Good day all!
I am using alglib3.16.0.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
Attachments: 
Alglib, MathCad, Me 2.png [ 422.09 KiB  Viewed 925 times ]

