Bending non-positive-definite (symmetric) matrices to positive-definite, using weighted and unweighted methods
The mbend package is used for bending a non-positive-definite matrix to a positive-definite (PD) matrix. For a matrix to be invertible, it has to be PD. The methods of Jorjani et al. (2003) and Schaeffer (2014) are used in this package. The unweighted method of Schaeffer (2014) is also extended to weighted bending, with the possibility of choosing between unweighted and weighted bending.
Start with loading the package library:
library(mbend)
Consider the following non-PD matrix (Jorjani et al., 2003):
V = matrix(nrow=5, ncol=5, c(
100, 95, 80, 40, 40,
95, 100, 95, 80, 40,
80, 95, 100, 95, 80,
40, 80, 95, 100, 95,
40, 40, 80, 95, 100))
The following command can be used to bend matrix V to a PD matrix:
bend(V)
#> Unweighted bending
#> max.iter = 10000
#> small.positive = 1e-04
#> method = hj
#> Initial eigenvalues:
#> 399.475996527606
#> 98.523499553598
#> 23.6468969357859
#> -3.12289346339295
#> -18.5234995535981
#> Convergence met after 1 iterations.
#> Final eigenvalues:
#> 399.475996527606
#> 98.5234995535977
#> 23.6468969357859
#> 9.99999999917733e-05
#> 9.99999999516409e-05
#> Range of deviation = -5.9296122425301 6.49734335968796
#> Upper.tri average deviation = 0.72347050463841
#> Minimum absolute deviation = 0.528781551653452 [1,3]
#> Maximum absolute deviation = 6.49734335968796 [4,4]
#> Upper.tri average absolute deviation = 3.37269167108133
#> Upper.tri correlation = 0.985632028210772
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 103.16918 90.83313 79.47122 44.53731 37.07254
#> [2,] 90.83313 106.49734 94.18961 74.07039 44.53731
#> [3,] 79.47122 94.18961 102.31355 94.18961 79.47122
#> [4,] 44.53731 74.07039 94.18961 106.49734 90.83313
#> [5,] 37.07254 44.53731 79.47122 90.83313 103.16918
The above command is equivalent to bend(inmat=V), where inmat is the argument that takes the matrix to be bended, or the following command:
bend(V, max.iter=10000, small.positive=0.0001, method="hj")
This runs the unweighted bending method of Jorjani et al. (2003) (method="hj"), with maximum 10000 number of iterations (max.iter=10000), and eigenvalues smaller than 0.0001 are replaced with this small positive value (small.positive=0.0001). Providing the default parameters, the corresponding arguments can be omitted (e.g., bend(V)).
There might be different precision involved with different elements of a non-PD matrix. In this case, a weighted bending is recommended. Jorjani et al. (2003) used the reciprocal of the number data points in common between pairs of variables, as weights. Considering the following matrix for the number of data points in common between variables (Jorjani et al., 2003):
W = matrix(nrow=5, ncol=5, c(
1000, 500, 20, 50, 200,
500, 1000, 500, 5, 50,
20, 500, 1000, 20, 20,
50, 5, 20, 1000, 200,
200, 50, 20, 200, 1000))
Matrix V is bended using the following command:
bend(inmat=V, wtmat=W, reciprocal=TRUE)
#> Weighted bending
#> reciprocal = TRUE
#> max.iter = 10000
#> small.positive = 1e-04
#> method = hj
#> Initial eigenvalues:
#> 399.475996527606
#> 98.523499553598
#> 23.6468969357859
#> -3.12289346339295
#> -18.5234995535981
#> Convergence met after 428 iterations.
