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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:
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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: $$p(Y=y|\alpha,\beta)=\frac{1}{2}f\left(\frac{y+1}{2}\right).$$ |
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Here is an attempt to further illustrate how to apply Néstor's suggestion (+1, btw) of using the beta distribution. The beta distribution has two parameters $\alpha$ and $\beta$. These determine the shape of the distribution - it can look like the distributions in your figure, like a box, like a straight line, and so on. The question, then, is which parameters you should use for your distributions. You want to get the right mean and the right shape of the distributions. If $X\sim \rm Beta(\alpha,\beta)$ then its mean is $\mu=\frac{\alpha}{\alpha+\beta}$. Thus $\beta=\alpha(\mu^{-1}-1)$. Recall that if $Y=2X-1$ then $E(Y)=2E(X)-1$. If you want your distribution on $[-1,1]$ to have mean $0.5$, then the beta distributed variable $X$ (which is on $[0,1]$) should have mean $\mu=0.75$, since $0.5=2*0.75-1$. Example: Set $\alpha=5$ (say). Then $\beta=5\cdot(1/0.75-1)=5/3$ yields $X$ with mean $0.75$. By trying different combinations of $\alpha$ and $\mu$ you can in this way find distributions with the right mean and the right shape. Here are some examples that resemble your figures:
Finally, from the illustration in your question it seems that what you've marked in red is the mode (i.e. the maximum of the density function) and not the mean of the distribution. The mode of the beta distribution is $\frac{\alpha-1}{\alpha+\beta-2}$. Thus, if the mode is $m$, we have $\beta=(\alpha-1)/m-a+2$. Using this, you can find distributions with the right shape and the right mode with experiments analogous to those above. |
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