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unexpected values in covariance matrix. lsfitlinearw http://forum.alglib.net/viewtopic.php?f=2&t=4363 
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Author:  Dmitrii [ Mon Nov 16, 2020 3:30 pm ]  
Post subject:  unexpected values in covariance matrix. lsfitlinearw  
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

Author:  Dmitrii [ Fri Nov 20, 2020 1:43 pm ] 
Post subject:  Re: unexpected values in covariance matrix. lsfitlinearw 
I have done a little research and can answer my own question. Y = X * b parameters with minimal RSS: b = (XT * X)1 * XT * Y covariance matrix for parameters b: exact form M = (XT * X)1 * X * K * X * (XT * X)1, where K is covariance matrix for Y. But if you do not know K and suppose that Kii = sigma^2 (homoscedastic case, K = sigma^2 * I, M = (XT * X)1 * XT * K * X * (XT * X)1 = sigma^2 * (XT * X)1 * XT * I * X * (XT * X)1 = sigma^2 * (XT * X)1) than you could calculate a tally for sigma^2 from RSS: sigma*^2 = RSS / ( n  2). So a tally 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. For weighted least square the same is true. You know about heteroscedasticity in your data. You choose W matrix, which in ideal case is equal to K1. b = (XT * W * X)1 * XT * W * Y and covariance matrix for parameters b is M = (XT * W * X)1. If you do no know the ecxact form of K, but have some good tally for K (or you know relation between errors for Yi) you can transform the original equation (Y = X * b) into a new homoscedastic one by multiplying by the W matrix. And the new equation Y_ = X_ * b will have the same solution (b) and like before (for OLS) you can tally M matrix: M = RSS / ( n  2) * (X_T * X_)1. Thus alglib evaluates the covariance matrix from the data itself. This is the typical way for a computing software. p.s. alglib do the same for nonelinear least square problem (evaluates the covariance matrix from the data itself), I cheked it. If you want tot use your tally for Kmatrix not only to solve equation but rather for covariance matrix of parameters evaluation you need calculate M by yourself. 
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