public static object LeastSquare
(double[,] As, double[] bs
, bool opt_get_stat = false
)
{
if (HDebug.Selftest())
{
/// >> A = [ 1,3,2, 1 ; 4,5,6, 1 ; 7,9,9, 1 ; 11,11,12, 1 ; 13,16,15, 1 ]
/// >> b = [1, 4, 6, 9, 12]'
/// >> x = inv(A' * A) * (A' * b)
/// 0.2171
/// 0.2125
/// 0.4205
/// -0.7339
/// >> esti = A * x
/// 0.9619
/// 3.7203
/// 6.4832
/// 9.0381
/// 11.7965
/// >> corr(esti,b)
/// 0.9976
/// >> mean( (b-esti).^2 )
/// 0.0712
double[,] _A = new double[5, 3] {
{ 1, 3, 2 }, { 4, 5, 6 }, { 7, 9, 9 }, { 11, 11, 12 }, { 13, 16, 15 }
};
double[] _b = new double[5] {
1, 4, 6, 9, 12
};
dynamic _out = LeastSquare(_A, _b, true);
double _matlab_corr = 0.9976;
double _matlab_mse = 0.0712;
double[] _matlab_x = new double[] { 0.2171, 0.2125, 0.4205, -0.7339 };
double[] _matlab_esti = new double[] { 0.9619, 3.7203, 6.4832, 9.0381, 11.7965 };
double err1 = Math.Abs(_matlab_corr - _out.opt_estimation_corr);
double err2 = Math.Abs(_matlab_mse - _out.opt_mean_square_err);
double err3 = (_matlab_x - (Vector)_out.x).ToArray().MaxAbs();
double err4 = (_matlab_esti - (Vector)_out.opt_estimation).ToArray().MaxAbs();
HDebug.Assert(err1 < 0.0001);
HDebug.Assert(err2 < 0.0001);
HDebug.Assert(err3 < 0.0001);
HDebug.Assert(err4 < 0.0001);
}
/// => A x = b
///
/// => At A x = At b
///
/// => AA * x = Ab
/// => x = inv(AA) * Ab
HDebug.Assert(As.GetLength(0) == bs.Length);
int n = As.GetLength(0);
int k = As.GetLength(1);
Matrix A = Matrix.Zeros(n, k + 1);
for (int c = 0; c < n; c++)
{
for (int r = 0; r < k; r++)
{
A[c, r] = As[c, r];
}
A[c, k] = 1;
}
Matrix AA = LinAlg.MtM(A, A);
Vector Ab = LinAlg.MtV(A, bs);
Vector x;
switch (k + 1)
{
case 2: { Matrix invAA = LinAlg.Inv2x2(AA.ToArray()); x = LinAlg.MV(invAA, Ab); } break;
case 3: { Matrix invAA = LinAlg.Inv3x3(AA.ToArray()); x = LinAlg.MV(invAA, Ab); } break;
case 4: { Matrix invAA = LinAlg.Inv4x4(AA.ToArray()); x = LinAlg.MV(invAA, Ab); } break;
default:
Matlab.PutMatrix("LinAlg_LeastSquare.AA", AA);
Matlab.PutVector("LinAlg_LeastSquare.Ab", Ab);
Matlab.Execute("LinAlg_LeastSquare.AA = inv(LinAlg_LeastSquare.AA);");
Matlab.Execute("LinAlg_LeastSquare.x = LinAlg_LeastSquare.AA * LinAlg_LeastSquare.Ab;");
x = Matlab.GetVector("LinAlg_LeastSquare.x");
Matlab.Execute("clear LinAlg_LeastSquare;");
break;
}
double?opt_mean_square_err = null;
double?opt_estimation_corr = null;
Vector opt_estimation = null;
if (opt_get_stat)
{
opt_estimation = new double[n];
double avg_err2 = 0;
for (int i = 0; i < n; i++)
{
double esti = 0;
for (int j = 0; j < k; j++)
{
esti += As[i, j] * x[j];
}
esti += x[k];
opt_estimation[i] = esti;
avg_err2 += (esti - bs[i]) * (esti - bs[i]);
}
avg_err2 /= n;
opt_mean_square_err = avg_err2;
opt_estimation_corr = HMath.HCorr(opt_estimation, bs);
}
return(new
{
x = x,
/// optional outputs
opt_mean_square_err = opt_mean_square_err,
opt_estimation_corr = opt_estimation_corr,
opt_estimation = opt_estimation,
});
}
public static double[] LeastSquare
(double[] As, double[] bs
, double[] mean_square_err = null
)
{
/// => A x = b
///
/// => [A1, 1] * [ x ] = [b1]
/// [A2, 1] [ t ] [b2]
/// [A3, 1] [b3]
/// [... ] [..]
///
/// => At A xt = At b
///
/// => [A1 A2 A3] * [A1, 1] * [ x ] = [A1 A2 A3] * [b1]
/// [ 1 1 1] [A2, 1] [ t ] [ 1 1 1] [b2]
/// [A3, 1] [b3]
/// [... ] [..]
///
/// => [A1^2 + A2^2 + A3^2 + ..., A1+A2+A3+...] * [ x ] = [A1*b1 + A2*b2 + A3*b3 + ...]
/// [A1+A2+A3+... , 1+1+1+... ] [ t ] = [b1+b2+b3+... ]
///
/// => [sumA2, sumA ] * [ x ] = [sumAb]
/// [sumA , sum1 ] [ t ] = [sumb ]
///
/// => AA * xt = Ab
/// => xt = inv(AA) * Ab
double[,] AA = new double[2, 2];
double[] Ab = new double[2];
int n = As.Length;
HDebug.Assert(n == As.Length);
HDebug.Assert(n == bs.Length);
for (int i = 0; i < n; i++)
{
double ai = As[i];
double bi = bs[i];
double Ai2 = ai * ai;
AA[0, 0] += Ai2;
AA[0, 1] += ai;
AA[1, 0] += ai;
AA[1, 1] += 1;
Ab[0] += ai * bi;
}
MatrixByArr invA = LinAlg.Inv2x2(AA);
Vector xt = LinAlg.MV(invA, Ab);
if (mean_square_err != null)
{
HDebug.Assert(mean_square_err.Length == 1);
double err2 = 0;
double x = xt[0];
double t = xt[1];
for (int i = 0; i < n; i++)
{
double nbi = As[i] * x + t;
double erri = (nbi - bs[i]);
err2 += erri * erri;
}
mean_square_err[0] = err2 / n;
}
return(xt);
}