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 (-{\frac {\mu }{\sigma }})\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, $\Phi (x)$ 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 ^[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

↑ Harva, M.; Kaban, A. (2007). "Variational learning for rectified factor analysis☆". Signal Processing. 87 (3): 509. doi:10.1016/j.sigpro.2006.06.006.
↑ 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.

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[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.

Probability distributions
List
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped