# Sparse structure estimation for multivariate count data with PLN-network

## Preliminaries

This vignette illustrates the standard use of the PLNnetwork function and the methods accompanying the R6 Classes PLNnetworkfamily and PLNnetworkfit.

### 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 is a variant of the Poisson Lognormal model of , 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 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 .

#### Covariates and offsets

Just like PLN, PLN-network generalizes to a formulation close to a multivariate generalized linear model where the main effect is due to a linear combination of $$d$$ covariates $$\mathbf{x}_i$$ and to a vector $$\mathbf{o}_i$$ of $$p$$ offsets in sample $$i$$. The latent layer then reads $\begin{equation} \mathbf{Z}_i \sim \mathcal{N}\left({\mathbf{o}_i + \mathbf{x}_i^\top\boldsymbol\Theta},\boldsymbol\Omega^{-1}\right), \qquad \|\boldsymbol\Omega\|_1 < c , \end{equation}$ where $$\boldsymbol\Theta$$ is a $$d\times p$$ matrix of regression parameters.

#### Alternating optimization

Regularization via sparsification of $$\boldsymbol\Omega$$ and visualization of the consecutive network is the main objective in PLN-network. To reach this goal, we need to first estimate the model parameters. Inference in PLN-network focuses on the regression parameters $$\boldsymbol\Theta$$ and the inverse covariance $$\boldsymbol\Omega$$. Technically speaking, we adopt a variational strategy to approximate the $$\ell_1$$-penalized log-likelihood function and optimize the consecutive sparse variational surrogate with an optimization scheme that alternates between two step

1. a gradient-ascent-step, performed with the CCSA algorithm of implemented in the C++ library , which we link to the package.
2. a penalized log-likelihood step, performed with the graphical-Lasso of , implemented in the package fastglasso .

More technical details can be found in

## Analysis of trichoptera data with a PLNnetwork model

In the package, the sparse PLN-network model is adjusted with the function PLNnetwork, which we review in this section. This function adjusts the model for a series of value of the penalty parameter controlling the number of edges in the network. It then provides a collection of objects with class PLNnetworkfit, corresponding to networks with different levels of density, all stored in an object with class PLNnetworkfamily.

### 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 = 3.585368
sparsifying penalty = 3.3117
sparsifying penalty = 3.058921
sparsifying penalty = 2.825436
sparsifying penalty = 2.609773
sparsifying penalty = 2.410571
sparsifying penalty = 2.226574
sparsifying penalty = 2.056622
sparsifying penalty = 1.899641
sparsifying penalty = 1.754643
sparsifying penalty = 1.620713
sparsifying penalty = 1.497005
sparsifying penalty = 1.38274
sparsifying penalty = 1.277197
sparsifying penalty = 1.17971
sparsifying penalty = 1.089663
sparsifying penalty = 1.00649
sparsifying penalty = 0.9296657
sparsifying penalty = 0.8587051
sparsifying penalty = 0.7931609
sparsifying penalty = 0.7326196
sparsifying penalty = 0.6766994
sparsifying penalty = 0.6250475
sparsifying penalty = 0.5773382
sparsifying penalty = 0.5332704
sparsifying penalty = 0.4925664
sparsifying penalty = 0.4549692
sparsifying penalty = 0.4202418
sparsifying penalty = 0.3881651
sparsifying penalty = 0.3585368
##  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
## --------------------------------------------------------
## ========================================================
##  - 30 penalties considered: from 0.3585368 to 3.585368
##  - Best model (greater BIC): lambda = 0.859
##  - Best model (greater EBIC): lambda = 1.09

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

network_models$criteria %>% head() %>% knitr::kable() param nb_param loglik BIC ICL n_edges EBIC pen_loglik density stability 3.585368 17 -1279.044 -1312.125 -2677.119 0 -1312.125 -1286.470 0 NA 3.311700 17 -1272.110 -1305.190 -2670.171 0 -1305.190 -1279.236 0 NA 3.058921 17 -1265.712 -1298.793 -2663.838 0 -1298.793 -1272.545 0 NA 2.825436 17 -1259.751 -1292.831 -2657.830 0 -1292.831 -1266.295 0 NA 2.609773 17 -1254.199 -1287.279 -2652.319 0 -1287.279 -1260.461 0 NA 2.410571 17 -1249.036 -1282.117 -2647.132 0 -1282.117 -1255.023 0 NA A diagnostic of the optimization process is available via the convergence field: network_models$convergence %>% head() %>% knitr::kable()
param nb_param objective convergence outer_iterations inner_iterations inner_status inner_message
out 3.585368 17 1286.470364 0.000006 6.000000 5 4 xtol_rel or xtol_abs was reached
elt 3.311700 17 1279.235553 0.000008 9.000000 6 4 xtol_rel or xtol_abs was reached
elt.1 3.058921 17 1272.544636 0.000003 2.000000 6 4 xtol_rel or xtol_abs was reached
elt.2 2.825436 17 1266.295402 0.000009 2.000000 6 4 xtol_rel or xtol_abs was reached
elt.3 2.609773 17 1260.461130 0.000004 2.000000 6 4 xtol_rel or xtol_abs was reached
elt.4 2.410571 17 1255.022953 0.000001 2.000000 6 4 xtol_rel or xtol_abs was reached

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) # 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 , 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 , 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 = 1.18) ## ================================================================== ## nb_param loglik BIC ICL n_edges EBIC pen_loglik density ## 18 -1214.626 -1249.653 -2614.656 1 -1252.109 -1218.525 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(), standard_error()
## * Additional fields for sparse network
##     $EBIC,$density, $penalty ## * Additional S3 methods for network ## plot.PLNnetworkfit() The plot method provides a quick representation of the inferred network, with various options (either as a matrix, a graph, and always send back the plotted object invisibly if users needs to perform additional analyses). my_graph <- plot(model_StARS, plot = FALSE) my_graph ## IGRAPH 5cf5b07 UNW- 17 1 -- ## + attr: name (v/c), label (v/c), label.cex (v/n), size (v/n), ## | label.color (v/c), frame.color (v/l), weight (e/n), color (e/c), ## | width (e/n) ## + edge from 5cf5b07 (vertex names): ##  Hfo--Hsp plot(model_StARS) plot(model_StARS, type = "support", output = "corrplot") We can finally check that the fitted value of the counts – even with sparse regularization of the covariance matrix – are close to the observed ones: data.frame( fitted = as.vector(fitted(model_StARS)), observed = as.vector(trichoptera$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