stepan
/
dynare
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity
dynare/matlab/posterior_density_estimate.m

178 lines
6.7 KiB
Matlab
Raw Blame History

function [abscissa,f,h] = posterior_density_estimate(data,number_of_grid_points,bandwidth,kernel_function)
%%
%% This function aims at estimating posterior univariate densities from realisations of a Metropolis-Hastings
%% algorithm. A kernel density estimator is used (see Silverman [1986]) and the main task of this function is
%% to obtain an optimal bandwidth parameter.
%%
%% * Silverman [1986], "Density estimation for statistics and data analysis".
%% * M. Skold and G.O. Roberts [2003], "Density estimation for the Metropolis-Hastings algorithm".
%%
%% The last section is adapted from Anders Holtsberg's matlab toolbox (stixbox).
%%
%% stephane.adjemian@cepremap.cnrs.fr [01/16/2004].
if size(data,2) > 1 & size(data,1) == 1;
data = data';
elseif size(data,2)>1 & size(data,1)>1;
error('density_estimate: data must be a one dimensional array.');
end;
test = log(number_of_grid_points)/log(2);
if ( abs(test-round(test)) > 10^(-12));
error('The number of grid points must be a power of 2.');
end;
n = size(data,1);
%% KERNEL SPECIFICATION...
if strcmp(kernel_function,'gaussian');
k = inline('inv(sqrt(2*pi))*exp(-0.5*x.^2)');
k2 = inline('inv(sqrt(2*pi))*(-exp(-0.5*x.^2)+(x.^2).*exp(-0.5*x.^2))'); % second derivate of the gaussian kernel
k4 = inline('inv(sqrt(2*pi))*(3*exp(-0.5*x.^2)-6*(x.^2).*exp(-0.5*x.^2)+(x.^4).*exp(-0.5*x.^2))'); % fourth derivate...
k6 = inline('inv(sqrt(2*pi))*(-15*exp(-0.5*x.^2)+45*(x.^2).*exp(-0.5*x.^2)-15*(x.^4).*exp(-0.5*x.^2)+(x.^6).*exp(-0.5*x.^2))'); % sixth derivate...
mu02 = inv(2*sqrt(pi));
mu21 = 1;
elseif strcmp(kernel_function,'uniform');
k = inline('0.5*(abs(x) <= 1)');
mu02 = 0.5;
mu21 = 1/3;
elseif strcmp(kernel_function,'triangle');
k = inline('(1-abs(x)).*(abs(x) <= 1)');
mu02 = 2/3;
mu21 = 1/6;
elseif strcmp(kernel_function,'epanechnikov');
k = inline('0.75*(1-x.^2).*(abs(x) <= 1)');
mu02 = 3/5;
mu21 = 1/5;
elseif strcmp(kernel_function,'quartic');
k = inline('0.9375*((1-x.^2).^2).*(abs(x) <= 1)');
mu02 = 15/21;
mu21 = 1/7;
elseif strcmp(kernel_function,'triweight');
k = inline('1.09375*((1-x.^2).^3).*(abs(x) <= 1)');
k2 = inline('(105/4*(1-x.^2).*x.^2-105/16*(1-x.^2).^2).*(abs(x) <= 1)');
k4 = inline('(-1575/4*x.^2+315/4).*(abs(x) <= 1)');
k6 = inline('(-1575/2).*(abs(x) <= 1)');
mu02 = 350/429;
mu21 = 1/9;
elseif strcmp(kernel_function,'cosinus');
k = inline('(pi/4)*cos((pi/2)*x).*(abs(x) <= 1)');
k2 = inline('(-1/16*cos(pi*x/2)*pi^3).*(abs(x) <= 1)');
k4 = inline('(1/64*cos(pi*x/2)*pi^5).*(abs(x) <= 1)');
k6 = inline('(-1/256*cos(pi*x/2)*pi^7).*(abs(x) <= 1)');
mu02 = (pi^2)/16;
mu21 = (pi^2-8)/pi^2;
end;
%% OPTIMAL BANDWIDTH PARAMETER....
if bandwidth == 0; % Rule of thumb bandwidth parameter (Silverman [1986] corrected by
% Skold and Roberts [2003] for Metropolis-Hastings).
sigma = std(data);
h = 2*sigma*(sqrt(pi)*mu02/(12*(mu21^2)*n))^(1/5); % Silverman's optimal bandwidth parameter.
