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three_species_driver.m
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three_species_driver.m
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% two stage algorithm, three species model
% S1 --> S2
% S2 --> S1
% S1+S2 --> S3
% S3 --> S1 + S2
% X(t) = (#S1(t), #S2(t))
% Y(t) = #S3(t)
T = 1;
t1 = 0.9*T
t2 = T-t1;
dy = 4;
sys = @three_species;
c = [0.5; 1; 0.1; 1];
nu = feval(sys,'nu');
[n, m] = size(nu);
%n_obs = n-n_unobs;
x0 = feval(sys,'x0');
%% ODE model for approx determinstic sys
tspan = [0 3*T];
[t,zt] = ode23(@(t,zt) three_species_ode(t,zt,c), tspan, x0);
plot(t,zt(:,1),'-bo',t,zt(:,2),'-.r', t,zt(:,3),'-g')
% local function "lin_prop_ode" can be found at the end of this script
%% piecewise constant lambda (0ne stage)
ds = 0.1;
ts = 0:ds:T;
dt = diff(ts);
% Divide the interval [0, T] into ns evenly spaced subinterval
ns = length(dt);
prop_mat = zeros(m, ns);
lambda_mat = zeros(m, ns);
% Estimate E[a(z(t_k))]
zt_mat = zeros(3, ns);
prop_mat(:,1) = feval(sys, 'prop', x0, c);
for i = 2:ns
[t,zt] = ode23(@(t,zt) three_species_ode(t,zt,c), [0, ts(i)], x0);
zt_mat(:,i) = zt(end,:)';
prop_mat(:,i) = feval(sys, 'prop', zt_mat(:,i), c);
end
% Estimate E[a(t1)]
lambdas = zeros(m, 100);
t2 = T -t1;
for k = 1:100
t = 0;
x = x0;
while(t<t1)
lambda = feval(sys,'prop',x,c);
lambda0 = sum(lambda);
%tau = exprnd(1/lambda0);
tau = log(1/rand)/lambda0;
if (t + tau <= t1)
r = rand*lambda0;
q = cumsum(lambda);
i=1;
while (r > q(i))
i = i+1;
end
x = x + nu(:,i);
t = t + tau;
else
t = t1;
end
lambdas(:,k) = lambda;
end
end
lambda = mean(lambdas, 2);
%% Chose lambda using optimization
% x = fmincon(fun,x0,A,b,Aeq,beq) minimizes fun subject to
% the linear equalities Aeq*x = beq and A*x <= b.
% If no inequalities exist, set A = [] and b = [].
% vectorize proposal intensity function lambda (m by ns matrix)
% into a 1 by m*ns row vector
prop_vec = zeros(1, m*ns);
for i = 1:m
ind_start = 1+(i-1)*ns;
ind_end = i*ns;
prop_vec(ind_start:ind_end)=prop_mat(i,:);
end
% ---- Optimization --------
objfunc = @(x) sum((x - prop_vec).^2);
x0_lambda = prop_vec;
A = [];
b = [];
% r3 - r4 = dy
Aeq = [zeros(1,ns), zeros(1,ns), dt, -dt];
beq = dy;
lb = zeros(1, m*ns);
ub = Inf*ones(1, m*ns);
x = fmincon(objfunc,x0_lambda,A,b,Aeq,beq, lb, ub);
%------------------------
lambda_vec = x;
for i = 1:m
ind_start = 1+(i-1)*ns;
ind_end = i*ns;
lambda_mat(i,:) = lambda_vec(ind_start:ind_end);
end
%% Simulation
Ns = 1000;
num_trial = 100;
V = zeros(n, Ns);
V0 = zeros(n, Ns);
x_naive = zeros(num_trial, Ns);
w_naive = zeros(num_trial, Ns);
x_ip1 = zeros(num_trial, Ns);
w_ip1 = zeros(num_trial, Ns);
x_ip2 = zeros(num_trial, Ns);
w_ip2 = zeros(num_trial, Ns);
x_gt1 = zeros(num_trial, Ns);
w_gt1 = zeros(num_trial, Ns);
x_gt2 = zeros(num_trial, Ns);
w_gt2 = zeros(num_trial, Ns);
w_poiss1 = zeros(num_trial,Ns);
l_dat1 = zeros(num_trial, Ns);
w_poiss2 = zeros(num_trial,Ns);
l_dat2 = zeros(num_trial, Ns);
w2_poiss1 = zeros(num_trial,Ns);
l2_dat1 = zeros(num_trial, Ns);
w2_poiss2 = zeros(num_trial,Ns);
l2_dat2 = zeros(num_trial, Ns);
fprintf('Naive methond. ')
tic;
for trial = 1:num_trial
%[V, w_naive(trial,:)] = naive(T, T, dy, sys, c, Ns);
x_naive(trial,:) = V(1,:);
end
toc;
fprintf('common lambda. ')
tic;
for trial = 1:num_trial
[V, w2_poiss1(trial,:), l2_dat1(trial,:), w_ip1(trial,:), V0] ....
