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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 69 "The Geometry of LDA and PCA Clas
sifiers Illustrated with 3D Examples." }}{PARA 19 "" 0 "" {TEXT -1 17 
"J. Ross Beveridge" }}{PARA 256 "" 0 "" {TEXT -1 23 "Technical Report \+
01-101" }}{PARA 256 "" 0 "" {TEXT -1 27 "Computer Science Department" 
}}{PARA 256 "" 0 "" {TEXT -1 25 "Colorado State University" }}{PARA 
256 "" 0 "" {TEXT -1 21 "ross@cs.colostate.edu" }}{PARA 256 "" 0 "" 
{TEXT -1 26 "Last Update, May 30,  2001" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 8 "Overview" }}{PARA 0 "" 0 "" 
{TEXT -1 376 "This report will help in developing a geometric interpre
tation of Fisher Linear Discrimants. The report builds upon an underst
anding of the connection between Principal Component Analysis and Gaus
sian Distributions. It contains a running example showing how Fisher D
iscriminants are computed and what they look like for an illustrative \+
3 class problem in 3 dimensional space. " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "Maple 6.0 was used to write this \+
report, and consequently it is both a document and a program. It is av
ailable in three forms:" }}{PARA 15 "" 0 "" {TEXT -1 197 "Maple 6.0 Wo
rksheet: This is the best format, since the reader may interact with t
he 3D plots. It is also simple to construct alternative examples by ma
king minor modifications to the Maple source." }}{PARA 15 "" 0 "" 
{TEXT -1 321 "HTML: This is the way most people will view the document
. The 3D plots are animated so perceived 3D structure is evident. Howe
ver, the user cannot alter the view or explore the data as is possible
 in Maple directly. The link to this version is: http://www.cs.colosta
te.edu/evalfacerec/papers/csuldareport/report01101.html" }}{PARA 15 "
" 0 "" {TEXT -1 138 "PDF (from LaTeX): This is a more standard version
 with only text, math and static figures. It is best if one wishes to \+
print the document." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 542 "When viewing the Maple or HTML versions, the first \"S
ection\" is not intended to read. It is, instead, a collection of help
er routines written to service the remainder of the document.  The rea
der will also note that mathematics appears in three forms. In line ma
th, input expressions, and the results of evaluating these expression.
 In line math appears is what one normally expects for type set mathem
atics. Inputs to expressions are set aside in blocks and are shown in \+
red. Results of these expression appear below the expression in blue. \+
" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Helper Maple Code" }}{PARA 
0 "" 0 "" {TEXT -1 197 "Here are the various procedures that simplify \+
the development of the examples below. This section is NOT intended to
 read unless one has an intimate interest in how the examples are buil
t in Maple." }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 35 "Library Calls and \+
Global Parameters" }}{PARA 0 "" 0 "" {TEXT -1 78 "This worksheet uses \+
statistics, 3D plotting and the new Linear Algebra package" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "restart():\nDigits := 32:\nwith(Lin
earAlgebra):\nwith(stats):\nwith(plots):" }}{PARA 7 "" 1 "" {TEXT -1 
116 "Warning, these names have been redefined: anova, describe, fit, i
mportdata, random, statevalf, statplots, transform\n" }}}{PARA 0 "" 0 
"" {TEXT -1 105 "Here are some global parameters used to determine suc
h things as the number of points to sample per class" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 48 "pointsPerClass   := 100:\nanimationSample
s := 60:" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Set Random Number Se
ed" }}{PARA 0 "" 0 "" {TEXT -1 69 "It is handy to have the same point \+
samples each time this file is run" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "randomize(1111356902):" }}}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 33 "General Purpose Helper Procedures" }}{PARA 0 "" 0 "" 
{TEXT -1 107 "Here are some procedures that are not directly related t
o Gaussian Random Variables, but which are helpful." }}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 48 "Reduce the precision of a Matrix to Aid Printin
g" }}{PARA 0 "" 0 "" {TEXT -1 163 "This procedure simply rounds floati
ng point numbers in Matrices to k digits so Maple pretty print looks b
etter. This should not be necessary, but it appears to be." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 366 "sigDigits := proc(f,k)\n  local s,
 fn, tempDigits;\n  if (abs(f) < 10**(-k)) then\n    fn := 0.0;\n  els
e\n    s := ilog10(abs(f)) + 1;\n    # print(\"s is equal to \", s);\n
    tempDigits := Digits;\n    Digits := max(s + k + 1,1);\n    # prin
t(\"Setting Digits to \", Digits);\n    fn := (round (f*10**k))/(10.0*
*k);\n    Digits := tempDigits;\n  end if;\n  return fn;\nend proc:\n \+
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "roundMatrix := proc(M,
k)\n  map(proc(x) sigDigits(x,k) end proc,M);\nend proc:" }}}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 38 "Generate
 and Manipulate a Data Matrix " }}{PARA 0 "" 0 "" {TEXT -1 67 "This se
ction includes procedures to generate sample data matrices. " }}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 49 "A Data Matrix of Gaussian Random Variab
le Samples" }}{PARA 0 "" 0 "" {TEXT -1 1245 "The following set of proc
edures provide a general way of rotation, scaling and translating the \+
points represented as columns in a homogeneous coordinates data matrix
.  Thus, the matrix is of size 1..4 by 1..pointsPerClass.  The affine \+
transformation construction procedures are general purpose. The suppor
t the final procedure, genGaussData, that takes 9 arguments. This proc
edure can construct point samples for any 3D Gaussian Random Variable \+
by starting with samples from one with mean zero and standard deviatio
n one.  The 9 arguments specify the transformation from the zero-one r
eference frame to the final reference frame. They also relate directly
 to the paramters of the Gaussian pdf in the final reference frame. Th
e first three are the standard deviation of a general form 3D Gaussian
 Random Variable when viewed in its principal coordinate representatio
n. This is another way of saying these are the x, y and z standard dev
iation before it is rotated to a new orientation. The next three argum
ents are the translation away from the origin. These are just the x, y
 and z mean values for the pdf.  The last three values specify the rot
ation of the distribution in Euler angles. In other words, rotation ab
out the x, y and then z axis. " }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 31 
"Generate a 4x4 Rotation Matrix " }}{PARA 0 "" 0 "" {TEXT -1 68 "Given
 three Euler Angles, return a 4x4 homogenious rotation Matrix. " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 466 "homRotation := proc(theta,p
hi,xi)\n   local Data, ROTx, ROTy, ROTz, R33, R44, i, j;\n   ROTx := M
atrix([[1,0,0],[0,cos(theta),-sin(theta)],[0,sin(theta),cos(theta)]]):
\n   ROTy := Matrix([[cos(phi),0,-sin(phi)],[0,1,0],[sin(phi),0,cos(ph
i)]]):\n   ROTz := Matrix([[cos(xi),-sin(xi),0],[sin(xi),cos(xi),0],[0
,0,1]]):\n   R33  := map(eval,Multiply(Multiply(ROTz,ROTy),ROTx)):\n  \+
 R44  := Matrix(4,4,0.0):  R44[4,4] := 1.0:\n   R44[1..3,1..3] := R33:
\n   return R44:\nend proc:\n" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 
27 "Generate a 4x4 Scale Matrix" }}{PARA 0 "" 0 "" {TEXT -1 76 "Given \+
an x, y and z axis scale factor, return a 4x4 homogenious scale Matrix
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "homScale := proc(sx, sy
, sz)\n  local S:\n  S := Matrix(4,4,0.0):\n  S[1,1] := sx:  S[2,2] :=
 sy: S[3,3] := sz: S[4,4] := 1.0:\n  return S:\nend proc:" }}}}{SECT 
1 {PARA 5 "" 0 "" {TEXT -1 33 "Generate a 4x4 Translation Matrix" }}
{PARA 0 "" 0 "" {TEXT -1 88 "Given translations along the x, y an z ax
is, return a 4x4 homogenious translation Matrix" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 206 "homTranslation := proc(tx, ty, tz)\n  local T
:\n  T            := Matrix(4,4):\n  T[1..4,1..4] := IdentityMatrix(4,
4):\n  T[1,4]       := tx: \n  T[2,4]       := ty: \n  T[3,4]       :=
 tz: \n  return T:\nend proc:" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 
50 "Generate a 4x4 Scale, Rotation, Translation Matrix" }}{PARA 0 "" 
0 "" {TEXT -1 173 "Given scale parameters, rotation parameters, and tr
anslation parameters, create a single 4x4 matrix that executes the sca
ling, followed by rotation, followed by translation." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 204 "homSRT := proc(sx,sy,sz,tx,ty,tz,theta,p
hi,xi)\n  local S, R, T:\n  S := homScale(sx,sy,sz):\n  R := homRotati
on(theta,phi,xi):\n  T := homTranslation(tx,ty,tz):\n  return Multiply
(T,Multiply(R,S)):\nend proc:\n" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 
63 "Generate a 4x4 homogenous Matrix from 3x3 specifying upper left" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "homPad := proc(M33)\n  loc
al M44;\n  M44 := Matrix(4,4, IdentityMatrix(4,4));\n  M44[1..3,1..3] \+
:= M33;\n  return M44;\nend proc:" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 
-1 83 "Generate a Gaussiant Distributed data matrix with mean zero, st
andard deviation one" }}{PARA 0 "" 0 "" {TEXT -1 189 "Using the global
 variable samplesPerClass, generate that many 3D points which are samp
led from a simple 3D Gaussian where the mean is zero and standard devi
ation is one along each dimension" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 315 "homGaussZeroOne := proc()\n  local n, D3, D4:  \n  n
 := pointsPerClass:\n  D3 := Matrix(3,n, [[stats[random,normald[0.0,1.
