PLNmodels: Poisson lognormal models

The Poisson lognormal model and variants can be used for analysis of mutivariate count data. This package implements efficient algorithms extracting meaningful data from difficult to interpret and complex multivariate count data. It has been built to scale on large datasets even though it has memory limitations. Possible fields of applications include

• Genomics (number of times a gene is expressed in a cell)
• Ecology (species abundances)

One main functionality is to normalize the count data to obtain more valuable data. It also analyse the significance of each variable and their correlation as well as the weight of covariates (if available).

Getting started

A notebook to get started can be found here. If you need just a quick view of the package, see the quickstart next.

🛠 Installation

pyPLNmodels is available on pypi. The development version is available on GitHub and GitLab.

Package installation

pip install pyPLNmodels

Statistical description

For those unfamiliar with the concepts of Poisson or Gaussian random variables, it is not necessary to delve into these statistical descriptions. The key takeaway is as follows: This package is designed to analyze multi-dimensional count data. It effectively extracts significant information, such as the mean, the relationships with covariates, and the correlation between count variables, in a manner appropriate for count data.

Consider $\mathbf Y$ a count matrix (denoted as endog in the package) consisting of $n$ rows and $p$ columns. It is assumed that each individual $\mathbf Y_i$, that is the $i^{\text{th}}$ row of $\mathbf Y$, is independent from the others and follows a Poisson lognormal distribution:

$$\mathbf Y_{i}\sim \mathcal P(\exp(\mathbf Z_{i})), \quad \mathbf Z_i \sim \mathcal N(\mathbf o_i + \mathbf B ^{\top} \mathbf x_i, \mathbf \Sigma),$$

where $\mathbf x_i \in \mathbb R^d$ (exog) and $\mathbf o_i \in \mathbb R^p$ (offsets) are user-specified covariates and offsets. The matrix $\mathbf B$ is a $d\times p$ matrix of regression coefficients and $\mathbf \Sigma$ is a $p\times p$ covariance matrix. The goal is to estimate the parameters $\mathbf B$ and $\mathbf \Sigma$, denoted as coef and covariance in the package, respectively. A normalization procedure adequate to count data can be applied by extracting the latent_variables $\mathbf Z_i$ once the parameters are learned.

⚡️ Quickstart

The package comes with an ecological data set to present the functionality:

import pyPLNmodels
from pyPLNmodels.models import PlnPCAcollection, Pln, ZIPln
from pyPLNmodels.oaks import load_oaks
oaks = load_oaks()

How to specify a model

Each model can be specified in two distinct manners:

• by formula (similar to R), where a data frame is passed and the formula is specified using the from_formula initialization:

model = Model.from_formula("endog ~ 1 + covariate_name ", data = oaks)# not run

We rely to the patsy package for the formula parsing.

• by specifying the endog, exog, and offsets matrices directly:

model = Model(endog = oaks["endog"], exog = oaks[["covariate_name"]], offsets = oaks[["offset_name"]])# not run

The parameters exog and offsets are optional. By default, exog is set to represent an intercept, which is a vector of ones. Similarly, offsets defaults to a matrix of zeros.

Unpenalized Poisson lognormal model (aka Pln)

This is the building-block of the models implemented in this package. It fits a Poisson lognormal model to the data:

pln = Pln.from_formula("endog ~ 1  + tree ", data = oaks)
pln.fit()
print(pln)
transformed_data = pln.transform()
pln.show()

Rank Constrained Poisson lognormal for Poisson Principal Component Analysis (aka PlnPCA and PlnPCAcollection)

This model excels in dimension reduction and is capable of scaling to high-dimensional count data ($p >> 1$). It represents a variant of the PLN model, incorporating a rank constraint on the covariance matrix. This can be interpreted as an extension of the probabilistic PCA for count data, where the rank determines the number of components in the probabilistic PCA. Users have the flexibility to define the rank of the covariance matrix via the rank keyword of the PlnPCA object. Furthermore, they can specify multiple ranks simultaneously within a single object (PlnPCAcollection), and then select the optimal model based on either the AIC (default) or BIC criterion:

pca_col =  PlnPCAcollection.from_formula("endog ~ 1  + tree ", data = oaks, ranks = [3,4,5])
pca_col.fit()
print(pca_col)
pca_col.show()
best_pca = pca_col.best_model()
best_pca.show()
transformed_data = best_pca.transform(project = True)
print('Original data shape: ', oaks["endog"].shape)
print('Transformed data shape: ', transformed_data.shape)

A correlation circle may be employed to graphically represent the relationship between the variables and the components:

best_pca.plot_pca_correlation_circle(["var_1","var_2"], indices_of_variables = [0,1])

Zero inflated Poisson Log normal Model (aka ZIPln)

The ZiPln model, a variant of the PLN model, is designed to handle zero inflation in the data. It is defined as follows:

$$Y_{ij}\sim \mathcal W_{ij} \times P(\exp(Z_{ij})), \quad \mathbf Z_i \sim \mathcal N(\mathbf o_i + \mathbf B ^{\top} \mathbf x_i, \mathbf \Sigma), \quad W_{ij} \sim \mathcal B(\sigma( \mathbf x_i^{0^{\top}}\mathbf B^0_j))$$

This model is particularly beneficial when the data contains a significant number of zeros. It incorporates additional covariates for the zero inflation coefficient, which are specified following the pipe | symbol in the formula or via the exog_inflation keyword. If not specified, it is set to the covariates for the Poisson part.

zi =  ZIPln.from_formula("endog ~ 1  + tree | 1 + tree", data = oaks)
zi.fit()
print(zi)
print("Transformed data shape: ", zi.transform().shape)
z_latent_variables, w_latent_variables = zi.transform(return_latent_prob = True)
print(r'$Z$ latent variables shape', z_latent_variables.shape)
print(r'$W$ latent variables shape', w_latent_variables.shape)

By default, the transformation of the data returns only the $\mathbf Z$ latent variable. However, if the return_latent_prob parameter is set to True, the transformed data will include both the latent variables $\mathbf W$ and $\mathbf Z$. Here, $\mathbf W$ accounts for the zero inflation, while $\mathbf Z$ accounts for the Poisson parameter.

Visualization

The package is equipped with a set of visualization functions designed to help the user interpret the data. The viz function conducts Principal Component Analysis (PCA) on the latent variables, while the viz_positions function carries out PCA on the latent variables, adjusted for covariates. Additionally, the viz_prob function provides a visual representation of the zero-inflation probability.

best_pca.viz(colors = oaks["tree"])
best_pca.viz_positions(colors = oaks["dist2ground"])
pln.viz(colors = oaks["tree"])
pln.viz_positions(colors = oaks["dist2ground"])
zi.viz(colors = oaks["tree"])
zi.viz_positions(colors = oaks["dist2ground"])
zi.viz_prob(colors = oaks["tree"])

👐 Contributing

Feel free to contribute, but read the CONTRIBUTING.md first. A public roadmap will be available soon.

⚡️ Citations

Please cite our work using the following references:

• J. Chiquet, M. Mariadassou and S. Robin: Variational inference for probabilistic Poisson PCA, the Annals of Applied Statistics, 12: 2674–2698, 2018. pdf

• B. Batardiere, J.Chiquet, M.Mariadassou: Zero-inflation in the Multivariate Poisson Lognormal Family. pdf