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{ ******************************************************************
Principal component analysis
****************************************************************** }
unit upca;
interface
uses
utypes, ujacobi;
procedure VecMean(X : PMatrix;
Lb, Ub, Nvar : Integer;
M : PVector);
{ ----------------------------------------------------------------------
Computes the mean vector (M) from matrix X
----------------------------------------------------------------------
Input : X[Lb..Ub, 1..Nvar] = matrix of variables
Output : M[1..Nvar] = mean vector
---------------------------------------------------------------------- }
procedure VecSD(X : PMatrix;
Lb, Ub, Nvar : Integer;
M, S : PVector);
{ ----------------------------------------------------------------------
Computes the vector of standard deviations (S) from matrix X
----------------------------------------------------------------------
Input : X, Lb, Ub, Nvar, M
Output : S[1..Nvar]
---------------------------------------------------------------------- }
procedure MatVarCov(X : PMatrix;
Lb, Ub, Nvar : Integer;
M : PVector;
V : PMatrix);
{ ----------------------------------------------------------------------
Computes the variance-covariance matrix (V) from matrix X
Input : X, Lb, Ub, Nvar, M
Output : V[1..Nvar, 1..Nvar]
---------------------------------------------------------------------- }
procedure MatCorrel(V : PMatrix;
Nvar : Integer;
R : PMatrix);
{ ----------------------------------------------------------------------
Computes the correlation matrix (R) from the variance-covariance
matrix (V)
Input : V, Nvar
Output : R[1..Nvar, 1..Nvar]
---------------------------------------------------------------------- }
procedure PCA(R : PMatrix;
Nvar : Integer;
MaxIter : Integer;
Tol : Float;
Lambda : PVector;
C, Rc : PMatrix);
{ ----------------------------------------------------------------------
Performs a principal component analysis of the correlation matrix R
----------------------------------------------------------------------
Input : R[1..Nvar] = Correlation matrix
MaxIter = Max. number of iterations
Tol = Required precision
Output : Lambda[1..Nvar] = Eigenvalues of the correlation matrix
(in descending order)
C[1..Nvar, 1..Nvar] = Eigenvectors of the correlation matrix
(stored as columns)
Rc[1..Nvar, 1..Nvar] = Correlations between principal factors
and variables (Rc^[I]^[J] is the
correlation coefficient between
variable I and factor J)
----------------------------------------------------------------------
NB : This procedure destroys the original matrix R
---------------------------------------------------------------------- }
procedure ScaleVar(X : PMatrix;
Lb, Ub, Nvar : Integer;
M, S : PVector;
Z : PMatrix);
{ ----------------------------------------------------------------------
Scales a set of variables by subtracting means and dividing by SD's
----------------------------------------------------------------------
Input : X, Lb, Ub, Nvar, M, S
Output : Z[Lb..Ub, 1..Nvar] = matrix of scaled variables
---------------------------------------------------------------------- }
procedure PrinFac(Z : PMatrix;
Lb, Ub, Nvar : Integer;
C, F : PMatrix);
{ ----------------------------------------------------------------------
Computes principal factors
----------------------------------------------------------------------
Input : Z[Lb..Ub, 1..Nvar] = matrix of scaled variables
C[1..Nvar, 1..Nvar] = matrix of eigenvectors from PCA
Output : F[Lb..Ub, 1..Nvar] = matrix of principal factors
---------------------------------------------------------------------- }
implementation
procedure VecMean(X : PMatrix;
Lb, Ub, Nvar : Integer;
M : PVector);
var
I, K, Nobs : Integer;
Sum : Float;
begin
Nobs := Ub - Lb + 1;
for I := 1 to Nvar do
begin
Sum := 0.0;
for K := Lb to Ub do
Sum := Sum + X^[K]^[I];
M^[I] := Sum / Nobs;
end;
end;
procedure VecSD(X : PMatrix;
Lb, Ub, Nvar : Integer;
M, S : PVector);
var
I, K, Nobs : Integer;
Sum : Float;
begin
Nobs := Ub - Lb + 1;
for I := 1 to Nvar do
begin
Sum := 0.0;
for K := Lb to Ub do
Sum := Sum + Sqr(X^[K]^[I] - M^[I]);
S^[I] := Sqrt(Sum / Nobs);
end;
end;
procedure MatVarCov(X : PMatrix;
Lb, Ub, Nvar : Integer;
M : PVector;
V : PMatrix);
var
I, J, K, Nobs : Integer;
Sum : Float;
begin
Nobs := Ub - Lb + 1;
for I := 1 to Nvar do
for J := I to Nvar do
begin
Sum := 0.0;
for K := Lb to Ub do
Sum := Sum + (X^[K]^[I] - M^[I]) * (X^[K]^[J] - M^[J]);
V^[I]^[J] := Sum / Nobs;
end;
for I := 2 to Nvar do
for J := 1 to Pred(I) do
V^[I]^[J] := V^[J]^[I];
end;
procedure MatCorrel(V : PMatrix;
Nvar : Integer;
R : PMatrix);
var
I, J : Integer;
P : Float;
begin
for I := 1 to Nvar do
begin
R^[I]^[I] := 1.0;
for J := Succ(I) to Nvar do
begin
P := V^[I]^[I] * V^[J]^[J];
if P > 0.0 then
R^[I]^[J] := V^[I]^[J] / Sqrt(P)
else
R^[I]^[J] := 0.0;
R^[J]^[I] := R^[I]^[J];
end;
end;
end;
procedure PCA(R : PMatrix;
Nvar : Integer;
MaxIter : Integer;
Tol : Float;
Lambda : PVector;
C, Rc : PMatrix);
var
I, J : Integer;
Rac : Float;
begin
{ Compute eigenvalues and eigenvectors of correlation matrix }
Jacobi(R, 1, Nvar, MaxIter, Tol, Lambda, C);
if MathErr <> MatOk then Exit;
{ Compute correlations between principal factors and reduced variables }
for J := 1 to Nvar do
begin
Rac := Sqrt(Lambda^[J]);
for I := 1 to Nvar do
Rc^[I]^[J] := C^[I]^[J] * Rac;
end;
end;
procedure ScaleVar(X : PMatrix;
Lb, Ub, Nvar : Integer;
M, S : PVector;
Z : PMatrix);
var
I, J : Integer;
begin
for I := Lb to Ub do
for J := 1 to Nvar do
Z^[I]^[J] := (X^[I]^[J] - M^[J]) / S^[J];
end;
procedure PrinFac(Z : PMatrix;
Lb, Ub, Nvar : Integer;
C, F : PMatrix);
var
I, J, K : Integer;
begin
for I := Lb to Ub do
for J := 1 to Nvar do
begin
F^[I]^[J] := 0.0;
for K := 1 to Nvar do
F^[I]^[J] := F^[I]^[J] + Z^[I]^[K] * C^[K]^[J];
end;
end;
end.
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