NIST PCA Example
# A Principal Components Analysis example
#
# walk through the NIST/Sematech PCA example data section 6.5
# http://www.itl.nist.gov/div898/handbook/
#
# This file is part of the Hume La Package
# (C)Copyright 2001, Hume Integration Software
#
# These calculations use the Hume Linear Algebra package, La
#
package require La
catch { namespace import La::*}
# Look at the bottom of this file to see the console output of this procedure.
proc nist_pca {} {
puts {Matrix of observation data X[10,3]}
set X {2 10 3 7 4 3 4 1 8 6 3 5 8 6 1 8 5 7 7 2 9 5 3 3 9 5 8 7 4 5 8 2 2}
puts [show $X]
puts {Compute column norms}
mnorms_br X Xmeans Xsigmas
puts "Means=[show $Xmeans]"
puts "Sigmas=[show $Xsigmas %.6f]"
puts {Normalize observation data creating Z[10,3]}
puts [show [set Z [mnormalize $X $Xmeans $Xsigmas]] %10.6f]\n
puts {Naive computation method that is sensitive to round-off error}
puts {ZZ is (Ztranspose)(Z) }
puts "ZZ=\n[show [set ZZ [mmult [transpose $Z] $Z]] %10.6f]"
puts {the correlation matrix, R = (Zt)(Z)/(n-1)}
puts "R=\n[show [set R [mscale $ZZ [expr 1/9.0]]] %8.4f]"
puts {Continuing the naive example use R for the eigensolution}
set V $R
mevsvd_br V evals
puts "eigenvalues=[show $evals %8.4f]"
puts "eigenvectors:\n[show $V %8.4f]\n"
puts "Good computation method- perform SVD of Z"
set U $Z
msvd_br U S V
puts "Singular values, S=[show $S]"
puts "square S and divide by (number of observations)-1 to get the eigenvalues"
set SS [mprod $S $S]
set evals [mscale $SS [expr 1.0/9.0]]
puts "eigenvalues=[show $evals %8.4f]"
puts "eigenvectors:\n[show $V %8.4f]"
puts "(the 3rd vector/column has inverted signs, thats ok the math still works)"
puts "\nverify V x transpose = I"
puts "As computed\n[show [mmult $V [transpose $V]] %20g]\n"
puts "Rounded off\n[show [mround [mmult $V [transpose $V]]]]"
#puts "Diagonal matrix of eigenvalues, L"
#set L [mdiag $evals]
#
puts "\nsquare root of eigenvalues as a diagonal matrix"
proc sqrt x { return [expr sqrt($x)] }
set evals_sr [mat_unary_op $evals sqrt]
set Lhalf [mdiag $evals_sr]
puts [show $Lhalf %8.4f]
puts "Factor Structure Matrix S"
set S [mmult $V $Lhalf]
puts [show $S %8.4f]\n
set S2 [mrange $S 0 1]
puts "Use first two eigenvalues\n[show $S2 %8.4f]"
# Nash discusses better ways to compute this than multiplying SSt
# the NIST method is sensitive to round-off error
set S2S2 [mmult $S2 [transpose $S2]]
puts "SSt for first two components, communality is the diagonal"
puts [show $S2S2 %8.4f]\n
# define reciprocal square root function
proc recipsqrt x { return [expr 1.0/sqrt($x)] }
# use it for Lrhalf calculation
set Lrhalf [mdiag [mat_unary_op $evals recipsqrt]]
set B [mmult $V $Lrhalf]
puts "Factor score coefficients B Matrix:\n[show $B %12.4f]"
puts "NIST shows B with a mistake, -.18 for -1.2 (-1.18 with a typo)"
puts "\nWork with first normalized observation row"
set z1 {2 3 0 0.065621795889 0.316227766017 -0.7481995314}
puts [show $z1 %.4f]
puts "compute the \"factor scores\" from multiplying by B"
set t [mmult [transpose $z1] $B]
puts [show $t %.4f]
puts "NIST implies you might chart these"
#set T2 [dotprod $t $t]
#puts "Hotelling T2=[format %.4f $T2]"
puts "Compute the scores using V, these are more familiar"
set t [mmult [transpose $z1] $V]
puts [show $t %.4f]
puts "Calculate T2 from the scores, sum ti*ti/evi"
set T2 [msum [transpose [mdiv [mprod $t $t] [transpose $evals]]]]
puts "Hotelling T2=[format %.4f $T2]"
puts "Fit z1 using first two principal components"
set p [mrange $V 0 1]
puts p=\n[show $p %10.4f]
set t [mmult [transpose $z1] $p]
set zhat [mmult $t [transpose $p]]
puts " z1=[show $z1 %10.6f]"
puts "zhat=[show $zhat %10.6f]"
set diffs [msub [transpose $z1] $zhat]
puts "diff=[show $diffs %10.6f]"
puts "Q statistic - distance from the model measurement for the first observation"
set Q [dotprod $diffs $diffs]
puts Q=[format %.4f $Q]
puts "Some experts would compute a corrected Q that is larger since the
observation was used in building the model. The Q calculation just used
properly applies to the analysis of new observations."
