# Negative log Likelihood functions for Poisson, negative binomial, Delaporte, Poisson-inverse Gaussian and Poisson-beta distributions The negative log Likelihood functions for Poisson, negative binomial, Delaporte, Poisson-inverse Gaussian and Poisson-beta distributions. Mixing two distributions of the same kind and/or adding zero-inflation allows to take characteristics of real data into account. Additionally, one population and two population mixtures - with and without zero-inflation - allow distribution fitting of the Poisson, negative binomial, Delaporte, Poisson-inverse Gaussian and the Poisson-beta distribution. ```r nlogL_pois(data, par.pois) nlogL_nb(data, par.nb) nlogL_del(data, par.del) nlogL_pig(data, par.pig) nlogL_pb(data, par.pb) nlogL_pois2(data, par.pois2) nlogL_nb2(data, par.nb2) nlogL_del2(data, par.del2) nlogL_pig2(data, par.pig2) nlogL_pb2(data, par.pb2) nlogL_zipois(data, par.zipois) nlogL_zinb(data, par.zinb) nlogL_zidel(data, par.zidel) nlogL_zipig(data, par.zipig) nlogL_zipb(data, par.zipb) nlogL_zipois2(data, par.zipois2) nlogL_zinb2(data, par.zinb2) nlogL_zidel2(data, par.zidel2) nlogL_zipig2(data, par.zipig2) nlogL_zipb2(data, par.zipb2) ``` ## Arguments - `data`: Vector containing the discrete observations - `par.pois`: Scalar containing the lambda parameter of the Poisson distribution - `par.nb`: Vector of length 2, containing the size and the mu parameter of the negative binomial distribution - `par.del`: Vector of length 3, containing the mu, sigma and the nu parameter of the Delaporte distribution - `par.pig`: Vector of length 2, containing the mu and the sigma parameter of the Poisson-inverse Gaussian distribution - `par.pb`: Vector of length 3, containing the alpha, beta and c parameter of the Poisson-beta distribution - `par.pois2, par.nb2, par.del2, par.pig2, par.pb2`: Vector containing the parameters of the two mixing distributions. First entry represents the fraction of the first distribution, followed by all parameters of the first, then all of the second distribution. - `par.zipois, par.zinb, par.zidel, par.zipig, par.zipb`: Vector containing the respective zero-inflated distribution parameters. The additional first entry is the inflation parameter for all cases. - `par.zipois2, par.zinb2, par.zidel2, par.zipig2, par.zipb2`: Parameters for the zero-inflated two population model. ## Details Functions nlogL_pois, nlogL_nb, nlogL_del, nlogL_pig, nlogL_pb compute the negative log-likelihood of Poisson, negative binomial, Poisson-inverse Gaussian and the Poisson-beta distributions given the data. Functions nlogL_pois2, nlogL_nb2, nlogL_del2, nlogL_pig2 and nlogL_pb2 compute the negative log-likelihood values for a two population mixture of distributions whereas nlogL_zipois, nlogL_zinb, nlogL_zidel, nlogL_zipig, nlogL_zipb compute the same for the zero-inflated distributions. Furthermore, nlogL_zipois2, nlogL_zinb2, nlogL_zidel2, nlogL_zipig2 and nlogL_zipb2 are for two population mixtures