Sparse structure estimation for multivariate count data with PLN-network

Requirements

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

library(PLNmodels)
library(ggplot2)

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 network model for multivariate count data that we introduce in Chiquet, Robin, and Mariadassou (2019) is a variant of the Poisson Lognormal model of Aitchison and Ho (1989), see the PLN vignette as a reminder. Compare to the standard PLN model we add a sparsity constraint on the inverse covariance matrix \({\boldsymbol\Sigma}^{-1}\triangleq \boldsymbol\Omega\) by means of the \(\ell_1\)-norm, such that \(\|\boldsymbol\Omega\|_1 < c\). PLN-network is the equivalent of the sparse multivariate Gaussian model (Banerjee, Ghaoui, and d’Aspremont 2008) in the PLN framework. It relates some \(p\)-dimensional observation vectors \(\mathbf{Y}_i\) to some \(p\)-dimensional vectors of Gaussian latent variables \(\mathbf{Z}_i\) as follows \[\begin{equation} \begin{array}{rcl} \text{latent space } & \mathbf{Z}_i \sim \mathcal{N}\left({\boldsymbol\mu},\boldsymbol\Omega^{-1}\right) & \|\boldsymbol\Omega\|_1 < c \\ \text{observation space } & Y_{ij} | Z_{ij} \quad \text{indep.} & Y_{ij} | Z_{ij} \sim \mathcal{P}\left(\exp\{Z_{ij}\}\right) \end{array} \end{equation}\]

The parameter \({\boldsymbol\mu}\) corresponds to the main effects and the latent covariance matrix \(\boldsymbol\Sigma\) describes the underlying structure of dependence between the \(p\) variables.

The \(\ell_1\)-penalty on \(\boldsymbol\Omega\) induces sparsity and selection of important direct relationships between entities. Hence, the support of \(\boldsymbol\Omega\) correspond to a network of underlying interactions. The sparsity level (\(c\) in the above mathematical model), which corresponds to the number of edges in the network, is controlled by a penalty parameter in the optimization process sometimes referred to as \(\lambda\). All mathematical details can be found in Chiquet, Robin, and Mariadassou (2019).

Adjusting a collection of network - a.k.a. a regularization path

PLNnetwork finds an hopefully appropriate set of penalties on its own. This set can be controlled by the user, but use it with care and check details in ?PLNnetwork. The collection of models is fitted as follows:

network_models <- PLNnetwork(Abundance ~ 1 + offset(log(Offset)), data = trichoptera)

## 
##  Initialization...
##  Adjusting 30 PLN with sparse inverse covariance estimation
##  Joint optimization alternating gradient descent and graphical-lasso
##  sparsifying penalty = 3.603979 
    sparsifying penalty = 3.328891 
    sparsifying penalty = 3.074799 
    sparsifying penalty = 2.840102 
    sparsifying penalty = 2.62332 
    sparsifying penalty = 2.423084 
    sparsifying penalty = 2.238132 
    sparsifying penalty = 2.067297 
    sparsifying penalty = 1.909502 
    sparsifying penalty = 1.763752 
    sparsifying penalty = 1.629126 
    sparsifying penalty = 1.504776 
    sparsifying penalty = 1.389918 
    sparsifying penalty = 1.283827 
    sparsifying penalty = 1.185833 
    sparsifying penalty = 1.09532 
    sparsifying penalty = 1.011715 
    sparsifying penalty = 0.9344916 
    sparsifying penalty = 0.8631626 
    sparsifying penalty = 0.7972782 
    sparsifying penalty = 0.7364226 
    sparsifying penalty = 0.6802121 
    sparsifying penalty = 0.6282921 
    sparsifying penalty = 0.5803351 
    sparsifying penalty = 0.5360386 
    sparsifying penalty = 0.4951232 
    sparsifying penalty = 0.4573309 
    sparsifying penalty = 0.4224232 
    sparsifying penalty = 0.39018 
    sparsifying penalty = 0.3603979 
##  Post-treatments
##  DONE!

Note the use of the formula object to specify the model, similar to the one used in the function PLN.

Structure of PLNnetworkfamily

The network_models variable is an R6 object with class PLNnetworkfamily, which comes with a couple of methods. The most basic is the show/print method, which sends a very basic summary of the estimation process:

network_models

## --------------------------------------------------------
## COLLECTION OF 30 POISSON LOGNORMAL MODELS
## --------------------------------------------------------
##  Task: Network Inference 
## ========================================================
##  - 30 penalties considered: from 0.3603979 to 3.603979 
##  - Best model (greater BIC): lambda = 0.495 
##  - Best model (greater EBIC): lambda = 0.736

One can also easily access the successive values of the criteria in the collection

network_models$criteria %>% head() %>% knitr::kable()

A diagnostic of the optimization process is available via the convergence field:

network_models$convergence %>% head() %>% knitr::kable()

An nicer view of this output comes with the option “diagnostic” in the plot method:

plot(network_models, "diagnostic")

Exploring the path of networks

By default, the plot method of PLNnetworkfamily displays evolution of the criteria mentioned above, and is a good starting point for model selection:

plot(network_models)

Note that we use the original definition of the BIC/ICL criterion (\(\texttt{loglik} - \frac{1}{2}\texttt{pen}\)), which is on the same scale as the log-likelihood. A popular alternative consists in using \(-2\texttt{loglik} + \texttt{pen}\) instead. You can do so by specifying reverse = TRUE:

plot(network_models, reverse = TRUE)

In this case, the variational lower bound of the log-likelihood is hopefully strictly increasing (or rather decreasing if using reverse = TRUE) with a lower level of penalty (meaning more edges in the network). The same holds true for the penalized counterpart of the variational surrogate. Generally, smoothness of these criteria is a good sanity check of optimization process. BIC and its extended-version high-dimensional version EBIC are classically used for selecting the correct amount of penalization with sparse estimator like the one used by PLN-network. However, we will consider later a more robust albeit more computationally intensive strategy to chose the appropriate number of edges in the network.