0]](n)],\n                     [stats[random,normald[0.0,1.0]](n)],\n \+
                    [stats[random,normald[0.0,1.0]](n)]]);\n  D4 := Ma
trix(4,n,1):\n  D4[1..3,1..n] := D3:\n  return D4;\nend proc:\n" }}}}
{SECT 1 {PARA 5 "" 0 "" {TEXT -1 67 "Generate a scaled, rotated and th
en translated Gaussian data matrix" }}{PARA 0 "" 0 "" {TEXT -1 282 "St
arting with a data matrix of 3D Gaussian Random Variable sample with m
ean zero and standard deviation one for all three axes, scale the samp
les along each axis by sx, sy, sz, then rotate the samples by Euler an
gles theta, phi, xi, and finally translate the samples by tx, ty, tz. \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "genGaussData := proc(sx
,sy,sz,tx,ty,tz,theta,phi,xi)\n  local D3, D4:\n  D4 := Multiply(homSR
T(sx,sy,sz,tx,ty,tz,theta,phi,xi),homGaussZeroOne());\n  # D3 := SubMa
trix(D4,[1..3],[1..ColumnDimension(D4)]);\n  return D4;\nend proc:" }}
}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "Three classes from means, stan
dard deviations and angles." }}{PARA 0 "" 0 "" {TEXT -1 304 "This proc
edure returns a three element list, where each element is the data mat
rix for a different class. The parameters for the class are specified \+
in three matrices, MU, SD, and EA. These contain in each successive co
lumn, the means, standard deviations, and Euler angles for each for th
e three classes" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "listClas
ses := proc(MU, SD, EA)\n  local n, i, L:\n  n := ColumnDimension(MU);
\n  L := []:\n  for i from 1 to n do\n    L := [op(L),genGaussData(SD[
1,i],SD[2,i],SD[3,i],\n                             MU[1,i],MU[2,i],MU
[3,i],\n                             EA[1,i],EA[2,i],EA[3,i])]:\n  od:
\n  return L:\nend proc:" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 57 "App
end Class Data Matrices to form one Common Data Matrix" }}{PARA 0 "" 
0 "" {TEXT -1 80 "This procedure will create a single common data matr
ix out of k distinct classes" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
263 "columnAppendDataMatrices := proc(C)\n  local A, i, k, r, c, n:\n \+
 k := nops(C);\n  n := ColumnDimension(C[1]):\n  r := RowDimension(C[1
]):\n  A := Matrix(r,k*n,1.0):\n  for i from 1 to k do\n    c := (i-1)
*n + 1;\n    A[1..r,c..c+n-1] := C[i]:\n  od:\n  return A:\nend proc:
" }}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 35 "Sample Mean and Covariance
 Matrices" }}{PARA 0 "" 0 "" {TEXT -1 154 "This section has two proced
ures, one to estimate the mean from a data matrix. The other to estima
te the covariance matrix by computing the scatter matrix." }}{SECT 1 
{PARA 20 "" 0 "" {TEXT -1 42 "Compute the Mean Vector from a Data Matr
ix" }}{PARA 0 "" 0 "" {TEXT -1 109 "This procedure sums the elements i
n the x, y and z dimensions of the data matrix and divides the totals \+
by n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "dataMatrixMean := \+
proc(A)\n  Vector(3,[seq(stats[describe,mean](convert(A,listlist)[i]),
i=1..3)]);\nend proc:" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 47 "Compu
te the x, y and z ranges for a Data Matrix" }}{PARA 0 "" 0 "" {TEXT 
-1 238 "Use the statistics package range facility to get the range of \+
values along each axis. Return as a sequence of three range speciffica
tions.  Thus, calling order would list three range variables on the le
ft side of the assignment statement." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 105 "dataMatrixRange := proc(A,dims)\n  seq(stats[describ
e,range](convert(A,listlist)[i]),i=1..dims);\nend proc:" }}}}{SECT 1 
{PARA 20 "" 0 "" {TEXT -1 36 "Sample Covariance from a Data Matrix" }}
{PARA 0 "" 0 "" {TEXT -1 65 "This procedure returns the 3x3 scatter ma
trix from a data matrix." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 263 
"scatterMatrix := proc(A)\n  local n, MU, NT, M, CV, D3:\n  n  := Colu
mnDimension(A):\n  MU := dataMatrixMean(A):\n  NT := homTranslation(-M
U[1],-MU[2],-MU[3]):\n  D3 := SubMatrix(Multiply(NT,A),[1..3],[1..n]):
\n  CV := Multiply(D3,Transpose(D3));\n  return CV;\nend proc:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "sampleCovariance := proc(A)
\n  return scatterMatrix(A) . (1/(ColumnDimension(A) -1));\nend proc:
\n  " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 20 "" 0 "" 
{TEXT -1 41 "Scatter Matrix within and Between Classes" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 388 "scatterWBClass := proc(P)\n  local
 SW, SB, mu, mus, n;\n  SW  := scatterMatrix(P[1]) + scatterMatrix(P[2
]) + scatterMatrix(P[3]):\n  mus := [seq(dataMatrixMean(P[i]),i=1..3)]
:\n  n   := [seq(ColumnDimension(P[i]),i=1..3)];\n  mu  := (1/eval(add
(n[i],i=1..3))) . eval(add(n[i].mus[i],i=1..3)):\n  SB  := eval(add(n[
i].OuterProductMatrix(mus[i]-mu,mus[i]-mu),i=1..3)): \n  return SW, SB
;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 31 "Diagonilize a Covariance Matrix" }}{PARA 0 "" 0 "
" {TEXT -1 188 "This is a simple routine that conforms that yields a r
otation and scale such that it represents the change in coordinates ba
ck to the PCA space for a given covariance (or scattter) matrix." }}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 55 "Return R and S Matrices given Sam
ple Covariance/Scatter" }}{PARA 0 "" 0 "" {TEXT -1 89 "Given the sampl
e covariance or scatter matrix, give the diagonal decomposition such t
hat " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Omega = R*S*S*R^t;" "6#/%&Omega
G**%\"RG\"\"\"%\"SGF'F(F')F&%\"tGF'" }{TEXT -1 204 "  This routine wil
l use Singular Value Decomposition and returns an expression sequence,
 thus two variables appear on the left hand side of the equals, one to
 hold the R matrix and the other the S matrix." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 297 "Diagonalize
RS := proc(omega)\n  local i, SV, RM, SM;\n  RM, SV := SingularValues(
omega,output=['U','S'],outputoptions['U']=[datatype=float]):\n  SM := \+
Matrix(RowDimension(omega),ColumnDimension(omega),0);\n  for i from 1 \+
to RowDimension(omega) do SM[i,i] := sqrt(SV[i]); od;\n  return(RM,SM)
:\nend proc:" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 47 "Routines for Plotting Classes and Discriminants" }}
{PARA 0 "" 0 "" {TEXT -1 500 "Here are routines that will plot three d
ata matrices of points along with 2 basis vectors. There are two routi
nes, one that plots only the points and the other that plots the point
s and the discriminants. In both cases, the points are color coded by \+
class and shown in a scatter plot. When shown, the discriminants are a
re drawn as line segments. The length of the basis vectors will be plu
s and minus one standard deviation. They are drawn with their origin a
t the centroid of the joint data matrix." }}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 19 "View Specification " }}{PARA 0 "" 0 "" {TEXT -1 193 "Retu
rn a sequence of three ranges suitable for use as the view argument to
 the plotting routine such that x, y and z are centered on the mean wi
th range sufficient to express largest variation." }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 326 "jointDataView := proc(CC,dims)\n   local m, \+
r, i, width, middle;\n   r := [dataMatrixRange(CC,dims)];\n   m := map
(proc(x) op(2,x) - op(1,x) end proc,r);\n   width  := max(seq(m[i],i=1
..dims));\n   middle := map(proc(x) (op(1,x) + op(2,x))/2.0 end proc,r
);\n   seq((middle[i]-(width/2))..(middle[i]+(width/2)),i=1..dims);\ne
nd proc:\n" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Basis vector endp
oints" }}{PARA 0 "" 0 "" {TEXT -1 597 "Return two endpoints for each o
f the two Fisher Basis Vectors. As arguments, take in the rotation mat
rix, the scale matrix, and the joint data matrix. The joint data matri
x is used to determine the joint mean. The resulting endpoints are enc
ode in a 4x4 matrix, with the first two columns representing the start
 and end point of the first discriminant, the third and fourth the sec