A = 0;
for i=1:n;
j = i;
while j<= n & data(j,1)==data(i,1);
j = j+1;
end;
A = A + 2*(j-i) - 1;
end;
A = A/n;
h = h*A^(1/5); % correction
elseif bandwidth == -1; % Adaptation of the Sheather and Jones [1991] plug-in estimation of the optimal bandwidth
% parameter for metropolis hastings algorithm.
if strcmp(kernel_function,'uniform') | ...
strcmp(kernel_function,'triangle') | ...
strcmp(kernel_function,'epanechnikov') | ...
strcmp(kernel_function,'quartic');
error('I can''t compute the optimal bandwidth with this kernel... Try the gaussian, triweight or cosinus kernels.');
end;
sigma = std(data);
A = 0;
for i=1:n;
j = i;
while j<= n & data(j,1)==data(i,1);
j = j+1;
end;
A = A + 2*(j-i) - 1;
end;
A = A/n;
Itilda4 = 8*7*6*5/(((2*sigma)^9)*sqrt(pi));
g3 = abs(2*A*k6(0)/(mu21*Itilda4*n))^(1/9);
Ihat3 = 0;
for i=1:n;
Ihat3 = Ihat3 + sum(k6((data(i,1)-data)/g3));
end;
Ihat3 = -Ihat3/((n^2)*g3^7);
g2 = abs(2*A*k4(0)/(mu21*Ihat3*n))^(1/7);
Ihat2 = 0;
for i=1:n;
Ihat2 = Ihat2 + sum(k4((data(i)-data)/g2));
end;
Ihat2 = Ihat2/((n^2)*g2^5);
h = (A*mu02/(n*Ihat2*mu21^2))^(1/5); % equation (22) in Skold and Roberts [2003] --> h_{MH}
elseif bandwidth == -2; % Bump killing... We construct local bandwith parameters in order to remove
% spurious bumps introduced by long rejecting periods.
if strcmp(kernel_function,'uniform') | ...
strcmp(kernel_function,'triangle') | ...
strcmp(kernel_function,'epanechnikov') | ...
strcmp(kernel_function,'quartic');
error('I can''t compute the optimal bandwidth with this kernel... Try the gaussian, triweight or cosinus kernels.');
end;
sigma = std(data);
A = 0;
T = zeros(n,1);
for i=1:n;
j = i;
while j<= n & data(j,1)==data(i,1);
j = j+1;
end;
T(i) = (j-i);
A = A + 2*T(i) - 1;
end;
A = A/n;
Itilda4 = 8*7*6*5/(((2*sigma)^9)*sqrt(pi));
g3 = abs(2*A*k6(0)/(mu21*Itilda4*n))^(1/9);
Ihat3 = 0;
for i=1:n;
Ihat3 = Ihat3 + sum(k6((data(i,1)-data)/g3));
end;
Ihat3 = -Ihat3/((n^2)*g3^7);
g2 = abs(2*A*k4(0)/(mu21*Ihat3*n))^(1/7);
Ihat2 = 0;
for i=1:n;
Ihat2 = Ihat2 + sum(k4((data(i)-data)/g2));
end;
Ihat2 = Ihat2/((n^2)*g2^5);
h = ((2*T-1)*mu02/(n*Ihat2*mu21^2)).^(1/5); % Note that h is a column vector (local banwidth parameters).
elseif bandwidth > 0;
h = bandwidth;
else;
error('density_estimate: bandwidth must be positive or equal to 0,-1 or -2.');
end;
%% COMPUTE DENSITY ESTIMATE, using the optimal bandwidth parameter.
%%
%% This section is adapted from Anders Holtsberg's matlab toolbox
%% (stixbox --> plotdens.m).
a = min(data) - (max(data)-min(data))/3;
b = max(data) + (max(data)-min(data))/3;
abscissa = linspace(a,b,number_of_grid_points)';
d = abscissa(2)-abscissa(1);
xi = zeros(number_of_grid_points,1);
xa = (data-a)/(b-a)*number_of_grid_points;
for i = 1:n;
indx = floor(xa(i));
temp = xa(i)-indx;
xi(indx+[1 2]) = xi(indx+[1 2]) + [1-temp,temp]';
end;
xk = [-number_of_grid_points:number_of_grid_points-1]'*d;
kk = k(xk/h);
kk = kk / (sum(kk)*d*n);
f = ifft(fft(fftshift(kk)).*fft([xi ;zeros(size(xi))]));
f = real(f(1:number_of_grid_points));
Reference in New Issue View Git Blame Copy Permalink