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    Metadata details:
    filename: covSM.m
    path of file: ./repos/Gaussian-Process-Regression/gpml-matlab-v4.2-2018-06-11/cov
    
    the code of the file until where you have to start completion: function [K,dK] = covSM(Q, hyp, x, z)

% Gaussian Spectral Mixture covariance function. The covariance function 
% parametrization depends on the sign of Q.
%
% Let t(Dx1) be an offset vector in dataspace e.g. t = x-z. Then w(DxP)
% are the weights and m(Dx|Q|) = 1/p, v(Dx|Q|) = (2*pi*ell)^-2 are spectral
% means (frequencies) and variances, where p is the period and ell the length
% scale of the Gabor function h(t2v,tm) given by the expression
%   h(t2v,tm) = exp(-2*pi^2*t2v)
    
function h(t2v,tm) given by the expression
%   h(t2v,tm) = exp(-2*pi^2*t2v).*cos(2*pi*tm)
%
% Then, the two covariances are obtained as follows:
%
% SM, spectral mixture:          Q>0 => P = 1
%   k(x,z) = w'*h((t.*t)'*v,t'*m), t = x-z
%
% SMP, spectral mixture product: Q<0 => P = D
%   k(x,z) = prod(w'*h(T*T*v,T*m)), T = diag(t), t = x-z
%
% Note that for D=1, the two modes +Q and -Q are exactly the same.
%
% The hyperparameters are:
%
% hyp = [ log(w(:))
%         log(m(:))
%         log(sqrt(v(:))) ]
%
% For more help on design of covariance functions, try "help covFunctions".
%
% Note that the spectral density H(s) = F[ h(t) ] of covGaboriso is given by
% H(s) = N(s|m,v)/2 + N(s|-m,v)/2 where m=1/p is the mean and v=(2*pi*ell)^-2
% is the variance of a symmetric Gaussian mixture. Hence the covGaboriso
% covariance forms a basis for the class of stationary covariances since a
% weighted sum of covGaboriso covariances corresponds to an isotropic
% location-scale mixture of a symmetric Gaussian mixture in the spectral domain.
%
% For more details, see 
% [1] SM: Gaussian Process Kernels for Pattern Discovery and Extrapolation,
%     ICML, 2013, by Andrew Gordon Wilson and Ryan Prescott Adams,
% [2] SMP: GPatt: Fast Multidimensional Pattern Extrapolation with GPs,
%     arXiv 1310.5288, 2013, by Andrew Gordon Wilson, Elad Gilboa, 
%     Arye Nehorai and John P. Cunningham, and
% [3] Covariance kernels for fast automatic pattern discovery and extrapolation
%     with Gaussian processes, Andrew Gordon Wilson, PhD Thesis, January 2014.
%     http://www.cs.cmu.edu/~andrewgw/andrewgwthesis.pdf
% [4] http://www.cs.cmu.edu/~andrewgw/pattern/.
%
% For Q>0, covSM corresponds to Eq. 4.7 in Ref [3].
% For Q<0, covSM corresponds to Eq. 14 in Ref [2] and Eq. 5.3 in Ref [3] but our
% w here corresponds to  w^2 in Eq. 14.
%
% Copyright (c) by Andrew Gordon Wilson and Hannes Nickisch, 2016-05-06.
%
% See also COVFUNCTIONS.M, COVGABORISO.M, COVGABORARD.M.

if nargin<1, error('You need to provide Q.'), end