//$$ example.cpp Example of use of matrix package
#define WANT_STREAM // include.h will get stream fns
#define WANT_MATH // include.h will get math fns
// newmatap.h will get include.h
#include "newmatap.h" // need matrix applications
#include "newmatio.h" // need matrix output routines
#ifdef use_namespace
using namespace NEWMAT; // access NEWMAT namespace
#endif
// demonstration of matrix package on linear regression problem
void test1(Real* y, Real* x1, Real* x2, int nobs, int npred)
{
cout << "\n\nTest 1 - traditional, bad\n";
// traditional sum of squares and products method of calculation
// but not adjusting means; maybe subject to round-off error
// make matrix of predictor values with 1s into col 1 of matrix
int npred1 = npred+1; // number of cols including col of ones.
Matrix X(nobs,npred1);
X.Column(1) = 1.0;
// load x1 and x2 into X
// [use << rather than = when loading arrays]
X.Column(2) << x1; X.Column(3) << x2;
// vector of Y values
ColumnVector Y(nobs); Y << y;
// form sum of squares and product matrix
// [use << rather than = for copying Matrix into SymmetricMatrix]
SymmetricMatrix SSQ; SSQ << X.t() * X;
// calculate estimate
// [bracket last two terms to force this multiplication first]
// [ .i() means inverse, but inverse is not explicity calculated]
ColumnVector A = SSQ.i() * (X.t() * Y);
// Get variances of estimates from diagonal elements of inverse of SSQ
// get inverse of SSQ - we need it for finding D
DiagonalMatrix D; D << SSQ.i();
ColumnVector V = D.AsColumn();
// Calculate fitted values and residuals
ColumnVector Fitted = X * A;
ColumnVector Residual = Y - Fitted;
Real ResVar = Residual.SumSquare() / (nobs-npred1);
// Get diagonals of Hat matrix (an expensive way of doing this)
DiagonalMatrix Hat; Hat << X * (X.t() * X).i() * X.t();
// print out answers
cout << "\nEstimates and their standard errors\n\n";
// make vector of standard errors
ColumnVector SE(npred1);
for (int i=1; i<=npred1; i++) SE(i) = sqrt(V(i)*ResVar);
// use concatenation function to form matrix and use matrix print
// to get two columns
cout << setw(11) << setprecision(5) << (A | SE) << endl;
cout << "\nObservations, fitted value, residual value, hat value\n";
// use concatenation again; select only columns 2 to 3 of X
cout << setw(9) << setprecision(3) <<
(X.Columns(2,3) | Y | Fitted | Residual | Hat.AsColumn());
cout << "\n\n";
}
void test2(Real* y, Real* x1, Real* x2, int nobs, int npred)
{
cout << "\n\nTest 2 - traditional, OK\n";
// traditional sum of squares and products method of calculation
// with subtraction of means - less subject to round-off error
// than test1
// make matrix of predictor values
Matrix X(nobs,npred);
// load x1 and x2 into X
// [use << rather than = when loading arrays]
X.Column(1) << x1; X.Column(2) << x2;
// vector of Y values
ColumnVector Y(nobs); Y << y;
// make vector of 1s
ColumnVector Ones(nobs); Ones = 1.0;
// calculate means (averages) of x1 and x2 [ .t() takes transpose]
RowVector M = Ones.t() * X / nobs;
// and subtract means from x1 and x1
Matrix XC(nobs,npred);
XC = X - Ones * M;
// do the same to Y [use Sum to get sum of elements]
ColumnVector YC(nobs);
Real m = Sum(Y) / nobs; YC = Y - Ones * m;
// form sum of squares and product matrix
// [use << rather than = for copying Matrix into SymmetricMatrix]
SymmetricMatrix SSQ; SSQ << XC.t() * XC;
// calculate estimate
// [bracket last two terms to force this multiplication first]
// [ .i() means inverse, but inverse is not explicity calculated]
ColumnVector A = SSQ.i() * (XC.t() * YC);
// calculate estimate of constant term
// [AsScalar converts 1x1 matrix to Real]
