Exponentially modified Gaussian distribution

EMG
Probability density function
Cumulative distribution function
Parameters	μ ∈ R — mean of Gaussian component σ² > 0 — variance of Gaussian component λ > 0 — rate of exponential component
Support	x ∈ R
PDF	$\frac{\lambda}{2} e^{\frac{\lambda}{2} (2 \mu + \lambda \sigma^2 - 2 x)} \operatorname{erfc} (\frac{\mu + \lambda \sigma^2 - x}{ \sqrt{2} \sigma})$
CDF	$\Phi(u, 0, v) - e^{-u + v^2/2+ \log(\Phi(u, v^2, v))\; }$ where $\Phi(x, \mu, \sigma)$ is the CDF of a Gaussian distribution $u = \lambda(x - \mu)$ $v = \lambda \sigma$
Mean	$\mu + 1/\lambda$
Mode	$x_m = \mu - \operatorname{sgn}\left(\tau\right)\sqrt{2}\sigma\operatorname{erfcxinv}\left(\frac{{{\|}}\tau{{\|}}}{\sigma}\sqrt{\frac{2}{\pi}}\right) + \frac{\sigma^2}{\tau}$ $f(x_m)=h\exp \left( -\frac {1}{2} \left( \frac {x_m-\mu}{\sigma} \right)^2\right)$
Variance	$\sigma^2 + 1/\lambda^2$
Skewness	$\frac{2}{\sigma^3 \lambda^3} \left( 1 + \frac{1}{\sigma^2 \lambda^2} \right)^{-3/2}$
Ex. kurtosis	$\frac{3 (1 + \frac{2}{\sigma^2 \lambda^2} + \frac{3}{\lambda^4 \sigma^4})}{\left( 1 + \frac{1}{\lambda^2 \sigma^2} \right)^2 } - 3$
MGF	$\left(1 - \frac{t}{\lambda}\right)^{-1}\,\exp \{ \mu t + \frac{1}{2}\sigma^2 t^2 \}$
CF	$\left(1 - \frac{it}{\lambda}\right)^{-1}\,\exp \{ i\mu t - \frac{1}{2}\sigma^2 t^2 \}$

In probability theory, an exponentially modified Gaussian (EMG) distribution (ExGaussian distribution) describes the sum of independent normal and exponential random variables. An exGaussian random variable Z may be expressed as Z = X + Y where X and Y are independent, X is Gaussian with mean μ and variance σ² and Y is exponential of rate λ. It has a characteristic positive skew from the exponential component.

It may also be regarded as a weighted function of a shifted exponential with the weight being a function of the normal distribution.

Definition

The probability density function (pdf) of the exponentially modified normal distribution is^[1]

f(x;\mu,\sigma,\lambda) = \frac{\lambda}{2} e^{\frac{\lambda}{2} (2 \mu + \lambda \sigma^2 - 2 x)} \operatorname{erfc} \left(\frac{\mu + \lambda \sigma^2 - x}{ \sqrt{2} \sigma}\right)

where erfc is the complementary error function defined as

\begin{align} \operatorname{erfc}(x) & = 1-\operatorname{erf}(x) \\ & = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2}\,dt. \end{align}

This density function is derived via convolution of the normal and exponential probability density functions.

The density function is a solution of the following differential equation:

\left\{\begin{array}{l} \sigma ^2 f''(x)+f'(x) \left(\lambda \sigma ^2-\mu +x\right)+\lambda f(x) (x-\mu)=0, \\[12pt] f(0)=\frac{1}{2} \lambda e^{\frac{1}{2} \lambda \left(\lambda \sigma ^2+2 \mu \right)} \text{erfc}\left(\frac{\lambda \sigma ^2+\mu }{\sqrt{2} \sigma}\right), \\[12pt] f'(0)=\frac{\lambda e^{-\frac{\mu ^2}{2 \sigma ^2}} \left(\sqrt{2}-\sqrt{\pi } \lambda \sigma e^{\frac{\left(\lambda \sigma ^2+\mu \right)^2}{2 \sigma ^2}} \text{erfc}\left(\frac{\lambda \sigma ^2+\mu }{\sqrt{2} \sigma }\right)\right)}{2 \sqrt{\pi } \sigma } \end{array} \right\}