with zero-inflation. ## Examples ```r x <- rpois(100, 11) nl1 <- nlogL_pois(x, 11) nl2 <- nlogL_pois(x, 13) x <- rnbinom(100, size = 13, mu = 9) nl <- nlogL_nb(x, c(13, 9)) x <- gamlss.dist::rDEL(100, mu = 5, sigma = 0.2, nu= 0.5) nl <- nlogL_del(x, c(5, 0.2, 0.5)) x <- gamlss.dist::rPIG(100, mu = 5, sigma = 0.2) nl <- nlogL_pig(x, c(5, 0.2)) x <- rpb(n = 1000, alpha=5, beta= 3, c=20) nl <- nlogL_pb(x, c(5, 3, 20)) s <- sample(x = c(0,1), size = 100, replace = TRUE, prob = c(0.3,0.7)) x <- s*rpois(100, 7) + (1-s)*rpois(100, 13) nl <- nlogL_pois2(x, c(0.3, 13, 7)) s <- sample(x = c(0,1), size = 100, replace = TRUE, prob = c(0.3,0.7)) x <- s*rnbinom(100, size = 13, mu = 9) + (1-s)*rnbinom(100, size = 17, mu = 29) nl <- nlogL_nb2(x, c(0.3, 17, 29, 13, 9)) s <- sample(x = c(0,1), size = 100, replace = TRUE, prob = c(0.3,0.7)) x <- s * gamlss.dist::rDEL(100, mu = 5, sigma = 0.2, nu = 0.5) + (1 - s) * gamlss.dist::rDEL(100, mu = 20, sigma = 2, nu = 0.1) nl <- nlogL_del2(x, c(0.7,5, 0.2, 20, 2)) s <- sample(x = c(0,1), size = 100, replace = TRUE, prob = c(0.3,0.7)) x <- s * gamlss.dist::rPIG(100, mu = 5, sigma = 0.2) + (1 - s) * gamlss.dist::rPIG(100, mu = 20, sigma = 2) nl <- nlogL_pig2(x, c(0.7, 5, 0.2, 20, 2)) s <- sample(x = c(0,1), size = 100, replace = TRUE, prob = c(0.3,0.7)) x <- s*rpb(100, 5, 3, 20) + (1-s)*rpb(100, 7, 13, 53) nl <- nlogL_pb2(x, c(0.7, 7, 13, 53, 5, 3, 20)) x <- c(rep(0, 10), rpois(90, 7)) nl <- nlogL_zipois(x, c(0.1, 7)) x <- c(rep(0,10), rnbinom(90, size = 13, mu = 9)) nl <- nlogL_zinb(x, c(0.1, 13, 9)) x <- c(rep(0,10), gamlss.dist::rDEL(90, mu = 13, sigma = 2, nu = 0.5)) nl <- nlogL_zidel(x, c(0.1, 13, 2, 0.5)) x <- c(rep(0,10), gamlss.dist::rPIG(90, mu = 13, sigma = 2)) nl <- nlogL_zipig(x, c(0.1, 13, 2)) x <- c(rep(0, 10), rpb(n = 90, alpha=5, beta= 3, c=20)) nl <- nlogL_zipb(x, c(0.1, 5, 3, 20)) s <- sample(x = c(0, 1), size = 90, replace = TRUE, prob = c(0.3, 0.7)) x <- c(rep(0, 10), s * rpois(90, 7) + (1 - s) * rpois(90, 13)) nl1 <- nlogL_zipois2(x, c(0.1, 0.63, 7, 13)) s <- sample(x = c(0, 1), size = 90, replace = TRUE, prob = c(0.3, 0.7)) x <- c(rep(0, 10), s * rnbinom(90, size = 13, mu = 9) + (1 - s) * rnbinom(90, size = 17, mu = 29)) nl <- nlogL_zinb2(x, c(0.1, 0.63, 13, 9, 17, 29)) s <- sample(x = c(0, 1), size = 90, replace = TRUE, prob = c(0.3, 0.7)) x <- c(rep(0, 10), s * gamlss.dist::rDEL(90, mu = 13, sigma = 9, nu = 0.5) + (1 - s) * gamlss.dist::rDEL(90, mu = 17, sigma = 29, nu = 0.1)) nl <- nlogL_zidel2(x, c(0.1, 0.63, 13, 9, 0.5, 17, 29, 0.1)) s <- sample(x = c(0,1), size = 90, replace = TRUE, prob = c(0.3, 0.7)) x <- c(rep(0, 10), s * gamlss.dist::rPIG(90, mu = 13, sigma = 0.2) + (1-s) * gamlss.dist::rPIG(90, mu = 17, sigma = 2)) nl <- nlogL_zipig2(x, c(0.1, 0.63, 13, 0.2, 17, 2)) s <- sample(x = c(0,1), size = 90, replace = TRUE, prob = c(0.3,0.7)) x <- c(rep(0,10), s*rpb(90, 5, 3, 20) + (1-s)*rpb(90, 7, 13, 53)) nl <- nlogL_zipb2(x, c(0.1, 0.63, 7, 13, 53, 5, 3, 20)) ```

nlogL function