ond. There are four rows because the endpoints are expressed in homoge
nious coordinates. This is in anticipation of our later desire to affi
ne transform the space in which both discriminants and points are expr
essed." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 564 "basisVectorEndPoi
nts := proc(R,S,P)\n  local delta1, delta2, mu, r33, result;\n  result
 := Matrix(4,4,1);\n  mu     := dataMatrixMean(columnAppendDataMatrice
s(P));\n  # delta1 := Transpose(ScalarMultiply(R[1,1..3],S[1,1]));\n  \+
# Changed scale to that of first vector for better viewing.\n  # delta
2 := Transpose(ScalarMultiply(R[2,1..3],S[1,1]));\n  delta1 := ScalarM
ultiply(R[1..3,1],S[1,1]);\n  delta2 := ScalarMultiply(R[1..3,2],S[1,1
]);\n  r33    := Matrix([(mu - delta1, mu + delta1, mu - delta2, mu + \+
delta2)]);\n  result[1..3,1..4] := r33;\n  return result;\nend proc:\n
" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Plot the points and basis v
ectors" }}{PARA 0 "" 0 "" {TEXT -1 118 "Generate a scatter plot of the
 different classes expressed by the points in the data matrices contai
ned in the list P." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "point
sToSeq := proc(P,row)\n  local i, c, s;\n  c := ColumnDimension(P);\n \+
 s := NULL;\n  for i from 1 to c do\n    s := s,[seq(P[j,i],j=1..row)]
;\n  od:\n  return s;\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 536 "pointPlotSpec := proc(P,row)\n  local i, lst,col,col
ors, cn;\n  colors := matrix(3,3,0.0);\n  colors[1,1] := 1.0; colors[2
,2] := 1.0; colors[3,3] := 1.0;\n  lst := [];\n  for i from 1 to nops(
P) do\n    col := COLOR(RGB,colors[1,i],colors[2,i],colors[3,i]);\n   \+
 cn  := ColumnDimension(P[i]);\n    if (row = 2) then\n       lst := [
op(lst),PLOT(POINTS(pointsToSeq(P[i],row)),SYMBOL(CIRCLE,10),col)];\n \+
   else\n       lst := [op(lst),PLOT3D(POINTS(pointsToSeq(P[i],row)),S
YMBOL(CIRCLE,10),col)];\n    fi;\n  od;\n  seq(lst[i],i=1..nops(P));\n
end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 405 "basisPlotSpec
 := proc(BV)\n  local i, j, pts, lst, col, colors;\n  lst := [];\n  co
lors := matrix(3,2,1.0);\n  colors[1,1] := 0.0; colors[2,2] := 0.0;\n \+
 j := 1;\n  for i from 1 to 3 by 2 do\n    pts := convert(Transpose(BV
[1..3,i..(i+1)]),listlist);\n    col := COLOR(RGB,colors[1,j],colors[2
,j],colors[3,j]);\n    j   := j + 1;\n    lst := [op(lst),PLOT3D(CURVE
S(pts),col)];\n  od;\n  seq(lst[i],i=1..2);\nend proc:\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 343 "plotPoints := proc(P,theta,phi,axe
sType)\n  local PP, v;\n  PP := columnAppendDataMatrices(P);\n  v  := \+
[jointDataView(PP,3)];\n  display(\{pointPlotSpec(P,3)\},insequence=fa
lse,axes=axesType,labels=[X,Y,Z],\n           axesfont=[TIMES,ROMAN,14
],labelfont=[TIMES,ROMAN,16],  \n           orientation=[theta,phi],sc
aling=CONSTRAINED,view=v);\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 297 "plotPoints2D := proc(P)\n  local PP, v;\n  PP := col
umnAppendDataMatrices(P);\n  v  := [jointDataView(PP,2)];\n  display(
\{pointPlotSpec(P,2)\},insequence=false,axes=FRAMED,labels=[X,Y],\n   \+
        axesfont=[TIMES,ROMAN,14],labelfont=[TIMES,ROMAN,16],  \n     \+
      scaling=CONSTRAINED,view=v);\nend proc:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 169 "pointsToSeq := proc(P,row)\n  local i, c, s;\n \+
 c := ColumnDimension(P);\n  s := NULL;\n  for i from 1 to c do\n    s
 := s,[seq(P[j,i],j=1..row)];\n  od:\n  return s;\nend proc:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 433 "plotPointsBasis := proc(P,B
V,theta,phi,axesType)\n  local PP, v, C1, C2, C3, B1, B2;\n  PP := col
umnAppendDataMatrices(P);\n  v  := [jointDataView(PP,3)];\n  C1,C2,C3 \+
:= pointPlotSpec(P,3);\n  B1,B2    := basisPlotSpec(BV);\n  display(\{
C1,C2,C3,B1,B2\},insequence=false,axes=axesType,labels=[X,Y,Z],\n     \+
      axesfont=[TIMES,ROMAN,14],labelfont=[TIMES,ROMAN,16],  \n       \+
    orientation=[phi,theta],scaling=CONSTRAINED,view=v);\nend proc:" }
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
56 "Compute R and S using the Generalized Eigenvector Method" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 412 "generalizedEigenvectorsRS :
= proc(SB,SW)\n  local i, R, S, result:\n  result := Eigenvectors(SB,S
W,output=list);\n  result := sort(result,proc(x,y) if (Re(x[1]) > Re(y
[1])) then true else false fi end proc);\n  R      := Matrix(3,3, 0.0)
;\n  S      := Matrix(3,3, 0.0);\n  for i from 1 to 3 do\n     R[1..3,
i] := result[i,3];\n     S[i,i]    := sqrt(result[i,1]);\n  od;\n  ret
urn(map(Re,R),map(Re,S));\nend proc:\n        \n" }}}}{SECT 1 {PARA 4 
"" 0 "" {TEXT -1 37 "Compute The Fisher Criterion Function" }}{PARA 0 
"" 0 "" {TEXT -1 253 "The criterion for the Fisher Basis Vectors maxim
izes a ration of two determinants. Here the arguement FB is a 3 row by
 2 column matrix with one Fisher Basis Vector per column.  SW is the w
ithin scatter matrix, and SB is the between class scatter matrix." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
235 "fischerCriterion := proc(FB,SW,SB)\n  local num, den, ret;\n  num
 := Multiply(Transpose(FB),Multiply(SB,FB));\n  den := Multiply(Transp
ose(FB),Multiply(SW,FB));\n  ret := Determinant(num) / Determinant(den
);\n  convert(ret,float);\nend proc:" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 36 "Project onto 2D Fisher Basis Vectors" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 130 "projectLD := proc(P,LD)\n  local Pp, LDP, PP
;\n  LDP := homPad(LD)[1..4,1..2];\n  PP  := [seq(Transpose(LDP).P[i],
i=1..3)];\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 
2775 "Principal Component Analysis (PCA) and Linear Discriminant Analy
sis (LDA) play a critical role in many pattern classification tasks. I
t is helpful to develop a geometric intuition for how each transforms \+
the coordinate reference frame in which data is classified. There are \+
limits to how far this can go, since these techniques are typically ap
plied to problems in higher dimensional spaces. However, conceding thi
s weakness, with some small effort many people can usefully extend a n
atural 3D intuition to higher dimensions. \n\nThere are several motiva
tions for this paper, but one in particular concerns whether LDA basis
 vectors are orthogonal. It has been reported in the literature that t
he LDA basis vectors are orthogonal: unfortunately this is false. It i
s easy show that LDA basis vectors are not orthogonal. As this paper w
ill show, one may conceive of an intermediate space in which LDA basis
 vectors are indeed orthogonal. Moreover, understanding how this inter
mediate space is subsequently transformed provides insight into how LD
A vectors are configured. The general mathematics of these transformat
ions are reviewed here and each step is illustrated with a 3D running \+
example. \n\nWe will start with some very basic properties of Gaussian
 point clouds in 3D and draw heavily upon the connection between these
 point clouds and PCA. For our purposes, a Gaussian point cloud is for
med by sampling a finite number of observations from a multivariate Ga
ussian random variable in 3D. Our choice of a Gaussian point cloud est
ablishes a well known and tight connection between PCA and Gaussian di
stributions. However, PCA classifiers are often successfully applied t
o non-Gaussian data: the choice should not be seen as limiting. The ad
vantage of fixing our attention on Gaussian points clouds is that it a
llows us to develop an intuition for the geometry underlying both PCA \+
and LDA spaces. \n\nIn particular, there is a tight coupling between s
catter matrices and covariance matrices. In moving from PCA to LDA, em
phasis shifts from the properties of a single scatter matrix to the pr
operties of two related scatter matrices: the between class and within
 class scatter matrices. Understanding in geometric terms what is happ
ening when one solves for the linear discriminants becomes more diffic