#> Final eigenvalues:
#> 387.637738787031
#> 102.03133648365
#> 12.299126553176
#> 7.66787308386723e-05
#> 1.21038807540936e-06
#> Range of deviation = -20.0007908829187 5.84554999273102
#> Upper.tri average deviation = -1.71605829547344
#> Minimum absolute deviation = 0.161540442682281 [1,1]
#> Maximum absolute deviation = 20.0007908829187 [2,4]
#> Upper.tri average absolute deviation = 3.62526845370067
#> Upper.tri correlation = 0.962320607917192
#> 15 Upper.tri elements; for the 15 elements that are not set to constant:
#> Upper.tri weighted average absolute deviation = 0.610010348402353
#> Upper.tri weighted correlation = 0.995512129407956
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 100.16154 94.51747 82.93843 43.56681 39.18292
#> [2,] 94.51747 100.62488 93.98150 59.99921 45.84555
#> [3,] 82.93843 93.98150 100.69547 84.89294 73.13431
#> [4,] 43.56681 59.99921 84.89294 100.30697 94.23170
#> [5,] 39.18292 45.84555 73.13431 94.23170 100.17942
Using wtmat=1/W, reciprocal=FALSE, the argument reciprocal could be omitted, because FALSE is the default parameter for the argument reciprocal. For the same reason, max.iter=10000, small.positive=0.0001, method="hj" are omitted, unless different parameters are provided to these arguments.
If there is high confidence about some elements of the non-PD matrix to remain unchanged after bending, the corresponding weights are set to zero. For example, to keep the first 2 × 2 block of V unchanged during the bending procedure:
W2 = W; W2[1:2, 1:2] = 0
bend(V, W2, reciprocal=TRUE)
#> Weighted bending
#> reciprocal = TRUE
#> max.iter = 10000
#> small.positive = 1e-04
#> method = hj
#> Initial eigenvalues:
#> 399.475996527606
#> 98.523499553598
#> 23.6468969357859
#> -3.12289346339295
#> -18.5234995535981
#> Convergence met after 475 iterations.
#> Final eigenvalues:
#> 387.658237440711
#> 102.15441895382
#> 11.4162918091265
#> 7.64681848664974e-05
#> 1.17480224900039e-07
#> Range of deviation = -20.4157775153382 6.15037904095363
#> Upper.tri average deviation = -1.73283635615101
#> Minimum absolute deviation = 0 [1,1]
#> Maximum absolute deviation = 20.4157775153382 [4,2]
#> Upper.tri average absolute deviation = 3.70526947045777
#> Upper.tri correlation = 0.959799073876244
#> 15 Upper.tri elements; for the 12 elements that are not set to constant:
#> Upper.tri weighted average absolute deviation = 0.770543658780136
#> Upper.tri weighted correlation = 0.995120937140909
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 100.00000 95.00000 83.70895 43.70489 39.12852
#> [2,] 95.00000 100.00000 93.89940 59.58422 46.15038
#> [3,] 83.70895 93.89940 100.73015 84.47456 72.91450
#> [4,] 43.70489 59.58422 84.47456 100.31566 94.21299
#> [5,] 39.12852 46.15038 72.91450 94.21299 100.18322
As it can be seen in the report, 3 of the 15 upper triangle elements are set to constant.
To bend V using the method of Schaeffer (2014):
bend(inmat=V, method="lrs")
#> Unweighted bending
#> max.iter = 10000
#> method = lrs
#> Initial eigenvalues:
#> 399.475996527606
#> 98.523499553598
#> 23.6468969357859
#> -3.12289346339295
#> -18.5234995535981
#> Convergence met after 1 iterations.
