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

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

A comparison of Gaussian distribution, rectified Gaussian distribution, and truncated Gaussian distribution.

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.

## Extension to general bounds

An extension to the rectified Gaussian distribution was proposed by Palmer et al.,[3] allowing rectification between arbitrary lower and upper bounds. For lower and upper bounds ${\displaystyle a}$ and ${\displaystyle b}$ respectively, the cdf, ${\displaystyle F_{R}(x|\mu ,\sigma ^{2})}$ is given by:

${\displaystyle F_{R}(x|\mu ,\sigma ^{2})={\begin{cases}0,&x

where ${\displaystyle \Phi (x|\mu ,\sigma ^{2})}$ is the cdf of a normal distribution with mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2))$. The mean and variance of the rectified distribution is calculated by first transforming the constraints to be acting on a standard normal distribution:

${\displaystyle c={\frac {a-\mu }{\sigma )),\qquad d={\frac {b-\mu }{\sigma )).}$

Using the transformed constraints, the mean and variance, ${\displaystyle \mu _{R))$ and ${\displaystyle \sigma _{R}^{2))$ respectively, are then given by:

${\displaystyle \mu _{t}={\frac {1}{\sqrt {2\pi ))}\left(e^{\left(-{\frac {c^{2)){2))\right)}-e^{\left(-{\frac {d^{2)){2))\right)}\right)+{\frac {c}{2))\left(1+{\textrm {erf))\left({\frac {c}{\sqrt {2))}\right)\right)+{\frac {d}{2))\left(1-{\textrm {erf))\left({\frac {d}{\sqrt {2))}\right)\right),}$
{\displaystyle {\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))}
${\displaystyle \mu _{R}=\mu +\sigma \mu _{t},}$
${\displaystyle \sigma _{R}^{2}=\sigma ^{2}\sigma _{t}^{2},}$

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.