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Last data update: 2014.03.03

R: Total Least Squares Composition Estimator

tlsce

R Documentation

Total Least Squares Composition Estimator

Description

estimates a matrix X for which:

(A+ε_A)X = B+ε_B

minimize ∑{ε_A^2 + ε_B^2}

∑{X_{i,}}=1 forall i

X>0

the elements of ε_A are NULL if the corresponding elements of A are NULL. A typically contains biomarker concentrations for several taxonomic groups, and B field measurements of the same biomarkers. X is then an estimate of the taxonomic composition of the field sample.

Usage

tlsce(A, B, Wa=NULL, Wb=NULL, minA=NULL, maxA=NULL,
       A_init=A, Xratios=TRUE, ...)

Arguments

`A`	a matrix or data frame. If A contains biomarker data for taxonomic groups, the biomarkers have to be organized per row, and the taxonomic groups per column.
`B`	a matrix or data frame. If B contains biomarker field data, the biomarkers have to be organized per row, and the samples per column.
`Wa`	weighting of A, a matrix with the same dimensions of A. If `Wa=NULL`, Wa defaults to 1. This parameter can be used to give more importance to elements of A or A in total compared to B. weights are implemented as proportional to 1/s (as opposed to 1/s^2) with s the standard deviation of the error term.
`Wb`	weighting of B, a matrix with the same dimensions of B. If `Wb=NULL`, Wb defaults to 1. This parameter can be used to give more importance to elements of B or B in total compared to A. weights are implemented as proportional to 1/s (as opposed to 1/s^2) with s the standard deviation of the error term.
`minA`	minimum values for A
`maxA`	maximum values for A
`A_init`	a matrix with the same structure as A. a general, non-linear optimization routine (default `nlminb`) is used to minimize the sum of squared residuals of A versus the fitted matrix A_fit (see falue). This optimization routine requires a set of starting values, by default the non-zero elements of A. This provides a good fit, but when in doubt about the convergence of the algorithm, one can provide different starting values for the optimization routine in A_init.
`Xratios`	TRUE or FALSE: are the colSums of the matrix X equal to 1? This is for example the case in a compositional matrix. (only if A and B are both expressed relative to the unit of biomass) if Xratios =TRUE, A has pigment concentrations per biomass unit, B has pigment concentrations per biomass unit per sample, and X contains ratios of biomass unit per sample. if Xratios =FALSE, A has pigment concentrations per biomass unit, B has pigment concentrations per sample, and X has biomass units per sample
`...`	Arguments to be passed to lsei() or to modFit()

Details

instead of a linear least squares regression, in which the elements of A would be fixed, the function tlsce includes the non-zero elements of A in the least squares regression. This is similar to other total least squares regression methods (also called orthogonal regression), with the main difference that only non-zero elements of A contain an error term.

Value

A list with the following elements:

`X`	Array with dimension c(ncol(`A`),ncol(`B`), `iter`) containing the species composition of each sample
`A_fit`	Array with same dimension as `A`, containing the best-fit values of the input biomarker data per taxonomic group
`B_fit`	Array with same dimension as `B`, containing the biomarker field data, corresponding to Afit
`solutionNorms`	a vector of 3 values: the value of the minimised quadratic function at the solution, in this case sum((Afit-A)Wa)^2 + (Bfit-B)^2)*, and the shares of this value attributed to A and to B
`convergence`	An integer code. '0' indicates successful convergence.

Author(s)

Karel Van den Meersche <k.vdmeersche@nioo.knaw.nl>, Karline Soetaert <k.soetaert@nioo.knaw.nl>

References

Van den Meersche, K., K. Soetaert and J.J. Middelburg (2008) A Bayesian compositional estimator for microbial taxonomy based on biomarkers, Limnology and Oceanography Methods 6, 190-199

Examples

A <- t(bceInput$Rat)
B <- t(bceInput$Dat)
tlsce(A,B)
## weighting Wa inversely proportional to A
tlsce(A,B,Wa=1/A)

Results


R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(BCE)
Loading required package: FME
Loading required package: deSolve

Attaching package: 'deSolve'