To pursue the analysis, we can represent the coefficient path (i.e., value of the edges in the network according to the penalty level) to see if some edges clearly come off. An alternative and more intuitive view consists in plotting the values of the partial correlations along the path, which can be obtained with the options corr = TRUE. To this end, we provide the S3 function coefficient_path:

coefficient_path(network_models, corr = TRUE) %>% 
  ggplot(aes(x = Penalty, y = Coeff, group = Edge, colour = Edge)) + 
    geom_line(show.legend = FALSE) +  coord_trans(x="log10") + theme_bw()

Model selection issue: choosing a network

To select a network with a specific level of penalty, one uses the getModel(lambda) S3 method. We can also extract the best model according to the BIC or EBIC with the method getBestModel().

model_pen <- getModel(network_models, network_models$penalties[20]) # give some sparsity
model_BIC <- getBestModel(network_models, "BIC")   # if no criteria is specified, the best BIC is used

An alternative strategy is to use StARS (Liu, Roeder, and Wasserman 2010), which performs resampling to evaluate the robustness of the network along the path of solutions in a similar fashion as the stability selection approach of Meinshausen and Bühlmann (2010), but in a network inference context.

Resampling can be computationally demanding but is easily parallelized: the function stability_selection integrates some features of the future package to perform parallel computing. We set our plan to speed the process by relying on 2 workers:

library(future)
plan(multisession, workers = 2)

We first invoke stability_selection explicitly for pedagogical purpose. In this case, we need to build our sub-samples manually:

n <- nrow(trichoptera)
subs <- replicate(10, sample.int(n, size = n/2), simplify = FALSE)
stability_selection(network_models, subsamples = subs)

## 
## Stability Selection for PLNnetwork: 
## subsampling: ++++++++++

Requesting ‘StARS’ in gestBestmodel automatically invokes stability_selection with 20 sub-samples, if it has not yet been run.

model_StARS <- getBestModel(network_models, "StARS")

When “StARS” is requested for the first time, getBestModel automatically calls the method stability_selection with the default parameters. After the first call, the stability path is available from the plot function:

plot(network_models, "stability")

When you are done, do not forget to get back to the standard sequential plan with future.

future::plan("sequential")

Structure of a PLNnetworkfit

The variables model_BIC, model_StARS and model_pen are other R6Class objects with class PLNnetworkfit. They all inherits from the class PLNfit and thus own all its methods, with a couple of specific one, mostly for network visualization purposes. Most fields and methods are recalled when such an object is printed:

model_StARS

## Poisson Lognormal with sparse inverse covariance (penalty = 1.39)
## ==================================================================
##  nb_param    loglik      BIC       ICL n_edges      EBIC pen_loglik density
##        35 -1200.764 -1268.87 -2584.115       1 -1271.327  -1205.678   0.007
## ==================================================================
## * Useful fields
##     $model_par, $latent, $latent_pos, $var_par, $optim_par
##     $loglik, $BIC, $ICL, $loglik_vec, $nb_param, $criteria
## * Useful S3 methods
##     print(), coef(), sigma(), vcov(), fitted()
##     predict(), predict_cond(), standard_error()
## * Additional fields for sparse network
##     $EBIC, $density, $penalty 
## * Additional S3 methods for network
##     plot.PLNnetworkfit()

The plot method provides a quick representation of the inferred network, with various options (either as a matrix, a graph, and always send back the plotted object invisibly if users needs to perform additional analyses).

my_graph <- plot(model_StARS, plot = FALSE)
my_graph

## IGRAPH 3df1d29 UNW- 17 1 -- 
## + attr: name (v/c), label (v/c), label.cex (v/n), size (v/n),
## | label.color (v/c), frame.color (v/l), weight (e/n), color (e/c),
## | width (e/n)
## + edge from 3df1d29 (vertex names):
## [1] Hfo--Hsp

plot(model_StARS)

plot(model_StARS, type = "support", output = "corrplot")

We can finally check that the fitted value of the counts – even with sparse regularization of the covariance matrix – are close to the observed ones:

data.frame(
  fitted   = as.vector(fitted(model_StARS)),
  observed = as.vector(trichoptera$Abundance)
) %>% 
  ggplot(aes(x = observed, y = fitted)) + 
    geom_point(size = .5, alpha =.25 ) + 
    scale_x_log10(limits = c(1,1000)) + 
    scale_y_log10(limits = c(1,1000)) + 
    theme_bw() + annotation_logticks()

fitted value vs. observation