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## -*- texinfo -*-
## @deftypefn  {} {@var{x} =} lscov (@var{A}, @var{b})
## @deftypefnx {} {@var{x} =} lscov (@var{A}, @var{b}, @var{V})
## @deftypefnx {} {@var{x} =} lscov (@var{A}, @var{b}, @var{V}, @var{alg})
## @deftypefnx {} {[@var{x}, @var{stdx}, @var{mse}, @var{S}] =} lscov (@dots{})
##
## Compute a generalized linear least squares fit.
##
## Estimate @var{x} under the model @var{b} = @var{A}@var{x} + @var{w}, where
## the noise @var{w} is assumed to follow a normal distribution with covariance
## matrix @math{{\sigma^2} V}.
##
## If the size of the coefficient matrix @var{A} is n-by-p, the size of the
## vector/array of constant terms @var{b} must be n-by-k.
##
## The optional input argument @var{V} may be an n-element vector of positive
## weights (inverse variances), or an n-by-n symmetric positive semi-definite
## matrix representing the covariance of @var{b}.  If @var{V} is not supplied,
## the ordinary least squares solution is returned.
##
## The @var{alg} input argument, a guidance on solution method to use, is
## currently ignored.
##
## Besides the least-squares estimate matrix @var{x} (p-by-k), the function
## also returns @var{stdx} (p-by-k), the error standard deviation of estimated
## @var{x}; @var{mse} (k-by-1), the estimated data error covariance scale
## factors (@math{\sigma^2}); and @var{S} (p-by-p, or p-by-p-by-k if k > 1),
## the error covariance of @var{x}.
##
## Reference: @nospell{Golub and Van Loan},
## @cite{Matrix Computations (3rd Ed.)}, Johns Hopkins, Section 5.6.3, 1996.
##
## @seealso{ols, gls, lsqnonneg}
## @end deftypefn

function [x, stdx, mse, S] = lscov (A, b, V = [], alg)

  if (nargin < 2)
    print_usage ();
  endif

  if (rows (A) != rows (b))
    error ("lscov: A and B must have the same number of rows");
  endif


  if (nargin == 4)
    warning ("lscov: algorithm selection input ALG is not yet implemented, using default");
  endif

  n = rows (A);
  p = columns (A);
  k = columns (b);

  if (! isempty (V))

    if (isvector (V))
      ## n-element vector of inverse variances
      if (numel (V) != n)
        error ("lscov: vector V must have n (number of row in A) elements ");
      endif

      v = diag (sqrt (V));
      A = v * A;
      b = v * b;
    else
      ## n-by-n covariance matrix
      if (size (V) != [n, n])
        error ("lscov: matrix V must be square with the same number of rows as A");
      endif

      try
        ## Ordinarily V will be positive definite
        B = chol (V)';
      catch
        ## If V is only positive semi-definite, use its
        ## eigendecomposition to find a factor B such that V = B*B'
        [B, lambda] = eig (V);
        image_dims = (diag (lambda) > 0);
        B = B(:, image_dims) * sqrt (lambda(image_dims, image_dims));
      end_try_catch
      A = B \ A;
      b = B \ b;
    endif
  endif

  pinv_A = pinv (A);

  x = pinv_A * b;

  if (nargout > 1)
    dof = n - p;  # degrees of freedom remaining after fit
    SSE = sumsq (b - A * x);
    mse = SSE / dof;

    s = pinv_A * pinv_A';
    stdx = sqrt (diag (s) * mse);

    if (nargout > 3)
      if (k == 1)
        S = mse * s;
      else
        S = NaN (p, p, k);
        for i = 1:k
          S(:, :, i) = mse(i) * s;
        endfor
      endif
    endif
  endif