The following object is masked from 'package:graphics':

    matplot

Loading required package: rootSolve
Loading required package: coda
Loading required package: limSolve
Loading required package: Matrix
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/BCE/tlsce.Rd_%03d_medium.png", width=480, height=480)
> ### Name: tlsce
> ### Title: Total Least Squares Composition Estimator
> ### Aliases: tlsce
> ### Keywords: models
> 
> ### ** Examples
> 
> A <- t(bceInput$Rat)
> B <- t(bceInput$Dat)
> tlsce(A,B)
$X
           sample1    sample2   sample3   sample4   sample5
species1 0.0000000 0.36739459 0.5247678 0.2243613 0.1940198
species2 0.0000000 0.00000000 0.0000000 0.3019997 0.0000000
species3 0.0000000 0.02169081 0.0000000 0.0000000 0.0000000
species4 0.7826177 0.34311315 0.2246568 0.1753205 0.2943653
species5 0.2173823 0.26780144 0.2505754 0.2983185 0.5116149

$A_fit
            species1   species2   species3   species4  species5
biomarker1 0.0000000 0.00000000 0.00000000 0.00000000 0.3120027
biomarker2 0.0000000 0.00000000 0.00000000 0.00000000 0.2450860
biomarker3 0.4213464 0.23308917 2.52898327 1.21352173 0.3044179
biomarker4 0.1900306 0.09612558 0.24963488 0.40726891 1.1221068
biomarker5 1.4678069 0.34531022 0.06323953 0.07593617 0.0000000
biomarker6 0.0000000 0.57903990 0.03360601 0.18730850 0.0000000
biomarker7 0.0000000 0.00000000 0.00000000 1.32118872 0.0000000
biomarker8 0.0000000 0.00000000 0.00000000 0.23776652 0.0000000

$B_fit
              sample1    sample2    sample3    sample4    sample5
biomarker1 0.06782386 0.08355477 0.07818019 0.09307617 0.15962521
biomarker2 0.05327735 0.06563437 0.06141251 0.07311367 0.12538962
biomarker3 1.01589865 0.70755492 0.57001456 0.46849537 0.59471300
biomarker4 0.56266201 0.51547212 0.47239000 0.47781320 0.73084207
biomarker5 0.05942899 0.56669072 0.78731742 0.44691581 0.30713655
biomarker6 0.14659095 0.06499695 0.04208012 0.20770892 0.05513713
biomarker7 1.03398570 0.45331723 0.29681399 0.23163141 0.38891218
biomarker8 0.18608029 0.08158082 0.05341586 0.04168534 0.06999023

$SS
    total         A         B 
0.5934600 0.2887776 0.3046824 

$fit
$par
 [1] 0.42134644 0.19003056 1.46780686 0.23308917 0.09612558 0.34531022
 [7] 0.57903990 2.52898327 0.24963488 0.06323953 0.03360601 1.21352173
[13] 0.40726891 0.07593617 0.18730850 1.32118872 0.23776652 0.31200268
[19] 0.24508595 0.30441787 1.12210682