Real a = m - (M * A).AsScalar();
// Get variances of estimates from diagonal elements of inverse of SSQ
// [ we are taking inverse of SSQ - we need it for finding D ]
Matrix ISSQ = SSQ.i(); DiagonalMatrix D; D << ISSQ;
ColumnVector V = D.AsColumn();
Real v = 1.0/nobs + (M * ISSQ * M.t()).AsScalar();
// for calc variance of const
// Calculate fitted values and residuals
int npred1 = npred+1;
ColumnVector Fitted = X * A + a;
ColumnVector Residual = Y - Fitted;
Real ResVar = Residual.SumSquare() / (nobs-npred1);
// Get diagonals of Hat matrix (an expensive way of doing this)
Matrix X1(nobs,npred1); X1.Column(1)<<Ones; X1.Columns(2,npred1)<<X;
DiagonalMatrix Hat; Hat << X1 * (X1.t() * X1).i() * X1.t();
// print out answers
cout << "\nEstimates and their standard errors\n\n";
cout.setf(ios::fixed, ios::floatfield);
cout << setw(11) << setprecision(5) << a << " ";
cout << setw(11) << setprecision(5) << sqrt(v*ResVar) << endl;
// make vector of standard errors
ColumnVector SE(npred);
for (int i=1; i<=npred; i++) SE(i) = sqrt(V(i)*ResVar);
// use concatenation function to form matrix and use matrix print
// to get two columns
cout << setw(11) << setprecision(5) << (A | SE) << endl;
cout << "\nObservations, fitted value, residual value, hat value\n";
cout << setw(9) << setprecision(3) <<
(X | Y | Fitted | Residual | Hat.AsColumn());
cout << "\n\n";
}
void test3(Real* y, Real* x1, Real* x2, int nobs, int npred)
{
cout << "\n\nTest 3 - Cholesky\n";
// traditional sum of squares and products method of calculation
// with subtraction of means - using Cholesky decomposition
Matrix X(nobs,npred);
X.Column(1) << x1; X.Column(2) << x2;
ColumnVector Y(nobs); Y << y;
ColumnVector Ones(nobs); Ones = 1.0;
RowVector M = Ones.t() * X / nobs;
Matrix XC(nobs,npred);
XC = X - Ones * M;
ColumnVector YC(nobs);
Real m = Sum(Y) / nobs; YC = Y - Ones * m;
SymmetricMatrix SSQ; SSQ << XC.t() * XC;
// Cholesky decomposition of SSQ
LowerTriangularMatrix L = Cholesky(SSQ);
// calculate estimate
ColumnVector A = L.t().i() * (L.i() * (XC.t() * YC));
// calculate estimate of constant term
Real a = m - (M * A).AsScalar();
// Get variances of estimates from diagonal elements of invoice of SSQ
DiagonalMatrix D; D << L.t().i() * L.i();
ColumnVector V = D.AsColumn();
Real v = 1.0/nobs + (L.i() * M.t()).SumSquare();
// Calculate fitted values and residuals
int npred1 = npred+1;
ColumnVector Fitted = X * A + a;
ColumnVector Residual = Y - Fitted;
Real ResVar = Residual.SumSquare() / (nobs-npred1);
// Get diagonals of Hat matrix (an expensive way of doing this)
Matrix X1(nobs,npred1); X1.Column(1)<<Ones; X1.Columns(2,npred1)<<X;
DiagonalMatrix Hat; Hat << X1 * (X1.t() * X1).i() * X1.t();
// print out answers
cout << "\nEstimates and their standard errors\n\n";
cout.setf(ios::fixed, ios::floatfield);
cout << setw(11) << setprecision(5) << a << " ";
cout << setw(11) << setprecision(5) << sqrt(v*ResVar) << endl;
ColumnVector SE(npred);
for (int i=1; i<=npred; i++) SE(i) = sqrt(V(i)*ResVar);
cout << setw(11) << setprecision(5) << (A | SE) << endl;
cout << "\nObservations, fitted value, residual value, hat value\n";
cout << setw(9) << setprecision(3) <<
(X | Y | Fitted | Residual | Hat.AsColumn());
cout << "\n\n";
}
void test4(Real* y, Real* x1, Real* x2, int nobs, int npred)
{
cout << "\n\nTest 4 - QR triangularisation\n";
// QR triangularisation method
// load data - 1s into col 1 of matrix
int npred1 = npred+1;
Matrix X(nobs,npred1); ColumnVector Y(nobs);
X.Column(1) = 1.0; X.Column(2) << x1; X.Column(3) << x2; Y << y;
// do Householder triangularisation
// no need to deal with constant term separately
