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#include <iostream>
#include <iomanip>
#include <vector>
#include <algorithm>
#include <cassert>
#include <cmath>
using namespace std;
const double SMALL = 1.0E-30; // used to stop divide-by-zero
using vec = vector<double>; // vector
using matrix = vector<vec>; // matrix
// Function prototypes
double poly( const vec &C, double x );
matrix transpose( const matrix &A );
matrix matmul( const matrix &A, const matrix &B );
vec matmul( const matrix &A, const vec &V );
bool solve( const matrix &A, const vec &B, vec &X );
bool polynomialRegression( const vec &X, const vec &Y, int degree, vec &C, double &Rsquared );
//======================================================================
double poly( const vec &C, double x )
{
double result = 0.0;
for ( int i = C.size() - 1; i >= 0; i-- ) result = C[i] + result * x;
return result;
}
//======================================================================
matrix transpose( const matrix &A ) // Transpose a matrix
{
int m = A.size(), n = A[0].size();
matrix AT( n, vec( m ) );
for ( int i = 0; i < n; i++ )
{
for ( int j = 0; j < m; j++ ) AT[i][j] = A[j][i];
}
return AT;
}
//======================================================================
matrix matmul( const matrix &A, const matrix &B ) // Matrix times matrix
{
int rowsA = A.size(), colsA = A[0].size();
int rowsB = B.size(), colsB = B[0].size();
assert( colsA == rowsB );
matrix C( rowsA, vec( colsB, 0.0 ) );
for ( int i = 0; i < rowsA; i++ )
{
for ( int j = 0; j < colsB; j++ )
{
for ( int k = 0; k < colsA; k++ ) C[i][j] += A[i][k] * B[k][j];
}
}
return C;
}
//======================================================================
vec matmul( const matrix &A, const vec &V ) // Matrix times vector
{
int rowsA = A.size(), colsA = A[0].size();
int rowsV = V.size();
assert( colsA == rowsV );
vec C( rowsA, 0.0 );
for ( int i = 0; i < rowsA; i++ )
{
for ( int k = 0; k < colsA; k++ ) C[i] += A[i][k] * V[k];
}
return C;
}
//======================================================================
bool solve( const matrix &A, const vec &B, vec &X )
//--------------------------------------
// Solve AX = B by Cholesky factorisation of A (i.e. A = L.LT)
// Requires A to be SYMMETRIC
//--------------------------------------
{
int n = A.size();
// Cholesky-factorise A
matrix L( n, vec( n, 0 ) );
for ( int i = 0; i < n; i++ )
{
// Diagonal value
L[i][i] = A[i][i];
for ( int j = 0; j < i; j++ ) L[i][i] -= L[i][j] * L[i][j];
L[i][i] = sqrt( L[i][i] );
if ( abs( L[i][i] ) < SMALL ) return false;
// Rest of the ith column of L
for ( int k = i + 1; k < n; k++ )
{
L[k][i] = A[k][i];
for ( int j = 0; j < i; j++ ) L[k][i] -= L[i][j] * L[k][j];
L[k][i] /= L[i][i];
}
}
// Solve LY = B, where L is lower-triangular and Y = LT.X
vec Y = B;
for ( int i = 0; i < n; i++ )
{
for ( int j = 0; j < i; j++ ) Y[i] -= L[i][j] * Y[j];
Y[i] /= L[i][i];
}
// Solve UX = Y, where U = LT is upper-triangular
X = Y;
for ( int i = n - 1; i >= 0; i-- )
{
for ( int j = i + 1; j < n; j++ ) X[i] -= L[j][i] * X[j];
X[i] /= L[i][i];
}
return true;
}
//======================================================================
bool polynomialRegression( const vec &X, const vec &Y, int degree, vec &C, double &Rsquared )
{
int N = X.size(); assert( Y.size() == N );
matrix A( N, vec( 1 + degree ) );
for ( int i = 0; i < N; i++ )
{
double xp = 1;
for ( int j = 0; j <= degree; j++ )
{
A[i][j] = xp;
xp *= X[i];
}
}
// Solve the least-squares problem for the polynomial coefficients C
matrix AT = transpose( A );
if ( !solve( matmul( AT, A ), matmul( AT, Y ), C ) ) return false;
// Calculate R^2
vec AC = matmul( A, C );
double sumYsq = 0, sumACsq = 0, sumY = 0.0;
for ( int i = 0; i < N; i++ )
{
sumY += Y[i];
sumYsq += Y[i] * Y[i];
sumACsq += AC[i] * AC[i];
}
Rsquared = 1.0 - ( sumYsq - sumACsq ) / ( sumYsq - sumY * sumY / N + SMALL );
return true;
}
//======================================================================
// Interface routine that takes a (pointer to) an array Y, of size N,
// and returns the R^2 parameter for a cubic polynomial fit to the points
// (0,Y0), (1,y1), ..., (N-1,yN-1)
// extern "C" __declspec( dllexport ) double __stdcall PolyFitFunction( double *Y, int N ) // VERY UNSURE ABOUT THIS
double PolyFitFunction( double *Y, int N )
{
vec X( N ); for ( int i = 0; i < N; i++ ) X[i] = i;
vec F( Y, Y + N );
vec C;
double Rsquared;
if ( polynomialRegression( X, F, 3, C, Rsquared ) ) return Rsquared;
else return 0.0;
}
//======================================================================
int main()
{
// Data
vec X = { 0, 1, 2, 3, 4, 5 };
int N = X.size();
vec Y( N );
// Exact cubic polynomial
for ( int i = 0; i < N; i++ ) Y[i] = 10 + 8 * X[i] + 6 * X[i] * X[i] + 4 * X[i] * X[i] * X[i];
// Fudge some values or you'll get perfect agreement
Y[0] -= 10; Y[N-1] += 20; // Comment this out and you should get the original cubic
vec C;
int degree = 3;
double Rsquared;
if ( polynomialRegression( X, Y, degree, C, Rsquared ) )
{
cout << "Coefficients in C0 + C1.X + C2.X^2 + ... : ";
for ( double e : C ) cout << e << " ";
cout << "\n\nCheck fit: (Xi, Yi, poly(Xi) )\n";
for ( int i = 0; i < X.size(); i++ ) cout << X[i] << '\t' << Y[i] << '\t' << poly( C, X[i] ) << '\n';
cout << "\n\nR^2 = " << Rsquared << '\n';
// Can we get the same answer via an interface function? (Assumes X is { 0, 1, 2, ... } )
cout << "\nR^2 (by interface) = " << PolyFitFunction( Y.data(), N ) << '\n';
}
else
{
cerr << "Unable to solve\n";
}
return 0;
}
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