$hessian
               [,1]          [,2]         [,3]          [,4]          [,5]
 [1,]  2.561320e+00  0.1651321911  0.083430047  8.936733e-02 -1.167770e-03
 [2,]  1.651322e-01  2.6734164410  0.367058559 -1.167759e-03  4.609390e-02
 [3,]  8.343005e-02  0.3670585587  2.262195471 -6.309703e-03  2.078047e-02
 [4,]  8.936733e-02 -0.0011677592 -0.006309703  2.120345e+00 -8.354491e-04
 [5,] -1.167770e-03  0.0460939005  0.020780466 -8.354491e-04  2.072268e+00
 [6,] -6.309708e-03  0.0207804673  0.016861134 -8.185894e-03  3.223670e-02
 [7,]  1.663401e-02  0.0473794641  0.034004738  2.312236e-02  7.394069e-02
 [8,]  1.452505e-03  0.0031272196  0.002310961 -3.421719e-10 -2.318084e-10
 [9,]  3.127215e-03  0.0094383581  0.005101986 -4.235526e-10 -1.360328e-09
[10,]  2.310956e-03  0.0051019839  0.003691620 -1.821776e-09 -1.756700e-09
[11,] -5.495352e-05  0.0003715646  0.000308663  1.549790e-09  5.377116e-10
[12,]  3.365259e-01  0.1066787931  0.057628311  6.984040e-02 -8.157381e-04
[13,]  1.066788e-01  0.4212162464  0.229701577 -8.157446e-04  3.736230e-02
[14,]  5.762830e-02  0.2297015384  0.167168203 -4.890153e-03  1.679883e-02
[15,] -1.214199e-02  0.0362745288  0.030180390  1.309440e-02  3.835931e-02
[16,] -1.517868e-01  0.1010860325  0.096748129 -4.720922e-02  6.554996e-03
[17,] -2.833738e-02  0.0141062084  0.015348026 -8.560917e-03  2.779164e-04
[18,]  5.504858e-02 -0.0989258599  0.087271571  5.948149e-03 -3.391825e-02
[19,]  4.288875e-02 -0.0789948718  0.067878210  4.656852e-03 -2.685975e-02
[20,]  4.490367e-01  0.1072539319  0.051199095  1.188328e-01 -1.456682e-03
[21,]  1.072540e-01  0.4657675830  0.253006200 -1.456690e-03  6.262120e-02
               [,6]          [,7]          [,8]          [,9]         [,10]
 [1,] -6.309708e-03  1.663401e-02  1.452505e-03  3.127215e-03  2.310956e-03
 [2,]  2.078047e-02  4.737946e-02  3.127220e-03  9.438358e-03  5.101984e-03
 [3,]  1.686113e-02  3.400474e-02  2.310961e-03  5.101986e-03  3.691620e-03
 [4,] -8.185894e-03  2.312236e-02 -3.421719e-10 -4.235526e-10 -1.821776e-09
 [5,]  3.223670e-02  7.394069e-02 -2.318084e-10 -1.360328e-09 -1.756700e-09
 [6,]  2.024475e+00  5.001313e-02  3.692749e-11 -2.917271e-10 -9.048561e-10
 [7,]  5.001313e-02  2.116462e+00 -2.620160e-10 -1.389465e-09 -2.927940e-09
 [8,]  3.692749e-11 -2.620160e-10  2.001084e+00  2.915374e-03  1.747592e-03
 [9,] -2.917271e-10 -1.389465e-09  2.915374e-03  2.008027e+00  4.708400e-03
[10,] -9.048561e-10 -2.927940e-09  1.747592e-03  4.708400e-03  2.002818e+00
[11,]  3.859625e-10  1.354197e-09 -1.170485e-04 -2.893648e-04 -1.654466e-04
[12,] -4.890181e-03  1.309439e-02  7.136634e-04  1.162089e-03  1.120730e-03
[13,]  1.679881e-02  3.835930e-02  1.162094e-03  4.004503e-03  1.926795e-03
[14,]  1.340950e-02  2.712939e-02  1.120733e-03  1.926811e-03  1.773209e-03
[15,]  2.712940e-02  6.306976e-02  2.195990e-05  5.474539e-04  4.065276e-04
[16,]  1.018835e-02  5.464998e-03  5.896188e-04  7.111406e-04  8.742117e-04
[17,]  1.457319e-03  8.683504e-05 -1.723947e-04 -6.338485e-04 -2.921551e-04
[18,]  1.193460e-02  1.956902e-02  9.305444e-04 -1.374096e-03  1.341500e-03
[19,]  9.284769e-03  1.515706e-02  7.236247e-04 -1.099468e-03  1.041933e-03
[20,] -8.349545e-03  2.221264e-02  1.121241e-03  2.450394e-03  1.785338e-03
[21,]  2.818663e-02  6.432303e-02  2.450406e-03  7.347314e-03  3.994769e-03
              [,11]         [,12]         [,13]         [,14]         [,15]
 [1,] -5.495352e-05  3.365259e-01  0.1066788261  0.0576282989 -0.0121419890
 [2,]  3.715646e-04  1.066788e-01  0.4212162464  0.2297015384  0.0362745288
 [3,]  3.086630e-04  5.762831e-02  0.2297015772  0.1671682034  0.0301803903
 [4,]  1.549790e-09  6.984040e-02 -0.0008157446 -0.0048901530  0.0130943974
 [5,]  5.377116e-10 -8.157381e-04  0.0373623037  0.0167988335  0.0383593129
 [6,]  3.859625e-10 -4.890181e-03  0.0167988066  0.0134094981  0.0271294048
 [7,]  1.354197e-09  1.309439e-02  0.0383592992  0.0271293924  0.0630697577
 [8,] -1.170485e-04  7.136634e-04  0.0011620944  0.0011207326  0.0000219599
 [9,] -2.893648e-04  1.162089e-03  0.0040045026  0.0019268112  0.0005474539
[10,] -1.654466e-04  1.120730e-03  0.0019267953  0.0017732093  0.0004065276
[11,]  2.000939e+00  2.197657e-05  0.0005474976  0.0004065327  0.0146427522
[12,]  2.197657e-05  3.205328e+00  0.3751095006  0.0332317258 -0.0726950096
[13,]  5.474976e-04  3.751095e-01  3.5198627231  0.2392290541  0.0882242857
[14,]  4.065327e-04  3.323173e-02  0.2392290541  3.3766277138  0.0326909387
[15,]  1.464275e-02 -7.269501e-02  0.0882242857  0.0326909387  3.7779847263
[16,] -1.701446e-03 -5.465976e-01  0.4575240886  0.0452865846 -0.1437937030
[17,] -2.744455e-04 -8.870060e-02  0.1126848076  0.0158285431 -0.0171838973
[18,]  1.454123e-04  7.544427e-02 -0.1161887163  0.0807629047  0.0314516578
[19,]  1.150617e-04  5.924766e-02 -0.0917223295  0.0628000323  0.0236810617
[20,] -4.718635e-05  6.366439e-01  0.1321963946  0.0402400951 -0.0241578851
[21,]  2.513351e-04  1.321964e-01  0.6230151416  0.2308528266  0.0700098332
              [,16]         [,17]         [,18]         [,19]         [,20]
 [1,] -0.1517868396 -2.833738e-02  0.0550485776  0.0428887513  4.490367e-01
 [2,]  0.1010860325  1.410621e-02 -0.0989258599 -0.0789948718  1.072539e-01
 [3,]  0.0967481290  1.534803e-02  0.0872715715  0.0678782102  5.119909e-02
 [4,] -0.0472092230 -8.560917e-03  0.0059481490  0.0046568522  1.188328e-01
 [5,]  0.0065549964  2.779164e-04 -0.0339182491 -0.0268597469 -1.456682e-03
 [6,]  0.0101883465  1.457319e-03  0.0119346025  0.0092847687 -8.349545e-03
 [7,]  0.0054649982  8.683504e-05  0.0195690153  0.0151570610  2.221264e-02
 [8,]  0.0005896188 -1.723947e-04  0.0009305444  0.0007236247  1.121241e-03
 [9,]  0.0007111406 -6.338485e-04 -0.0013740959 -0.0010994685  2.450394e-03
[10,]  0.0008742117 -2.921551e-04  0.0013414996  0.0010419331  1.785338e-03
[11,] -0.0017014456 -2.744455e-04  0.0001454123  0.0001150617 -4.718635e-05
[12,] -0.5465975911 -8.870060e-02  0.0754442681  0.0592476605  6.366439e-01
[13,]  0.4575240886  1.126848e-01 -0.1161887163 -0.0917223295  1.321964e-01
[14,]  0.0452865846  1.582854e-02  0.0807629047  0.0628000323  4.024010e-02
[15,] -0.1437937030 -1.718390e-02  0.0314516578  0.0236810617 -2.415789e-02
[16,]  2.7520406176 -1.831918e-01  0.1009046818  0.0794318533 -3.246192e-01
[17,] -0.1831918177  3.774486e+00  0.0199721401  0.0153455017 -6.672246e-02
[18,]  0.1009046818  1.997214e-02  3.0212934117 -0.0353009674  7.184994e-02
[19,]  0.0794318533  1.534550e-02 -0.0353009674  3.0366933945  5.683182e-02
[20,] -0.3246191787 -6.672246e-02  0.0718499399  0.0568318236  2.701141e+00
[21,]  0.1771645036  7.285593e-03 -0.1437438118 -0.1109389198  1.931741e-01
              [,21]
 [1,]  0.1072539897
 [2,]  0.4657675830
 [3,]  0.2530061998
 [4,] -0.0014566900
 [5,]  0.0626211962
 [6,]  0.0281866293
 [7,]  0.0643230311
 [8,]  0.0024504055
 [9,]  0.0073473136
[10,]  0.0039947687
[11,]  0.0002513351
[12,]  0.1321964186
[13,]  0.6230151416
[14,]  0.2308528266
[15,]  0.0700098332
[16,]  0.1771645036
[17,]  0.0072855934
[18,] -0.1437438118
[19,] -0.1109389198
[20,]  0.1931741402
[21,]  2.8048510810