= two_stage_three_species(t1, T, dy, sys, c, Ns, 1, lambda);
x_ip1(trial,:) = V(1,:);
end
toc;
fprintf('individual lambda. ')
tic;
for trial = 1:num_trial
[V, w2_poiss2(trial,:), l2_dat2(trial,:), w_ip2(trial,:), V0] ...
= two_stage_three_species(t1, T, dy, sys, c, Ns, 2, lambda);
x_ip2(trial,:) = V(1,:);
end
toc;
fprintf('piecewise constant lambda1.')
tic;
for trial = 1:num_trial
[V, w_poiss1(trial,:), l_dat1(trial,:), w_gt1(trial,:)] = targeting(T, T, prop_mat, dt, sys, dy, c, Ns);
x_gt1(trial,:) = V(1,:);
end
toc;
fprintf('piecewise constant lambda2.')
tic;
for trial = 1:num_trial
[V, w_poiss2(trial,:), l_dat2(trial,:), w_gt2(trial,:)] = targeting(T, T, lambda_mat, dt, sys, dy, c, Ns);
x_gt2(trial,:) = V(1,:);
end
toc;
%%
xmin = 0;
xmax = x0(1)+x0(2)+2*x0(3);
x_range = xmin:xmax;
x_prob_naive = zeros(num_trial, xmax-xmin+1);
x_prob_ip1 = zeros(num_trial, xmax-xmin+1);
x_prob_ip2 = zeros(num_trial, xmax-xmin+1);
x_prob_gt1 = zeros(num_trial, xmax-xmin+1);
x_prob_gt2 = zeros(num_trial, xmax-xmin+1);
for i = 1:num_trial
x_prob_naive(i,:) = get_hist(x_naive(i,:), w_naive(i,:), x_range);
x_prob_ip1(i,:) = get_hist(x_ip1(i,:), w_ip1(i,:), x_range);
x_prob_ip2(i,:) = get_hist(x_ip2(i,:), w_ip2(i,:), x_range);
x_prob_gt1(i,:) = get_hist(x_gt1(i,:), w_gt1(i,:), x_range);
x_prob_gt2(i,:) = get_hist(x_gt2(i,:), w_gt2(i,:), x_range);
end
%% Kolmogorov eqn
is_run_ode = 1;
if is_run_ode == 1
fprintf('Solving Kolmogorov ODE.\n')
tic;
base = x0(1)+x0(2)+2*x0(3)+1;
num_node = base^3;
% Setting up matrix A in Kolmogorov's eqn dp/dt = A*p
%global A_global
A = sparse(num_node, num_node);
%ind_state = zeros(num_node, 3);
for i=1:num_node
x = ind2state(i,base);
A(i,i) = -sum(prop(x, c));
%ind_state(i,:)=x';
for reac=1:4
x_in = x - nu(:,reac);
if prod (x_in>=0 & x_in < base)
j = state2ind(x_in, base);
%if j< 1 || j > num_node
% fprintf('j=%d\n',j)
%end
%if j >=1 && j <= num_node
prop_in = prop(x_in, c);
A(i,j)=prop_in(reac);
%end
end
end
end
tspan = [0, T];
index0 = state2ind(x0, base);
p0 = zeros(num_node, 1);
p0(index0) = 1;
[t, p] = ode23(@(t, p) kolmogorov(t, p, A), tspan, p0);
toc;
p_final = p(end,:)';
%%
x3 = x0(3)+dy;
[pi1, pi2] = p1p2_given_x3(p_final, base, x3);
[t, p] = ode23(@(t, p) kolmogorov(t, p, A), [0,t1], p0);
p_t0 = p(end,:)';
%%
prob_y = marginal_x3(p_final, base);
Fy = cumsum(prob_y);
%quantile = 0.01
%quantile = 0.5
%quantile = 0.99
%quantile = 0.05
%yT_quantile = min(x_range(Fy >= quantile))
mean_prop = 0;
for i = 1:num_node
x = ind2state(i, base);
