# Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (constant 0) and a continuous distribution (a truncated Gaussian distribution with interval ${\displaystyle (0,\infty )}$) as a result of censoring.

## Density function

The probability density function of a rectified Gaussian distribution, for which random variables X having this distribution, derived from the normal distribution ${\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2}),}$ are displayed as ${\displaystyle X\sim {\mathcal {N}}^{\textrm {R}}(\mu ,\sigma ^{2})}$, is given by

${\displaystyle f(x;\mu ,\sigma ^{2})=\Phi (-{\frac {\mu }{\sigma }})\delta (x)+{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\;e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}{\textrm {U}}(x).}$

Here, ${\displaystyle \Phi (x)}$ is the cumulative distribution function (cdf) of the standard normal distribution:

${\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt\quad x\in \mathbb {R} ,}$

${\displaystyle \delta (x)}$ is the Dirac delta function

${\displaystyle \delta (x)={\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}}$

and, ${\displaystyle {\textrm {U}}(x)}$ is the unit step function:

${\displaystyle {\textrm {U}}(x)={\begin{cases}0,&x\leq 0,\\1,&x>0.\end{cases}}}$

## Mean and variance

Since the unrectified normal distribution has mean ${\displaystyle \mu }$ and since in transforming it to the rectified distribution some probability mass has been shifted to a higher value (from negative values to 0), the mean of the rectified distribution is greater than ${\displaystyle \mu .}$

Since the rectified distribution is formed by moving some of the probability mass toward the rest of the probability mass, the rectification is a mean-preserving contraction combined with a mean-changing rigid shift of the distribution, and thus the variance is decreased; therefore the variance of the rectified distribution is less than ${\displaystyle \sigma ^{2}.}$

## Generating values

To generate values computationally, one can use

${\displaystyle s\sim {\mathcal {N}}(\mu ,\sigma ^{2}),\quad x={\textrm {max}}(0,s),}$

and then

${\displaystyle x\sim {\mathcal {N}}^{\textrm {R}}(\mu ,\sigma ^{2}).}$

## Application

A rectified Gaussian distribution is semi-conjugate to the Gaussian likelihood, and it has been recently applied to factor analysis, or particularly, (non-negative) rectified factor analysis. Harva [1] proposed a variational learning algorithm for the rectified factor model, where the factors follow a mixture of rectified Gaussian; and later Meng [2] proposed an infinite rectified factor model coupled with its Gibbs sampling solution, where the factors follow a Dirichlet process mixture of rectified Gaussian distribution, and applied it in computational biology for reconstruction of gene regulatory networks.

## References

1. Harva, M.; Kaban, A. (2007). "Variational learning for rectified factor analysis☆". Signal Processing. 87 (3): 509. doi:10.1016/j.sigpro.2006.06.006.
2. Meng, Jia; Zhang, Jianqiu (Michelle); Chen, Yidong; Huang, Yufei (2011). "Bayesian non-negative factor analysis for reconstructing transcription factor mediated regulatory networks". Proteome Science. 9 (Suppl 1): S9. doi:10.1186/1477-5956-9-S1-S9. ISSN 1477-5956. PMC 3289087.