$residuals
 [1] -0.0303464435 -0.1200305597 -0.0638068633 -0.0010891675 -0.0151255825
 [6] -0.0063102169 -0.0150398951 -0.0008933799 -0.0024438696 -0.0014417753
[11]  0.0001018530 -0.1435217263 -0.4012689050 -0.0639361728  0.0026914986
[16] -0.0931887234 -0.1157665218  0.0419973236  0.0249140457 -0.0694178661
[21] -0.2321068215 -0.0347402472 -0.0236457314 -0.1443577612 -0.3668251719
[26] -0.0046735702  0.0824883868 -0.1070399456 -0.0895607348  0.0081778815
[31]  0.0003077380 -0.0411887970 -0.1126685807 -0.0664751074  0.0046954445
[36] -0.0240697048 -0.0189216917  0.0182723589  0.0107212650 -0.0106306047
[41] -0.0621722138 -0.0329590821 -0.0178277119 -0.0119417745 -0.0203168612
[46]  0.0329903321  0.0216115272 -0.0036076330 -0.0500838007 -0.0208953463
[51] -0.0498018145  0.0041636036 -0.0484434190  0.0643871128  0.0407335329
[56] -0.0454986417 -0.1792108363 -0.0896888480 -0.1723760956  0.0026687933
[61] -0.0887399339