mean_prop = mean_prop + prop(x,c)*p_t0(i);
end
end %endif is_run_ode
%%
figure
hold on
errorbar(x_range, mean(x_prob_naive)-pi1', 2*std(x_prob_naive)/sqrt(num_trial), '-b')
errorbar(x_range, mean(x_prob_ip1)-pi1', 2*std(x_prob_ip1)/sqrt(num_trial), '-r')
errorbar(x_range, mean(x_prob_ip2)-pi1', 2*std(x_prob_ip2)/sqrt(num_trial), '-g')
errorbar(x_range, mean(x_prob_gt1)-pi1', 2*std(x_prob_gt1)/sqrt(num_trial), '-m')
errorbar(x_range, mean(x_prob_gt2)-pi1', 2*std(x_prob_gt2)/sqrt(num_trial), '-c')
%plot(x_range, pi1, '-ok')
lgd = legend('naive', 'common lambda', 'lambda_i', 'lambda1', 'lambda2');
lgd.Location = 'northeast';
xlabel('#S1(T)')
ylabel('Bias in conditional distribution')
xlim([10, 30])
saveas(gcf, 'pi_two_stage_three_species_diff.png')
hold off
figure
hold on
errorbar(x_range, mean(x_prob_naive), 2*std(x_prob_naive)/sqrt(num_trial), '-b')
errorbar(x_range, mean(x_prob_ip1), 2*std(x_prob_ip1)/sqrt(num_trial), '-r')
errorbar(x_range, mean(x_prob_ip2), 2*std(x_prob_ip2)/sqrt(num_trial), '-g')
errorbar(x_range, mean(x_prob_gt1), 2*std(x_prob_gt1)/sqrt(num_trial), '-m')
errorbar(x_range, mean(x_prob_gt2), 2*std(x_prob_gt2)/sqrt(num_trial), '-c')
plot(x_range, pi1, '-ok')
lgd = legend('naive', 'common lambda', 'lambda_i', 'lambda1', 'lambda2', 'ODE');
lgd.Location = 'northeast';
xlabel('#S1(T)')
ylabel('conditional distribution')
%title('t=0.8, z0=(10,0), yT')
xlim([10, 35])
saveas(gcf, 'pi_two_stage_three_species.png')
hold off
%% Compare two batches
%{
nr = num_trial/2;
x_prob_naive_1 = x_prob_naive(1:nr,:);
x_prob_naive_2 = x_prob_naive(nr+1:num_trial,:);
x_prob_ip1_1 = x_prob_ip1(1:nr,:);
x_prob_ip1_2 = x_prob_ip1(nr+1:num_trial,:);
x_prob_ip2_1 = x_prob_ip2(1:nr,:);
x_prob_ip2_2 = x_prob_ip2(nr+1:num_trial,:);
xconf= [x_range, x_range(end:-1:1)];
yconf_naive_up_1 = mean(x_prob_naive_1) - pi1' + 2*std(x_prob_naive_1)/sqrt(nr);
yconf_naive_low_1 = mean(x_prob_naive) -pi1'- 2*std(x_prob_naive_1)/sqrt(nr);
yconf_naive_1 = [yconf_naive_up_1, yconf_naive_low_1(end:-1:1)];
yconf_ip1_up_1 = mean(x_prob_ip1_1) - pi1'+ 2*std(x_prob_ip1_1)/sqrt(nr);
yconf_ip1_low_1 = mean(x_prob_ip1_1) -pi1'- 2*std(x_prob_ip1_1)/sqrt(nr);
yconf_ip1_1 = [yconf_ip1_up_1, yconf_ip1_low_1(end:-1:1)];
yconf_ip2_up_1 = mean(x_prob_ip2_1) -pi1' + 2*std(x_prob_ip2_1)/sqrt(nr);
yconf_ip2_low_1 = mean(x_prob_ip2_1)-pi1'- 2*std(x_prob_ip2_1)/sqrt(nr);
yconf_ip2_1 = [yconf_ip2_up_1, yconf_ip2_low_1(end:-1:1)];
yconf_naive_up_2 = mean(x_prob_naive_2) -pi1'+ 2*std(x_prob_naive_2)/sqrt(nr);
