272 -0.010163 -0.015305
V8 V9 V10
V1 0.0221736 0.0091646 0.0080119
V2 -0.0139440 0.0050390 -0.0166802
V3 0.0048882 -0.0050417 0.0220282
V4 -0.0007556 0.0018101 -0.0136779
V5 -0.0153812 -0.0261666 0.0012715
V6 0.0133343 0.0198578 -0.0101626
V7 0.0036355 0.0143446 -0.0153054
V8 0.1521926 -0.0309098 -0.0127246
V9 -0.0309098 0.1960516 0.0006932
V10 -0.0127246 0.0006932 0.1052154
##################################
##################################
# large data data set
# with 100 variables (columns)
# and 20 repetitions (rows)
X <- as.matrix(read.table("largedata.txt")) # tab-delimited data
dim(X) # 20x100 matrix
[1] 20 100
# load "corpcor" R package
library("corpcor")
# estimate 100x100 covariance matrix
s1 <- cov(X) # usual unbiased sample estimate
s2 <- cov.shrink(X) # shrinkage estimate
Estimating optimal shrinkage intensity lambda.var (variance vector): 0.7772
Estimating optimal shrinkage intensity lambda (correlation matrix): 0.8851
# shrinkage intensity variances: 0.7772
# shrinkage intensity correlations: 0.8851
# compare ranks and conditions
rank.condition(s1) # rank=19, condition=Inf
$rank
[1] 19
$condition
[1] Inf
$tol
[1] 2.421e-11
rank.condition(s2) # rank=100, condition=2.8671
$rank
[1] 100
$condition
[1] 2.867
$tol
[1] 4.673e-12
# compare positive definiteness
is.positive.definite(s1) # FALSE
[1] FALSE
is.positive.definite(s2) # TRUE
[1] TRUE
# which estimator can be inverted?
solve(s1) # throws error because s1 is singular
Error: system is computationally singular: reciprocal condition number =
4.18644e-21
solve(s2) # no problem at all (s2 has full rank and is positive definite)
V1 V2 V3 V4 V5 V6
V1 1.078e-02 7.678e-05 -1.893e-04 9.135e-05 4.895e-05 2.216e-04
V2 7.678e-05 1.039e-02 -1.188e-04 7.194e-05 -4.463e-07 5.647e-05
V3 -1.893e-04 -1.188e-04 9.594e-03 -9.408e-05 -1.975e-04 -2.253e-04
V4 9.135e-05 7.194e-05 -9.408e-05 1.197e-02 3.032e-04 -3.030e-04
V5 4.895e-05 -4.463e-07 -1.975e-04 3.032e-04 1.155e-02 2.000e-05
V6 2.216e-04 5.647e-05 -2.253e-04 -3.030e-04 2.000e-05 1.004e-02
V7 -1.946e-04 -2.332e-04 1.553e-04 -3.054e-04 2.568e-04 4.680e-06
V8 3.843e-05 2.350e-04 -5.949e-05 6.559e-05 -2.349e-04 6.128e-05
V9 -1.210e-04 3.525e-04 -2.677e-05 -3.545e-04 2.791e-04 4.824e-05
V10 -2.815e-04 2.377e-04 -1.268e-04 5.671e-05 -1.668e-04 1.619e-05
V11 -3.066e-04 5.202e-05 -1.530e-05 -9.352e-05 1.661e-04 1.690e-04
V12 -3.553e-04 2.817e-04 6.540e-05 -6.038e-05 7.215e-05 1.292e-04