$info
[1] 1

$message
[1] "Relative error in the sum of squares is at most `ftol'."

$iterations
[1] 12

$rsstrace
 [1] 1.1852456 0.7757644 0.6903279 0.6154719 0.5971540 0.5940798 0.5935597
 [8] 0.5934757 0.5934624 0.5934604 0.5934600 0.5934600 0.5934600

$ssr
[1] 0.59346

$diag
     diag1      diag2      diag3      diag4      diag5      diag6      diag7 
0.47681962 0.23752279 1.56105207 0.24125487 0.09932342 0.34741621 0.59565976 
     diag8      diag9     diag10     diag11     diag12     diag13     diag14 
2.54648475 0.27284062 0.06512500 0.03468434 1.55642253 1.01119030 0.10653174 
    diag15     diag16     diag17     diag18     diag19     diag20     diag21 
0.28162183 1.54978773 0.33430910 0.46104919 0.35308761 0.35374576 1.32880167 

$ms
[1] 0.009728852

$var_ms_unscaled
[1] NA

$var_ms_unweighted
[1] NA

$var_ms
[1] NA

$rank
[1] 21

$df.residual
[1] 40

attr(,"class")
[1] "modFit"

> ## weighting Wa inversely proportional to A
> tlsce(A,B,Wa=1/A)
$X
            sample1   sample2    sample3    sample4    sample5
species1 0.00000000 0.3316064 0.48024422 0.21454415 0.17824956
species2 0.00000000 0.0000000 0.00000000 0.28217109 0.00000000
species3 0.05720514 0.0560328 0.02346464 0.01805893 0.04097266
species4 0.59250216 0.2654179 0.17729545 0.14255547 0.21723831
species5 0.35029270 0.3469429 0.31899569 0.34267038 0.56353947

$A_fit
             species1   species2   species3    species4  species5
biomarker1 0.00000000 0.00000000 0.00000000 0.000000000 0.3402311
biomarker2 0.00000000 0.00000000 0.00000000 0.000000000 0.2645218
biomarker3 0.39806496 0.23221154 2.63861113 1.238611578 0.2422926
biomarker4 0.07078116 0.08112961 0.25022076 0.006014673 1.2094134
biomarker5 1.57849416 0.33998549 0.06185610 0.012014071 0.0000000
biomarker6 0.00000000 0.57017478 0.03371573 0.190947872 0.0000000
biomarker7 0.00000000 0.00000000 0.00000000 1.510810954 0.0000000
biomarker8 0.00000000 0.00000000 0.00000000 0.124760432 0.0000000