endfunction


%!test <49040>
%! ## Longley data from the NIST Statistical Reference Dataset
%! Z = [  60323    83.0   234289   2356     1590    107608  1947
%!        61122    88.5   259426   2325     1456    108632  1948
%!        60171    88.2   258054   3682     1616    109773  1949
%!        61187    89.5   284599   3351     1650    110929  1950
%!        63221    96.2   328975   2099     3099    112075  1951
%!        63639    98.1   346999   1932     3594    113270  1952
%!        64989    99.0   365385   1870     3547    115094  1953
%!        63761   100.0   363112   3578     3350    116219  1954
%!        66019   101.2   397469   2904     3048    117388  1955
%!        67857   104.6   419180   2822     2857    118734  1956
%!        68169   108.4   442769   2936     2798    120445  1957
%!        66513   110.8   444546   4681     2637    121950  1958
%!        68655   112.6   482704   3813     2552    123366  1959
%!        69564   114.2   502601   3931     2514    125368  1960
%!        69331   115.7   518173   4806     2572    127852  1961
%!        70551   116.9   554894   4007     2827    130081  1962 ];
%! ## Results certified by NIST using 500 digit arithmetic
%! ## b and standard error in b
%! V = [  -3482258.63459582         890420.383607373
%!         15.0618722713733         84.9149257747669
%!        -0.358191792925910E-01    0.334910077722432E-01
%!        -2.02022980381683         0.488399681651699
%!        -1.03322686717359         0.214274163161675
%!        -0.511041056535807E-01    0.226073200069370
%!         1829.15146461355         455.478499142212 ];
%! rsd =  304.854073561965;
%! y = Z(:,1); X = [ones(rows(Z),1), Z(:,2:end)];
%! alpha = 0.05;
%! [b, stdb, mse] = lscov (X, y);
%! assert (b, V(:,1), 3e-6);
%! assert (stdb, V(:,2), -1.e-5);
%! assert (sqrt (mse), rsd, -1E-6);

%!test
%! ## Adapted from example in Matlab documentation
%! x1 = [.2 .5 .6 .8 1.0 1.1]';
%! x2 = [.1 .3 .4 .9 1.1 1.4]';
%! X = [ones(size(x1)) x1 x2];
%! y = [.17 .26 .28 .23 .27 .34]';
%! [b, se_b, mse, S] = lscov(X, y);
%! assert (b, [0.1203 0.3284 -0.1312]', 1E-4);
%! assert (se_b, [0.0643 0.2267 0.1488]', 1E-4);
%! assert (mse, 0.0015, 1E-4);
%! assert (S, [0.0041 -0.0130 0.0075; -0.0130 0.0514 -0.0328; 0.0075 -0.0328 0.0221], 1E-4);
%! w = [1 1 1 1 1 .1]';
%! [bw, sew_b, msew] = lscov (X, y, w);
%! assert (bw, [0.1046 0.4614 -0.2621]', 1E-4);
%! assert (sew_b, [0.0309 0.1152 0.0814]', 1E-4);
%! assert (msew, 3.4741e-004, -1E-4);
%! V = .2*ones (length (x1)) + .8*diag (ones (size (x1)));
%! [bg, sew_b, mseg] = lscov (X, y, V);
%! assert (bg, [0.1203 0.3284 -0.1312]', 1E-4);
%! assert (sew_b, [0.0672 0.2267 0.1488]', 1E-4);
%! assert (mseg, 0.0019, 1E-4);
%! y2 = [y 2*y];
%! [b2, se_b2, mse2, S2] = lscov (X, y2);
%! assert (b2, [b 2*b], 2*eps);
%! assert (se_b2, [se_b 2*se_b], 3*eps);
%! assert (mse2, [mse 4*mse], eps);
%! assert (S2(:, :, 1), S, eps);
%! assert (S2(:, :, 2), 4*S, 3*eps);

%!test
%! ## Artificial example with positive semi-definite weight matrix
%! x = (0:0.2:2)';
%! y = round (100*sin (x) + 200*cos (x));
%! X = [ones(size(x)) sin(x) cos(x)];
%! V = eye (numel (x));
%! V(end, end-1) = V(end-1, end) = 1;
%! [b, seb, mseb, S] = lscov (X, y, V);
%! assert (b, [0 100 200]', 0.2);

## Test input validation
%!error <Invalid call> lscov ()
%!error <Invalid call> lscov (1)
%!error <A and B must have the same number of rows> lscov (ones (2,2),1)
%!warning <algorithm selection input ALG is not yet implemented>
%! lscov (1,1, [], "chol");
%!error <vector V must have n .* elements> lscov (1,2, [1, 2])
%!error <matrix V must be square> lscov (1,2, [1 2 3; 4 5 6])