V13 -1.212e-04 -2.866e-04 1.392e-04 3.590e-04 -2.048e-04 3.172e-04
V14 1.491e-04 5.445e-05 3.050e-04 8.245e-05 1.513e-04 -3.980e-05
V15 1.355e-04 1.161e-04 2.350e-04 -1.639e-04 6.249e-05 -9.825e-05
V16 6.666e-05 4.436e-05 2.833e-04 1.573e-04 -1.661e-04 -3.580e-04
V17 3.179e-04 3.272e-06 -4.800e-05 -1.912e-04 -6.380e-05 -1.602e-04
V18 5.755e-05 3.736e-05 3.292e-04 -1.381e-04 8.882e-06 -1.261e-05
V19 -2.136e-04 -1.485e-04 1.088e-04 2.120e-05 1.643e-04 7.019e-05
V20 -1.811e-04 -3.189e-04 -1.999e-05 -2.818e-05 2.641e-04 -1.591e-04
V21 -9.469e-05 -8.604e-06 2.652e-04 2.956e-05 3.722e-04 2.649e-04
V22 -3.488e-05 -3.795e-05 9.302e-06 1.526e-04 -3.110e-04 -2.719e-04
V23 4.899e-05 -7.190e-05 -6.615e-05 5.603e-05 1.587e-04 -9.222e-05
V24 3.107e-05 -5.028e-05 1.757e-05 2.225e-04 -9.624e-05 2.356e-04
V25 -7.805e-05 2.068e-04 -2.778e-05 -2.715e-04 1.283e-04 -1.756e-04
V26 -5.140e-05 -1.104e-04 1.476e-04 -2.331e-04 3.428e-04 7.345e-05
V27 -2.345e-04 1.832e-04 -4.403e-04 -2.647e-05 -4.390e-04 -2.891e-04
V28 -3.997e-06 3.454e-05 -1.800e-04 -2.035e-04 -4.577e-05 -8.220e-05
V29 -2.229e-04 -1.763e-04 -2.778e-04 2.476e-04 -3.673e-05 1.738e-04
V30 -3.034e-04 2.019e-04 -4.064e-05 -2.753e-04 9.812e-05 -4.690e-05
V31 1.594e-05 -2.246e-04 1.535e-06 6.165e-05 3.472e-04 4.697e-05
V32 -1.280e-04 -1.055e-04 1.938e-04 -2.392e-04 4.255e-04 3.413e-04
V33 8.650e-05 1.002e-04 -6.321e-05 5.231e-05 -5.843e-04 -6.785e-05
V34 1.848e-04 3.893e-05 -2.279e-04 1.931e-04 -1.496e-04 1.654e-05
V35 -5.761e-05 -2.686e-04 -1.653e-04 -1.926e-04 1.222e-04 -1.382e-04
V36 -1.798e-04 2.907e-04 -7.012e-05 -4.296e-05 -6.243e-05 2.763e-04
V37 -3.178e-04 -1.231e-05 1.502e-04 -3.094e-04 3.071e-04 2.141e-04
V38 2.662e-04 -2.394e-04 1.348e-04 3.605e-04 7.818e-05 -1.855e-05
V39 -1.649e-04 5.325e-05 -1.155e-04 2.358e-04 -1.760e-05 4.233e-04
V40 -9.771e-05 2.123e-04 -2.047e-04 1.545e-06 2.329e-04 -1.519e-04
V41 3.961e-04 5.593e-05 8.286e-05 3.062e-05 -1.427e-04 -1.969e-04
V42 -3.485e-04 9.717e-05 -2.938e-04 -1.163e-04 -1.889e-04 1.762e-04
V43 1.080e-04 -1.345e-04 8.920e-05 8.426e-05 -6.524e-06 -1.027e-04
V44 7.222e-05 -3.947e-04 -5.109e-05 1.280e-04 -5.144e-05 8.092e-05
V45 9.400e-06 1.103e-05 3.212e-04 3.340e-04 -1.282e-04 -2.747e-05
V46 1.965e-04 -1.802e-05 1.772e-04 -2.017e-05 3.890e-05 -4.973e-06
V47 -2.845e-05 -1.115e-04 1.336e-05 8.635e-05 -5.104e-05 -1.677e-04
V48 1.948e-04 -3.364e-05 -1.571e-04 3.663e-05 1.219e-04 -2.922e-04
V49 1.578e-04 -5.110e-05 2.220e-04 2.019e-05 -3.335e-04 3.168e-04
