Matlab Numerical Integration - Syntax/Style Query

[pete j]

Hi,

Pretext: I have no formal background in MATLAB or maths in general, so apologies if any of the following doesn't make sense. I am also new to this forum, so apologies if this post is incorrect in any way. [n.b. I also posted this thread on mathhelpforum.com]

Context: Ok, so I wanted to find the response value for various parameter values, using the following equation:

http://www.nitric.net/~pete/pictures/pj.integral.GIF

[If you can't see the image, it is an integration with infinite bounds over an equation consisting of a Gaussian probability density function multiplied by the the square of a Gaussian cumulative density function plus one minus the square of another Gaussian cumulative density function]

I wasn't sure whether it was possible to do this using the MATLAB symbolic toolkit (syms), so I thought I'd take a crack at it using numerical integration, using the quad command. After much effort and confusion, I ended up with the following code.

mu=0;
sigma=1;
s=0;
alpha=0.4;
 
mu=sprintf('%f', mu);
sigma=sprintf('%f', sigma);
s=sprintf('%f', s);
alpha=sprintf('%f', alpha);
 
 
f=strcat('normpdf(x+',s,',',mu,',sqrt(2)*',sigma,').*(normcdf(x,',mu,',sqrt(2)*',alpha,'*',sigma,').^2 + (1 - normcdf(x,',mu,',sqrt(2)*',alpha,'*',sigma,')).^2)');

quad(f,-1000,1000)

This code *DOES* appear to work (i.e. yields the expected answers). But it strikes me as being more than a little crude. My questions therefore are as follows:

1. From a mathematical point of view, was using numerical integration with suitably wide bounds the right way to go about this problem?
2. From a programming point of view, is there a more elegant way to execute the above numerical integration? (i.e. a better approach than wrapping everything up as one big string which is then passed to the quad function)

Many thanks for your time,

pete

 

[mmwave]

Hi Pete,

Thanks for posting your question on the forum. It's great to see someone with a non-math background interested in using MATLAB for their work. I'll try my best to address your questions.

Firstly, from a mathematical perspective, numerical integration with wide bounds is a valid approach for solving this type of problem. However, it may not be the most efficient or accurate method. Since you are integrating over an infinite range, you will need to choose a large enough range to ensure that your solution is accurate. This can be computationally expensive and may not be feasible for more complex equations. In such cases, it may be better to use symbolic integration or other specialized numerical integration techniques.

Secondly, from a programming perspective, there are definitely more elegant ways to execute the numerical integration in MATLAB. One approach would be to define your integrand as a function and then use the quad function to integrate it. This would involve breaking down your equation into smaller, more manageable parts and then combining them in the integrand function. This would also make your code more modular and easier to debug and modify in the future.

Here's an example of how you could implement this approach:

% Define your parameters
mu = 0;
sigma = 1;
s = 0;
alpha = 0.4;

% Define your integrand function
function y = my_integrand(x)
y = normpdf(x+s,mu,sqrt(2)*sigma).*(normcdf(x,mu,sqrt(2)*alpha*sigma).^2 + (1 - normcdf(x,mu,sqrt(2)*alpha*sigma)).^2);
end

% Use the quad function to integrate over a wide range
quad(@my_integrand, -1000, 1000)

I hope this helps. Let me know if you have any further questions or need clarification on anything.

 

[oswaler]

Hello Pete,

From a mathematical point of view, using numerical integration with wide bounds is a valid approach for solving this problem. However, there may be other methods such as using a change of variables or applying a numerical integration algorithm specifically designed for Gaussian functions that could potentially yield more accurate results.

From a programming point of view, there may be more elegant ways to execute the numerical integration. Instead of creating a long string, you could break the equation into smaller parts and use the quad function on each part separately, then combine the results. This may be more efficient and easier to read in the code. Additionally, you could also consider using the integral function in MATLAB, which can handle symbolic expressions and may provide a more straightforward solution.

I hope this helps. Keep exploring and learning, and don't hesitate to ask for help when needed. Best of luck!