yconf_naive_low_2 = mean(x_prob_naive_2) -pi1'- 2*std(x_prob_naive_2)/sqrt(nr);
yconf_naive_2 = [yconf_naive_up_2, yconf_naive_low_2(end:-1:1)];
yconf_ip1_up_2 = mean(x_prob_ip1_2) -pi1'+ 2*std(x_prob_ip1_2)/sqrt(nr);
yconf_ip1_low_2 = mean(x_prob_ip1_2) -pi1'- 2*std(x_prob_ip1_2)/sqrt(nr);
yconf_ip1_2 = [yconf_ip1_up_2, yconf_ip1_low_2(end:-1:1)];
yconf_ip2_up_2 = mean(x_prob_ip2_2) -pi1' + 2*std(x_prob_ip2_2)/sqrt(nr);
yconf_ip2_low_2 = mean(x_prob_ip2_2)-pi1'- 2*std(x_prob_ip2_2)/sqrt(nr);
yconf_ip2_2 = [yconf_ip2_up_2, yconf_ip2_low_2(end:-1:1)];
%%
figure
hold on
p1 = fill(xconf,yconf_ip1_1,'red');
p1.FaceColor = [1 0.5 0.5];
p1.EdgeColor = 'none';
p2 = fill(xconf,yconf_ip2_1, 'green');
p2.FaceColor = [0.7 1 0.7];
p2.EdgeColor = 'none';
p = fill(xconf,yconf_naive_1,'blue');
p.FaceColor = [0.5 0.7 1];
p.EdgeColor = 'none';
plot(x_range, mean(x_prob_naive_1)-pi1','-b')
plot(x_range, mean(x_prob_ip1_1)-pi1', '-r')
plot(x_range, mean(x_prob_ip2_1)-pi1', '-g')
legend('common lambda', 'lambda_i', 'naive')
xlabel('#S(T)')
ylabel('conditional distribution')
saveas(gcf,'pi_naive_gt_fill_three_species_1.png')
hold off
%%
figure
hold on
p1 = fill(xconf,yconf_ip1_2,'red');
p1.FaceColor = [1 0.8 0.8];
%p1.FaceColor = [0.5 0.7 1];
p1.EdgeColor = 'none';
p2 = fill(xconf,yconf_ip2_2, 'green');
p2.FaceColor = [0.7 1 0.7];
p2.EdgeColor = 'none';
p = fill(xconf,yconf_naive_2,'blue');
%p.FaceColor = [1 0.8 0.8];
p.FaceColor = [0.5 0.7 1];
p.EdgeColor = 'none';
plot(x_range, mean(x_prob_naive_2)-pi1','-b')
plot(x_range, mean(x_prob_ip1_2)-pi1', '-r')
plot(x_range, mean(x_prob_ip2_2)-pi1', '-g')
legend('common lambda', 'lambda_i', 'naive')
xlabel('#S(T)')
ylabel('conditional distribution')
%saveas(gcf,'pi_naive_gt_fill_three_species_2.png')
hold off
%%
figure
p = fill(xconf,yconf_naive_1,'blue');
p.FaceColor = [0.5 0.5 1];
p.EdgeColor = 'none';
hold on
p = fill(xconf,yconf_naive_2,'blue');
p.FaceColor = [0.6 0.8 1];
p.EdgeColor = 'none';
plot(x_range, mean(x_prob_naive_1)-pi1','-b')
plot(x_range, mean(x_prob_naive_2)-pi1','-b')
hold off
saveas(gcf,'naive_batch_s3.png')
%%
figure
p1 = fill(xconf,yconf_ip1_1,'red');
p1.FaceColor = [1 0.5 0.5];
p1.EdgeColor = 'none';
hold on
p1 = fill(xconf,yconf_ip1_2,'red');
p1.FaceColor = [1 0.8 0.8];
p1.EdgeColor = 'none';
plot(x_range, mean(x_prob_ip1_1)-pi1', '-r')
plot(x_range, mean(x_prob_ip1_2)-pi1', '-r')
hold off
saveas(gcf,'common_lambda_batch_s3.png')
%%
figure
p2 = fill(xconf,yconf_ip2_1, 'green');
