Generating samples on a 2-D composite binomial distribution

[Swamp Thing]

TL;DR

How to generate (X,Y) samples having a distribution specified in terms of the sum of a few normal 2-D distributions?

​

I would like to generate (X,Y) pairs such that they would follow a distribution something like this:

This is the sum of three normal distributions. Each distribution could have a different taper along the X and the Y, plus an offset along X and/or Y. So the parameters of these three distributions would be the input for my process.

I don't need to meet any rigorous test of randomness or compliance with the ideal model -- it just needs to look qualitatively like the model and be really EASY to understand and implement. Speed is not too much of an issue either.

What would be a simple algorithm?

 

[StoneTemplePython]

that picture doesn't look like a 2-d normal --i.e. it isn't symmetric about the axis that goes from top left to bottom right. This rather obvious skew is a sign of (approximation?) problems looming.
- - - - -
An easy way of generating 2-d normals is start with
$\mathbf x$ which is a d dimensional vector with iid components that are standard normals i.e. N(0,1) each

Then consider the 2-d multivariate gaussian given by
$\mathbf y:= A\mathbf x + \mathbf b$
where $\mathbf b \in \mathbb R^\text{2 x 1}$ and translates the mean and $\mathbf A \in \mathbb R^\text{2 x d}$ which controls the (co)variance which will be given by $AA^T$

edit:
maybe I misread this OP actually wants a mixture model. Frequently people here say adding distributions when they mean adding random variables, so hard to tell. The fact that the thread is titled 2-d composite binomial made me think we are actually dealing with adding normal random variables, not a mixture.

 

[Stephen Tashi]

What would be a simple algorithm?

Suppose $f_1, f_2,f_3$ are probability density functions and you want to generate random samples from a probability density function $g = k_1 f_1 + k_2 f_2 + k_3 f_3$, where the $0 \leq k_i \leq 1_i$ are constants. Since $g$ is a probability density, it must be that $k_1 + k_2 + k_3 = 1$. So you can use $k_1,k_2,k_3$ as probabilities. To generate a random sample from $g$, first pick which density $f_i$ to use. Pick $f_i$ with probability $k_i$, Then generate a random sample $(x,y)$ from the density that was selected.

To deal with your example, we need to know how to generate random samples from a 2-D gaussian distribution. Do you know how to do that?

 

[Swamp Thing]

To generate a random sample from gg, first pick which density fif_i to use. Pick fif_i with probability kik_i, Then generate a random sample (x,y)(x,y) from the density that was selected.

Thanks, that's pretty elegant. I wish I had thought of it . And it works the way sub-populations would actually contribute to the total stats according to their relative size.

To deal with your example, we need to know how to generate random samples from a 2-D gaussian distribution. Do you know how to do that?

I think that part should be OK.

Thanks.