Added new function. This function computes the mean of a prior
distribution (gamma, inverse gamma 1 and 2, beta) given the mode and the variance. git-svn-id: https://www.dynare.org/svn/dynare/trunk@2535 ac1d8469-bf42-47a9-8791-bf33cf982152time-shift
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function [mu, parameters] = mode_and_variance_to_mean(m,s2,distribution,lower_bound,upper_bound)
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% This function computes the mean of a distribution given the mode and variance of this distribution.
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%
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% INPUTS
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% m [double] scalar, mode of the distribution.
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% s2 [double] scalar, variance of the distribution.
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% distribution [string] name of the distribution ("gamma","inv-gamma-2","inv-gamma-1","beta")
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% lower_bound [double] scalar, lower bound of the random variable support (optional).
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% upper_bound [double] scalar, upper bound of the random variable support (optional).
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%
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% OUTPUT
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% mu [double] scalar, mean of the distribution.
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% parameters [double] 2*1 vector, parameters of the distribution.
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% info [integer] scalar. If info=1 we have a multiplicity of solutions.
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% If info=2 we have no solution.
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% ALGORITHM
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% Described in "Prior Distribution in Dynare".
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% Copyright (C) 2009 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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% Check input aruments.
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if ~(nargin==3 || nargin==5 || nargin==4)
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error('mode_and_variance_to mean:: 3 or 5 input arguments are needed!')
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end
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if ~ischar(distribution)
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error(['mode_and_variance_to_mean:: Third argument must be a string!'])
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end
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% Set defaults bounds.
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if nargin==3
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switch distribution
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case 'gamma'
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lower_bound = 0;
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upper_bound = Inf;
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case 'inv-gamma-1'
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lower_bound = 0;
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upper_bound = Inf;
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case 'inv-gamma-2'
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lower_bound = 0;
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upper_bound = Inf;
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case 'beta'
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lower_bound = 0;
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upper_bound = 1;
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otherwise
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disp(['mode_and_variance_to mean:: ' distribution ' is an unknown distribution...'])
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disp(' distribution is equal to ''beta'', ''gamma'',')
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disp(' ''inv-gamma-1'' or ''inv-gamma-2'' ')
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error()
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end
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end
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if nargin==4
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switch distribution
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case 'gamma'
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upper_bound = Inf;
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case 'inv-gamma-1'
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upper_bound = Inf;
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case 'inv-gamma-2'
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upper_bound = Inf;
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case 'beta'
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upper_bound = 1;
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otherwise
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disp(['mode_and_variance_to mean:: ' distribution ' is an unknown distribution...'])
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disp(' distribution is equal to ''beta'', ''gamma'',')
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disp(' ''inv-gamma-1'' or ''inv-gamma-2'' ')
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error()
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end
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end
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if strcmpi(distribution,'gamma')
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if m<lower_bound
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error('mode_and_variance_to_mean:: The mode has to be greater than the lower bound!')
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end
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if (m-lower_bound)<1e-12
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error('mode_and_variance_to_mean:: The gamma distribution should be specified with the mean and variance.')
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end
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m = m - lower_bound ;
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tmp = 1-sqrt(4*s2/(m*m));
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alpha = 1 - 2/tmp;
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beta = -.5*m*tmp;
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parameters(1) = alpha;
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parameters(2) = beta;
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mu = alpha*beta + lower_bound ;
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return
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end
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if strcmpi(distribution,'inv-gamma-2')
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if m<lower_bound+2*eps
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error('mode_and_variance_to_mean:: The mode has to be greater than the lower bound!')
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end
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m = m - lower_bound ;
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if isinf(s2)
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nu = 2;
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s = 4*m;
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else
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delta = 2*m*m/s2;
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poly = [ 1 , -6 , 12-delta , -8-2*delta ];
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all_roots = roots(poly);
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real_roots = all_roots(find(abs(imag(all_roots))<2*eps));
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nu = real_roots(find(real_roots>2));
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s = m*(nu+2);
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end
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parameters(1) = nu;
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parameters(2) = s;
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mu = s/(nu-2) + lower_bound;
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return
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end
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if strcmpi(distribution,'inv-gamma-1')
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if m<lower_bound+2*eps
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error('mode_and_variance_to_mean:: The mode has to be greater than the lower bound!')
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end
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m = m - lower_bound ;
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if isinf(s2)
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nu = 2;
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s = 1/(m*m);
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else
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[mu, parameters] = mode_and_variance_to_mean(m,s2,'inv-gamma-2');
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nu = sqrt(parameters(1));
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nu2 = 2*nu;
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nu1 = 2;
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tmp = s2*m*m;
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err = tmp - (nu-1)/(nu-2) + .5*(nu-1)*(gamma((nu-1)/2)/gamma(nu/2))^2;
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while abs(nu2-nu1) > 1e-12
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if err > 0
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nu1 = nu;
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if nu < nu2
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nu = nu2;
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else
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nu = 2*nu;
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nu2 = nu;
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end
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else
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nu2 = nu;
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end
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nu = (nu1+nu2)/2;
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err = tmp - (nu-1)/(nu-2) + .5*(nu-1)*(gamma((nu-1)/2)/gamma(nu/2))^2;
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end
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s = (nu-1)/m^2 ;
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end
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parameters(1) = nu;
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parameters(2) = s;
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mu = sqrt(.5*s)*gamma(.5*(nu-1))/gamma(.5*nu) + lower_bound ;
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return
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end
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if strcmpi(distribution,'beta')
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if m<lower_bound
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error('mode_and_variance_to_mean:: The mode has to be greater than the lower bound!')
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end
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if m>upper_bound
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error('mode_and_variance_to_mean:: The mode has to be less than the upper bound!')
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end
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if (m-lower_bound)<1e-12
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error('mode_and_variance_to_mean:: The beta distribution should be specified with the mean and variance.')
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end
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if (upper_bound-m)<1e-12
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error('mode_and_variance_to_mean:: The beta distribution should be specified with the mean and variance.')
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end
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ll = upper_bound-lower_bound;
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m = (m-lower_bound)/ll ;
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s2 = s2/(ll*ll) ;
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poly = NaN(1,4);
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poly(1) = 1/m^3;
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poly(2) = (7*m*s2-3*s2+m^3-m^2)/(m^3*s2);
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poly(3) = (16*m^2*s2-14*m*s2+3*s2-2*m^3+m^2)/(m^3*s2);
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poly(4) = 12*m^3-16*m^2-7*m-1;
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all_roots = roots(poly);
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real_roots = all_roots(find(abs(imag(all_roots))<2*eps));
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idx = find(real_roots>1);
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if length(idx)>1
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error('mode_and_variance_to_mean:: Multiplicity of solutions for the beta distribution specification.')
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elseif isempty(idx)
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disp('mode_and_variance_to_mean:: No solution for the beta distribution specification.')
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disp(' You should reduce the variance.');
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error();
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end
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alpha = real_roots(idx);
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beta = ((1-m)*alpha+2*m-1)/m;
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parameters(1) = alpha;
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parameters(2) = beta;
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mu = alpha/(alpha+beta)*(upper_bound-lower_bound)+lower_bound;
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return
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end
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