$B_fit
              sample1    sample2    sample3    sample4    sample5
biomarker1 0.11918046 0.11804077 0.10853224 0.11658711 0.19173364
biomarker2 0.09266006 0.09177397 0.08438131 0.09064378 0.14906848
biomarker3 0.96969550 0.69266102 0.54997296 0.45817374 0.58468118
biomarker4 0.44152630 0.45868589 0.42672761 0.45788440 0.70572772
biomarker5 0.01065685 0.53009347 0.76164417 0.43742048 0.28651021
biomarker6 0.11506574 0.05257016 0.03464532 0.18871637 0.04286262
biomarker7 0.89515875 0.40099623 0.26785990 0.21537436 0.32820601
biomarker8 0.07392083 0.03311365 0.02211946 0.01778528 0.02710275

$SS
    total         A         B 
0.8357650 0.2281971 0.6075679 

$fit
$par
 [1] 0.398064957 0.070781156 1.578494158 0.232211542 0.081129615 0.339985492
 [7] 0.570174778 2.638611127 0.250220761 0.061856103 0.033715726 1.238611578
[13] 0.006014673 0.012014071 0.190947872 1.510810954 0.124760432 0.340231066
[19] 0.264521801 0.242292649 1.209413387

$hessian
              [,1]          [,2]        [,3]          [,4]          [,5]
 [1,] 1.315496e+01   0.158800381 0.126940045  0.0039032406  7.355217e-03
 [2,] 1.588004e-01 408.660168570 0.295297953  0.0073552140  3.235365e-02
 [3,] 1.269400e-01   0.295297953 1.238401161  0.0069966532  1.547563e-02
 [4,] 3.903241e-03   0.007355214 0.006996653 37.1640921549  1.376275e-02
 [5,] 7.355217e-03   0.032353647 0.015475632  0.0137627532  3.048947e+02
 [6,] 6.996662e-03   0.015475648 0.012825554  0.0109770516  2.927524e-02
 [7,] 1.619972e-02   0.035167779 0.029613402  0.0253239940  6.644840e-02
 [8,] 6.280412e-03   0.012497240 0.010762862  0.0003660395  8.075616e-04
 [9,] 1.249726e-02   0.037753190 0.022969082  0.0008075537  3.670505e-03
[10,] 1.076287e-02   0.022969075 0.018668738  0.0006707298  1.713786e-03
[11,] 2.764979e-04   0.001800762 0.002297497  0.0015487524  3.890896e-03
[12,] 3.809179e-02   0.078398128 0.065708290  0.0026315996  5.078527e-03
[13,] 7.839812e-02   0.241912498 0.145071727  0.0050785313  2.245914e-02
[14,] 6.570831e-02   0.145071747 0.114768610  0.0047320149  1.070027e-02
[15,] 7.164069e-03   0.028414393 0.023975009  0.0109519890  2.431228e-02
[16,] 4.920665e-02   0.113316378 0.085335704  0.0023064277  6.348841e-03
[17,] 2.248008e-03   0.003183939 0.003554888 -0.0003709653 -2.297651e-03
[18,] 6.779171e-02  -0.113702854 0.087311612  0.0090525859 -3.565578e-02
[19,] 5.230461e-02  -0.089772227 0.067109946  0.0069666408 -2.808110e-02
[20,] 6.696163e-02   0.136866744 0.115496573  0.0063810632  1.248566e-02
[21,] 1.368667e-01   0.425452548 0.253838107  0.0124856385  5.538417e-02
               [,6]         [,7]          [,8]          [,9]         [,10]
 [1,]  0.0069966625  0.016199716  0.0062804118  0.0124972550  1.076287e-02
 [2,]  0.0154756482  0.035167779  0.0124972396  0.0377531898  2.296908e-02
 [3,]  0.0128255540  0.029613402  0.0107628621  0.0229690824  1.866874e-02
 [4,]  0.0109770516  0.025323994  0.0003660395  0.0008075537  6.707298e-04
 [5,]  0.0292752410  0.066448401  0.0008075616  0.0036705049  1.713786e-03
 [6,] 17.4240005156  0.047714173  0.0006707354  0.0017137922  1.258042e-03
 [7,]  0.0477141728  6.397213067  0.0015487529  0.0038908910  2.896654e-03
 [8,]  0.0006707354  0.001548753  0.3364054488  0.0623792798  1.364268e-02
 [9,]  0.0017137922  0.003890891  0.0623792798 32.9037271798  4.099308e-02
[10,]  0.0012580417  0.002896654  0.0136426835  0.0409930801  5.237270e+02
[11,]  0.0028966584  0.006671877  0.0015741269  0.0119702246  1.154486e-02
[12,]  0.0047320228  0.010951992 -0.0023076954 -0.0150977743 -1.674546e-03
[13,]  0.0107002779  0.024312277 -0.0150977810 -0.0390773637 -7.072872e-03
[14,]  0.0087032026  0.020086952 -0.0016745436 -0.0070728655  7.235290e-02
[15,]  0.0200869556  0.046376480  0.0003853838 -0.0007391000 -4.901740e-03
[16,]  0.0043824441  0.010075006 -0.0032078511 -0.0162590151  1.236689e-03
[17,] -0.0008630337 -0.001940848 -0.0177568113 -0.0509122697 -1.252867e-02