ult. It is well know that the problem of finding the linear discrimina
nts is equivalent to solving a generalized Eigenproblem. However,  quo
ting [Strang], ''Geometrically, this has a meaning which we do not und
erstand very well.''  In this statement, Strang is addressing all gene
ralized Eigenproblems. Fortunately, both PCA and LDA induce positive d
efinite and symmetric scatter matrices. These restrictions in turn len
d themselves to a much clearer geometric interpretation. " }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 50 "Gaussian Random Variables and Principal C
omponents" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 "Axis Aligned Gaussia
ns" }}{PARA 0 "" 0 "" {TEXT -1 105 "Consider the equation that defines
 the probability density function for a Gaussian random variable in 3D
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "p(
x);" "6#-%\"pG6#%\"xG" }{TEXT -1 6 "  =   " }{XPPEDIT 18 0 "1/((2*Pi)^
(3/2)*Omega^(1/2));" "6#*&\"\"\"F$*&)*&\"\"#F$%#PiGF$*&\"\"$F$F(!\"\"F
$)%&OmegaG*&F$F$F(F,F$F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp;" "6#%$e
xpG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(-1/2)*(x-mu)^t*Omega^(-1)*(x-mu)
;" "6#**,$*&\"\"\"F&\"\"#!\"\"F(F&),&%\"xGF&%#muGF(%\"tGF&)%&OmegaG,$F
&F(F&,&F+F&F,F(F&" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 5 "where" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 200 "eq1 := x = <x,y,z>:\neq2 := mu = <mu[x],mu[y],mu[z]>
:\neq3 := Omega = Matrix(3,3, [[sigma[xx], sigma[xy], sigma[xz]],[sigm
a[yx], sigma[yy], sigma[yz]],[sigma[zx], sigma[zy], sigma[zz]]]):\neq1
, eq2, eq3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"xG-%'RTABLEG6$\"*C$
f/l-%'MATRIXG6#7%7#F$7#%\"yG7#%\"zG/%#muG-F&6$\"*WxW]'-F*6#7%7#&F36#F$
7#&F36#F/7#&F36#F1/%&OmegaG-F&6$\"*?gV_'-F*6#7%7%&%&sigmaG6#%#xxG&FM6#
%#xyG&FM6#%#xzG7%&FM6#%#yxG&FM6#%#yyG&FM6#%#yzG7%&FM6#%#zxG&FM6#%#zyG&
FM6#%#zzG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 100 "In the special case that the cross terms in the covariance mat
rix are zero, then the pdf reduces to:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 195 "Vx := <x,y,z>:\nMs := Matrix(3,3, [[sigma[xx], 0, 0]
,[0, sigma[yy], 0],[0, 0, sigma[zz]]]):\np(x) = (1/((2*Pi)^(3/2) * Det
erminant(Ms)^(1/2))) * e^((-1/2)*(Transpose(Vx) . MatrixInverse(Ms) . \+
Vx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,$*&*&-%%sqrtG6
#\"\"#\"\"\")%\"eG,(*&*$)F'F.F/F/&%&sigmaG6#%#xxG!\"\"#F:F.*&#F/F.F/*&
*$)%\"yGF.F/F/&F76#%#yyGF:F/F:*&#F/F.F/*&*$)%\"zGF.F/F/&F76#%#zzGF:F/F
:F/F/*&)%#PiG#\"\"$F.F/-F,6#*(F6F/FBF/FKF/F/F:#F/\"\"%" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 17 "Collecting terms:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "p(x
);" "6#-%\"pG6#%\"xG" }{TEXT -1 4 " =  " }{XPPEDIT 18 0 "1/((2*Pi)^(1/
2)*sqrt(sigma[xx]));" "6#*&\"\"\"F$*&)*&\"\"#F$%#PiGF$*&F$F$F(!\"\"F$-
%%sqrtG6#&%&sigmaG6#%#xxGF$F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "e^(-1*x
^2/(2*sigma[xx]));" "6#)%\"eG,$*(\"\"\"F'*$%\"xG\"\"#F'*&F*F'&%&sigmaG
6#%#xxGF'!\"\"F0" }{TEXT -1 5 "     " }{XPPEDIT 18 0 "1/((2*Pi)^(1/2)*
sqrt(sigma[yy]));" "6#*&\"\"\"F$*&)*&\"\"#F$%#PiGF$*&F$F$F(!\"\"F$-%%s
qrtG6#&%&sigmaG6#%#yyGF$F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "e^(-1*y^2/
(2*sigma[yy]));" "6#)%\"eG,$*(\"\"\"F'*$%\"yG\"\"#F'*&F*F'&%&sigmaG6#%
#yyGF'!\"\"F0" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "1/((2*Pi)^(1/2)*sqrt(s
igma[zz]));" "6#*&\"\"\"F$*&)*&\"\"#F$%#PiGF$*&F$F$F(!\"\"F$-%%sqrtG6#
&%&sigmaG6#%#zzGF$F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "e^(-1*z^2/(2*sig
ma[zz]));" "6#)%\"eG,$*(\"\"\"F'*$%\"zG\"\"#F'*&F*F'&%&sigmaG6#%#zzGF'
!\"\"F0" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 406 "When the cross terms in the covariance matrix are
 zero, then the pdf is simply the product of the independent probabili
ties along each of the three axes.  This also means, from a practical \+
standpoint, that samples from an axis-aligned 3D Gaussian random varia
ble may be generated by independently calling a 1D Gaussian random num
ber generator three times, once for each of the three independent comp
onents." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 35 "Example of an Axis Aligned Gaussian" }}{PARA 0 "" 0 "" 
{TEXT -1 91 "Consider the Gaussian random variable defined by the foll
owing mean and standard deviation:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 125 "MU := Matrix(3,1, [[5],[25],[50]]):\nSD := Matrix(3,
1, [[1],[3],[10]]):\nEA := Matrix(3,1, [[0],[0],[0]]):\nmu = MU, sigma
 = SD;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%#muG-%'RTAB
LEG6$\"*[*RCl-%'MATRIXG6#7%7#\"\"&7#\"#D7#\"#]/%&sigmaG-F&6$\"*/yV_'-F
*6#7%7#\"\"\"7#\"\"$7#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 97 "Note the s
igmas along the diagonal of the covariance matrix have been placed in \+
a column vector. " }}{PARA 0 "" 0 "" {TEXT -1 60 "Here is a plot of 10
0 points sampled from this distribution:" }}{SECT 0 {PARA 5 "" 0 "" 
{TEXT -1 17 "3D Plot of Points" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 86 "P := listClasses(MU,SD,EA):\ntheta := -45: phi := 45:\nplotPoint
s(P,theta,phi,FRAMED);\n\n" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 
{PLOTDATA 3 "6)-%'POINTSG6bq7%$\"3Z4PC\"H+bb'!#<$\"39yH*Hi:!QG!#;$\"3/
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0 "" 0 "" {TEXT -1 161 "One sees from these plots that the distributio
n shown has minimal variation along the x dimension, modest variation \+
along y, and the greatest variation along z. " }}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 55 "The Scattter Matrix, Sample Mean and Sample Covarian
ce " }}{PARA 0 "" 0 "" {TEXT -1 192 "To begin to draw the connection b
etween our sampling from a Gaussian random variable and classification
 using PCA and LDA, let us begin by defining a data matrix A as the co
llection of points." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "eq5 \+
:= A = Matrix(1,4, [p[1], p[2], `...` , p[n]]):\neq6 := A = Transpose(
Matrix(4,3, [[x[1], y[1], z[1]],[x[2], y[2], z[2]],[`...`, `...`, `...
`],[x[n], y[n], z[n]]])):\neq5; eq6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/%\"AG-%'RTABLEG6$\"*cn\\_'-%'MATRIXG6#7#7&&%\"pG6#\"\"\"&F/6#\"\"#%
$...G&F/6#%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'RTABLEG6$
\"*[&)[_'-%'MATRIXG6#7%7&&%\"xG6#\"\"\"&F/6#\"\"#%$...G&F/6#%\"nG7&&%
\"yGF0&F;F3F5&F;F77&&%\"zGF0&F@F3F5&F@F7" }}}{PARA 0 "" 0 "" {TEXT -1 
65 "The sample mean and sample covariance for this set of points are:
" }}{PARA 256 "" 0 "" {TEXT -1 1 "\n" }{XPPEDIT 18 0 "mu[s];" "6#&%#mu
G6#%\"sG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/n;" "6#*&\"\"\"F$%\"nG!
\"\"" }{XPPEDIT 18 0 "sum(p[i],i = 1 .. n);" "6#-%$sumG6$&%\"pG6#%\"iG
/F);\"\"\"%\"nG" }{TEXT -1 5 "     " }{XPPEDIT 18 0 "Sigma[s];" "6#&%&
SigmaG6#%\"sG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(n-1);" "6#*&\"\"\"
F$,&%\"nGF$F$!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "S;" "6#%\"SG" }
{TEXT -1 7 "       " }{XPPEDIT 18 0 "S;" "6#%\"SG" }{TEXT -1 4 " =  " 
}{XPPEDIT 18 0 "sum((p[i]-mu[s])*(p[i]-mu[s])^t,i = 1 .. n);" "6#-%$su
mG6$*&,&&%\"pG6#%\"iG\"\"\"&%#muG6#%\"sG!\"\"F,),&&F)6#F+F,&F.6#F0F1%
\"tGF,/F+;F,%\"nG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 196 "The matrix S is commonly called the scat
ter matrix or moment matrix for the set of points.  For the specific p
oints in the example above, the sample mean, scatter matrix and covari
ance matrix are:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "MUs := \+
dataMatrixMean(P[1]): Ss := scatterMatrix(P[1]): OMEGA := sampleCovari
ance(P[1]):\nmu[s] = roundMatrix(MUs,1), S = roundMatrix(Ss,1), Omega[
s] = roundMatrix(OMEGA,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%#muG6
#%\"sG-%'RTABLEG6$\"*K*G2l-%'MATRIXG6#7%7#$\"$5&!\"#7#$\"%5DF37#$\"%S
\\F3/%\"SG-F)6$\"*g'pCl-F-6#7%7%$\"%](*F3$!%]8F3$!%I'*F37%FE$\"&]t*F3$
\"&!3NF37%FGFL$\"'qW$*F3/&%&OmegaGF&-F)6$\"*K(pCl-F-6#7%7%$\"#5!\"\"$!