p2.FaceColor = [0.4 1 0.4];
p2.EdgeColor = 'none';
hold on
p2 = fill(xconf,yconf_ip2_2, 'green');
p2.FaceColor = [0.8 1 0.8];
p2.EdgeColor = 'none';
plot(x_range, mean(x_prob_ip2_1)-pi1', '-g')
plot(x_range, mean(x_prob_ip2_2)-pi1', '-g')
hold off
saveas(gcf, 'lambda_i_batch_s3.png')
%%
%}
%% Statistics
tve_naive = zeros(num_trial, 1);
tve_ip1 = zeros(num_trial, 1);
tve_ip2 = zeros(num_trial, 1);
tve_gt1 = zeros(num_trial, 1);
tve_gt2 = zeros(num_trial, 1);
ess_naive = zeros(num_trial, 1);
ess_ip1 = zeros(num_trial, 1);
ess_ip2 = zeros(num_trial, 1);
ess_gt1 = zeros(num_trial, 1);
ess_gt2 = zeros(num_trial, 1);
ess_poiss1 = zeros(num_trial, 1);
ess_girsanov1 = zeros(num_trial, 1);
ess_overall1 = zeros(num_trial, 1);
ess_poiss2 = zeros(num_trial, 1);
ess_girsanov2 = zeros(num_trial, 1);
ess_overall2 = zeros(num_trial, 1);
ess2_poiss1 = zeros(num_trial, 1);
ess2_girsanov1 = zeros(num_trial, 1);
ess2_overall1 = zeros(num_trial, 1);
ess2_poiss2 = zeros(num_trial, 1);
ess2_girsanov2 = zeros(num_trial, 1);
ess2_overall2 = zeros(num_trial, 1);
for i = 1:num_trial
tve_naive(i) = sum(abs(x_prob_naive(i,:)-pi1'));
tve_ip1(i) = sum(abs(x_prob_ip1(i,:)-pi1'));
tve_ip2(i) = sum(abs(x_prob_ip2(i,:)-pi1'));
tve_gt1(i) = sum(abs(x_prob_gt1(i,:)-pi1'));
tve_gt2(i) = sum(abs(x_prob_gt2(i,:)-pi1'));
ess_naive(i) = norm(w_naive(i,:), 1)^2/norm(w_naive(i,:), 2)^2;
ess_ip1(i) = norm(w_ip1(i,:), 1)^2/norm(w_ip1(i,:), 2)^2;
ess_ip2(i) = norm(w_ip2(i,:), 1)^2/norm(w_ip2(i,:), 2)^2;
ess_gt1(i) = norm(w_gt1(i,:), 1)^2/norm(w_gt1(i,:), 2)^2;
ess_gt2(i) = norm(w_gt2(i,:), 1)^2/norm(w_gt2(i,:), 2)^2;
ess_poiss1(i) = norm(w_poiss1(i,:), 1)^2/norm(w_poiss1(i,:), 2)^2;
ess_girsanov1(i) = norm(l_dat1(i,:), 1)^2/norm(l_dat1(i,:), 2)^2;
ess_overall1(i) = norm(w_gt1(i,:), 1)^2/norm(w_gt1(i,:), 2)^2;
ess_poiss2(i) = norm(w_poiss2(i,:), 1)^2/norm(w_poiss2(i,:), 2)^2;
ess_girsanov2(i) = norm(l_dat2(i,:), 1)^2/norm(l_dat2(i,:), 2)^2;
ess_overall2(i) = norm(w_gt2(i,:), 1)^2/norm(w_gt2(i,:), 2)^2;
ess2_poiss1(i) = norm(w2_poiss1(i,:), 1)^2/norm(w2_poiss1(i,:), 2)^2;
ess2_girsanov1(i) = norm(l2_dat1(i,:), 1)^2/norm(l2_dat1(i,:), 2)^2;
ess2_overall1(i) = norm(w_ip1(i,:), 1)^2/norm(w_ip1(i,:), 2)^2;
ess2_poiss2(i) = norm(w2_poiss2(i,:), 1)^2/norm(w2_poiss2(i,:), 2)^2;
ess2_girsanov2(i) = norm(l2_dat2(i,:), 1)^2/norm(l2_dat2(i,:), 2)^2;
ess2_overall2(i) = norm(w_ip2(i,:), 1)^2/norm(w_ip2(i,:), 2)^2;
end
fprintf('---------TVE----------------\n')
fprintf('naive: %2.4f, [%2.4f, %2.4f] \n', ...