#> Final eigenvalues:
#> 399.475996527605
#> 98.5234995535977
#> 23.6468969357859
#> 0.203583282231235
#> 0.0774047769998617
#> Range of deviation = -5.93704734709341 6.54176982926614
#> Upper.tri average deviation = 0.732955037925687
#> Minimum absolute deviation = 0.563235099819792 [3,1]
#> Maximum absolute deviation = 6.54176982926614 [4,4]
#> Upper.tri average absolute deviation = 3.40870716448891
#> Upper.tri correlation = 0.98545110017304
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 103.18978 90.82704 79.43676 44.56754 37.06769
#> [2,] 90.82704 106.54177 94.13680 74.06295 44.56754
#> [3,] 79.43676 94.13680 102.46429 94.13680 79.43676
#> [4,] 44.56754 74.06295 94.13680 106.54177 90.82704
#> [5,] 37.06769 44.56754 79.43676 90.82704 103.18978
The method of Schaeffer (2014) does not require the argument small.positive, and this argument is ignored. This method is originally an unweighted bending method. However, in this package, it is extended to accommodate weighted bending (i.e., a combination of Schaeffer (2014) and Jorjani et al. (2003) methods). Weighted bending of V with reciprocals of W using the method of Schaeffer (2014):
bend(V, W, reciprocal=TRUE, method="lrs")
#> Weighted bending
#> reciprocal = TRUE
#> max.iter = 10000
#> method = lrs
#> Initial eigenvalues:
#> 399.475996527606
#> 98.523499553598
#> 23.6468969357859
#> -3.12289346339295
#> -18.5234995535981
#> Convergence met after 787 iterations.
#> Final eigenvalues:
#> 387.636539857614
#> 102.023398559989
#> 12.2826080278304
#> 0.0324216138054008
#> 0.000443532672897638
#> Range of deviation = -19.9923416420825 5.8637972845693
#> Upper.tri average deviation = -1.71580553899355
#> Minimum absolute deviation = 0.161942434558412 [1,1]
#> Maximum absolute deviation = 19.9923416420825 [4,2]
#> Upper.tri average absolute deviation = 3.63380255067224
#> Upper.tri correlation = 0.962240055098133
#> 15 Upper.tri elements; for the 15 elements that are not set to constant:
#> Upper.tri weighted average absolute deviation = 0.612202615505032
#> Upper.tri weighted correlation = 0.995488405316724
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 100.16194 94.51554 82.97457 43.57119 39.18071
#> [2,] 94.51554 100.62754 93.97669 60.00766 45.86380
#> [3,] 82.97457 93.97669 100.69806 84.86261 73.10529
#> [4,] 43.57119 60.00766 84.86261 100.30785 94.22946
#> [5,] 39.18071 45.86380 73.10529 94.22946 100.18003
Function bend automatically considers any matrix with all diagonal elements equal to one, as a correlation matrix. Considering the correlation matrix V2 (V converted to a correlation matrix):
V2 = cov2cor(V)
bend(V2, W, reciprocal=TRUE)
#> Weighted bending
#> reciprocal = TRUE
#> max.iter = 10000
#> small.positive = 1e-04
#> method = hj
#> Found a correlation matrix.
#> Initial eigenvalues:
#> 3.99475996527606
#> 0.985234995535981
#> 0.23646896935786
#> -0.0312289346339294
#> -0.185234995535981
#> Convergence met after 286 iterations.
#> Final eigenvalues:
#> 3.86884346904565
#> 1.01542046498059
#> 0.115656626525816
#> 7.74077131621184e-05
#> 2.03173477736353e-06
#> Statistics excluding diagonal elements:
#> Range of deviation = -0.199535820290176 0.0630373480880824
#> Upper.tri average deviation = -0.0283824839213205
#> Minimum absolute deviation = 0.00524637203616529 [1,1]
#> Maximum absolute deviation = 0.199535820290176 [2,4]
#> Upper.tri average absolute deviation = 0.0553854809191007
#> Upper.tri correlation = 0.946312783301257
#> 10 Upper.tri elements; for the 10 elements that are not set to constant:
#> Upper.tri weighted average absolute deviation = 0.0141695879562372
#> Upper.tri weighted correlation = 0.994254131862483
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1.0000000 0.9447536 0.8342673 0.4377103 0.3912583
#> [2,] 0.9447536 1.0000000 0.9385087 0.6004642 0.4630373
#> [3,] 0.8342673 0.9385087 1.0000000 0.8394178 0.7248761
#> [4,] 0.4377103 0.6004642 0.8394178 1.0000000 0.9418815
#> [5,] 0.3912583 0.4630373 0.7248761 0.9418815 1.0000000
As indicated above, diagonal elements are not used in the reporting statistics. This happens to correlation matrices.