#5F3$FinFgn7%Fhn$\"$!)*F3$\"$]$F37%FjnF^o$\"%S%*F3" }}}{PARA 0 "" 0 "
" {TEXT -1 127 "The standard deviation along the x, y and z axis are t
he square roots of the diagonal elements of the sample covariance matr
ix:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "sigma[xx] = sigDigit
s(sqrt(OMEGA[1,1]),1), sigma[yy] = sigDigits(sqrt(OMEGA[2,2]),1), sigm
a[zz] = sigDigits(sqrt(OMEGA[3,3]),1);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%/&%&sigmaG6#%#xxG$\"#5!\"\"/&F%6#%#yyG$\"$5$!\"#/&F%6#%#zzG$\"$q
*F1" }}}{PARA 0 "" 0 "" {TEXT -1 98 "These values are close to the ori
ginal. How close depends in part on how many points are sampled. " }}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 60 "Example of a Gaussian Rotated wit
h respect to Principal Axes" }}{PARA 0 "" 0 "" {TEXT -1 112 "Now let u
s consider rotating the distribution about the x axis by 45 degrees an
d about the z axis by 45 degrees." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 148 "MU := Matrix(3,1, [[5],[25],[50]]):\nSD := Matrix(3,
1, [[1],[3],[10]]):\nEA := Matrix(3,1, [[0.7854],[0],[0.7854]]):\nmu =
 MU, sigma = SD, angles = EA;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%#muG-%'RTABLEG6$\"*C@Z_'-%
'MATRIXG6#7%7#\"\"&7#\"#D7#\"#]/%&sigmaG-F&6$\"*[SZ_'-F*6#7%7#\"\"\"7#
\"\"$7#\"#5/%'anglesG-F&6$\"*'z^Cl-F*6#7%7#$\"%ay!\"%7#\"\"!FI" }}}
{PARA 0 "" 0 "" {TEXT -1 100 "Again, the sigmas along the diagonal of \+
the covariance matrix have been placed in a column vector.  " }}{PARA 
0 "" 0 "" {TEXT -1 60 "Here is a plot of 100 points sampled from this \+
distribution:" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 18 "3D plot of point
s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "P := listClasses(MU,SD
,EA):\ntheta := -45: phi := 45:\nplotPoints(P,theta,phi,FRAMED);\n\n" 
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m$\"#XFggm\"\"\"" 1 2 0 1 10 0 2 1 1 3 1 1.000000 45.000000 
-45.000000 1 0 "Curve 1" }}}}}{PARA 0 "" 0 "" {TEXT -1 47 "Here is an \+
animation of the same set of points." }}{SECT 1 {PARA 5 "" 0 "" {TEXT 
-1 28 "3D annimated plot of points." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 139 "samples :=animationSamples:\ndisplay3d(seq(plotPoint
s(P,theta+i*(360/samples),phi+i*(720/samples),BOXED),i=0..(samples-1))
,insequence=true);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 
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\"$t&F[hmFf[n7)F'F`hmFdhmF`imFgimF[jm-F`[n6%F\\bn$\"$&eF[hmFf[n7)F'F`h
mFdhmF`imFgimF[jm-F`[n6%$\"$J#F[hm$\"$(fF[hmFf[n7)F'F`hmFdhmF`imFgimF[
jm-F`[n6%Fcbn$\"$4'F[hmFf[n7)F'F`hmFdhmF`imFgimF[jm-F`[n6%$\"$V#F[hm$
\"$@'F[hmFf[n7)F'F`hmFdhmF`imFgimF[jm-F`[n6%Fhbn$\"$L'F[hmFf[n7)F'F`hm
FdhmF`imFgimF[jm-F`[n6%$\"$b#F[hm$\"$X'F[hmFf[n7)F'F`hmFdhmF`imFgimF[j
m-F`[n6%F_cn$\"$d'F[hmFf[n7)F'F`hmFdhmF`imFgimF[jm-F`[n6%$\"$n#F[hm$\"
$p'F[hmFf[n7)F'F`hmFdhmF`imFgimF[jm-F`[n6%Fdcn$\"$\"oF[hmFf[n7)F'F`hmF
dhmF`imFgimF[jm-F`[n6%$\"$z#F[hm$\"$$pF[hmFf[n7)F'F`hmFdhmF`imFgimF[jm
-F`[n6%F[dn$\"$0(F[hmFf[n7)F'F`hmFdhmF`imFgimF[jm-F`[n6%$\"$\"HF[hm$\"
$<(F[hmFf[n7)F'F`hmFdhmF`imFgimF[jm-F`[n6%F`dn$\"$H(F[hmFf[n7)F'F`hmFd
hmF`imFgimF[jm-F`[n6%$\"$.$F[hm$\"$T(F[hmFf[n7)F'F`hmFdhmF`imFgimF[jm-
F`[n6%Fgdn$\"$`(F[hmFf[n" 1 2 0 1 10 0 2 1 1 1 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}}{PARA 0 "" 0 "" {TEXT -1 249 "Now we see \+
the point cloud is moving up in a diagonal direction relative to the x
, y and z coordinates. As we will see in the next section, we can anal
yze the data matrix of this new set of points to recover the principal
 axes of this distribution. " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 56 "
Scatter and Covariance Matrices for Rotated Distribution" }}{PARA 0 "
" 0 "" {TEXT -1 133 "In the same fashion as above, we can compute the \+
sample mean vector, the scatter matrix, and the associated sample cova
riance matrix." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "MUs := da
taMatrixMean(P[1]): Ss := scatterMatrix(P[1]): OMEGA := sampleCovarian
ce(P[1]):\nmu[s] = roundMatrix(MUs,1), S = roundMatrix(Ss,1), Omega[s]
 = roundMatrix(OMEGA,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%#muG6#%
\"sG-%'RTABLEG6$\"*?'y2l-%'MATRIXG6#7%7#$\"$+&!\"#7#$\"%!\\#F37#$\"%+]
F3/%\"SG-F)6$\"*k\\]_'-F-6#7%7%$\"'!)oGF3$!'!\\!HF3$\"'5ELF37%FE$\"'5a
JF3$!'!HZ$F37%FGFL$\"'S-bF3/&%&OmegaGF&-F)6$\"*sZ4c'-F-6#7%7%$\"%+HF3$
!%IHF3$\"%gLF37%Fgn$\"%!>$F3$!%5NF37%FinF^o$\"%gbF3" }}}{PARA 0 "" 0 "
" {TEXT -1 213 "While the sample mean has not been changed, the scatte
r matrix is very different; the off diagonal elements now have signifi
cant value indicating the variance along one axis is correlated with t
hat along another. " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 51 "Recoverin
g the Rotation and Original Principal Axes" }}{PARA 0 "" 0 "" {TEXT 
-1 143 "By construction, the covariance and scatter matrices are symme
tric positive definite.   Thus, they may be diagonalized in the follow
ing manner:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Omega[s];" "6#&%&OmegaG6
#%\"sG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "R*Delta*R^t;" "6#*(%\"RG\"\"
\"%&DeltaGF%)F$%\"tGF%" }}{PARA 0 "" 0 "" {TEXT -1 250 "Here, R is an \+
orthogonal matrix consisting of unit length row (and column) vectors. \+
Thus, it is essentially a rotation matrix. The only reason we need qua
lify our statement is that it may include reflection about axes as wel
l as rotation. The matrix " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }
{TEXT -1 21 " is a digonal matrix:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "eq10 := Delta = Matrix([[sigma[xx]^2, 0, 0], [0, sigm
a[yy]^2, 0], [0, 0, sigma[zz]^2]]): eq10;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%&DeltaG-%'RTABLEG6$\"*'z)4c'-%'MATRIXG6#7%7%*$)&%&sig
maG6#%#xxG\"\"#\"\"\"\"\"!F67%F6*$)&F16#%#yyGF4F5F67%F6F6*$)&F16#%#zzG
F4F5" }}}{PARA 0 "" 0 "" {TEXT -1 60 "For reasons that will become app
arent shortly, let us write " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }
{TEXT -1 3 " as" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Delta;" "6#%&DeltaG
" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "S*S;" "6#*&%\"SG\"\"\"F$F%" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "So now our diagonalizatio
n may be written as" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Omega[s];" "6#&%
&OmegaG6#%\"sG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "R*S*S*R^t;" "6#**%\"
RG\"\"\"%\"SGF%F&F%)F$%\"tGF%" }}{PARA 0 "" 0 "" {TEXT -1 65 "The actu
al R and S matrices for our sample covariance matrix are:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "ROMEG
A,SOMEGA := DiagonalizeRS(OMEGA):\nR = roundMatrix(ROMEGA,2), S = roun
dMatrix(SOMEGA,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"RG-%'RTABLEG
6$\"*W]5c'-%'MATRIXG6#7%7%$!$+&!\"$$\"$q%F0$!$I(F07%$\"$?&F0$!$5&F0$!$
!oF07%$!$!pF0$!$?(F0$\"\"!FB/%\"SG-F&6$\"*Od5c'-F*6#7%7%$\"&5.\"F0FAFA
7%FA$\"%5IF0FA7%FAFA$\"%?5F0" }}}{PARA 0 "" 0 "" {TEXT -1 135 "The dia
gonal terms of the S matrix are in fact the sample standard deviations
 for the set of points after they have been rotated by R. " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "sigma[xx] = sigDigits(SOMEGA[1,1],
1), sigma[yy] = sigDigits(SOMEGA[2,2],1), sigma[zz] = sigDigits(SOMEGA
[3,3],1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%&sigmaG6#%#xxG$\"%I5!
