distributionsnormal distribution
I'm currently using Gaussian distribution as a mutation operator for my genetic algorithm. However, I only want to obtain values between -1 and 1. I also don't wish to truncate my Gaussian distribution, which leaves me with lots of 1's and -1's.
What type of probability density function can I use to obtain values between -1 and 1, based on a mean value between -1 and 1?
Here's an image for the distribution that I'm looking for with mean values of 0, -0.5, and 0.5:
There are many such.
- One example (symmetric):
http://en.wikipedia.org/wiki/Logistic_distribution
- another (asymmetric):
http://en.wikipedia.org/wiki/Gumbel_distribution
- another (symmetric, finite range)
The $\text{Beta}(2,2)$ distribution
(Indeed, the $\text{Beta}(m,n)$ with $m>1$ and $n$ integer and $>1$ should all have closed form cdf, so lots of symmetric and asymmetric cases here)
(Edit: this won't match 'infinitely differentiable' at the bounds; I believe the others do)
- another (symmetric, finite range)
The raised cosine.
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In considering other possible examples:
a) when you say 'finite moments', do you mean "finite moments of all positive integer orders" or do you mean "at least some finite moments"?
e.g. if you have finite moments up to say order 4, is that what you seek, or must moments of all orders exist?
b) when you say "smooth", what kind of smoothness is it? Clearly, from your mention of the Laplace, continuity is insufficient. Is continuous first derivative sufficient, as with this example, which has quadratic ends and a uniform center:
or do you seek something 'stronger' - i.e. with higher order continuity?
You may be looking for distribution known under the names of generalized normal (version 1), Subbotin distribution, or exponential power distribution. It is parametrized by location $\mu$, scale $\sigma$ and shape $\beta$ with pdf
$$ \frac{\beta}{2\sigma\Gamma(1/\beta)} \exp\left[-\left(\frac{|x-\mu|}{\sigma}\right)^{\beta}\right] $$
as you can notice, for $\beta=1$ it resembles and converges to Laplace distribution, with $\beta=2$ it converges to normal, and when $\beta = \infty$ to uniform distribution.
If you are looking for software that has it implemented, you can check normalp library for R (Mineo and Ruggieri, 2005). What is nice about this package is that, among other things, it implements regression with generalized normally distributed errors, i.e. minimizing $L_p$ norm.
Mineo, A. M., & Ruggieri, M. (2005). A software tool for the exponential power distribution: The normalp package. Journal of Statistical Software, 12(4), 1-24.
Best Answer
A beta distribution seems to suit your needs, but you'll have to perform a transformation in order to change its $(0,1)$ (finite) support to $(-1,1)$ support.
Let $X$ be distributed with a beta distribution, then the random variable $Y$ given by the transformation $$Y=(b-a)X+a$$ is beta distributed and the PDF has finite support in $(a,b)$. In your case, $a=-1$ and $b=1$. The PDF of this linear transformation is given by:$$p(Y=y|\alpha,\beta,a,b)=f\left(\frac{y-a}{b-a}\right)\frac{1}{b-a},$$ where $f(x)$ is the PDF of the beta distribution given in the wiki page that I cited, and $\alpha$ and $\beta$ are it's parameters. In your case, with $a=-1$ and $b=1$ we get: $$p(Y=y|\alpha,\beta)=\frac{1}{2}f\left(\frac{y+1}{2}\right).$$