Alternative forms for computation

An alternative but equivalent form of the EMG distribution is used for description of peak shape in chromatography.^[2] This is as follows

f(x; h, \mu, \sigma, \tau )=\frac{h\sigma}{\tau} \sqrt{\frac{\pi}{2}}\exp \left( \frac {1}{2} \left( \frac {\sigma}{\tau} \right)^2 - \frac {x-\mu}{\tau} \right) \operatorname{erfc} \left( \frac {1}{\sqrt{2}}\ \left( \frac {\sigma}{\tau} - \frac {x-\mu}{\sigma} \right) \right ) ,

(1)

where

h

= the amplitude of Gaussian,

\tau=\frac{1}{\lambda}

exponent relaxation time.

This function cannot be calculated for some values of parameters (for example, τ=0) because of arithmetic overflow. Alternative, but equivalent form of writing the function was proposed by Delley^[3]

$f(x; h, \mu, \sigma, \tau )=h\exp \left( -\frac {1}{2} \left( \frac {x-\mu}{\sigma} \right)^2\right) \frac{\sigma}{\tau} \sqrt{\frac{\pi}{2}}\operatorname{erfcx} \left( \frac {1}{\sqrt{2}}\ \left( \frac {\sigma}{\tau} - \frac {x-\mu}{\sigma} \right) \right ) ,$ (2)

where $\operatorname{erfcx} \left( t \right)=\exp{ \left( t^2 \right)}\operatorname{erfc} \left( t \right)$ is a scaled complementary error function

In the case of this formula arithmetic overflow is also possible, region of overflow is different from the first formula, except for very small τ.

For small τ it is reasonable to use asymptotic form of the second formula,

$f(x; h, \mu, \sigma, \tau )=\frac{h\exp \left( -\frac {1}{2} \left( \frac {x-\mu}{\sigma} \right)^2\right)} {\left(1- \frac {\left(x-\mu\right)\tau}{\sigma^2} \right)} ,$ (3)

Decision on formula usage is made on the basis of the parameter

$z = \frac{1}{\sqrt{2}}\left( \frac {\sigma}{\tau}-\frac {x-\mu}{\sigma} \right)$

for z<0 computation should be made^[2] according to the first formula,

for 0<=z<=6.71·10⁷ (in the case of Double-precision floating-point format) according to the second formula,

and for z >6.71·10⁷ according to the third formula.

Mode (position of apex, most probable value) is calculated^[2] using derivative of formula 2; inverse of scaled complementary error function erfcxinv() is used for calculation. The apex is always located on the original (unmodified) Gaussian.

Parameter estimation

There are three parameters: the mean of the normal distribution (μ), the standard deviation of the normal distribution (σ) and the exponential decay parameter ( τ = 1 / λ ). A fourth parameter — the shape K = τ / σ — is sometimes used also to characterise the distribution. Depending on the values of the parameters the distribution may vary in shape from almost normal to almost exponential.

The parameters of the distribution can be estimated from the sample data with the method of moments as follows:,^[4]^[5]

m = \mu + \tau

s^2 = \sigma^2 + \tau^2

\gamma_1 = \frac{ 2 \tau^{ 3 } } { ( \sigma^2 + \tau^2 )^{3/2} }

where m is the sample mean, s is the sample standard deviation and γ₁ is the skewness.

Solving these for the parameters gives

\hat{ \mu } = m - s \left( \frac{\gamma_1} {2} \right)^{ 1 / 3 }

\hat{ \sigma^2 } = s^2 \left[ 1 - \left( \frac{ \gamma_1 } { 2 } \right)^{ 2 / 3 } \right]

\hat{ \tau } = s \left( \frac{ \gamma_1 } { 2 } \right)^{ 1 / 3 }.

Recommendations

Ratcliff has suggested that there be at least 100 data points in the sample before the parameter estimates should be regarded as reliable.^[6] Vincent averaging may be used with smaller samples as this procedure only modestly distorts the shape of the distribution.^[7] These point estimates may be used as initial values that can be refined with more powerful methods including maximum likelihood.