V50 -1.810e-04 -1.376e-04 -2.283e-04 1.923e-04 -1.346e-04 -3.960e-05
V51 3.129e-04 -1.911e-04 5.285e-05 -3.050e-04 -7.830e-05 1.548e-04
V52 -1.701e-04 3.705e-04 3.162e-04 1.053e-04 1.720e-04 2.385e-04
V53 -9.597e-05 9.841e-05 -2.390e-04 -5.097e-04 1.761e-04 -2.140e-04
V54 -2.730e-04 -1.244e-04 -3.387e-04 5.385e-05 -1.477e-04 2.904e-05
V55 2.272e-05 3.959e-04 4.637e-05 5.621e-05 -9.065e-05 9.848e-05
V56 -2.336e-04 9.335e-05 2.556e-04 2.688e-04 -1.996e-04 1.600e-04
V57 -1.520e-04 1.269e-04 -1.658e-04 5.443e-05 1.994e-04 -8.627e-05
V58 -1.192e-04 -2.483e-05 5.881e-05 8.142e-05 2.457e-05 3.239e-04
V59 1.028e-04 1.133e-04 -1.123e-04 -2.991e-04 -1.369e-04 -1.034e-04
V60 3.869e-04 -1.380e-04 -3.050e-05 -3.752e-04 1.517e-04 -1.650e-04
V61 -6.205e-05 6.357e-05 -1.730e-04 1.588e-04 -1.298e-04 -2.878e-04
V62 -4.611e-05 1.686e-04 -2.670e-05 -3.375e-04 2.798e-04 -1.470e-04
V63 -2.688e-04 -1.833e-04 -1.215e-04 1.211e-04 -6.632e-05 4.901e-05
V64 1.157e-04 -1.936e-04 4.183e-05 -1.571e-04 -1.063e-04 -8.905e-07
V65 -2.969e-05 -5.533e-08 -1.484e-05 5.846e-07 3.395e-04 -1.382e-04
V66 7.513e-05 -1.185e-04 1.513e-04 -3.452e-06 4.519e-05 1.129e-04
V67 2.732e-04 -3.140e-05 2.201e-04 -1.792e-05 2.712e-04 -2.533e-05
V68 -1.591e-04 4.553e-05 1.122e-04 -6.503e-05 2.696e-04 -5.197e-05
V69 -6.125e-05 -2.083e-04 1.938e-04 1.674e-04 -3.820e-04 1.408e-04
V70 1.005e-04 3.208e-04 1.857e-04 1.978e-05 4.320e-04 -7.913e-05
V71 -1.748e-04 -2.065e-04 1.341e-04 1.441e-04 -2.486e-04 1.213e-04
V72 2.709e-06 2.311e-04 -1.662e-05 -3.312e-05 -6.167e-05 -4.285e-05
V73 2.049e-04 6.589e-05 -5.000e-05 3.754e-05 -2.159e-04 -2.123e-04
V74 -2.195e-04 -2.071e-04 -1.562e-04 -8.750e-05 4.295e-05 -3.169e-04
V75 -4.032e-06 1.366e-04 -6.251e-05 -4.095e-04 -7.689e-05 -6.900e-05
V76 1.430e-04 2.407e-04 9.597e-05 2.979e-04 1.110e-04 2.123e-05
V77 8.892e-05 4.062e-05 2.022e-04 9.066e-05 5.967e-05 3.034e-04
V78 7.389e-05 -2.456e-04 -1.816e-04 -1.890e-04 6.859e-05 7.523e-06
V79 7.880e-05 -3.006e-04 -3.369e-04 -7.358e-05 -2.141e-04 -3.532e-04
V80 2.758e-04 1.773e-04 1.080e-04 2.798e-04 -2.080e-04 5.803e-05
V81 1.225e-04 5.128e-06 3.860e-04 1.271e-04 1.577e-04 5.208e-05
V82 -6.442e-05 2.327e-04 -2.121e-04 3.255e-04 -7.556e-05 -2.758e-05
V83 -1.461e-04 1.976e-04 -8.941e-05 -1.814e-04 -2.974e-04 9.553e-05
V84 -5.408e-05 3.202e-06 1.194e-04 1.425e-05 9.185e-05 8.705e-05
V85 3.228e-04 5.324e-05 2.004e-04 -2.499e-04 -2.446e-04 3.383e-05
