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Main_1D_ultimate.m
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% Main 1D
clear all;
close all;
path(path,'utils')
% Random seeds used in paper: 6,5,5,5 for example 0, 2, 6, 11
rs = 5;
% rng(rs);
example = 0;
N = 1000;
N_rs = 1;
weight = 0.0;
% N = 1600;
switch example
case 0
z_marginal = 'uniform';
zstar = [0.21,0.75];
rs = 6;
xl = [-0.8,0.8];
yl = [0,6];
case 2
z_marginal = 'beta';
zstar = [0.4,0.8];
rs = 5;
xl = [-2.5,3.5];
yl = [0,2.8];
case 6
z_marginal = 'uniform';
zstar = [0.2,0.9];
rs = 5;
xl = [-1.3,3.8];
yl = [0,3.5];
case 10
z_marginal = 'categorical2';
zstar = [0.0,0.75];
rs = 5;
xl = [-5,10];
yl = [0,0.3];
case 12
1
z_marginal = 'uniform';
zstar = [0.0,0.75];
rs = 5;
xl = [-22,22];
yl = [0,1];
case 13
z_marginal = 'uniform';
zstar = [0.49,0.51];
rs = 5;
xl = [-22,22];
yl = [0,1];
case 14
z_marginal = 'uniform';
zstar = [0.2,0.75];
rs = 5;
xl = [-10,5];
yl = [0,2];
end
a = 2.5;
viz = true;
% zstar = [0, 0.75];
nums_to_do = [1:6];
% nums_to_do = 1:5;
% nums_to_do = [1,2,3,5,6,9];
center_x = true;
[x,z, xstar, fstar_true, z_type, range_star] = generate_data_1D(example, N, z_marginal, viz, zstar, rs, a);
x = x(1:N);
z = z(1:N);
if z_type == 'd'
zstar = unique(z);
end
G = @(x,h,m) 1./(sqrt(2*pi*h.^2)).*exp( -1/2*( x - m').^2./h./h);
L = size(z,1);
hz = cell(1,6);
hx = cell(1,6);
lklhdval = zeros(1,6);
if sum(1 == nums_to_do)
% Proposal 1: benchmark: both rule-of-thumb
[hx{1},hz{1},output1, lklhdval(1), KCDE_fun{1}, KCDE_fun_loo{1}] = find_both_h2(x,z,z_type,'none','constant',center_x);
end
if sum(2 == nums_to_do)
% Proposal 2: rule-of-thumb hz, variable hx
[hx{2},hz{2},output2, lklhdval(2), KCDE_fun{2}, KCDE_fun_loo{2}] = find_both_h2(x,z,z_type,'none','variable',center_x);
end
if sum(3 == nums_to_do)
% Proposal 3: variable hx as a function of hz; hz constant chosen via LOO-CV
[hx{3},hz{3},output3, lklhdval(3), KCDE_fun{3}, KCDE_fun_loo{3}] = find_both_h2(x,z,z_type,'z','variable',center_x);
end
if sum(4 == nums_to_do)
% Proposal 4: Solve for hz and then hx4 separately
[hx{4},hz{4},output4, lklhdval(4), KCDE_fun{4}, KCDE_fun_loo{4}] = find_both_h2(x,z,z_type,'both','constant',center_x);
end
if sum(5 == nums_to_do)
% Proposal 5: Solve an optimization problem for two variables at once
[hx{5}, hz{5},output5, lklhdval(5), KCDE_fun{5}, KCDE_fun_loo{5}] = find_both_h2(x,z,z_type,'both','variable',center_x);
end
if sum(6 == nums_to_do)
% Proposal 6: Solve an optimization problem for each zstar
hx{6} = cell(1,length(zstar));
hz{6} = cell(1,length(zstar));
KCDE_fun{6} = cell(1,length(zstar));
KCDE_fun_loo{6} = cell(1,length(zstar));
for i = 1:length(zstar)
[hx{6}{i}, hz{6}{i},output6, lklhdval(6,i), KCDE_fun{6}{i}, KCDE_fun_loo{6}{i}] = find_both_h2_ultimate(x,z,z_type,'both','variable',center_x, zstar(i));
end
end
score_true = sum( log(fstar_true(x,z)))/N;
linestyles = {'-','--','-.','-','-.',':'};
%%
calc_score = false;
if calc_score
for i = 1:size(hz,2)
if size(hz{i},1)>0
score(i) = 0;
score_loo(i) = 0;
if i ~= 6
for j = 1:N
score(i) = score(i) + log( KCDE_fun{i}(x(j), z(j)));
score_loo(i) = score_loo(i) + log( KCDE_fun_loo{i}(x(j), z(j), j));
end
else
for j = 1:N
j/N
[~, ~,~, ~, KCDE_fun6, KCDE_fun_loo6] = find_both_h2_ultimate(x,z,z_type,'both','variable',center_x, z(j));
score(i) = score(i) + log( KCDE_fun6(x(j), z(j)));
score_loo(i) = score_loo(i) + log( KCDE_fun_loo6(x(j), z(j), j));
end
%
%
end
score(i) = score(i)/N;
