Analyzing multivariate count data with the Poisson log-normal model

knitr::opts_chunk$set(
  screenshot.force = FALSE, 
  echo = TRUE,
  rows.print = 5,
  message = FALSE, 
  warning = FALSE)

Preliminaries

This vignette illustrates the use of the PLN function and the methods accompaning the R6 class PLNfit.

From the statistical point of view, the function PLN adjusts a multivariate Poisson lognormal model to a table of counts, possibly after correcting for effects of offsets and covariates. PLN is the building block for all the multivariate models found in the PLNmodels package: having a basic understanding of both the mathematical background and the associated set of R functions is a good place to start.

Requirements

The packages required for the analysis are PLNmodels plus some others for data manipulation and representation:

library(PLNmodels)
library(ggplot2)
library(corrplot)

Data set

We illustrate our point with the trichoptera data set, a full description of which can be found in the corresponding vignette. Data preparation is also detailed in the specific vignette.

data(trichoptera)
trichoptera <- prepare_data(trichoptera$Abundance, trichoptera$Covariate)

The trichoptera data frame stores a matrix of counts (trichoptera$Abundance), a matrix of offsets (trichoptera$Offset) and some vectors of covariates (trichoptera$Wind, trichoptera$Temperature, etc.)

Mathematical background

The multivariate Poisson lognormal model (in short PLN, see @AiH89) relates some $p$-dimensional observation vectors $\mathbf{Y}_i$ to some $p$-dimensional vectors of Gaussian latent variables $\mathbf{Z}_i$ as follows

\begin{equation} \begin{array}{rcl} \text{latent space } & \mathbf{Z}i \sim \mathcal{N}({\boldsymbol\mu},\boldsymbol\Sigma), \ \text{observation space } & Y{ij} | Z_{ij} \quad \text{indep.} & \mathbf{Y}_i | \mathbf{Z}_i\sim\mathcal{P}\left(\exp{\mathbf{Z}_i}\right). \end{array} \end{equation}

The parameter ${\boldsymbol\mu}$ corresponds to the main effects and the latent covariance matrix $\boldsymbol\Sigma$ describes the underlying residual structure of dependence between the $p$ variables. The following figure provides insights about the role played by the different layers

library(grid)
library(gridExtra)
library(dplyr)

set.seed(20171110)
x <- rnorm(100)
y <- rnorm(100)
b <- data.frame(x = x + y, y = y) / 1
mu <- 0
##
data.perfect <- as.data.frame((b + matrix(rep(mu, each = length(x)), ncol = 2)))
p.latent <- ggplot(data.perfect, aes(x, y)) + geom_point() + ggtitle(expression(Latent~Space~(Z)))
.rpois <- function(lambda) {
  unlist(lapply(exp(lambda), function(x) {rpois(1, x)}))
}
observation <- as.data.frame(lapply(data.perfect, .rpois))
mapped.parameter <- as.data.frame(lapply(data.perfect, exp))
## segment between mapped and observed data
segment.data <- cbind(mapped.parameter, observation)
names(segment.data) <- c("x", "y", "xend", "yend")
## Mapped parameters
p.mapped <- ggplot(mapped.parameter, aes(x, y)) + geom_point(col = "red") + ggtitle(expression(Observation~Space~(exp(Z))))
## Observations only
obs <- group_by(observation, x, y)
obs <- dplyr::summarize(obs, count = n())
p.observation.only <- ggplot(obs, aes(x, y)) +
  geom_point(aes(size = count)) +
  ggtitle(Observation~Space~(Y)~+'noise') +
  theme(legend.position = c(.95, .95), legend.justification = c(1, 1),
        legend.background = element_rect(color = "transparent"),
        legend.box.background = element_blank())
## Observations and latent parameters
p.observation.mixed <- p.observation.only +
  geom_point(data = mapped.parameter, color = "red", alpha = 0.5) +
  geom_segment(data = segment.data, aes(xend = xend, yend = yend), color = "black", alpha = 0.2) +
  ggtitle(Observation~Space~(Y==P(exp(Z)))~+'noise')

grid.arrange(p.latent + labs(x = "species 1", y = "species 2"),
             p.mapped  + labs(x = "species 1", y = "species 2"),
             p.observation.mixed + labs(x = "species 1", y = "species 2"),
             p.observation.only + labs(x = "species 1", y = "species 2"),
             ncol = 2)