[18,]  0.0096960466  0.024341678  0.0137592458 -0.0147203100  1.249422e-02
[19,]  0.0073823761  0.018571789  0.0106665720 -0.0116209472  9.544201e-03
[20,]  0.0114953794  0.026599270  0.0093216066  0.0163509386  1.252100e-02
[21,]  0.0263276457  0.059814376  0.0163509290  0.0514117471  2.857458e-02
              [,11]         [,12]         [,13]         [,14]         [,15]
 [1,]  2.764979e-04  0.0380917945  7.839812e-02  6.570831e-02  0.0071640695
 [2,]  1.800762e-03  0.0783981282  2.419125e-01  1.450717e-01  0.0284143932
 [3,]  2.297497e-03  0.0657082896  1.450717e-01  1.147686e-01  0.0239750093
 [4,]  1.548752e-03  0.0026315996  5.078531e-03  4.732015e-03  0.0109519890
 [5,]  3.890896e-03  0.0050785265  2.245914e-02  1.070027e-02  0.0243122836
 [6,]  2.896658e-03  0.0047320228  1.070028e-02  8.703203e-03  0.0200869556
 [7,]  6.671877e-03  0.0109519918  2.431228e-02  2.008695e-02  0.0463764804
 [8,]  1.574127e-03 -0.0023076954 -1.509778e-02 -1.674544e-03  0.0003853838
 [9,]  1.197022e-02 -0.0150977743 -3.907736e-02 -7.072866e-03 -0.0007391000
[10,]  1.154486e-02 -0.0016745461 -7.072872e-03  7.235290e-02 -0.0049017400
[11,]  1.760261e+03  0.0003853967 -7.390725e-04 -4.901733e-03  0.1103708626
[12,]  3.853967e-04  1.8953728900  3.244702e-01  4.805521e-02  0.0129557450
[13,] -7.390725e-04  0.3244702310  5.555646e+04  1.625444e-01  0.0724515218
[14,] -4.901733e-03  0.0480552130  1.625444e-01  1.388970e+04  0.0356686805
[15,]  1.103709e-01  0.0129557450  7.245152e-02  3.566868e-02 56.4471165303
[16,] -1.073351e-02  0.1862810907  4.334232e-01  9.100195e-02 -0.0846566025
[17,] -8.197979e-03  0.0485231762  1.306116e-01  3.274832e-02  0.0090573517
[18,]  4.393458e-03  0.0924301402 -1.154563e-01  6.381205e-02  0.0224514423
[19,]  3.187177e-03  0.0717986757 -9.012743e-02  4.921511e-02  0.0169669621
[20,]  2.210052e-03  0.0916994111  1.817609e-01  7.200428e-02  0.0174604735
[21,]  8.360884e-03  0.1817608433  5.339889e-01  1.812521e-01  0.0684331997
             [,16]         [,17]        [,18]        [,19]        [,20]
 [1,]  0.049206647  2.248008e-03  0.067791706  0.052304608  0.066961627
 [2,]  0.113316378  3.183939e-03 -0.113702854 -0.089772227  0.136866744
 [3,]  0.085335704  3.554888e-03  0.087311612  0.067109946  0.115496573
 [4,]  0.002306428 -3.709653e-04  0.009052586  0.006966641  0.006381063
 [5,]  0.006348841 -2.297651e-03 -0.035655775 -0.028081102  0.012485658
 [6,]  0.004382444 -8.630337e-04  0.009696047  0.007382376  0.011495379
 [7,]  0.010075006 -1.940848e-03  0.024341678  0.018571789  0.026599270
 [8,] -0.003207851 -1.775681e-02  0.013759246  0.010666572  0.009321607
 [9,] -0.016259015 -5.091227e-02 -0.014720310 -0.011620947  0.016350939
[10,]  0.001236689 -1.252867e-02  0.012494219  0.009544201  0.012521003
[11,] -0.010733514 -8.197979e-03  0.004393458  0.003187177  0.002210052
[12,]  0.186281091  4.852318e-02  0.092430140  0.071798676  0.091699411
[13,]  0.433423220  1.306116e-01 -0.115456341 -0.090127427  0.181760866
[14,]  0.091001951  3.274832e-02  0.063812052  0.049215108  0.072004279
[15,] -0.084656603  9.057352e-03  0.022451442  0.016966962  0.017460474
[16,]  1.582695552 -2.340490e-02  0.065933764  0.051203216  0.115221845
[17,] -0.023404903  1.354577e+02  0.009970989  0.007496675 -0.003756779
[18,]  0.065933764  9.970989e-03 17.469273065 -0.040191405  0.111903668
[19,]  0.051203216  7.496675e-03 -0.040191405 28.962407120  0.087570464
[20,]  0.115221845 -3.756779e-03  0.111903668  0.087570464 36.370523425
[21,]  0.254856527 -1.520942e-02 -0.237385845 -0.182292587  0.336883909
             [,21]
 [1,]  0.136866742
 [2,]  0.425452548
 [3,]  0.253838107
 [4,]  0.012485638
 [5,]  0.055384171
 [6,]  0.026327646
 [7,]  0.059814376
 [8,]  0.016350929
 [9,]  0.051411747
[10,]  0.028574583
[11,]  0.008360884
[12,]  0.181760843
[13,]  0.533988906
[14,]  0.181252056
[15,]  0.068433200
[16,]  0.254856527
[17,] -0.015209422
[18,] -0.237385845
[19,] -0.182292587
[20,]  0.336883909
[21,]  3.534930353