score_loo(i) = score_loo(i)/N;
end
end
end
%%
% figure;
for l = 1:size(zstar,2)
%xl = [min(x)-std(x), max(x)+std(x)];
xgrid = linspace(xl(1),xl(2),1e3);
zstarl = zstar(l);%*ones(1,length(xgrid));
fstar = cell(1,size(hz,2));
figure;
for i = 1:size(hz,2)
if size(hz{i},1)>0
if i == 6
fstar{i} = KCDE_fun{6}{l}(xgrid, zstarl);
else
fstar{i} = KCDE_fun{i}(xgrid, zstarl);
end
plot(xgrid,fstar{i},linestyles{i},'linewidth',2), hold on,
end
end
plot(xgrid,fstar_true(xgrid,zstar(l)),'k','linewidth',2)
set(gca,'fontsize',16)
ylim(yl);
xlim(xl);
% legs{1} = ['rule-of-thumb hz & hx (',num2str(score(1),'%.2f'),')'];
% legs{2} = ['rule-of-thumb hz, variable hx (',num2str(score(2),'%.2f'),')'];
% legs{3} = ['ML constant hz, variable hx (',num2str(score(3),'%.2f'),')'];
% legs{4} = ['ML constant hz, constant hx, both optimized (',num2str(score(4),'%.2f'),')'];
% legs{5} = ['ML constant hz, variable hx (\alpha), both optimized (',num2str(score(5),'%.2f'),')'];
% legs{6} = ['ML constant hz, ultimate hx (\alpha etc), both optimized (',num2str(score(6),'%.2f'),')'];
%
% legs_used = cell(1,length(nums_to_do));
% counter = 1;
% for i = 1:6
% if sum(i == nums_to_do)
% legs_used{counter} = legs{i};
% counter = counter + 1;
% end
% end
% leg = legend(legs_used);
title(['$\rho(x|$',num2str(zstar(l)),'$)$'],'interpreter','latex')
legend('Approach 1','Approach 2','Approach 3','Approach 4','Approach 5','Approach 6')
% set(leg,'location','southoutside','fontsize',14)
xlabel('$x$','interpreter','latex')
% set(gcf,'position',[-647 1047 611 618]);
set(gcf,'position',[270 1230 632 328]);
end
if calc_score
score
score_loo
score_true
lklhdval/N
end
%%
zgrid = sort(z);
switch z_type
case 'c'
zgrid = linspace(0,1,50);
case 'd'
zgrid = unique(z);
end
Hdist = zeros( size(hz,2), length(zgrid));
ISEdist = zeros( size(hz,2), length(zgrid));
for i = 1:size(hz,2)
if size(hz{i},1)>0
for j = 1:length(zgrid)
if i == 6
j/length(zgrid)
[~, ~, output6, ~, KCDE_fun6, KCDE_fun_loo6] = find_both_h2_ultimate(x,z,z_type,'both','variable',center_x, zgrid(j));
hz6(j) = abs(output6(2));
alpha6(j) = abs(output6(1));
f_est = @(xi) KCDE_fun6(xi, zgrid(j));
figure(19);
subplot(1,2,1)
switch z_type
case 'c'
scatter(zgrid(j), hz6(j),'k','filled'); hold on,
hold on,plot(linspace(0,1,100),output5(2)*ones(1,100),'b-.','linewidth',2)
case 'd'
scatter(zgrid(j), (1/2+1/2*cos(2*pi*hz6(j)).^2),'k','filled'); hold on,
hold on,plot(linspace(0,1,100),(1/2+1/2*cos(2*pi*output5(2)).^2)*ones(1,100),'b-.','linewidth',2)
end
set(gca,'fontsize',16)
xlabel('$z$','interpreter','latex')
title('$h_z(z)$','interpreter','latex')
xlim([min(z),max(z)]);
subplot(1,2,2)
scatter(zgrid(j), alpha6(j),'k','filled'); hold on,
set(gca,'fontsize',16)
xlabel('$z$','interpreter','latex')
title('$\alpha(z)$','interpreter','latex')
xlim([min(z),max(z)]);
set(gcf,'position',[21 452 853 286])
hold on,plot(linspace(0,1,100),output5(1)*ones(1,100),'b-.','linewidth',2)
else
f_est = @(xi) KCDE_fun{i}(xi, zgrid(j));
end
f_true = @(xi) fstar_true(xi,zgrid(j));
Hdist(i, j) = pqdistances(f_est, f_true, 1,'Hellinger', range_star(zgrid(j)) );
% ISEdist(i, j) = pqdistances(f_est, f_true, 1,'ISE',range_star(zgrid(j)));
end
end
end
mean(Hdist')
%%
figure;
for i = 1:size(hz,2)
if size(hz{i},1)>0
plot( zgrid, Hdist(i,:),linestyles{i},'linewidth',2), hold on
end
end
legs{1} = ['rule-of-thumb hz & hx (',num2str(mean(Hdist(1,:)),'%.2f'),')'];
legs{2} = ['rule-of-thumb hz, variable hx (',num2str(mean(Hdist(2,:)),'%.2f'),')'];
