# Zero Truncated Poisson Lognormal Distribution

A compound Poisson-lognormal distribution (PLN) is a Poisson probability distribution where its parameter \(\lambda\) is a random variable with lognormal distribution, that is to say \(log \lambda\) are normally distributed with mean \(\mu\) and variance \(\sigma^2\) (Bulmer 1974). The density function is

\[ \mathcal{PLN} (k ; \mu, \sigma) = \int_0^\infty {Pois}(k; \lambda) \times \mathcal{N}(log\lambda; \mu, \sigma) d\lambda　\\ = \frac{1}{\sqrt{2\pi\sigma^2}k!}\int^\infty_0\lambda^{k}exp(-\lambda)exp(\frac{-(log\lambda-\mu)^2}{2\sigma^2})d\lambda, \; \text{where} \; k = 0, 1, 2, ... \; \;\;(1). \]

## ZTPLN - type 1

The zero-truncated Poisson-lognormal distribution (ZTPLN) at least have two different forms. First, it can be derived from a Poisson-lognormal distribution,

\[ \mathcal{PLN}_{zt}(k ; \mu, \sigma) = \frac{\mathcal{PLN}(k ; \mu, \sigma)}{1-\mathcal{PLN}(0 ; \mu, \sigma)}, \; \text{where} \; k = 1, 2, 3, ... \;\;(2). \]

Random samples from ZTPLN (eq 2.) requires the inverse cumulative distribution function of ZTPLN (eq 2.),

\[ G_{zt}(x; \lambda) =\sum_{i=1}^k \mathcal{PLN}_{zt}(i ; \mu, \sigma) \;\;(3). \]

Random sampling using \(G_{zt}^{-1}\) is not implemented at the moment. Instead, zero values will be discarded from random variates that are generated by the inverse cumulative distribution of PLN.

## ZTPLN - type 2

An alternative form of ZTPLN can be directly derived from a mixture of zero-truncated Poisson distribution and lognormal distribution,

\[ \mathcal{PLN}_{zt2} (k ; \mu, \sigma) = \int_0^\infty \frac{{Pois}(k; \lambda)}{1 - Pois(0; \lambda)} \times \mathcal{N}(log\lambda; \mu, \sigma) d\lambda　\\ = \frac{1}{\sqrt{2\pi\sigma^2}k!}\int^\infty_0 \frac{\lambda^{k}}{exp(\lambda) - 1}exp(\frac{-(log\lambda-\mu)^2}{2\sigma^2})d\lambda, \; \text{where} \; k = 1, 2, 3,... \; \;\;(4). \]

Random samples from ZTPLN (eq. 3) can be generated by inverse transform sampling with the cumulative distribution function of a zero-truncated Poisson distribution (\(F_{zt}\)),

\[ F_{zt2}^{-1}(x; \lambda) = F^{-1}((1 - Pois(0; \lambda)) x + Pois(0; \lambda); \lambda) \\ = F^{-1}((1 - e^{-\lambda}) x + e^{-\lambda}; \lambda) \;\;\;\;(5) \]

where \(F^{-1}_{zt2}\) and \(F^{-1}\) are inverse cumulative distribution function for a zero-truncated Poisson distribution and a Poisson distribution, respectively.

## Mixture models

A Poisson lognormal (\(\mathcal{PLN}\)) mixture model with K components for variables \(\boldsymbol y = (y_1, \ldots, y_N)\) is parameterized by the mixture component weight vector \(\boldsymbol \theta = (\theta_1, \ldots, \theta_K)\) such that \(0 \le \theta_{k} \le 1\) and \(\sum\theta_{k} = 1\), and the mean \(\boldsymbol \mu = (\mu_1, \ldots, \mu_K)\) and standard deviation \(\boldsymbol \sigma = (\sigma_1, \ldots, \sigma_K)\) of the variable’s natural logarithm. The probability mass function (\(p\)) for a Poisson lognormal mixture is

\[ p(\boldsymbol y; {\boldsymbol \mu}, {\boldsymbol \sigma}, {\boldsymbol \theta}) = \sum_{k=1}^{K}\theta_k \mathcal{PLN}(\boldsymbol{y} ; \mu_k, \sigma_k) \; \; (6). \]

Given that each mixture component is zero-truncated, the probability mass function for the zero-truncated Poisson lognormal mixture (\(p_{zt}\)) is

\[ p_{zt}(\boldsymbol y; {\boldsymbol \mu}, {\boldsymbol \sigma}, {\boldsymbol \theta}) = \sum_{k=1}^{K}\theta_k \frac{\mathcal{PLN}(\boldsymbol{y} ; \mu_k, \sigma_k)} {1 - \mathcal{PLN}(0 ; \mu_k, \sigma_k)} \; \; (7) \]

for ZTPLN type 1, and

\[ p_{zt2}(\boldsymbol y; {\boldsymbol \mu}, {\boldsymbol \sigma}, {\boldsymbol \theta}) = \sum_{k=1}^{K}\theta_k \mathcal{PLN}_{zt2}(\boldsymbol{y} ; \mu_k, \sigma_k) \; \; (8) \]

for ZTPLN type 2.

