Im trying to generate random integers with logarithmic distribution. I use the following formula:

idx = Math.floor(Math.log((Math.random() * Math.pow(2.0, max)) + 1.0) / Math.log(2.0));

This works well and produces sequence like this for 1000 iterations (each number represents how many times that index was generated):

[525, 261, 119, 45, 29, 13, 5, 1, 1, 1]

Fiddle

I am now trying to adjust the slope of this distribution so it doesn't drop as quickly and produces something like:

[150, 120, 100, 80, 60, ...]

Blindly playing with coefficients didn't give me what I wanted. Any ideas how to achieve it?

Im trying to generate random integers with logarithmic distribution. I use the following formula:

idx = Math.floor(Math.log((Math.random() * Math.pow(2.0, max)) + 1.0) / Math.log(2.0));

This works well and produces sequence like this for 1000 iterations (each number represents how many times that index was generated):

[525, 261, 119, 45, 29, 13, 5, 1, 1, 1]

Fiddle

I am now trying to adjust the slope of this distribution so it doesn't drop as quickly and produces something like:

[150, 120, 100, 80, 60, ...]

Blindly playing with coefficients didn't give me what I wanted. Any ideas how to achieve it?

• javascript
• algorithm
• math
• statistics
• distribution

Share Improve this question Follow asked Jun 8, 2015 at 22:24 sergserg 111k7878 gold badges325325 silver badges337337 bronze badges 3

• 1 set every 2.0 in you formula closer to 1.0 e,g. 1.3 – zmii Commented Jun 8, 2015 at 22:28
• @zmii that works well if max=10, but if max=2 it produces even sharper slope: [830, 170] vs [744, 256] for 2.0 – serg Commented Jun 8, 2015 at 22:32
• I do it like this: pseudo random numbers with a predefined distribution – Spektre Commented Jun 9, 2015 at 6:26

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1 Answer 1

Sorted by: Reset to default 7

You mention a logarithmic distribution, but it looks like your code is designed to generate a truncated geometric distribution instead, although it is flawed. There is more than one distribution called a logarithmic distribution and none of them are that mon. Please clarify if you really do mean one of them.

You pute floor[log_2 U] where U is uniformly distributed from 1 to (2^max)+1. This has a 1/2^max chance to produce max, but you clamp that to max-1. So, you have a 1/2^max chance to produce 0, 2/2^max chance to produce 1, 4/2^max chance to produce 2, ... up to a 1/2 + 1/2^max chance to produce max-1.

Present in your code, but missing from the description in the question, is that you are flipping the puted index around with

idx = (max-idx) - 1

After this, your chance to produce 0 is 1/2 + 1/2^max, and your chance to produce a value of k is 1/2^(k+1).

I think it is a mistake to let U be uniform on [1,2^max+1]. Instead, I think you want U to be uniform on [1,2^max]. Then your chance to generate idx=k is 2^(max-k-1)/((2^max)-1).

idx = Math.floor(Math.log((Math.random()*(Math.pow(2.0, max)-1.0)) + 1.0) / Math.log(2.0));

zmii's ment that you could get a flatter distribution by replacing both 2.0s with a value closer to 1.0 is good. The reason it produced unsatisfactory results for small values of max was that you were sampling uniformly from [1,1.3^max+1] instead of [1,1.3^max]. The extra +1 made a larger difference when max was smaller and the base was smaller. Try the following:

var zmii = 1.3;
idx = Math.floor(Math.log((Math.random()*(Math.pow(zmii, max)-1.0))+1.0) / Math.log(zmii));