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<html>
<head>
<title>NIST PCA Example</title>
</head>
<P>
<H2>NIST PCA Example</H2>
<pre>
# 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
##########################################################################
Console Printout
##########################################################################
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)
</PRE>
</html>
\ No newline at end of file
lib/la1.0/NIST.tcl 0 → 100755
View file @ f60cfa7c
# 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 Services
#
# 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
##########################################################################
Console Printout
##########################################################################
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)
\ No newline at end of file
lib/la1.0/defs 0 → 100755
View file @ f60cfa7c
# This file contains support code for the Tcl test suite. It is
# normally sourced by the individual files in the test suite before
# they run their tests. This improved approach to testing was designed
# and initially implemented by Mary Ann May-Pumphrey of Sun Microsystems.
# This file is NOT Copyright by Hume Integration
#
set VERBOSE 0
set TESTS {}
set auto_noexec 1
# If tests are being run as root, issue a warning message and set a
# variable to prevent some tests from running at all.
set user {}
catch {set user [exec whoami]}
# Some of the tests don't work on some system configurations due to
# configuration quirks, not due to Tcl problems; in order to prevent
# false alarms, these tests are only run in the master source directory
# at Berkeley. The presence of a file "Berkeley" in this directory is
# used to indicate that these tests should be run.
set atBerkeley [file exists Berkeley]
proc print_verbose {test_name test_description contents_of_test code answer} {
puts stdout "\n"
puts stdout "==== $test_name $test_description"
puts stdout "==== Contents of test case:"
puts stdout "$contents_of_test"
if {$code != 0} {
if {$code == 1} {
puts stdout "==== Test generated error:"
puts stdout $answer
} elseif {$code == 2} {
puts stdout "==== Test generated return exception; result was:"
puts stdout $answer
} elseif {$code == 3} {
puts stdout "==== Test generated break exception"
} elseif {$code == 4} {
puts stdout "==== Test generated continue exception"
} else {
puts stdout "==== Test generated exception $code; message was:"
puts stdout $answer
}
} else {
puts stdout "==== Result was:"
puts stdout "$answer"
}
}
proc test {test_name test_description contents_of_test passing_results} {
global VERBOSE
global TESTS
if {[string compare $TESTS ""] != 0} then {
set ok 0
foreach test $TESTS {
if [string match $test $test_name] then {
set ok 1
break
}
}
if !$ok then return
}
set code [catch {uplevel $contents_of_test} answer]
if {$code != 0} {
print_verbose $test_name $test_description $contents_of_test \
$code $answer
} elseif {[string compare $answer $passing_results] == 0} then {
if $VERBOSE then {
print_verbose $test_name $test_description $contents_of_test \
$code $answer
puts stdout "++++ $test_name PASSED"
}
} else {
print_verbose $test_name $test_description $contents_of_test $code \
$answer
puts stdout "---- Result should have been:"
puts stdout "$passing_results"
puts stdout "---- $test_name FAILED"
}
}
proc dotests {file args} {
global TESTS
set savedTests $TESTS
set TESTS $args
source $file
set TESTS $savedTests
}
lib/la1.0/index.html 0 → 100755
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<html>
<head>
<title>La Package Documentation</title>
</head>
<frameset cols="120,140,*" frameborder="yes" border=2 framespacing=2>
<frame name="products" src="products.html" marginwidth="6" ></frame>
<frame name="cmds" marginwidth="6" src="la_cmds.html"></frame>
<frame name="man" src="la.html"></frame>
</frameset>
</html>
lib/la1.0/la.html 0 → 100755
View file @ f60cfa7c
This diff is collapsed.
lib/la1.0/la.tcl 0 → 100755
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This diff is collapsed.
lib/la1.0/la.test 0 → 100755
View file @ f60cfa7c
# Part of the Hume La package
# (C)Copyright 2001, Hume Integration Software
source la.tcl
catch { namespace import La::*}
if { [info commands dim] == {} } {
puts "Cannot find La package procedures like dim"
return
}
if {[string compare test [info procs test]] == 1} then \
{source defs}
test mathprec {compute precision of math calculations} {
set eps [mathprec]
expr $eps + 1 > 1
} {1}
test dim-1 {various good args} {
set list {}
foreach arg { {} {2.3} {2 1 0 2.3} {2 2 2 1.1 1.2 1.3 1.4} {3 1 1 1 1}} {
lappend list [dim $arg]
}
set list
} {{} 0 1 2 3}
test dim-2 {bad vector arg} {
catch {dim {2 0 3 1 2 3}} txt
set txt
} {improper length of matrix}
test dim-3 {bad scalar arg} {
catch {dim {2 0 0 3.14}} txt
set txt
} {improper length of vector}
test dim-4 {bad vector arg} {
catch {dim {2 3 0 1 2 3 4}} txt
set txt
} {improper length of vector}
test dim-5 {overlong matrix arg} {
catch {dim {2 3 1 1 2 3 4}} txt
set txt
} {improper length of matrix}
test dim-6 {short matrix arg} {
catch {dim {2 3 1 1 2}} txt
set txt
} {improper length of matrix}
test dim-7 {incomplete vector/matrix} {
catch {dim {2 1}} txt
set txt
} {improper La operand format}
test dim-8 {wrong type arg} {
catch {dim {hello world}} txt
set txt
} {improper La operand format}
test dim-9 {wrong type arg} {
catch {dim {hello}} txt
set txt
} {improper La operand format}
test dotprod-1 {vxv good} {
dotprod {2 3 0 1 2 3} {2 3 0 1 2 3}
} {14.0}
test dotprod-2 {v x v bad} {
catch {dotprod {2 3 0 1 2 3} {2 4 0 1 2 3 4}} txt
set txt
} {Assumed length does not seem proper.}
test dotprod-3 {v x Nx1 good} {
dotprod {2 3 0 1 2 3} {2 3 1 1 2 3}
} {14.0}
test dotprod-4 {v x 1XN good} {
dotprod {2 3 0 1 2 3} {2 1 3 1 2 3}
} {14.0}
test dotprod-5 {1xN x Nx1 good} {
dotprod {2 1 3 1 2 3} {2 3 1 1 2 3}