Matrix X1 = X; // Want copy of matrix
ColumnVector Y1 = Y;
UpperTriangularMatrix U; ColumnVector M;
QRZ(X1, U); QRZ(X1, Y1, M); // Y1 now contains resids
ColumnVector A = U.i() * M;
ColumnVector Fitted = X * A;
Real ResVar = Y1.SumSquare() / (nobs-npred1);
// get variances of estimates
U = U.i(); DiagonalMatrix D; D << U * U.t();
// Get diagonals of Hat matrix
DiagonalMatrix Hat; Hat << X1 * X1.t();
// print out answers
cout << "\nEstimates and their standard errors\n\n";
ColumnVector SE(npred1);
for (int i=1; i<=npred1; i++) SE(i) = sqrt(D(i)*ResVar);
cout << setw(11) << setprecision(5) << (A | SE) << endl;
cout << "\nObservations, fitted value, residual value, hat value\n";
cout << setw(9) << setprecision(3) <<
(X.Columns(2,3) | Y | Fitted | Y1 | Hat.AsColumn());
cout << "\n\n";
}
void test5(Real* y, Real* x1, Real* x2, int nobs, int npred)
{
cout << "\n\nTest 5 - singular value\n";
// Singular value decomposition method
// load data - 1s into col 1 of matrix
int npred1 = npred+1;
Matrix X(nobs,npred1); ColumnVector Y(nobs);
X.Column(1) = 1.0; X.Column(2) << x1; X.Column(3) << x2; Y << y;
// do SVD
Matrix U, V; DiagonalMatrix D;
SVD(X,D,U,V); // X = U * D * V.t()
ColumnVector Fitted = U.t() * Y;
ColumnVector A = V * ( D.i() * Fitted );
Fitted = U * Fitted;
ColumnVector Residual = Y - Fitted;
Real ResVar = Residual.SumSquare() / (nobs-npred1);
// get variances of estimates
D << V * (D * D).i() * V.t();
// Get diagonals of Hat matrix
DiagonalMatrix Hat; Hat << U * U.t();
// print out answers
cout << "\nEstimates and their standard errors\n\n";
ColumnVector SE(npred1);
for (int i=1; i<=npred1; i++) SE(i) = sqrt(D(i)*ResVar);
cout << setw(11) << setprecision(5) << (A | SE) << endl;
cout << "\nObservations, fitted value, residual value, hat value\n";
cout << setw(9) << setprecision(3) <<
(X.Columns(2,3) | Y | Fitted | Residual | Hat.AsColumn());
cout << "\n\n";
}
int main()
{
cout << "\nDemonstration of Matrix package\n";
cout << "\nPrint a real number (may help lost memory test): " << 3.14159265 << "\n";
// Test for any memory not deallocated after running this program
Real* s1; { ColumnVector A(8000); s1 = A.Store(); }
{
// the data
#ifndef ATandT
Real y[] = { 8.3, 5.5, 8.0, 8.5, 5.7, 4.4, 6.3, 7.9, 9.1 };
Real x1[] = { 2.4, 1.8, 2.4, 3.0, 2.0, 1.2, 2.0, 2.7, 3.6 };
Real x2[] = { 1.7, 0.9, 1.6, 1.9, 0.5, 0.6, 1.1, 1.0, 0.5 };
#else // for compilers that do not understand aggregrates
Real y[9], x1[9], x2[9];
y[0]=8.3; y[1]=5.5; y[2]=8.0; y[3]=8.5; y[4]=5.7;
y[5]=4.4; y[6]=6.3; y[7]=7.9; y[8]=9.1;
x1[0]=2.4; x1[1]=1.8; x1[2]=2.4; x1[3]=3.0; x1[4]=2.0;
x1[5]=1.2; x1[6]=2.0; x1[7]=2.7; x1[8]=3.6;
x2[0]=1.7; x2[1]=0.9; x2[2]=1.6; x2[3]=1.9; x2[4]=0.5;
x2[5]=0.6; x2[6]=1.1; x2[7]=1.0; x2[8]=0.5;
#endif
int nobs = 9; // number of observations
int npred = 2; // number of predictor values
// we want to find the values of a,a1,a2 to give the best
// fit of y[i] with a0 + a1*x1[i] + a2*x2[i]
// Also print diagonal elements of hat matrix, X*(X.t()*X).i()*X.t()
// this example demonstrates five methods of calculation
Try
{
test1(y, x1, x2, nobs, npred);
test2(y, x1, x2, nobs, npred);
test3(y, x1, x2, nobs, npred);
test4(y, x1, x2, nobs, npred);
test5(y, x1, x2, nobs, npred);
}
CatchAll { cout << BaseException::what(); }
}
#ifdef DO_FREE_CHECK
FreeCheck::Status();
#endif
Real* s2; { ColumnVector A(8000); s2 = A.Store(); }
cout << "\n\nThe following test does not work with all compilers - see documentation\n";
cout << "Checking for lost memory: "
<< (unsigned long)s1 << " " << (unsigned long)s2 << " ";
if (s1 != s2) cout << " - error\n"; else cout << " - ok\n";
return 0;
}
