# The Bernoulli-Gamma distribution Density, distribution function, quantile function and random generation for the Bernoulli-Gamma distribution with parameters `prob`, `shape`, and `scale`. ```r dberngamma(x, prob, scale, shape) pberngamma(q, prob, scale, shape) qberngamma(p, prob, scale, shape) rberngamma(n, prob, scale, shape) ``` ## Arguments - `x,q`: vector of quantiles. - `p`: vector of probabilities. - `prob`: probability of non-zero event. - `n`: number of random samples. - `scale,shape`: shape and scale parameters of the gamma distribution. ## Details Mixture of the Bernoulli and the Gamma distribution. The Bernoulli distribution is used to model the occurrence of zero values with the probability of `1-prob`. Non-zero values follow the Gamma distribution with `shape` and `scale` parameters. The probability density function (PDF) is defined as: $$ g(x) = \left\{\begin{array}{l l}\pi*\gamma(x) & \quad \mbox{if } x > 0 \\1- \pi & \quad \mbox{if } x \leq 0\end{array} \right.g(x) = if(x > 0) pi*gamma(x) else 1-pi $$ where $gamma(x)$ is the probability density function of the gamma distribution and $\pi$ is probability of a non-zero event. The cumulative distribution function (CDF) is defined as: $$ G(x) = \left\{\begin{array}{l l}1 - \pi + \pi*\Gamma(x) & \quad \mbox{if } x > 0 \\1 - \pi & \quad \mbox{if } x \leq 0\end{array} \right.G(x) = if(x > 0) 1- pi + pi*Gamma(x) else 1-pi $$ where $Gamma(x)$ is the cumulative distribution function of the gamma distribution. The quantile function (inverse of the CDF) is defined as $$ G^{-1}(p) = \left\{\begin{array}{l l}\Gamma^{-1}\left(\frac{p-1+\pi}{\pi}\right) & \quad \mbox{if } \pi > 1-p \\0 & \quad \mbox{if } p \leq 1-p\end{array} \right.G^{-1}(x) = if(p > 1-p) Gamma^{-1}((p-1+pi)/pi) else 0 $$ where $Gamma^{-1}(p)$ is the inverse CDF of the gamma distribution and $p$ is a probability. ## Returns `dberngamma` gives the density (pdf), `pberngamma` gives the distribution function (cdf), `qberngamma` gives the quantile function (inverse cdf), and `rberngamma` generates random deviates. ## Note The implementation is largely based on the `bgamma` family in the `CaDENCE`-package (Cannon, 2012) that was only available as test version at time of implementation (Mar. 2012). For further details and meteorological application of Bernoulli-Gamma distributions (also referred to as 'Mixed Gamma' distribution) see Burger et al. 2012, Cannon 2008, Li et al. 2010, Mooley 1973, Piani et al. 2010, Thom 1968, Sloughter et al. 2007. ## References Burger, G.; Murdock, T. Q.; Werner, A. T.; Sobie, S. R. & Cannon, A. J. Downscaling extremes - an intercomparison of multiple statistical methods for present climate. Journal of Climate, American Meteorological Society, early online release, 2012, . Cannon, A. J. Probabilistic Multisite Precipitation Downscaling by an Expanded Bernoulli-Gamma Density Network. Journal of Hydrometeorology, American Meteorological Society, 2008, 9, 1284-1300, . Cannon, A. J. Neural networks for probabilistic environmental prediction: Conditional Density Estimation Network Creation and Evaluation (CaDENCE) in R. Computers & Geosciences, 2012, 41, 126 - 135, . Li, H.; Sheffield, J. & Wood, E. F. Bias correction of monthly precipitation and temperature fields from Intergovernmental Panel on Climate Change AR4 models using equidistant quantile matching. J. Geophys. Res., AGU, 2010, 115, D10101, . Mooley, D. A. Gamma Distribution Probability Model for Asian Summer Monsoon Monthly Rainfall. Monthly Weather Review, 1973, 101, 160-176, 2.3.CO;2>. Piani, C.; Haerter, J. & Coppola, E. Statistical bias correction for daily precipitation in regional climate models over Europe. Theoretical and Applied Climatology, 2010, 99, 187-192, . Thom, H. C. S. Approximate convolution of the gamma and mixed gamma distributions. Monthly Weather Review, 1968, 96, 883-886, 2.0.CO;2>. Sloughter, J. M. L.; Raftery, A. E.; Gneiting, T. & Fraley, C. Probabilistic Quantitative Precipitation Forecasting Using Bayesian Model Averaging. Monthly Weather Review, 2007, 135, 3209-3220, . ## Author(s) Lukas Gudmundsson ## See Also `GammaDist` ## Examples ```r data(obsprecip) (ts <- startberngamma(obsprecip[,1])) hist(obsprecip[,1],freq=FALSE) lines(seq(0,20),dberngamma(0:20, prob=ts$prob, scale=ts$scale, shape=ts$shape), col="red") pp <- seq(0.01,0.99,by=0.01) qq <-quantile(obsprecip[,1],probs=pp) plot(qq,pp) lines(qberngamma(pp, prob=ts$prob, scale=ts$scale, shape=ts$shape), pp,col="red") plot(qq,pp) lines(qq, pberngamma(qq, prob=ts$prob, scale=ts$scale, shape=ts$shape), col="red") hist(rberngamma(1000, prob=ts$prob, scale=ts$scale, shape=ts$shape),freq=FALSE) ```

berngamma function