[Rd] Principal Components Analysis (PR#8320)

[ripley@stats.ox.ac.uk]

Please do RTFM:  ?princomp says

Note:

      The signs of the columns of the loadings and scores are arbitrary,
      and so may differ between different programs for PCA, and even
      between different builds of R.

and ?prcomp has a similar comment.  Equally, the signs of eigenvectors are 
arbitrary, and ?eigen says

           Recall that the eigenvectors are only defined up to a
           constant: even when the length is specified they are still
           only defined up to a scalar of modulus one (the sign for real
           matrices).

If you don't know why these are true, please ask your advisors for help.


On Thu, 17 Nov 2005 sarkar at mail.utexas.edu wrote:

> Full_Name: Sahotra Sarkar
> Version: 2.2.0
> OS: Windows XP Professional
> Submission from: (NULL) (146.6.130.180)
>
>
> The following two commands should give the same results for the eigenvectors but
> do not (there is a sign reversal for the first one):
>
>> summary(princomp(bumpus),loading = TRUE)
> Importance of components:
>                          Comp.1    Comp.2     Comp.3      Comp.4     Comp.5
> Standard deviation     6.2801653 2.4285636 1.13995042 0.560733747 0.03458915
> Proportion of Variance 0.8399662 0.1256084 0.02767525 0.006696272 0.00002548
> Cumulative Proportion  0.8399662 0.9655746 0.99324988 0.999946156 0.99997164
>                             Comp.6       Comp.7       Comp.8       Comp.9
> Standard deviation     3.025628e-02 1.405339e-02 1.147326e-02 9.339938e-03
> Proportion of Variance 1.949623e-05 4.206121e-06 2.803451e-06 1.857837e-06
> Cumulative Proportion  9.999911e-01 9.999953e-01 9.999981e-01 1.000000e+00
>
> Loadings:
>   Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9
> V1  0.497  0.851  0.171
> V2  0.853 -0.515
> V3  0.149  0.103 -0.965  0.189
> V4               -0.180 -0.981
> V5                             -0.288         0.568  0.327  0.693
> V6                             -0.399         0.565  0.154 -0.705
> V7                             -0.867        -0.475 -0.107
> V8                                            0.352 -0.926  0.118
> V9                                     0.994
>> eigen(cov(bumpus))
> $values
> [1] 3.973263e+01 5.941610e+00 1.309113e+00 3.167514e-01 1.205271e-03
> 9.222233e-04
> [7] 1.989606e-04 1.326107e-04 8.788062e-05
>
> $vectors
>              [,1]          [,2]         [,3]         [,4]          [,5]
> [1,] -0.496897319  0.8505093640  0.171388489  0.018744742 -0.0002708851
> [2,] -0.852790441 -0.5152155401  0.081515933  0.025301353  0.0031275939
> [3,] -0.149011771  0.1032588064 -0.965116017  0.188813583  0.0029039523
> [4,] -0.059981091  0.0228574697 -0.180065518 -0.981007482  0.0301789269
> [5,] -0.002474454 -0.0009781164 -0.003402721 -0.011404980 -0.2884751568
> [6,] -0.002255070 -0.0003234929 -0.002776659 -0.014386855 -0.3990284747
> [7,] -0.003505091 -0.0005603692 -0.006526439 -0.022328724 -0.8671382015
> [8,] -0.001124313  0.0005858447 -0.003031985 -0.006957916 -0.0674635788
> [9,] -0.003810707  0.0004531193 -0.006127501 -0.009908937 -0.0122192249
>              [,6]          [,7]          [,8]          [,9]
> [1,] -0.001028476  1.638253e-06  0.0004887328 -0.0003561342
> [2,] -0.002350593 -8.313864e-04 -0.0003929312  0.0005026715
> [3,] -0.004636198  8.031165e-05  0.0015963693  0.0010649875
> [4,] -0.011239114 -5.568762e-03  0.0035062479  0.0003785351
> [5,]  0.083901242  5.675807e-01  0.3265320650 -0.6934139460
> [6,]  0.032231827  5.646388e-01  0.1541661011  0.7049350072
> [7,] -0.058318858 -4.748060e-01 -0.1070487286 -0.0849487969
> [8,]  0.014813290  3.516922e-01 -0.9260185177 -0.1182046449
> [9,]  0.994055719 -9.937481e-02 -0.0249932592  0.0324567705
>
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-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
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