Covariates and offsets

This model generalizes naturally to a formulation closer to a multivariate generalized linear model, where the main effect is due to a linear combination of $d$ covariates $\mathbf{x}_i$ (including a vector of intercepts). We also let the possibility to add some offsets for the $p$ variables in in each sample, that is $\mathbf{o}_i$. Hence, the previous model generalizes to

\begin{equation} \mathbf{Y}_i | \mathbf{Z}_i \sim \mathcal{P}\left(\exp{\mathbf{Z}_i}\right), \qquad \mathbf{Z}_i \sim \mathcal{N}({\mathbf{o}_i + \mathbf{x}_i^\top\boldsymbol\Theta},\boldsymbol\Sigma), \ \end{equation} where $\boldsymbol\Theta$ is a $d\times p$ matrix of regression parameters. When all individuals $i=1,\dots,n$ are stacked together, the data matrices available to feed the model are

Inference in PLN then focuses on the regression parameters $\boldsymbol\Theta$ and on the covariance matrix $\boldsymbol\Sigma$.

Optimization by Variational inference

Technically speaking, we adopt in PLNmodels a variational strategy to approximate the log-likelihood function and optimize the consecutive variational surrogate of the log-likelihood with a gradient-ascent-based approach. To this end, we rely on the CCSA algorithm of @Svan02 implemented in the C++ library [@nlopt], which we link to the package.

Analysis of trichoptera data with a PLN model

The standard PLN model described above is adjusted with the function PLN. We now review its usage on a the trichoptera data set.

A PLN model with latent main effects

Adjusting a fit

In order to become familiar with the function PLN and its outputs, let us first fit a simple PLN model with just an intercept for each species:

myPLN <- PLN(Abundance ~ 1, trichoptera)

Note the use of the formula object to specify the model: the vector $\boldsymbol\mu$ of main effects in the mathematical formulation (one per column species) is specified in the call with the term ~ 1 in the right-hand-side of the formula. Abundance is a variable in the data frame trichoptera correspondings to a matrix of 17 columns and the response in the model, occurring on the left-hand-side of the formula.

The PLNfit object

myPLN is an R6 object with class PLNfit, which comes with a couple of methods, as recalled when printing/showing such an object in the R console:

myPLN

See also ?PLNfit for more comprehensive information.

Field access

Accessing public fields of a PLNfit object can be done just like with a traditional list, e.g.,

c(myPLN$loglik, myPLN$BIC, myPLN$ICL, myPLN$R_squared)
myPLN$criteria

GLM-like interface

We provide a set of S3-methods for PLNfit that mimic the standard (G)LM-like interface of R::stats, which we present now.

One can access the fitted value of the counts (Abundance -- $\hat{\mathbf{Y}}$) and check that the algorithm basically learnt correctly from the data^[We use a log-log scale in our plot in order not to give an excessive importance to the higher counts in the fit]:

data.frame(
  fitted   = as.vector(fitted(myPLN)),
  observed = as.vector(trichoptera$Abundance)
) %>% 
  ggplot(aes(x = observed, y = fitted)) + 
    geom_point(size = .5, alpha =.25 ) + 
    scale_x_log10() + 
    scale_y_log10() + 
    theme_bw() + annotation_logticks()

We can also reach the matrix of regression parameters $\mathbf{\Theta}$ and the residual variance/covariance matrix $\boldsymbol{\Sigma}$ of the latent variable with the traditional functions found in R for (G)LM manipulation: for the regression coefficents, we can use the coef (or coefficients) method. Approximated standard errors of the coefficients are also accessible via standard_error:

data.frame(
  rbind(t(coef(myPLN)), t(standard_error(myPLN))), 
  row.names = c("effect", "stderr")
 ) %>% select(1:5) %>% knitr::kable()

