Bounnded Gaussian Generator

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Sylvain Ductor

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Jun 15, 2018, 2:41:05 PM6/15/18
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Hi,

I' not expert in statistic but I have a problem to solve.
I need to produce a distribution of number within the [0,1] range, like RandBasis.uniform is doing.
However, I want to specify a mean and a standard deviation to this distribution.
Is there a trick to do it?

Regards,
S.D.
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Sylvain Ductor

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Jun 15, 2018, 5:20:14 PM6/15/18
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Here as far as I went, does not seems correct though…


 
case class ApproxGaussian(
  val μ /*mean*/: Double, val σ /*standard deviation*/: Double
) ) {

    private val distrib =  breeze.stats.distributions.
    Gaussian(μ, σ)
 
   /*
   We want to normalize so that the results is within 0,1
   In this class we exploit the fact that 97%
    of the values are within  [μ-3σ,μ+3σ]
    we so map this interval map to 0 and compress the remaining values to [0,1]
   */

    /* Resolving :
   a * (μ - 3σ) + b = 0
    a * (μ + 3σ) + b = 1
   */
  val a = 1 / (6 * σ)
  val b = 0.5 - / (6 * σ))

    def draw  = {
    val comp = a * distrib.draw - b
    if (comp < 0 || 1 < comp) {
      println(s"Ha! $comp")
      Math.max(1,Math.min(0,comp))
    } else {
      comp
    }
  } ensuring {v => v >= 0 && v <= 1}
 
 }

  case class FiniteGaussian(
  val μ /*mean*/: Double, val σ /*standard deviation*/: Double
)(
  nbElems : Int
) {
  /*
   We want to normalize so that the results is within 0,1
   In this class we generate the set of elements and take its max and min in order to normalize
   */
  private val distrib =
    Gaussian(μ, σ).sample(nbElems)

    val max = distrib.max
  val min = distrib.min

    /* Resolving
   a * max + b = 1
   a * min + b = 0
   */
  val a = 1 / (max - min)
  val b = -min / (max - min)

    /* */

    val distribIt = distrib.iterator

    def draw  = (a * distribIt.next - b) ensuring {v => v >= 0 && v <= 1}
 
 }

Sylvain Ductor

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Jun 15, 2018, 5:29:08 PM6/15/18
to Scala Breeze
my bad … i put «- b» and not «+ b»
Did someone used to statistic can please confirm the soundness of those solutions?
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Jeremy

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Jun 15, 2018, 6:19:49 PM6/15/18
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Hey Sylvain,

The Beta Distribution (wikipedia page) from the breeze library does just that. It's always between [0,1] and it takes two parameters a and b that define the shape of the graph. You can mess around with this widget to get a feel for what the parameters do. 
If you would like to supply a mean and standard deviation instead of the parameters, you'll have to do some math. The mean (or Expected value E(X)) is a / (a +b) and the variance V(X) is a*b/( (a+b+1)(a+b)^2 ). With some tricky algebra, you can solve for a and b in terms of the mean and standard deviation. 

You should get:
a = mean * (mean*(1-mean) / sd^2 -1) and b = (1-mean) * (mean*(1-mean)/sd^2 - 1)

Plug in your desired mean and standard deviation and you'll get the right parameters. I have tested a few values and it seems to work great. 

One thing to note about the beta distribution is that since it's bounded between 0 and 1, the maximum variance (assuming everything is thrown in the tails) is 1/4, so don't try to give it a standard deviation larger than 1/2. Also, as the parameters a and b get large, it looks/behaves very similar to a Gaussian curve. Although the beta distribution is not an exact "gaussian generator", it's pretty darn close, and you won't have to worry about truncating values outside of [0,1]. Let me know if you have any other questions.

Jeremy 

David Hall

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Jun 15, 2018, 6:24:59 PM6/15/18
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Do you actually need a distribution? if you just need to sample (i.e. you need a Rand), you can do Gaussian(µ, σ).filter(x => x < 1 && x > 0), which uses rejection sampling.  Otherwise, you could implement https://en.wikipedia.org/wiki/Truncated_normal_distribution. We have the building blocks in terms of the special functions (erf is available in numerics)

-- David

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Sylvain Ductor

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Jun 15, 2018, 6:32:33 PM6/15/18
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Thank you David and Jeremy!!
Both your answer are really usefull for me.
I will process those information!
Thank you!


Em sexta-feira, 15 de junho de 2018 15:41:05 UTC-3, Sylvain Ductor escreveu:
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