Because the argument method is not provided, the default parameter "hj" is used. To do the same using the method of Schaeffer (2014):
bend(V2, W, reciprocal=TRUE, method="lrs")
#> Weighted bending
#> reciprocal = TRUE
#> max.iter = 10000
#> method = lrs
#> Found a correlation matrix.
#> Initial eigenvalues:
#> 3.99475996527606
#> 0.985234995535981
#> 0.23646896935786
#> -0.0312289346339294
#> -0.185234995535981
#> Convergence met after 700 iterations.
#> Final eigenvalues:
#> 3.86881998612449
#> 1.01533779687542
#> 0.115577591310937
#> 0.000261768738811671
#> 2.85695033471629e-06
#> Statistics excluding diagonal elements:
#> Range of deviation = -0.19940766680859 0.0631892792397414
#> Upper.tri average deviation = -0.0283856208324125
#> Minimum absolute deviation = 0.00526174379216604 [1,1]
#> Maximum absolute deviation = 0.19940766680859 [4,2]
#> Upper.tri average absolute deviation = 0.0554774992677458
#> Upper.tri correlation = 0.946234975233815
#> 10 Upper.tri elements; for the 10 elements that are not set to constant:
#> Upper.tri weighted average absolute deviation = 0.0142076326019521
#> Upper.tri weighted correlation = 0.994232700482272
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1.0000000 0.9447383 0.8345316 0.4377385 0.3912447
#> [2,] 0.9447383 1.0000000 0.9384618 0.6005923 0.4631893
#> [3,] 0.8345316 0.9384618 1.0000000 0.8392102 0.7245638
#> [4,] 0.4377385 0.6005923 0.8392102 1.0000000 0.9418734
#> [5,] 0.3912447 0.4631893 0.7245638 0.9418734 1.0000000
Bending the same correlation matrix using the unweighted Schaeffer (2014):
bend(V2, method="lrs")
#> Unweighted bending
#> max.iter = 10000
#> method = lrs
#> Found a correlation matrix.
#> Initial eigenvalues:
#> 3.99475996527606
#> 0.985234995535981
#> 0.23646896935786
#> -0.0312289346339294
#> -0.185234995535981
#> Convergence met after 39 iterations.
#> Final eigenvalues:
#> 3.90836387072011
#> 0.918358999335951
#> 0.165663862030552
#> 0.0071038703941073
#> 0.000509397519274219
#> Statistics excluding diagonal elements:
#> Range of deviation = -0.0788188733585764 0.0716891552890742
#> Upper.tri average deviation = -0.0203850283712788
#> Minimum absolute deviation = 0.0205864131251348 [1,1]
#> Maximum absolute deviation = 0.0788188733585764 [4,2]
#> Upper.tri average absolute deviation = 0.0490606904667602
#> Upper.tri correlation = 0.985404338381732
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1.0000000 0.8935779 0.7794136 0.4716892 0.3599505
#> [2,] 0.8935779 1.0000000 0.9128285 0.7211811 0.4716892
#> [3,] 0.7794136 0.9128285 1.0000000 0.9128285 0.7794136
#> [4,] 0.4716892 0.7211811 0.9128285 1.0000000 0.8935779
#> [5,] 0.3599505 0.4716892 0.7794136 0.8935779 1.0000000
Jorjani, H., Klie. L., & Emanuelson, U. (2000). A simple method for weighted bending of genetic (co)variance matrices. J. Dairy Sci. 86(2): 677–679. https://doi.org/10.3168/jds.S0022-0302(03)73646-7
Schaeffer, L. R. (2014). Making covariance matrices positive definite. Available at: http://animalbiosciences.uoguelph.ca/~lrs/ELARES/PDforce.pdf