V86 -1.340e-04 1.137e-05 -3.069e-05 -1.815e-04 5.928e-05 -2.399e-04
V87 3.350e-04 -2.370e-04 -3.669e-05 -2.726e-04 -1.399e-04 -2.069e-04
V88 3.020e-04 -9.671e-05 -1.380e-04 -1.717e-04 -1.762e-05 9.278e-06
V89 8.990e-06 -1.718e-04 -7.355e-05 -2.190e-04 -5.938e-05 2.267e-04
V90 -7.408e-05 -3.237e-04 -3.142e-04 3.722e-04 -1.445e-04 3.643e-06
V91 -3.332e-04 -1.219e-04 -1.904e-04 1.653e-04 -2.281e-05 3.505e-04
V92 -8.897e-05 -3.001e-04 -1.053e-04 -7.095e-05 -1.284e-05 1.146e-04
V93 -2.526e-05 8.170e-05 -2.624e-05 -3.276e-04 -1.028e-04 6.789e-05
V94 3.981e-05 4.486e-04 -2.535e-05 4.841e-06 -2.028e-04 -1.160e-04
V95 3.471e-05 3.519e-04 2.566e-04 -1.731e-05 9.296e-05 1.312e-04
V96 2.182e-05 -1.528e-04 2.959e-04 1.058e-04 7.034e-05 7.566e-06
V97 -3.115e-04 1.697e-05 -3.590e-04 -1.642e-04 -1.973e-04 -1.032e-04
V98 3.166e-05 -1.439e-04 -8.764e-05 2.075e-04 -3.089e-04 1.301e-04
V99 -1.879e-04 2.903e-04 4.927e-05 -4.865e-04 1.262e-04 -2.440e-04
V100 2.316e-04 -1.053e-04 -7.536e-06 -3.380e-04 1.042e-04 -2.502e-04
V7 V8 V9 V10 V11 V12
V1 -1.946e-04 3.843e-05 -1.210e-04 -2.815e-04 -3.066e-04 -3.553e-04
V2 -2.332e-04 2.350e-04 3.525e-04 2.377e-04 5.202e-05 2.817e-04
V3 1.553e-04 -5.949e-05 -2.677e-05 -1.268e-04 -1.530e-05 6.540e-05
V4 -3.054e-04 6.559e-05 -3.545e-04 5.671e-05 -9.352e-05 -6.038e-05
V5 2.568e-04 -2.349e-04 2.791e-04 -1.668e-04 1.661e-04 7.215e-05
V6 4.680e-06 6.128e-05 4.824e-05 1.619e-05 1.690e-04 1.292e-04
V7 1.284e-02 3.910e-04 -5.795e-05 -1.573e-04 -1.042e-04 1.686e-04
V8 3.910e-04 1.086e-02 7.881e-05 4.216e-04 -3.732e-05 1.316e-04
V9 -5.795e-05 7.881e-05 1.219e-02 -2.693e-05 -5.327e-05 -3.108e-04
V10 -1.573e-04 4.216e-04 -2.693e-05 1.096e-02 -1.527e-04 -1.975e-04
V11 -1.042e-04 -3.732e-05 -5.327e-05 -1.527e-04 1.068e-02 -2.069e-04
V12 1.686e-04 1.316e-04 -3.108e-04 -1.975e-04 -2.069e-04 1.085e-02
V13 8.115e-05 -1.885e-04 2.333e-04 1.583e-04 -1.218e-04 -2.137e-04
V14 -1.309e-04 1.770e-04 3.079e-04 -1.763e-04 1.809e-05 -1.309e-04
V15 -1.840e-04 1.439e-04 -2.947e-04 1.918e-04 3.025e-05 -1.288e-04
V16 -3.945e-05 2.552e-05 2.401e-04 -1.029e-05 3.602e-04 -4.116e-05
V17 2.291e-04 -2.988e-04 -3.756e-05 5.071e-04 1.882e-04 2.105e-04
V18 -1.601e-04 1.792e-04 2.281e-04 -2.559e-04 -9.237e-05 -7.723e-05
V19 -1.173e-04 3.993e-04 -9.422e-06 -8.818e-06 3.787e-06 -2.681e-04
V20 -1.580e-05 -2.280e-04 -1.143e-05 4.391e-04 -1.910e-04 1.659e-04
V21 -1.841e-04 3.580e-05 -2.897e-04 1.762e-04 -1.835e-04 -1.103e-05