## Functions

- `dztpln(x, mu, sig, log = FALSE, type1 = TRUE)`: gives the (log) density of a zero truncated poisson lognormal distribution
- `rztpln(n, mu, sig, type1 = TRUE)`: random draw from a zero truncated poisson lognormal distribution
- `dztplnm(x, mu, sig, theta, log = FALSE, type1 = TRUE)`: gives the (log) density of a zero truncated poisson lognormal distribution mixture.
- `ztplnm(n, mu, sig, theta, type1 = TRUE)`: random draw from a zero truncated poisson lognormal distribution mixture.

## Examples

### Random variates

We can generate ZTPLN random variates.

    library(dplyr)
    #>
    #> Attaching package: 'dplyr'
    #> The following objects are masked from 'package:stats':
    #>
    #>     filter, lag
    #> The following objects are masked from 'package:base':
    #>
    #>     intersect, setdiff, setequal, union
    library(tidyr)
    library(ggplot2)
    set.seed(123)

    rztpln(n = 10, mu = 0, sig = 1)
    #>  [1] 1 1 1 1 2 6 1 7 2 1

    rztpln(n = 10, mu = 6, sig = 4)
    #>  [1]    20   155     5    24   208   714  6221  2207    36 13624

We can also generate mixture of ZTPLN random variates.

    rztplnm(n = 100,
      mu = c(0, 4),
      sig = c(0.5, 0.5),
      theta = c(0.2, 0.8)) %>%
      log %>% hist(main = "", xlab = "k")

### Maximum likelihood estimation

Type 1 model (`dztpln(..., type1 = TRUE)`) fits better for type 1 random variates (`rztpln(..., type1 = TRUE)`).

    y <- rztpln(n = 100, mu = 2, sig = 5, type1 = TRUE)
    sum(dztpln(y, mu = 2, sig = 5, log = TRUE, type1 = TRUE))
    #> [1] -764.0654
    sum(dztpln(y, mu = 2, sig = 5, log = TRUE, type1 = FALSE))
    #> [1] -790.5514

Type 2 model (`dztpln(..., type1 = FALSE)`) fits better for type 2 random variates (`rztpln(..., type1 = FALSE)`).

    y2 <- rztpln(n = 100, mu = 2, sig = 5, type1 = FALSE)
    sum(dztpln(y2, mu = 2, sig = 5, log = TRUE, type1 = TRUE))
    #> [1] -571.571
    sum(dztpln(y2, mu = 2, sig = 5, log = TRUE, type1 = FALSE))
    #> [1] -545.3733

### Probability density plots

Let’s consider a simple example where random variates follows the same parameters of type 1 and type 2 ZTPLN. We define two different sets of random variates, \(\mathcal{PLN}_{zt}(k ;1, 2)\) and \(\mathcal{PLN}_{zt2}(k ; 1, 2)\).

In this example, when \(k\) is small, type 2 ZTPLN shows greater likelihood.

    k1 <- 1
    k <- 1000
    dat <- tibble(type1 = dztpln(k1:k, mu = 1, sig = 2, type1 = TRUE),
             type2 = dztpln(k1:k, mu = 1, sig = 2, type1 = FALSE),
             x = k1:k) %>%
      pivot_longer(-x, names_to = "type", values_to = "y")

    ggplot(dat %>% dplyr::filter(x <= 10), aes(x = x, y = y, col = type)) +
      geom_point() +
      geom_line() +
      scale_y_log10() +
      xlab("k") +
      ylab("Likelihood") +
      theme_bw()

When \(k\) is large, type 1 ZTPLN shows greater likelihood.

    ggplot(dat, aes(x = x, y = y, col = type)) +
      geom_point() +
      geom_line() +
      scale_y_log10() +
      xlab("k") +
      ylab("Likelihood") +
      theme_bw()

## Reference

Bulmer, M. G. 1974. On Fitting the Poisson Lognormal Distribution to Species-Abundance Data. Biometrics 30:101-110.

Inouye, D., E. Yang, G. Allen, and P. Ravikumar. 2017. A Review of Multivariate Distributions for Count Data Derived from the Poisson Distribution. Wiley interdisciplinary reviews. Computational statistics 9:e1398.