$residuals
 [1] -0.0180689440 -0.0111593663 -0.1242835882 -0.0009118192 -0.0016001807
 [6] -0.0029070572 -0.0109481883 -0.0437172900 -0.0122567173 -0.0009442096
[11] -0.0002332161 -0.1575809143 -0.0024454390 -0.0011725832 -0.0049887988
[16] -0.2303020795 -0.0226264894  0.0388952925  0.0202896269 -0.0310325493
[21] -0.3588914460  0.0166163557  0.0157369787 -0.1905609052 -0.4879608780
[26] -0.0534457138  0.0509631750 -0.2458668908 -0.2017202010  0.0426638814
[31]  0.0264473374 -0.0560827016 -0.1694548175 -0.1030723548 -0.0077313434
[36] -0.0763907032 -0.0673888644  0.0486244089  0.0336900696 -0.0306721964
[41] -0.1078346016 -0.0586323246 -0.0252625183 -0.0408958570 -0.0516132625
[46]  0.0565012704  0.0391416388 -0.0139292689 -0.0700125996 -0.0303906756
[51] -0.0687943607 -0.0120934511 -0.0723434735  0.0964955404  0.0644123908
[56] -0.0555304557 -0.2043251909 -0.1103151872 -0.1846506124 -0.0580373716
[61] -0.1316274140

$info
[1] 1

$message
[1] "Relative error in the sum of squares is at most `ftol'."

$iterations
[1] 7

$rsstrace
[1] 1.1852456 0.8421130 0.8359591 0.8357774 0.8357659 0.8357651 0.8357650
[8] 0.8357650

$ssr
[1] 0.835765

$diag
   diag1    diag2    diag3    diag4    diag5    diag6    diag7    diag8 
1.020899 1.014766 1.249979 1.000992 1.002057 1.003558 1.020019 1.101723 
   diag9   diag10   diag11   diag12   diag13   diag14   diag15   diag16 
1.029206 1.001585 1.000375 1.205744 1.002794 1.001202 1.014488 1.397125 
  diag17   diag18   diag19   diag20   diag21 
1.028299 1.042713 1.025559 1.033233 1.623961 

$ms
[1] 0.01370107

$var_ms_unscaled
[1] NA

$var_ms_unweighted
[1] NA

$var_ms
[1] NA

$rank
[1] 21

$df.residual
[1] 40

attr(,"class")
[1] "modFit"

> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>