Define a density function and draw N samples of it

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Orr Streicher 2021-4-26

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评论： Jeff Miller 2021-4-27

Hi,

So i want to define a density function and draw N samples of it. does anyone know how can i do it?

for example if i want a gaussian mixture (with M=4 gaussians with diffrent expectations mu's) i defined:

f1=(1/(2*pi*sigma2*M))*(exp((-1/(2*sigma2))*((vecnorm((X-mu(1,:)).').').^2))+...

exp((-1/(2*sigma2))*((vecnorm((X-mu(2,:)).').').^2))+...

exp((-1/(2*sigma2))*((vecnorm((X-mu(3,:)).').').^2))+...

exp((-1/(2*sigma2))*((vecnorm((X-mu(4,:)).').').^2)));

But im not sure how to chose X such as the samples will represent the distrebution (that is more samples in the center of the gauusians and less in its sides).

The gaussian mixture is just an exaple i am looking for a way to do it to any density function.

Thanks

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Answer 1

Jeff Miller 2021-4-26

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I don't think that density function is right--seems like you at least need to include the mixture probabilities in there--e.g., 1/4 * each density if the 4 gaussians are equally likely.

You might have a look at Cupid which supports densities and random number generation for mixtures of lots of density functions. It might already do what you want; if not, you might get some useful formulas.

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Orr Streicher 2021-4-27

As far as i understand i did the normalization you talked about, with the coeffetiont (1/(2*pi*sigma2*M)) (M=4).

Cupid looks great, i will take a look. thanks!

Jeff Miller 2021-4-27

M=4 seems right--I did not notice that.

If you know the cdf's of all your distributions, then the cdf of the mixture at each x is the sum of the cdf_i(x)*p*i values, and you can use the inverse transform sampling technique suggested by Paul.

Alternatively, you can generate random numbers by first choosing one of the distributions randomly with probabilities p_i and then choosing a random number from that distribution. (This is what Cupid does for mixtures.)

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Answer 2

Paul 2021-4-27

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The Statistics and Machine Learning Toolbox has lots of typical densities that can be used to generate samples of random variables. If you have a custom-defined probability density function for which you can define the CDF, you can try an inverse transform sampling technique, which can be implemented numerically if you don't have a closed form expression for the CDF.