Obs. from Arbitrary PDF

Generate a sample from an arbitrary PDF.

Description

This program assumes you have created an arbitrary PDF and can represent it via an arbitrarily high number of x,y points.

As an aside, if you only have observations from the PDF, I find the combination of ksdensity and a smoothing spline are a great way to create cfit function that you can then run feval on and then generate the arbitrarily high x,y pairs mentioned above.

Here's the surprisingly simple part: imagine this, take your y (heights) and lay them end-to-end? what you then have is a line composed of ?segments? each of length y(x). Now generate a uniform random number of the range of 0 to the length of the segmented line (e.g. sum(y)*rand()). Now, wherever that uniform random number falls, look down at which y ?segment? you landed on and generate the corresponding x as the output!

I wish I could remember where I read this trick because I just codified it in MATLAB. This seems to be a common problem with an illusively simple and elegant solution.

NB: In general, this technique seems NOT to be perfect as the samples rarely represent the extremes of the original PDF. If there is a sophisticated way to induce some kurtosis to properly account for this, I'd be very interested to learn of it.

• MATLAB release: MATLAB 5.2 (R10)
• Other requirements: I alluded to some tools in other toolboxes in the comments above, but they are NOT required to use this function.

[Craig Chippindale]

Exactly what I was looking for.

[Hong Zhang]

Extrmely simply yet efficient.

[ibrahim karauz]

 

[Ameya Deoras]

This method is called CDF inversion - Generate a standard uniform random number and apply the inverse CDF of the distribution D to that number to obtain a new random number drawn from distribution D. This is available in MATLAB for a number of densities: eg: expinv, binoinv etc..
http://www.mathworks.com/access/helpdesk/help/toolbox/stats/bqttfc1.html#bqucf5l-2

[Alex Lu]

Thank you very much .for your excellent work, good luck

[Ahmed Fasih]

Valuable code to have.

Code can be made faster by vectorizing the while loop, and also to return multiple draws from the distribution. Just make sure to validate your modifications with a non-symmetric distribution like a beta.

This technique of random number generation is discussed here: http://en.wikipedia.org/wiki/Inverse_transform_sampling

[Seth]