Improve running time of code with integrals

1 次查看（过去 30 天）
MGee
MGee 2023-9-22
评论： Dyuman Joshi 2023-9-23
Hello,
I'm aiming to create the following covariance matrix,
,
where the set of possible combinations is given by
.
This results in different combinations, meaning Σ is of dimension .
The integrand is given by with some parameters .
Below is my created function, which takes as input an array of parameters , the output is the resulting covariance matrix Σ.
Because the covariance matrix is symmetric, I just compute the lower triangular part and add the upper triangular part in the last step.
Computing this takes very long, after 5 minutes I just got the first 200 lines of the matrix. Regarding that all this is just one step in computing a Likelihood-function, which I aim to maximize for numerically, the running time has to be reduced. The problem is obviously the part where I'm integrating in the middle of the code, can anyone give me an advice how to optimize this part?
function Sigma = covariance_matrix(P)
sig = @(t,x) (P(1) + P(2).*exp(P(3).*(x+t))) .* (P(4)+t) .* exp(-P(5).*t);
i = 0;
for x1 = 30:95
for t1 = 0:95-x1
i = i+1;
j = 1;
for x2 = 30:x1
t2 = 0;
while j <= i && t2 <= 95-x2
Sigma(i,j) = integral(@(s) integral(@(u) sig(u,x1-1+s), ...
t1+1-s,t1+2-s).*integral(@(u) sig(u,x2-1+s), ...
t2+1-s,t2+2-s),0,1,'ArrayValued',true);
t2 = t2+1;
j = j+1;
end
end
end
end
Sigma = Sigma + (Sigma-diag(diag(Sigma)))';
end

请先登录，再回答此问题。

采纳的回答

Harald
Harald 2023-9-22
Hi,
it may be an alternative to calculate Sigma analytically using Symbolic Math Toolbox (please excuse any typos :) )
syms k c r a b tau x s t1 t2 u t x1 x2
sig(tau, s) = (k + c*exp(r*(tau+x)))*(a+t)*exp(-b*tau);
S = int( int(sig(u, x1-1+s), u, t1+1-s, t1+2-s) ...
.* int(sig(u, x2-1+s), u, t2+1-s, t2+2-s), s, 0, 1)
Using matlabFunction, you can convert this to a function handle.
This is quite a long expression and may take a bit to evaluate as well, but an advantage I see is that you can likely evaluate this for several t2 in a vectorized operation.
Best wishes,
Harald
  5 个评论
显示 3更早的评论
MGee
MGee 2023-9-23
As Harald mentioned in his answer, using vectorized operation reduces computing time again. The covariance matrix and mean vector is now computed in one function, running time is 20 seconds.
syms k c r a b tau x s t1 t2 u x1 x2
sig(tau,x) = (k+c*exp(r*(tau+x)))*(a+tau)*exp(-b*tau);
sig1(u,s) = sig(u,x1-1+s);
sig2(u,s) = sig(u,x2-1+s);
int1 = int(sig1,u,t1+1-s,t1+2-s);
int2 = int(sig2,u,t2+1-s,t2+2-s);
int3 = int(sig1,u,0,t1+1-s);
Sig = int(int1*int2,s,0,1);
Mean = int(int1*int3,s,0,1);
Sfh = matlabFunction(Sig);
Mfh = matlabFunction(Mean);
function [Sigma,M] = covariance_mean(P,Mfh,Sfh)
M = zeros(2211,1);
Sigma = zeros(2211);
Comb = zeros(2211,2211,4);
i = 0;
for x1 = 30:95
for t1 = 0:95-x1
i = i+1;
j = 1;
for x2 = 30:x1
t2 = 0;
while j <= i && t2 <= 95-x2
Comb(i,j,1) = t1;
Comb(i,j,2) = t2;
Comb(i,j,3) = x1;
Comb(i,j,4) = x2;
t2 = t2+1;
j = j+1;
end
end
end
end
Comb = Comb + tril(NaN(2211),-1)';
M = Mfh(P(1),P(2),P(3),P(4),P(5),Comb(:,1,1),Comb(:,1,3));
Sigma = Sfh(P(1),P(2),P(3),P(4),P(5),Comb(:,:,1),Comb(:,:,2), ...
Comb(:,:,3),Comb(:,:,4));
Sigma(isnan(Sigma)) = 0;
Sigma = Sigma + tril(Sigma,-1)';
M = M + 0.5.*diag(Sigma);
end
Dyuman Joshi
Dyuman Joshi 2023-9-23
Another modification can be to combine this
Comb(i,j,1) = t1;
Comb(i,j,2) = t2;
Comb(i,j,3) = x1;
Comb(i,j,4) = x2;
to
Comb(i,j,1:4) = [t1 t2 x1 x2];

请先登录，再进行评论。

更多回答（0 个）

类别

在 Help Center 和 File Exchange 中查找有关 Numerical Integration and Differentiation 的更多信息

标签

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by