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% STK_DISTRIB_NORMAL_CRPS computes the CRPS for Gaussian predictive distributions
%
% CALL: CRPS = stk_distrib_normal_crps (Z, MU, SIGMA)
%
% computes the Continuous Ranked Probability Score (CRPS) of Z with respect
% to a Gaussian predictive distribution with mean MU and standard deviation
% SIGMA.
%
% The CRPS is defined as the integral of the Brier score for the event
% {Z <= z}, when z ranges from -inf to +inf:
%
% CRPS = int_{-inf}^{+inf} [Phi((z - MU)/SIGMA) - u(z - Z)]^2 dz,
%
% where Phi is the normal cdf and u the Heaviside step function. The CRPS
% is equal to zero if, and only if, the predictive distribution is a Dirac
% distribution (SIGMA = 0) and the observed value is equal to the predicted
% value (Z = MU).
%
% REFERENCE
%
% [1] Tilmann Gneiting and Adrian E. Raftery, "Strictly proper scoring
% rules, prediction, and estimation", Journal of the American
% Statistical Association, 102(477):359-378, 2007.
%
% See also: stk_distrib_normal_cdf, stk_predict_leaveoneout
% Copyright Notice
%
% Copyright (C) 2018 CentraleSupelec
% Copyright (C) 2017 CentraleSupelec & LNE
%
% Authors: Remi Stroh <remi.stroh@lne.fr>
% Julien Bect <julien.bect@centralesupelec.fr>
% Copying Permission Statement
%
% This file is part of
%
% STK: a Small (Matlab/Octave) Toolbox for Kriging
% (http://sourceforge.net/projects/kriging)
%
% STK is free software: you can redistribute it and/or modify it under
% the terms of the GNU General Public License as published by the Free
% Software Foundation, either version 3 of the License, or (at your
% option) any later version.
%
% STK is distributed in the hope that it will be useful, but WITHOUT
% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
% or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
% License for more details.
%
% You should have received a copy of the GNU General Public License
% along with STK. If not, see <http://www.gnu.org/licenses/>.
function crps = stk_distrib_normal_crps(z, mu, sigma)
%% Center and reduce the data
if nargin > 1 && ~ isempty (mu)
delta = bsxfun (@minus, z, mu); % compute z - m
else
% Default: mu = 0;
delta = z;
end
if nargin > 2 && ~ isempty (sigma)
sigma(sigma < 0) = nan;
else
% Default: sigma = 1
sigma = 1;
end
% Check size
[delta, sigma] = stk_commonsize (delta, sigma);
%% Formula for CRPS
crps = nan (size (delta));
b0 = ~ (isnan (delta) | isnan (sigma));
b1 = (sigma > 0);
% Compute the CRPS where sigma > 0
b = b0 & b1;
if any (b)
u = delta(b) ./ sigma(b); % (z - m)/s
crps(b) = sigma(b) .* (2 * stk_distrib_normal_pdf (u)...
+ u .* (2 * stk_distrib_normal_cdf (u) - 1)) - sigma(b) / (sqrt (pi));
end
% Compute the CRPS where sigma == 0: CRPS = abs(z - mu)
b = b0 & (~ b1);
crps(b) = abs (delta(b));
% Correct numerical inaccuracies
crps(crps < 0) = 0;
end
% Check particular values
%!assert (stk_isequal_tolabs (stk_distrib_normal_crps (0.0, 0.0, 0.0), 0.0))
%!assert (stk_isequal_tolabs (stk_distrib_normal_crps (0.0, 0.0, 1.0), (sqrt(2) - 1)/sqrt(pi)))
% Compute Continuous Ranked Probability Score (CRPS)
%!shared n, x_obs, mu, sigma, crps, crps_exp
%! x_obs = [ 1.78; -2.29; -1.62; -5.89; 2.88; 0.65; 2.74; -3.42]; % observations
%! mu = [-0.31; -0.59; 1.48; -1.57; -0.05; -0.27; 1.05; 1.27]; % predictions
%! sigma = [ 2.76; 6.80; 1.63; 1.19; 4.98; 9.60; 5.85; 2.24]; % standard dev
%! n = size(x_obs, 1);
%! crps = stk_distrib_normal_crps (x_obs, mu, sigma);
%!assert (isequal (size (crps), [n, 1]))
%!assert (all (crps >= 0))
%!assert (stk_isequal_tolabs (crps, stk_distrib_normal_crps(mu, x_obs, sigma)))
%!assert (stk_isequal_tolabs (stk_distrib_normal_crps (x_obs, mu, 0), abs (x_obs - mu)))
% % Numerical integration to get the reference results used below
% crps_ref = nan (n, 1);
% for k = 1:n
% x1 = linspace (mu(k) - 6*sigma(k), x_obs(k), 2e6);
% x2 = linspace (x_obs(k), mu(k) + 6*sigma(k), 2e6);
% F1 = stk_distrib_normal_cdf (x1, mu(k), sigma(k)) .^ 2;
% F2 = stk_distrib_normal_cdf (mu(k), x2, sigma(k)) .^ 2;
% crps_ref(k) = trapz ([x1 x2], [F1 F2]);
% end
%! crps_ref = [ ...
%! 1.247856605928301 ...
%! 1.757798727719891 ...
%! 2.216236225997414 ...
%! 3.648696666764968 ...
%! 1.832355265287495 ...
%! 2.278618297947438 ...
%! 1.560544734359158 ...
%! 3.455697443411153 ];
%! assert (stk_isequal_tolabs (crps, crps_ref, 1e-10));
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