set corr [expr 10.0/(10.0 - 2 -1)] ;# N/(N - Ncomp - 1)
puts "Corrected Q=[format %.4f [expr $Q * $corr]] (factor=[format %.4f $corr])"
}
return
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Console Printout
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Matrix of observation data X[10,3]
7 4 3
4 1 8
6 3 5
8 6 1
8 5 7
7 2 9
5 3 3
9 5 8
7 4 5
8 2 2
Compute column norms
Means=6.9 3.5 5.1
Sigmas=1.523884 1.581139 2.806738
Normalize observation data creating Z[10,3]
0.065622 0.316228 -0.748200
-1.903032 -1.581139 1.033228
-0.590596 -0.316228 -0.035629
0.721840 1.581139 -1.460771
0.721840 0.948683 0.676942
0.065622 -0.948683 1.389513
-1.246814 -0.316228 -0.748200
1.378058 0.948683 1.033228
0.065622 0.316228 -0.035629
0.721840 -0.948683 -1.104485
Naive computation method that is sensitive to round-off error
ZZ is (Ztranspose)(Z)
ZZ=
9.000000 6.017916 -0.911824
6.017916 9.000000 -2.591349
-0.911824 -2.591349 9.000000
the correlation matrix, R = (Zt)(Z)/(n-1)
R=
1.0000 0.6687 -0.1013
0.6687 1.0000 -0.2879
-0.1013 -0.2879 1.0000
Continuing the naive example use R for the eigensolution
eigenvalues= 1.7688 0.9271 0.3041
eigenvectors:
0.6420 0.3847 -0.6632
0.6864 0.0971 0.7207
-0.3417 0.9179 0.2017
Good computation method- perform SVD of Z
Singular values, S=3.98985804571 2.88854344819 1.65449373616
square S and divide by (number of observations)-1 to get the eigenvalues
eigenvalues= 1.7688 0.9271 0.3041
eigenvectors:
0.6420 0.3847 0.6632
0.6864 0.0971 -0.7207
-0.3417 0.9179 -0.2017
(the 3rd vector/column has inverted signs, thats ok the math still works)
verify V x transpose = I
As computed
1 -5.55112e-017 1.38778e-016
-5.55112e-017 1 5.55112e-017
1.38778e-016 5.55112e-017 1
Rounded off
1.0 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0
square root of eigenvalues as a diagonal matrix
1.3300 0.0000 0.0000
0.0000 0.9628 0.0000
0.0000 0.0000 0.5515
Factor Structure Matrix S
0.8538 0.3704 0.3658
0.9128 0.0935 -0.3975
-0.4544 0.8838 -0.1112
Use first two eigenvalues
0.8538 0.3704
0.9128 0.0935
-0.4544 0.8838
SSt for first two components, communality is the diagonal
0.8662 0.8140 -0.0606
0.8140 0.8420 -0.3321
-0.0606 -0.3321 0.9876
Factor score coefficients B Matrix:
0.4827 0.3995 1.2026
0.5161 0.1009 -1.3069
-0.2569 0.9533 -0.3657
NIST shows B with a mistake, -.18 for -1.2 (-1.18 with a typo)
Work with first normalized observation row
0.0656 0.3162 -0.7482
compute the "factor scores" from multiplying by B
0.3871 -0.6552 -0.0608
NIST implies you might chart these
Compute the scores using V, these are more familiar
0.5148 -0.6308 -0.0335
Calculate T2 from the scores, sum ti*ti/evi
Hotelling T2=0.5828
Fit z1 using first two principal components
p=
0.6420 0.3847
0.6864 0.0971
-0.3417 0.9179
z1= 0.065622 0.316228 -0.748200
zhat= 0.087847 0.292075 -0.754958
diff= -0.022225 0.024153 0.006758
Q statistic - distance from the model measurement for the first observation
Q=0.0011
Some experts would compute a corrected Q that is larger since the
observation was used in building the model. The Q calculation just used
properly applies to the analysis of new observations.
Corrected Q=0.0016 (factor=1.4286)