\"#/&F%6#%#yyG$\"$+$F*/&F%6#%#zzG$\"$+\"F*" }}}{PARA 0 "" 0 "" {TEXT 
-1 95 "We can see this visually if we actually rotate and plot the poi
nts in the new coordinate system" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 
56 "3D Plot of Points after Rotation to Principal Components" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "Q := homPad(ROMEGA) . P[1]:
\ntheta := -45: phi := 45:\nplotPoints([Q],theta,phi,FRAMED);" }}
{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6)-%'POINTSG6bq7%$
!3(oi))fI-*yT!#;$!3_W#R,A7[?%F)$!3uN'G(47v\"3#F)7%$!3'H\\v!=]tT:!#<$!3
#3.3`biN&\\F)$!3:o*z4(*zR=#F)7%$!32$yTXwM*>=F)$!3%opzC!Hg\"o%F)$!3'y]w
U:%zk@F)7%$!3KNXV0RfS?F)$!3=\"*=k(z&))))[F)$!3Y1X6N)>'e>F)7%$!3_#\\=`%
fmL<F)$!3.?d5g&3Cu%F)$!3%RAt=%*z,A#F)7%$!3#HPIu<PV!QF)$!3%)GH**p\"*45R
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bn$\"$L'FjgmFe[n7)F'F_hmFchmF_imFfimFjim-F_[n6%$\"$b#Fjgm$\"$X'FjgmFe[
n7)F'F_hmFchmF_imFfimFjim-F_[n6%F^cn$\"$d'FjgmFe[n7)F'F_hmFchmF_imFfim
Fjim-F_[n6%$\"$n#Fjgm$\"$p'FjgmFe[n7)F'F_hmFchmF_imFfimFjim-F_[n6%Fccn
$\"$\"oFjgmFe[n7)F'F_hmFchmF_imFfimFjim-F_[n6%$\"$z#Fjgm$\"$$pFjgmFe[n
7)F'F_hmFchmF_imFfimFjim-F_[n6%Fjcn$\"$0(FjgmFe[n7)F'F_hmFchmF_imFfimF
jim-F_[n6%$\"$\"HFjgm$\"$<(FjgmFe[n7)F'F_hmFchmF_imFfimFjim-F_[n6%F_dn
$\"$H(FjgmFe[n7)F'F_hmFchmF_imFfimFjim-F_[n6%$\"$.$Fjgm$\"$T(FjgmFe[n7
)F'F_hmFchmF_imFfimFjim-F_[n6%Ffdn$\"$`(FjgmFe[n" 1 2 0 1 10 0 2 1 1 
1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}}{PARA 0 "" 0 "" 
{TEXT -1 131 "As further confirmation, here is the sample mean, the sc
atter matrix, and the sample covariance matrix for the new set of poin
ts Q." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "MUs := dataMatrixM
ean(Q): Ss := scatterMatrix(Q): OMEGA := sampleCovariance(Q):\nmu[s] =
 roundMatrix(MUs,1), S = roundMatrix(Ss,1), Omega[s] = roundMatrix(OME
GA,1);\nsigma[xx] = sigDigits(sqrt(OMEGA[1,1]),1), sigma[yy] = sigDigi
ts(sqrt(OMEGA[2,2]),1), sigma[zz] = sigDigits(sqrt(OMEGA[3,3]),1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%#muG6#%\"sG-%'RTABLEG6$\"*g,+^'-%'M
ATRIXG6#7%7#$!%IF!\"#7#$!%IWF37#$!%!4#F3/%\"SG-F)6$\"*c+7c'-F-6#7%7%$
\"(I![5F3$!&q=&F3$!&q1%F37%FE$\"&!\\#*F3$\"%+_F37%FGFL$\"&5?\"F3/&%&Om
egaGF&-F)6$\"*S$4hl-F-6#7%7%$\"&!f5F3$!$?&F3$!$5%F37%Fgn$\"$I*F3$\"#]F
37%FinF^o$\"$?\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%&sigmaG6#%#xx
G$\"%I5!\"#/&F%6#%#yyG$\"$5$F*/&F%6#%#zzG$\"$5\"F*" }}}{PARA 0 "" 0 "
" {TEXT -1 11 "The matrix " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 
320 " defines the principal components of the point cloud generated by
 our rotated Gaussian distribution. The row vectors in R are also the \+
Eigenvectors of the covariance or scatter matrix. The covariance and s
catter matrices may be used interchangeably to find R since these matr
ices only differ by a constant scale factor. " }}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 30 "Interpreting Variance as Scale" }}{PARA 0 "" 0 "" 
{TEXT -1 11 "The matrix " }{XPPEDIT 18 0 "S;" "6#%\"SG" }{TEXT -1 396 
" may be described as a diagonal matrix containing the sample estimate
s for the standard deviation of the points along each of the principal
 components. Another way to view this is to observe that scaling the p
oints by the inverse of this matrix will generate a new point set with
 unit variance along each of the principals axes. To illustrate, let u
s apply this transformation to the data matrix " }{XPPEDIT 18 0 "P;" "
6#%\"PG" }{TEXT -1 70 " containing our sample points. The result will \+
be a new set of points " }{XPPEDIT 18 0 "Q;" "6#%\"QG" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Q = S^(-1)*R*P;" "6#/%\"QG*()%\"SG,
$\"\"\"!\"\"F)%\"RGF)%\"PGF)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 44 "3D Plot of Points after
 Rotation and Scaling" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "Q \+
:= MatrixInverse(homPad(SOMEGA)) . homPad(ROMEGA) . P[1]:\ntheta := -4
5: phi := 45:\nplotPoints([Q],theta,phi,FRAMED);" }}{PARA 13 "" 1 "" 
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-1 168 "Looking at the sample mean, scatter matrix and sample covarian
ce for the resulting set of points Q we will see the sample standard d
eviations are now all close to one. " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 292 "MUs := dataMatrixMean(Q): Ss := scatterMatrix(Q): OM
EGA := sampleCovariance(Q):\nmu[s] = roundMatrix(MUs,1), S = roundMatr
ix(Ss,1), Omega[s] = roundMatrix(OMEGA,1);\nsigma[xx] = sigDigits(sqrt
(OMEGA[1,1]),1), sigma[yy] = sigDigits(sqrt(OMEGA[2,2]),1), sigma[zz] \+
= sigDigits(sqrt(OMEGA[3,3]),1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&
%#muG6#%\"sG-%'RTABLEG6$\"*3u7^'-%'MATRIXG6#7%7#$!$q#!\"#7#$!%q9F37#$!
%]?F3/%\"SG-F)6$\"*!)yU`'-F-6#7%7%$\"%g)*F3$!%q;F3$!%qQF37%FE$\"&+-\"F
3$\"%!p\"F37%FGFL$\"&]:\"F3/&%&OmegaGF&-F)6$\"*;27c'-F-6#7%7%$\"#5!\"
\"$!#?F3$!#SF37%Fhn$\"$+\"F3$\"#?F37%FjnF_o$\"$?\"F3" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6%/&%&sigmaG6#%#xxG$\"#5!\"\"/&F%6#%#yyG$\"$+\"!\"#/&F
%6#%#zzG$\"$5\"F1" }}}{PARA 0 "" 0 "" {TEXT -1 202 "In essence, what w
e have done with the rotation and scale derived from the original cova
riance matrix is create a transformation to a new space in which the p
oints have unit variance in all directions. " }}}{SECT 0 {PARA 4 "" 0 
"" {TEXT -1 47 "Principal Components Subpace Maximizes Variance" }}
{PARA 0 "" 0 "" {TEXT -1 509 "Often we do not bother to make explicit \+
the criteria that the principal components optimize. However, because \+
it will play a role latter in how the Fisher Linear Discriminants are \+
defined, it is worthwhile to review this basic material. Commonly it i
s stated that principal components represent axes of maximal variance.