mean(tve_naive), mean(tve_naive)-2*std(tve_naive)/sqrt(num_trial), ...
mean(tve_naive)+2*std(tve_naive)/sqrt(num_trial));
fprintf('IP1: two stage: %2.4f, [%2.4f, %2.4f] \n', ...
mean(tve_ip1), mean(tve_ip1)-2*std(tve_ip1)/sqrt(num_trial), ...
mean(tve_ip1)+2*std(tve_ip1)/sqrt(num_trial));
fprintf('IP2: two stage individual lambda: %2.4f, [%2.4f, %2.4f] \n', ...
mean(tve_ip2), mean(tve_ip2)-2*std(tve_ip2)/sqrt(num_trial), ...
mean(tve_ip2)+2*std(tve_ip2)/sqrt(num_trial));
fprintf('GT1: lambda = E[a(x)]: %2.4f, [%2.4f, %2.4f] \n', ...
mean(tve_gt1), mean(tve_gt1)-2*std(tve_gt1)/sqrt(num_trial), ...
mean(tve_gt1)+2*std(tve_gt1)/sqrt(num_trial));
fprintf('GT2: lambda from optimazation: %2.4f, [%2.4f, %2.4f] \n', ...
mean(tve_gt2), mean(tve_gt2)-2*std(tve_gt2)/sqrt(num_trial), ...
mean(tve_gt2)+2*std(tve_gt2)/sqrt(num_trial));
fprintf('---------ESS----------------\n')
fprintf('naive: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess_naive), mean(ess_naive)-2*std(ess_naive)/sqrt(num_trial), ...
mean(ess_naive)+2*std(ess_naive)/sqrt(num_trial));
fprintf('IP1: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess_ip1), mean(ess_ip1)-2*std(ess_ip1)/sqrt(num_trial), ...
mean(ess_ip1)+2*std(ess_ip1)/sqrt(num_trial));
fprintf('IP2: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess_ip2), mean(ess_ip2)-2*std(ess_ip2)/sqrt(num_trial), ...
mean(ess_ip2)+2*std(ess_ip2)/sqrt(num_trial));
fprintf('GT1: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess_gt1), mean(ess_gt1)-2*std(ess_gt1)/sqrt(num_trial), ...
mean(ess_gt1)+2*std(ess_gt1)/sqrt(num_trial));
fprintf('GT2: %5.3f, [%5.2f, %5.2f] \n', ...
mean(ess_gt2), mean(ess_gt2)-2*std(ess_gt2)/sqrt(num_trial), ...
mean(ess_gt2)+2*std(ess_gt2)/sqrt(num_trial));
fprintf('-------targeting (piecewise constant intensities-----\n')
fprintf('Poisson weight 1: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess_poiss1)/Ns, mean(ess_poiss1)-2*std(ess_poiss1)/sqrt(num_trial), ...
mean(ess_poiss1)+2*std(ess_poiss1)/sqrt(num_trial));
fprintf('Girsanov weight 1: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess_girsanov1)/Ns, mean(ess_girsanov1)-2*std(ess_girsanov1)/sqrt(num_trial), ...
mean(ess_girsanov1)+2*std(ess_girsanov1)/sqrt(num_trial));
fprintf('Overall weight 1: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess_overall1)/Ns, mean(ess_overall1)-2*std(ess_overall1)/sqrt(num_trial), ...
mean(ess_overall1)+2*std(ess_overall1)/sqrt(num_trial));
fprintf('Poisson weight 2: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess_poiss2)/Ns, mean(ess_poiss2)-2*std(ess_poiss2)/sqrt(num_trial), ...
mean(ess_poiss2)+2*std(ess_poiss2)/sqrt(num_trial));
fprintf('Girsanov weight 2: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess_girsanov2)/Ns, mean(ess_girsanov2)-2*std(ess_girsanov2)/sqrt(num_trial), ...