V22 -6.228e-05 -2.945e-05 4.108e-04 5.353e-05 3.011e-05 1.114e-04
V23 -2.899e-04 3.988e-04 -1.801e-04 -7.244e-05 5.469e-04 1.827e-04
V24 3.233e-04 9.675e-05 1.087e-04 -1.629e-04 -9.629e-05 -2.875e-04
V25 -1.680e-04 1.433e-04 -3.223e-04 -2.915e-05 -6.860e-05 -1.060e-04
V26 -2.021e-04 2.146e-05 -4.796e-04 1.628e-04 -9.056e-05 -8.886e-05
V27 3.423e-04 -2.350e-04 -1.290e-04 -1.963e-04 1.624e-04 -1.383e-04
V28 -1.016e-04 -8.444e-05 2.789e-05 1.702e-05 3.078e-06 1.851e-04
V29 4.843e-04 -2.699e-04 2.113e-04 3.574e-04 1.865e-04 -1.448e-04
V30 1.710e-04 -2.125e-04 -1.332e-04 1.736e-05 -1.955e-04 -3.131e-04
V31 -2.077e-05 2.629e-05 1.673e-04 2.616e-04 2.074e-04 2.407e-04
V32 -7.746e-05 3.564e-05 -1.525e-04 2.694e-04 1.243e-04 -3.005e-04
V33 1.930e-04 -3.115e-04 2.195e-04 7.013e-05 3.516e-04 1.050e-04
V34 -4.077e-05 -9.565e-05 -6.045e-05 -9.585e-06 2.747e-04 2.274e-04
V35 -1.842e-04 -8.624e-06 9.524e-05 8.135e-05 1.600e-04 -7.065e-05
V36 2.121e-05 -4.178e-04 -3.805e-04 1.708e-04 -1.001e-04 -6.807e-05
V37 -1.041e-04 2.432e-04 -4.630e-04 1.581e-04 -2.051e-04 -2.988e-04
V38 2.429e-04 1.910e-04 1.227e-04 2.509e-04 1.578e-04 1.278e-04
V39 1.761e-04 -4.651e-04 8.933e-05 1.507e-04 -1.794e-04 -3.410e-05
V40 3.495e-04 8.185e-05 -1.321e-04 -1.340e-04 1.421e-04 -2.786e-04
V41 7.514e-05 1.173e-04 -2.211e-04 -3.472e-05 3.943e-05 1.563e-04
V42 3.395e-04 1.105e-04 -2.188e-04 -2.283e-04 -2.786e-04 -3.773e-04
V43 2.776e-04 1.165e-04 2.545e-04 -1.295e-04 -1.820e-04 -4.698e-05
V44 -1.967e-04 1.686e-04 1.411e-04 3.847e-05 -1.006e-04 2.051e-04
V45 1.098e-04 -2.178e-04 4.999e-05 6.737e-05 3.712e-04 -1.131e-04
V46 -1.253e-05 1.120e-04 2.078e-04 -7.940e-05 1.801e-04 -4.790e-05
V47 6.479e-05 2.980e-04 -2.920e-04 -1.194e-04 6.258e-05 3.944e-05
V48 1.693e-04 9.331e-05 1.659e-04 -1.153e-05 3.535e-04 8.670e-05
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V50 1.513e-04 6.753e-05 2.476e-04 -4.471e-05 -6.362e-06 -2.172e-04
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V37 V38 V39 V40 V41 V42
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V43 V44 V45 V46 V47 V48
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V79 V80 V81 V82 V83 V84
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V85 V86 V87 V88 V89 V90
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V42 5.318e-05 2.643e-04 2.790e-04 1.928e-04 -2.780e-04 7.250e-05
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V91 V92 V93 V94 V95 V96
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