 To put this more precisely, if one sought a single dimension over whi
ch the variance of the data in a data matrix was maximized, it would b
e the axis that maximized the following function " }{XPPEDIT 18 0 "V;
" "6#%\"VG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "V(W) = W*
Omega*W^t;" "6#/-%\"VG6#%\"WG*(F'\"\"\"%&OmegaGF))F'%\"tGF)" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "For a data matrix of points in \+
3D space, this product becomes." }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "V(W)
;" "6#-%\"VG6#%\"WG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "vector([w[x], w
[y], w[z]]);" "6#-%'vectorG6#7%&%\"wG6#%\"xG&F(6#%\"yG&F(6#%\"zG" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[sigma[xx], sigma[xy], sigma[xz
]], [sigma[yx], sigma[yy], sigma[yz]], [sigma[zx], sigma[zy], sigma[zz
]]]);" "6#-%'matrixG6#7%7%&%&sigmaG6#%#xxG&F)6#%#xyG&F)6#%#xzG7%&F)6#%
#yxG&F)6#%#yyG&F)6#%#yzG7%&F)6#%#zxG&F)6#%#zyG&F)6#%#zzG" }{TEXT -1 1 
" " }{XPPEDIT 18 0 "matrix([[w[x]], [w[y]], [w[z]]]);" "6#-%'matrixG6#
7%7#&%\"wG6#%\"xG7#&F)6#%\"yG7#&F)6#%\"zG" }}{PARA 0 "" 0 "" {TEXT -1 
34 "Looking at the general case where " }{XPPEDIT 18 0 "Omega;" "6#%&O
megaG" }{TEXT -1 80 " is not diagonal, it is not obvious what unit len
gth basis vector to choose for " }{XPPEDIT 18 0 "W;" "6#%\"WG" }{TEXT 
-1 30 ". However, for the case where " }{XPPEDIT 18 0 "Omega;" "6#%&Om
egaG" }{TEXT -1 42 " is diagonal, the choice is obvious given." }}
{PARA 0 "" 0 "" {XPPEDIT 18 0 "V(W) = matrix([[w[x], w[y], w[z]]]).mat
rix([[sigma[xx], 0, 0], [0, sigma[yy], 0], [0, 0, sigma[zz]]]).matrix(
[[w[x]], [w[y]], [w[z]]]);" "6#/-%\"VG6#%\"WG-%\".G6%-%'matrixG6#7#7%&
%\"wG6#%\"xG&F16#%\"yG&F16#%\"zG-F,6#7%7%&%&sigmaG6#%#xxG\"\"!FB7%FB&F
?6#%#yyGFB7%FBFB&F?6#%#zzG-F,6#7%7#&F16#F37#&F16#F67#&F16#F9" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Assume that  " }{XPPEDIT 18 0 "
sigma[xx];" "6#&%&sigmaG6#%#xxG" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "sig
ma[yy];" "6#&%&sigmaG6#%#yyG" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "sigma[
zz];" "6#&%&sigmaG6#%#zzG" }{TEXT -1 12 " , then the " }{XPPEDIT 18 0 
"W;" "6#%\"WG" }{TEXT -1 16 " that maximizes " }{XPPEDIT 18 0 "V(W);" 
"6#-%\"VG6#%\"WG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "vector([1, 0, 0])
;" "6#-%'vectorG6#7%\"\"\"\"\"!F(" }{TEXT -1 62 ".  Now recall our ear
lier diagonalization of the general case." }}{PARA 0 "" 0 "" {XPPEDIT 
18 0 "Omega;" "6#%&OmegaG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "R*S*S*R^t
;" "6#**%\"RG\"\"\"%\"SGF%F&F%)F$%\"tGF%" }{TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 17 "We observed that " }{XPPEDIT 18 0 "R;" "6#%\"RG" }
{TEXT -1 81 " is a rotation matrix that shifts us from the original co
ordinates system, where " }{XPPEDIT 18 0 "Omega;" "6#%&OmegaG" }{TEXT 
-1 180 " is not diagonal, to the principal components space, where the
 covariance matrix is diagonal. So, the axis in our original space tha
t maximizes variance is the backward mapping of " }{XPPEDIT 18 0 "vect
or([1, 0, 0])" "6#-%'vectorG6#7%\"\"\"\"\"!F(" }{TEXT -1 52 " into the
 original space. Putting this more simply, " }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "W = R^t*matrix([[1], [0], [0]]);" "6#/%\"WG*&)%\"RG%\"t
G\"\"\"-%'matrixG6#7%7#F)7#\"\"!7#F0F)" }}{PARA 0 "" 0 "" {TEXT -1 54 
"and even more simply, we see that the first column of " }{XPPEDIT 18 
0 "R;" "6#%\"RG" }{TEXT -1 163 " is the unit length basis vector that \+
defines the axis of maximum variance. This axis is what we typically c
all the first prinicipal component of the data matrix. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The extension to " }
{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 79 " axes that collectively max
imize variance generalizes the criterion function to" }}{PARA 0 "" 0 "
" {XPPEDIT 18 0 "V(W) = abs(W*Omega*W^t);" "6#/-%\"VG6#%\"WG-%$absG6#*
(F'\"\"\"%&OmegaGF,)F'%\"tGF," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 45 "Now we are maximizing the determinant of the " }{XPPEDIT 
18 0 "k;" "6#%\"kG" }{TEXT -1 1 "x" }{XPPEDIT 18 0 "k;" "6#%\"kG" }
{TEXT -1 35 " matrix formed from product of the " }{XPPEDIT 18 0 "k;" 
"6#%\"kG" }{TEXT -1 1 "x" }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 8 " \+
matrix " }{XPPEDIT 18 0 "W;" "6#%\"WG" }{TEXT -1 27 " and the covarian
ce matrix " }{XPPEDIT 18 0 "Omega;" "6#%&OmegaG" }{TEXT -1 113 ". By a
 generalization of the argument made above, the basis vectors that max
imize this determinant are the first " }{XPPEDIT 18 0 "k;" "6#%\"kG" }
{TEXT -1 12 " columns of " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 1 "
." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 20 "Fisher Discriminants" }}{PARA 0 "" 0 "" {TEXT -1 49 "Ficher's L
inear Discriminants are defined as the " }{XPPEDIT 18 0 "k;" "6#%\"kG
" }{TEXT -1 20 " basis vectors in a " }{XPPEDIT 18 0 "k;" "6#%\"kG" }
{TEXT -1 1 "x" }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 8 " matrix " }
{XPPEDIT 18 0 "W;" "6#%\"WG" }{TEXT -1 38 " that maximizes the followi
ng function" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "J(W) = abs(W*M[B]*W^t)/a
bs(W*M[W]*W^t);" "6#/-%\"JG6#%\"WG*&-%$absG6#*(F'\"\"\"&%\"MG6#%\"BGF-
)F'%\"tGF-F--F*6#*(F'F-&F/6#F'F-)F'F3F-!\"\"" }}{PARA 0 "" 0 "" {TEXT 
-1 6 "Where " }{XPPEDIT 18 0 "M[B];" "6#&%\"MG6#%\"BG" }{TEXT -1 41 " \+
is the between class scatter matrix and " }{XPPEDIT 18 0 "M[W];" "6#&%
\"MG6#%\"WG" }{TEXT -1 97 " is the within class scatter matrix [Duda].
 More particularly, the within class scatter matrix is" }}{PARA 0 "" 
0 "" {XPPEDIT 18 0 "M[W] = sum(M[i],i = 1 .. c);" "6#/&%\"MG6#%\"WG-%$
sumG6$&F%6#%\"iG/F-;\"\"\"%\"cG" }{TEXT -1 11 "    where  " }{XPPEDIT 
18 0 "M[i] = sum(p[j]-(mu[i].((p[j]-mu[i])^t)),j = 1 .. n[i]);" "6#/&%
\"MG6#%\"iG-%$sumG6$,&&%\"pG6#%\"jG\"\"\"-%\".G6$&%#muG6#F'),&&F-6#F/F
0&F56#F'!\"\"%\"tGF=/F/;F0&%\"nG6#F'" }}{PARA 0 "" 0 "" {TEXT -1 6 "wh
ere " }{XPPEDIT 18 0 "n[i];" "6#&%\"nG6#%\"iG" }{TEXT -1 36 " is the n
umber of elements in class " }{XPPEDIT 18 0 "i;" "6#%\"iG" }{TEXT -1 
2 ", " }{XPPEDIT 18 0 "p[j];" "6#&%\"pG6#%\"jG" }{TEXT -1 8 " is the \+
" }{XPPEDIT 18 0 "j;" "6#%\"jG" }{TEXT -1 18 "th point in class " }
{XPPEDIT 18 0 "i;" "6#%\"iG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "mu[i
];" "6#&%#muG6#%\"iG" }{TEXT -1 30 " is the mean vector for class " }
{XPPEDIT 18 0 "i;" "6#%\"iG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 35 "The between class scatter matrix is" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "M[B] = sum(n[i].(mu[i]-mu).((mu[i]-mu)^t),i = 1 .. c);
" "6#/&%\"MG6#%\"BG-%$sumG6$-%\".G6%&%\"nG6#%\"iG,&&%#muG6#F1\"\"\"F4!
\"\"),&&F46#F1F6F4F7%\"tG/F1;F6%\"cG" }}{PARA 0 "" 0 "" {TEXT -1 245 "
Now, quoting from [Duda], \"The problem of finding a rectangular matri
x W that minimizes J(.) is tricky.\" They, and  other common reference
s, state without elaboration that the optimal W may be found by solvin
g the generalized eigenvector problem" }}{PARA 0 "" 0 "" {XPPEDIT 18 
0 "M[B]*w[i] = lambda[i]*M[W]*w[i];" "6#/*&&%\"MG6#%\"BG\"\"\"&%\"wG6#
%\"iGF)*(&%'lambdaG6#F-F)&F&6#%\"WGF)&F+6#F-F)" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 77 "Specifically, the optimal W consists of t
he eigenvectors associated with the " }{XPPEDIT 18 0 "k;" "6#%\"kG" }
{TEXT -1 23 " largest eigenvalues.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 608 "The problem with stopping at this poin
t is two fold. First, many of us have no geometric intuition for what \+
it means to solve this general eigenvector problem. Second, while gene
ral, this approach may not always be the most efficient/robust means o
f finding the Fisher Discriminants. This latter observation has been m
ade by [Zhao] and others. They provide a general means of transforming
 a generalized eigenvector problem to a symmetric eigenvector problem,
 and this transformation is the basis for developing a geometric intui
tion for what is actually taking place when computing the Fisher Discr
iminants. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 98 "Here is the transformation with many of the intermediate steps \+
omitted from [Zhao] made explicit. " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 48 "Transformation to a Standard Eigenvector Problem" }}{PARA 0 "" 
0 "" {TEXT -1 48 "Begin with the generalized eigen vector problem:" }}
{PARA 0 "" 0 "" {XPPEDIT 18 0 "M[b].W = M[w].W.Lambda;" "6#/-%\".G6$&%
\"MG6#%\"bG%\"WG-F%6%&F(6#%\"wGF+%'LambdaG" }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 48 "Now use singular value decomposition to express " 
}{XPPEDIT 18 0 "M[w];" "6#&%\"MG6#%\"wG" }{TEXT -1 128 " in diagonal f
orm, and indeed, go one step further and explicitly identify the squar
e root of the diagonal as a scale matrix. So" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "M[w] = R[w].S[w].S[w].(R[w]^T);" "6#/&%\"MG6#%\"wG-%\".