Confidence intervals

There are currently no published tables available for significance testing with this distribution. The distribution can be simulated by forming the sum of two random variables one drawn from a normal distribution and the other from an exponential.

Skew

The value of the nonparametric skew

\frac{\text{mean} - \text{median}}{\text{standard deviation}}

of this distribution lies between 0 and 0.31^[8]^[9] The lower limit is approached when the normal component dominates and the upper when the exponential component dominates.

Usage

The distribution is used as a theoretical model for the shape of chromatographic peaks.^[1]^[2]^[10] It has been proposed as a statistical model of intermitotic time in dividing cells.^[11]^[12] It is also used in modelling cluster ion beams.^[13] It is commonly used in psychology and other brain sciences in the study of response times.^[14]^[15]

Related distributions

This family of distributions is a special or limiting case of the normal-exponential-gamma distribution. The distribution is a compound probability distribution in which the mean of a normal distribution varies randomly as a shifted exponential distribution.

Laplace-normal distribution

References

1 2 Grushka, Eli (1972). "Characterization of Exponentially Modified Gaussian Peaks in Chromatography". Analytical Chemistry 44 (11): 1733–1738. doi:10.1021/ac60319a011.
1 2 3 4 Kalambet, Y.; Kozmin, Y.; Mikhailova, K.; Nagaev, I.; Tikhonov, P. (2011). "Reconstruction of chromatographic peaks using the exponentially modified Gaussian function". Journal of Chemometrics 25 (7): 352. doi:10.1002/cem.1343.
↑ Delley, R (1985). "Series for the Exponentially Modified Gaussian Peak Shape". Anal. Chem. 57: 388. doi:10.1021/ac00279a094.
↑ Dyson, N.A. (1998). Chromatographic Integration Methods. Royal Society of Chemistry, Information Services. p. 27. ISBN 9780854045105. Retrieved 2015-05-15.
↑ Olivier J and Norberg MM (2010) Positively skewed data: Revisiting the Box−Cox power transformation. Int J Psych Res 3 (1) 68−75
↑ Ratcliff R (1979). Group reaction time distributions and an analysis of distribution statistics. Psych Bull 86, 446−461
↑ Vincent SB (1912) The functions of the vibrissae in the behaviour of the white rat. Behaviour Monographs 1, 7−81
↑ Heathcote A (1996). RTSYS: A DOS application for the analysis of reaction time data. Behavioural Research Methods, Instruments, & Computers 28, 427−445
↑ Ulrich R, & Miller J (1994) Effects of outlier exclusion on reaction time analysis. J Exp Psych: General 123, 34−80
↑ Gladney, HM; Dowden, BF; Swalen, JD (1969). "Computer-Assisted Gas-Liquid Chromatography". Anal. Chem. 41 (7): 883–888. doi:10.1021/ac60276a013.
↑ Golubev, A. (2010). "Exponentially modified Gaussian (EMG) relevance to distributions related to cell proliferation and differentiation". Journal of Theoretical Biology 262 (2): 257–266. doi:10.1016/j.jtbi.2009.10.005. PMID 19825376.
↑ Tyson, D. R.; Garbett, S. P.; Frick, P. L.; Quaranta, V. (2012). "Fractional proliferation: A method to deconvolve cell population dynamics from single-cell data". Nature Methods 9 (9): 923–928. doi:10.1038/nmeth.2138. PMC 3459330. PMID 22886092.
↑ Nicolaescu, D.; Takaoka, G. H.; Ishikawa, J. (2006). "Multiparameter characterization of cluster ion beams". Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures 24 (5): 2236. doi:10.1116/1.2335433.
↑ Palmer EM and Horowitz Todd S, Torralba A, Wolfe JM (2011) What are the shapes of response time distributions in visual search? J Exp Psychol 37(1) 58–71
↑ Rohere D, Wixted JT (1994) An analysis of latency and interresponse time in free recall. Memory & Cognition 22 (5) 511–524

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 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 Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet 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

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