legs{3} = ['ML constant hz, variable hx (',num2str(mean(Hdist(3,:)),'%.2f'),')'];
legs{4} = ['ML constant hz, constant hx, both optimized (',num2str(mean(Hdist(4,:)),'%.2f'),')'];
legs{5} = ['ML constant hz, variable hx (\alpha), both optimized (',num2str(mean(Hdist(5,:)),'%.2f'),')'];
legs{6} = ['ML constant hz, ultimate hx (\alpha etc), both optimized (',num2str(mean(Hdist(6,:)),'%.2f'),')'];
legs_used = cell(1,length(nums_to_do));
counter = 1;
for i = 1:6
if sum(i == nums_to_do)
legs_used{counter} = legs{i};
counter = counter + 1;
end
end
% leg = legend(legs_used);
%
% set(leg,'location','southoutside','fontsize',14)
xlabel('$z$','interpreter','latex')
title('squared Hellinger distance vs $z$','interpreter','latex')
set(gca,'fontsize',16)
legend('Approach 1','Approach 2','Approach 3','Approach 4','Approach 5','Approach 6')
% set(leg,'location','southoutside','fontsize',14)
xlabel('$x$','interpreter','latex')
set(gcf,'position',[-647 1047 611 618]);
set(gcf,'position',[270 1230 632 328]);
%%
% if strcmp(z_marginal, 'categorical')
%
%
%
% hx_vals = zeros( size(hz,2), length(zgrid));
%
% for i = 1:size(hz,2)
% if size(hz{i},1)>0
% for j = 1:length(zgrid)
% hx_vals(i,j) = hx{i}(zgrid(j));
% end
%
% end
% end
%
%
%
% % end
%
% figure;
% for i = 1:size(hz,2)
% if size(hz{i},1)>0
%
% plot( zgrid, hx_vals(i,:),'linewidth',2), hold on
% end
% end
% legs{1} = ['rule-of-thumb hz & hx (',num2str(mean(hx_vals(1,:)),'%.2f'),')'];
% legs{2} = ['rule-of-thumb hz, variable hx (',num2str(mean(hx_vals(2,:)),'%.2f'),')'];
% legs{3} = ['ML constant hz, variable hx (',num2str(mean(hx_vals(3,:)),'%.2f'),')'];
% legs{4} = ['ML constant hz, constant hx, both optimized (',num2str(mean(hx_vals(4,:)),'%.2f'),')'];
% legs{5} = ['ML constant hz, variable hx (\alpha), both optimized (',num2str(mean(hx_vals(5,:)),'%.2f'),')'];
% legs{6} = ['ML constant hz, ultimate hx (\alpha etc), both optimized (',num2str(mean(hx_vals(6,:)),'%.2f'),')'];
%
% legs_used = cell(1,length(nums_to_do));
% counter = 1;
% for i = 1:6
% if sum(i == nums_to_do)
% legs_used{counter} = legs{i};
% counter = counter + 1;
% end
% end
% leg = legend(legs_used);
%
% set(leg,'location','southoutside','fontsize',14)
% xlabel('$z$','interpreter','latex')
% title('$hx(z)$ vs $z$','interpreter','latex')
% set(gca,'fontsize',16)
%
% mean(Hdist')
%
%
% %%
%
xl = [min(x)-std(x), max(x)+std(x)];
xgrid = linspace(xl(1),xl(2),100);
zgrid = sort(z);
switch z_type
case 'c'
zgrid = linspace(0,1,20);
case 'd'
zgrid = unique(z);
end
% xgrid = linspace(min(xgrid),max(xgrid),100);
ftrue_array = zeros(length(zgrid),length(xgrid));
for i = 1:length(zgrid)
ftrue_array(i,:) = fstar_true(xgrid, zgrid(i));
end
figure;
% if z_type == 'c'
% [X,Z] = meshgrid(xgrid, zgrid);
% s=surf(X,Z,ftrue_array,'FaceAlpha',0.4);
% s.EdgeColor='none';
%
% else
for i = 1:length(zgrid)
p=fill3( xgrid, zgrid(i)+0*xgrid, ftrue_array(i,:), zgrid(i) );hold on,
p.FaceAlpha = 0.4;
p.EdgeColor = 'none';
end
% % [X,Z] = meshgrid(xgrid, zgrid);
% % s=surf(X,Z,ftrue_array,'FaceAlpha',0.4);
% %s.EdgeColor='none';
% end
xlabel('$x$','interpreter','latex')
ylabel('$z$','interpreter','latex')
set(gca,'fontsize',16)
title('$\rho^{true}(x|z)$','interpreter','latex')
hold on,
scatter(x,z,10,'k','filled')
xl = xlim;
if z_type == 'csdfsdf'
for l = 1:size(zstar,2)
plot(linspace(xl(1),xl(2),N),zstar(l)*ones(1,N),'k-.','linewidth',2)
end
end
xlim(xl);
box on
grid on