# 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 $(0,\infty )$) 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 ${\mathcal {N}}(\mu ,\sigma ^{2}),$ are displayed as $X\sim {\mathcal {N}}^{\textrm {R}}(\mu ,\sigma ^{2})$, is given by $f(x;\mu ,\sigma ^{2})=\Phi {\left(-{\frac {\mu }{\sigma }}\right)}\delta (x)+{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\;e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}{\textrm {U}}(x).$
Here, $\Phi (x)$ is the cumulative distribution function (cdf) of the standard normal distribution: $\Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt\quad x\in \mathbb {R} ,$ $\delta (x)$ is the Dirac delta function $\delta (x)={\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}$ and, ${\textrm {U}}(x)$ is the unit step function: ${\textrm {U}}(x)={\begin{cases}0,&x\leq 0,\\1,&x>0.\end{cases}}$

## Mean and variance

Since the unrectified normal distribution has mean $\mu$ 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 $\mu .$
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 $\sigma ^{2}.$

## Generating values

To generate values computationally, one can use
    $s\sim {\mathcal {N}}(\mu ,\sigma ^{2}),\quad x={\textrm {max}}(0,s),$
and then
    $x\sim {\mathcal {N}}^{\textrm {R}}(\mu ,\sigma ^{2}).$

## 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 proposed a variational learning algorithm for the rectified factor model, where the factors follow a mixture of rectified Gaussian; and later Meng 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.

## Extension to general bounds

An extension to the rectified Gaussian distribution was proposed by Palmer et al., allowing rectification between arbitrary lower and upper bounds. For lower and upper bounds $a$ and $b$ respectively, the cdf, $F_{R}(x|\mu ,\sigma ^{2})$ is given by:
    $F_{R}(x|\mu ,\sigma ^{2})={\begin{cases}0,&x<a,\\\Phi (x|\mu ,\sigma ^{2}),&a\leq x<b,\\1,&x\geq b,\\\end{cases}}$
where $\Phi (x|\mu ,\sigma ^{2})$ is the cdf of a normal distribution with mean $\mu$ and variance $\sigma ^{2}$. The mean and variance of the rectified distribution is calculated by first transforming the constraints to be acting on a standard normal distribution:
    $c={\frac {a-\mu }{\sigma }},\qquad d={\frac {b-\mu }{\sigma }}.$
Using the transformed constraints, the mean and variance, $\mu _{R}$ and $\sigma _{R}^{2}$ respectively, are then given by:
    $\mu _{t}={\frac {1}{\sqrt {2\pi }}}\left(e^{\left(-{\frac {c^{2}}{2}}\right)}-e^{\left(-{\frac {d^{2}}{2}}\right)}\right)+{\frac {c}{2}}\left(1+{\textrm {erf}}\left({\frac {c}{\sqrt {2}}}\right)\right)+{\frac {d}{2}}\left(1-{\textrm {erf}}\left({\frac {d}{\sqrt {2}}}\right)\right),$
    ${\begin{aligned}\sigma _{t}^{2}&={\frac {\mu _{t}^{2}+1}{2}}\left({\textrm {erf}}\left({\frac {d}{\sqrt {2}}}\right)-{\textrm {erf}}\left({\frac {c}{\sqrt {2}}}\right)\right)-{\frac {1}{\sqrt {2\pi }}}\left(\left(d-2\mu _{t}\right)e^{\left(-{\frac {d^{2}}{2}}\right)}-\left(c-2\mu _{t}\right)e^{\left(-{\frac {c^{2}}{2}}\right)}\right)\\&+{\frac {\left(c-\mu _{t}\right)^{2}}{2}}\left(1+{\textrm {erf}}\left({\frac {c}{\sqrt {2}}}\right)\right)+{\frac {\left(d-\mu _{t}\right)^{2}}{2}}\left(1-{\textrm {erf}}\left({\frac {d}{\sqrt {2}}}\right)\right),\end{aligned}}$
    $\mu _{R}=\mu +\sigma \mu _{t},$
    $\sigma _{R}^{2}=\sigma ^{2}\sigma _{t}^{2},$
where erf is the error function. This distribution was used by Palmer et al. for modelling physical resource levels, such as the quantity of liquid in a vessel, which is bounded by both 0 and the capacity of the vessel.