G6&&%\"RG6#F'&%\"SG6#F'&F/6#F')&F,6#F'%\"TG" }{TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 42 "The original problem may now be written as" }}
{PARA 0 "" 0 "" {XPPEDIT 18 0 "M[b].W = R[w].S[w].S[w].(R[w]^T).W.Lamb
da;" "6#/-%\".G6$&%\"MG6#%\"bG%\"WG-F%6(&%\"RG6#%\"wG&%\"SG6#F1&F36#F1
)&F/6#F1%\"TGF+%'LambdaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
65 "Take the inverse scale and rotation and right multiply both sides
" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "S[w]^i.(R[w]^T).M[b].W = S[w].(R[w]
^T).W.Lambda;" "6#/-%\".G6&)&%\"SG6#%\"wG%\"iG)&%\"RG6#F+%\"TG&%\"MG6#
%\"bG%\"WG-F%6&&F)6#F+)&F/6#F+F1F6%'LambdaG" }{TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 88 "Define a new matrix that will become the eigenvec
tors of a symmetric eigenvector problem" }}{PARA 0 "" 0 "" {XPPEDIT 
18 0 "V = S[w].(R[w]^T).W;" "6#/%\"VG-%\".G6%&%\"SG6#%\"wG)&%\"RG6#F+%
\"TG%\"WG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "and substitut
e " }{XPPEDIT 18 0 "V;" "6#%\"VG" }{TEXT -1 23 " on the right hand sid
e" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "S[w]^i.(R[w]^T).M[b].W = V.Lambda;
" "6#/-%\".G6&)&%\"SG6#%\"wG%\"iG)&%\"RG6#F+%\"TG&%\"MG6#%\"bG%\"WG-F%
6$%\"VG%'LambdaG" }{TEXT -1 143 "\nTo simplify the left hand side, int
roduce a single matrix G that combines the scale and rotation derived \+
from the within class scatter matrix." }}{PARA 0 "" 0 "" {XPPEDIT 18 
0 "G = S[w]^i.(R[w]^T);" "6#/%\"GG-%\".G6$)&%\"SG6#%\"wG%\"iG)&%\"RG6#
F,%\"TG" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "G^T = R[w].(S[w]^i);" "6
#/)%\"GG%\"TG-%\".G6$&%\"RG6#%\"wG)&%\"SG6#F-%\"iG" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 130 "Expand the left hand side by left multip
lying be a sequence or scales and rotations that together are the iden
tity transformation." }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "S[w]^i.(R[w]^T)
.M[b].R[w].(S[w]^i).S[w].(R[w]^T).W = V.Lambda;" "6#/-%\".G6*)&%\"SG6#
%\"wG%\"iG)&%\"RG6#F+%\"TG&%\"MG6#%\"bG&F/6#F+)&F)6#F+F,&F)6#F+)&F/6#F
+F1%\"WG-F%6$%\"VG%'LambdaG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 43 "Now substitute G and V in where appropriate" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "G.M[b].(G^T).V = V.Lambda;" "6#/-%\".G6&%\"GG&%\"MG6#%
\"bG)F'%\"TG%\"VG-F%6$F.%'LambdaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 55 "The final result is a symmetric eigenvector problem in " 
}{XPPEDIT 18 0 "N[b];" "6#&%\"NG6#%\"bG" }{TEXT -1 7 " where " }
{XPPEDIT 18 0 "N[b] = G.M[b].(G^T);" "6#/&%\"NG6#%\"bG-%\".G6%%\"GG&%
\"MG6#F')F+%\"TG" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "N[b
].V = V.Lambda;" "6#/-%\".G6$&%\"NG6#%\"bG%\"VG-F%6$F+%'LambdaG" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Once we solve for " }
{XPPEDIT 18 0 "V;" "6#%\"VG" }{TEXT -1 37 ", W is found directly by th
e equality" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "W = R[w].(S[w]^i).V;" "6#
/%\"WG-%\".G6%&%\"RG6#%\"wG)&%\"SG6#F+%\"iG%\"VG" }{TEXT -1 2 "  " }}
{PARA 0 "" 0 "" {TEXT -1 206 "We will see below that G is a transforma
tion that takes us into a space where the fisher discriminants are rel
ated directly to the eigenvectors of the scatter matrix associated wit
h the transformed points. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 52 "How this Transformation Acts Upon Fishe
r's Criterion" }}{PARA 0 "" 0 "" {TEXT -1 125 "Now we can draw the con
nection between the general algebraic manipulation above and the Fishe
r Criterion we wish to optimize." }}{PARA 0 "" 0 "" {TEXT -1 26 "Recal
l Fisher's criterion " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "J(W) = abs(W*M
[B]*W^t)/abs(W*M[W]*W^t)" "6#/-%\"JG6#%\"WG*&-%$absG6#*(F'\"\"\"&%\"MG
6#%\"BGF-)F'%\"tGF-F--F*6#*(F'F-&F/6#F'F-)F'F3F-!\"\"" }{TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 251 "Consider how much easier this problem \+
would be to solve if the denominator were a constant with respect to o
ur choice of W.  We can map to a new space where this is true by explo
iting the rotation and scale transformations obtained from diagonalizi
ng " }{XPPEDIT 18 0 "M[W];" "6#&%\"MG6#%\"WG" }{TEXT -1 69 ". Thus, re
iterating the diagonalization used in the previous section." }}{PARA 
0 "" 0 "" {XPPEDIT 18 0 "M[w] = R[w]*S[w]*S[w]*R[w]^T;" "6#/&%\"MG6#%
\"wG**&%\"RG6#F'\"\"\"&%\"SG6#F'F,&F.6#F'F,)&F*6#F'%\"TGF," }{TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Applying the transformation " }
{XPPEDIT 18 0 "G;" "6#%\"GG" }{TEXT -1 197 " to our points before crea
ting the within and between class scatter matrices leads to a new prob
lem where only the numerator varies as a function of W.  Showing the a
lgebra of this transformation, " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "J(V)
 = abs(V*R^t*S[w]^(-1)*M[B]*R*S[w]^(-1)*V^t)/abs(V*R^t*S[w]^(-1)*R[w]*
S[w]*S[w]*R[w]^T*R*S[w]^(-1)*V^t);" "6#/-%\"JG6#%\"VG*&-%$absG6#*0F'\"
\"\")%\"RG%\"tGF-)&%\"SG6#%\"wG,$F-!\"\"F-&%\"MG6#%\"BGF-F/F-)&F36#F5,
$F-F7F-)F'F0F-F--F*6#*6F'F-)F/F0F-)&F36#F5,$F-F7F-&F/6#F5F-&F36#F5F-&F
36#F5F-)&F/6#F5%\"TGF-F/F-)&F36#F5,$F-F7F-)F'F0F-F7" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The denominator of this
 new criterion function collapses to " }{XPPEDIT 18 0 "V*V^t;" "6#*&%
\"VG\"\"\")F$%\"tGF%" }{TEXT -1 67 ", and this further collapses and t
he entire denominator becomes 1. " }}{PARA 0 "" 0 "" {TEXT -1 410 "In \+
geometric terms, we have transformed our problem to a space where the \+
covariance of the within class scatter is one in all principal directi
ons. This is exactly the same as we illustrated above when we showed t
hat the R and S derived from the diagonalization of a covariance matri
x could be used to remap a data matrix into a space where the points f
ormed a compact ball with variance 1 in all directtions. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 463 "One loose end is \+
why maximizing J in the transformed space is equivalent to maximizing \+
it in the original space. The answer lies in observing that the rotati
ons leave the determinant unchanged, and the two scale matrices alter \+
both the numerator and denominator by the same constant factor. Thus, \+
maximizing one ratio is equivalent to maximizing the other.  We are no
w ready to illustrate this method of finding the Fisher Discriminants \+
on a specific 3D example. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 32 "Example of a Three Class Problem" }}
{PARA 0 "" 0 "" {TEXT -1 112 "Consider a three class recognition probl
em where samples from each class are drawn from distinct distributions
. " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "Different Covariance Struct
ures" }}{PARA 0 "" 0 "" {TEXT -1 163 "Here are three distinct tri-vari
ate Gaussian random variables and sample points from each. These three
 classes give rise to two Fisher Linear Discriminants in 3D. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
219 "MU := Transpose(Matrix(3,3, [[0,0,0],[20,5,30],[50,10,70]])):\nSD
 := Transpose(Matrix(3,3, [[10,3,1],[10,3,1],[10,3,1]])):\nEA := Trans
pose(Matrix(3,3, [[.4,0,0],[0,0.7,0],[0,0,1.2]])):\n[mu = MU, sigma = \+
SD, angles = EA];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/%#muG-%'RTABLE
G6$\"*sp7c'-%'MATRIXG6#7%7%\"\"!\"#?\"#]7%F/\"\"&\"#57%F/\"#I\"#q/%&si
gmaG-F'6$\"*7r7c'-F+6#7%7%F4F4F47%\"\"$FBFB7%\"\"\"FDFD/%'anglesG-F'6$
\"*'\\Hhl-F+6#7%7%$\"\"%!\"\"F/F/7%F/$\"\"(FPF/7%F/F/$\"#7FP" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "P := listClasses(MU,SD,EA);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG7%-%'RTABLEG6+\"*SJ8c'&%&flo
atG6#\"\")%'MatrixG%,rectangularG%.Fortran_orderG7\"\"\"#;\"\"\"\"\"%;
F4\"$+\"-F'6+\"*sL__'F*F.F/F0F1F2F3F6-F'6+\"*3f9c'F*F.F/F0F1F2F3F6" }}
}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 36 "3D Plot of Points from Three Cla
sses" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "theta := -45: phi :=
 45:\nplotPoints(P,theta,phi,FRAMED);" }}{PARA 13 "" 1 "" {GLPLOT3D 
400 300 300 {PLOTDATA 3 "6+-%'POINTSG6bq7%$!3gP2nD48A5!#<$!32qR%p[U;))
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