The residual covariance matrix better displays as an image matrix:

corrplot(sigma(myPLN), is.corr = FALSE)

Observation weights

It is also possible to use observation weights like in standard (G)LMs:

myPLN_weighted <- 
  PLN(
    Abundance ~ 1, 
    data    = trichoptera, 
    weights = runif(nrow(trichoptera)),
    control = list(trace = 0)
  )
data.frame(
  unweighted = as.vector(fitted(myPLN)),
  weighted   = as.vector(fitted(myPLN_weighted))
) %>% 
  ggplot(aes(x = unweighted, y = weighted)) + 
    geom_point(size = .5, alpha =.25 ) + 
    scale_x_log10() + 
    scale_y_log10() + 
    theme_bw() + annotation_logticks()

Accounting for covariates and offsets

For ecological count data, it is generally a good advice to include the sampling effort via an offset term whenever available, otherwise samples are not necessarily comparable:

myPLN_offsets <- 
  PLN(Abundance ~ 1 + offset(log(Offset)), 
      data = trichoptera, control = list(trace = 0))

Note that we use the function offset with a log-transform of the total counts since it acts in the latent layer of the model. Obviously the model with offsets is better since the log-likelihood is higher with the same number of parameters^[In PLNmodels the R-squared is a pseudo-R-squared that can only be trusted between model where the same offsets term was used]:

rbind(
  myPLN$criteria,
  myPLN_offsets$criteria
) %>% knitr::kable()

Let us try to correct for the wind effect in our model:

myPLN_wind <- PLN(Abundance ~ 1 + Wind + offset(log(Offset)), data = trichoptera)

When we compare the models, the gain is clear in terms of log-likelihood. However, the BIC choses not to include this variable:

rbind(
  myPLN_offsets$criteria,
  myPLN_wind$criteria
) %>% knitr::kable()

Covariance models (full, diagonal, spherical)

It is possible to change a bit the parametrization used for modelling the residual covariance matrix $\boldsymbol\Sigma$, and thus reduce the total number of parameters used in the model. By default, the residual covariance is fully parameterized (hence $p \times (p+1)/2$ parameters). However, we can chose to only model the variances of the species and not the covariances, by means of a diagonal matrix $\boldsymbol\Sigma_D$ with only $p$ parameters. In an extreme situation, we may also chose a single variance parameter for the whole matrix $\boldsymbol\Sigma = \sigma \mathbf{I}_p$. This can be tuned in PLN with the control argument, a list controlling various aspects of the underlying optimization process:

myPLN_diagonal <- 
  PLN(
    Abundance ~ 1 + offset(log(Offset)),
    data = trichoptera, control = list(covariance = "diagonal", trace = 0)
  )
myPLN_spherical <- 
  PLN(
    Abundance ~ 1 + offset(log(Offset)),
    data = trichoptera, control = list(covariance = "spherical", trace = 0)
  )

Note that, by default, the model chosen is \texttt{covariance = "spherical"}, so that the two following calls are equivalents:

myPLN_default <- 
  PLN(Abundance ~ 1, data = trichoptera, )
myPLN_full <- 
  PLN(Abundance ~ 1, data = trichoptera, control = list(covariance = "full"))

Different covariance models can then be compared with the usual criteria: it seems that the gain brought by passing from a diagonal matrix to a fully parameterized covariance is not worth having so many additional parameters:

rbind(
  myPLN_offsets$criteria,
  myPLN_diagonal$criteria,
  myPLN_spherical$criteria
) %>% 
  as.data.frame(row.names = c("full", "diagonal", "spherical")) %>%
  knitr::kable()

A final model that we can try is the diagonal one with the wind as a covariate, which gives a slight improvement.

myPLN_final <- 
  PLN(
    Abundance ~ 1 + Wind + offset(log(Offset)),
    data    = trichoptera, control = list(covariance = "diagonal", trace = 0)
  )
rbind(
  myPLN_wind$criteria,
  myPLN_diagonal$criteria,
  myPLN_final$criteria
) %>% knitr::kable()

References



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PLNmodels documentation built on Jan. 28, 2020, 1:07 a.m.