Good day all!
I am using alglib-3.16.0.csharp.
There are relatively simple equations to solve unweighted linearly
least square problem Y = X * b:
b = (XT * X)-1 * XT * Y
covariance matrix for parameters b is M = RSS / ( n - 2) * (XT * X)-1. RSS - Residual Sum of Squares. The numerator, n−p, is the statistical degrees of freedom. In case for line n - 2.
and for the weighted linearly least square
b = (XT * W * X)-1 * XT * W * Y, where W - diagonal matrix
covariance matrix for parameters b is M = (XT * W * X)-1
In both cases, the variance of the parameter bi is given by Sqrt(Mii).
https://en.wikipedia.org/wiki/Weighted_least_squares
For unweighted case Alglib's results are equal whith results from equations before: parameters b and covariance matrix M. Сomplete match.
But for weighted case only results for parameters b are equal whith results from equations before. The covariance matrix, in turn, differs from the equation above.
Can someone tell me what's the matter?
// This is my task
//------------------------------------------------------------------------------------------------------------------------------------------
// data points Y
double[] y = new double[] { 62, 55, 59, 70, 67, 68, 65, 56, 83, 49, 60, 59, 74, 61, 57, 67, 55, 62, 67, 63, 65, 64, 66, 50, 66, 55, 62, 74, 63, 62, 63, 53, 70, 60, 55, 65, 69, 58, 61, 59, 67, 58, 52, 66, 46, 55, 62, 60, 56, 51 };
// data points X from 0 to 49.
double[,] x = new double[y.Length, 1];
for (int i = 0; i < y.Length; i++)
{
x[i, 0] = (double)(i);
}
// data points will be fitted by line function: Y = c[0] + c[1] * X
// Fmatrix forming
double[,] fmatrix = new double[y.Length, 2];
// all fmatrix[i, 0] = 1. This is for c[0]
for (int i = 0; i < y.Length; i++)
{
fmatrix[i, 0] = 1;
}
// fmatrix[i, 1] = X[i]. This is for c[1]
for (int i = 0; i < y.Length; i++)
{
fmatrix[i, 1] = x[i, 0];
}
int info;
double[] c;
alglib.lsfitreport rep;
//
// Linear fitting without weights
//
alglib.lsfitlinear(y, fmatrix, out info, out c, out rep);
System.Console.WriteLine("{0}", info); // EXPECTED: 1
System.Console.WriteLine("{0}", alglib.ap.format(c, 4));
//
// Linear fitting with individual weights.
//
//
// in my case variance(Y[i]) = Y[i], so I combine vector w[i] as
double[] w = new double[y.Length];
for (int i = 0; i < y.Length; i++)
{
w[i] = Math.Sqrt(1 / y[i]);
}
alglib.lsfitlinearw(y, w, fmatrix, out info, out c, out rep);
System.Console.WriteLine("{0}", info); // EXPECTED: 1
System.Console.WriteLine("{0}", alglib.ap.format(c, 4));
System.Console.ReadLine();
//-----------------------------------------------------------------------------------------------------------
Best regards
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