mean(ess_girsanov2)+2*std(ess_girsanov2)/sqrt(num_trial));
fprintf('Overall weight 2: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess_overall2)/Ns, mean(ess_overall2)-2*std(ess_overall2)/sqrt(num_trial), ...
mean(ess_overall2)+2*std(ess_overall1)/sqrt(num_trial));
fprintf('--------two stage algorithm----------\n');
fprintf('Poisson weight 1: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess2_poiss1)/Ns, mean(ess2_poiss1)-2*std(ess2_poiss1)/sqrt(num_trial), ...
mean(ess2_poiss1)+2*std(ess2_poiss1)/sqrt(num_trial));
fprintf('Girsanov weight 1: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess2_girsanov1)/Ns, mean(ess2_girsanov1)-2*std(ess2_girsanov1)/sqrt(num_trial), ...
mean(ess2_girsanov1)+2*std(ess2_girsanov1)/sqrt(num_trial));
fprintf('Overall weight 1: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess2_overall1)/Ns, mean(ess2_overall1)-2*std(ess2_overall1)/sqrt(num_trial), ...
mean(ess2_overall1)+2*std(ess2_overall1)/sqrt(num_trial));
fprintf('Poisson weight 2: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess2_poiss2)/Ns, mean(ess2_poiss2)-2*std(ess2_poiss2)/sqrt(num_trial), ...
mean(ess2_poiss2)+2*std(ess2_poiss2)/sqrt(num_trial));
fprintf('Girsanov weight 2: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess2_girsanov2)/Ns, mean(ess2_girsanov2)-2*std(ess2_girsanov2)/sqrt(num_trial), ...
mean(ess2_girsanov2)+2*std(ess2_girsanov2)/sqrt(num_trial));
fprintf('Overall weight 2: %5.2f, [%5.2f, %5.2f] \n', ...
mean(ess2_overall2)/Ns, mean(ess2_overall2)-2*std(ess2_overall2)/sqrt(num_trial), ...
mean(ess2_overall2)+2*std(ess2_overall1)/sqrt(num_trial));
%%
function dxdt = three_species_ode(t,x,c)
% S1 --> S2
% S2 --> S1
% S1+S2 --> S3
% S3 --> S1 + S2
% dx1/dt = -c1*x1 + c2*x2 - c3*x1*x2 + c4*x3
% dx2/dt = c1*x1 - c2*x2 - c3*x1*x2 + c4*x3
% dx3/dt = c3*x1*x3 -c4*x3
dxdt = [-c(1)*x(1) + c(2)*x(2) - c(3)*x(1)*x(2) + c(4)*x(3);...
c(1)*x(1) - c(2)*x(2) - c(3)*x(1)*x(2) + c(4)*x(3);...
c(3)*x(1)*x(2) - c(4)*x(3)];
end
%% local functions dealing with conversions
function index = state2ind(x, base)
% n - conservative quantity, n=x1+x2+2*x3
% base = n+1 as xi takes value {0, 1, ..., n}
% state (0,0,0) maps to index 1, (0,0,1) to 2,
% (0, 0, n) maps to base
% (0, 1, 0) maps to base + 1, etc.
index = x(1)*base^2 + x(2)*base + x(3) + 1;
end
function x = ind2state(index, base)
% inverse conversion of state2ind
x = zeros(3,1);
num = index-1;
x(1)= floor(num/base^2);
num = num-x(1)*base^2;
x(2) = floor(num/base);
num = num-x(2)*base;
x(3) = num;
end
function a = prop(x,c)
a = [c(1)*x(1); c(2)*x(2); c(3)*x(1)*x(2); c(4)*x(3)];
end
function [p1, p2] = p1p2_given_x3(p, base, x3)
p1 = zeros(base, 1);
p2 = zeros(base, 1);
num_node = length(p);
for i=1:num_node
x = ind2state(i, base);
if (x(3)== x3)
p1(x(1)+1)=p1(x(1)+1) + p(i);
p2(x(2)+1)=p2(x(2)+1) + p(i);
end
end
p1 = p1/sum(p1);
p2 = p2/sum(p2);
end
function p_x3 = marginal_x3(p, base)
p_x3 = zeros(base,1);
num_node = length(p);
for i=1:num_node
x = ind2state(i, base);
p_x3(x(3)+1) = p_x3(x(3)+1) + p(i);
end
end