Author: PLN team
Date: 2021-03-16

# Dimension reduction of multivariate count data with PLN-PCA

#### PLN team

#### 2021-03-16

## Preliminaries

This vignette illustrates the standard use of the `PLNPCA` function and the methods accompanying the R6 Classes `PLNPCAfamily` and `PLNPCAfit`.

### Requirements

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

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

The main function `PLNPCA` integrates some features of the future package to perform parallel computing: we set our plan now to speed the fit by relying on 2 workers:

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

### 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

In the vein of Tipping and Bishop (1999), we introduce in Chiquet, Mariadassou, and Robin (2018) a probabilistic PCA model for multivariate count data which is a variant of the Poisson Lognormal model of Aitchison and Ho (1989) (see the PLN vignette as a reminder). Indeed, it can be viewed as a PLN model with an additional rank constraint on the covariance matrix \(\boldsymbol\Sigma\) such that \(\mathrm{rank}(\boldsymbol\Sigma)= q\).

This PLN-PCA model can be written in a hierarchical framework where a sample of \(p\)-dimensional observation vectors \(\mathbf{Y}_i\) is related to some \(q\)-dimensional vectors of latent variables \(\mathbf{W}_i\) as follows: \[\begin{equation} \begin{array}{rcl} \text{latent space } & \mathbf{W}_i \quad \text{i.i.d.} & \mathbf{W}_i \sim \mathcal{N}(\mathbf{0}_q, \mathbf{I}_q) \\ \text{parameter space } & \mathbf{Z}_i = {\boldsymbol\mu} + \mathbf{B}^\top \mathbf{W}_i & \\ \text{observation space } & Y_{ij} | Z_{ij} \quad \text{indep.} & Y_{ij} | Z_{ij} \sim \mathcal{P}\left(\exp\{Z_{ij}\}\right) \end{array} \end{equation}\]

The parameter \({\boldsymbol\mu}\in\mathbb{R}^p\) corresponds to the main effects, the \(p\times q\) matrix \(\mathbf{B}\) to the loadings in the parameter spaces and \(\mathbf{W}_i\) to the scores of the \(i\)-th observation in the low-dimensional latent subspace of the parameter space. The dimension of the latent space \(q\) corresponds to the number of axes in the PCA or, in other words, to the rank of \(\mathbf{B}\mathbf{B}^\intercal\). An hopefully more intuitive way of writing this model is the following: \[\begin{equation} \begin{array}{rcl} \text{latent space } & \mathbf{Z}_i \sim \mathcal{N}({\boldsymbol\mu},\boldsymbol\Sigma), \qquad \boldsymbol\Sigma = \mathbf{B}\mathbf{B}^\top \\ \text{observation space } & Y_{ij} | Z_{ij} \quad \text{indep.} & Y_{ij} | Z_{ij} \sim \mathcal{P}\left(\exp\{Z_{ij}\}\right), \end{array} \end{equation}\] where the interpretation of PLN-PCA as a rank-constrained PLN model is more obvious.

#### Covariates and offsets

Just like PLN, PLN-PCA 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 \[\begin{equation} \mathbf{Z}_i \sim \mathcal{N}({\mathbf{o}_i + \mathbf{x}_i^\top\boldsymbol\Theta},\boldsymbol\Sigma), \qquad \boldsymbol\Sigma = \mathbf{B}\mathbf{B}^\top, \end{equation}\] where \(\boldsymbol\Theta\) is a \(d\times p\) matrix of regression parameters.

#### Optimization by Variational inference

Dimension reduction and visualization is the main objective in (PLN)-PCA. To reach this goal, we need to first estimate the model parameters. Inference in PLN-PCA focuses on the regression parameters \(\boldsymbol\Theta\) and on the covariance matrix \(\boldsymbol\Sigma\). Technically speaking, we adopt 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 Svanberg (2002) implemented in the C++ library (Johnson 2011), which we link to the package. Technical details can be found in Chiquet, Mariadassou, and Robin (2018).

## Analysis of trichoptera data with a PLNPCA model

In the package, the PLNPCA model is adjusted with the function `PLNPCA`, which we review in this section. This function adjusts the model for a series of value of \(q\) and provides a collection of objects `PLNPCAfit` stored in an object with class `PLNPCAfamily`.

The class `PLNPCAfit` inherits from the class `PLNfit`, so we strongly recommend the reader to be comfortable with `PLN` and `PLNfit` before using `PLNPCA` (see the PLN vignette).

### A model with latent main effects for the Trichoptera data set

#### Adjusting a collection of fits

We fit a collection of \(q\) models as follows:

    PCA_models <- PLNPCA(
      Abundance ~ 1 + offset(log(Offset)),
      data  = trichoptera,
      ranks = 1:5
    )

    ##
    ##  Initialization...
    ##
    ##  Adjusting 5 PLN models for PCA analysis.
    ##   Rank approximation = 1
         Rank approximation = 3
         Rank approximation = 5
         Rank approximation = 4
         Rank approximation = 2
    ##  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 `PLNPCAfamily`

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

    PCA_models

    ## --------------------------------------------------------
    ## COLLECTION OF 5 POISSON LOGNORMAL MODELS
    ## --------------------------------------------------------
    ##  Task: Principal Component Analysis
    ## ========================================================
    ##  - Ranks considered: from 1 to 5
    ##  - Best model (greater BIC): rank = 4
    ##  - Best model (greater ICL): rank = 3

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

    PCA_models$criteria %>% knitr::kable()

param | nb_param | loglik | BIC | ICL
--- | --- | --- | --- | ---
1 | 34 | -1042.0491 | -1108.2100 | -1120.2627
2 | 50 | -729.6907 | -826.9863 | -860.9509
3 | 65 | -637.7419 | -764.2261 | -824.1529
4 | 79 | -596.1259 | -749.8528 | -837.8156
5 | 92 | -578.7050 | -757.7287 | -888.9477

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

    PCA_models$convergence  %>% knitr::kable()

| param | nb_param | iterations | status | message
--- | --- | --- | --- | --- | ---
out | 1 | 34 | 575 | 3 | ftol_rel or ftol_abs was reached
elt | 2 | 50 | 635 | 3 | ftol_rel or ftol_abs was reached
elt.1 | 3 | 65 | 626 | 3 | ftol_rel or ftol_abs was reached
elt.2 | 4 | 79 | 2601 | 3 | ftol_rel or ftol_abs was reached
elt.3 | 5 | 92 | 3346 | 3 | ftol_rel or ftol_abs was reached

Comprehensive information about `PLNPCAfamily` is available via `?PLNPCAfamily`.

#### Model selection of rank \(q\)

The `plot` method of `PLNPCAfamily` displays evolution of the criteria mentioned above, and is a good starting point for model selection:

    plot(PCA_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(PCA_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 the number of axes (or subspace dimension). Also note the (approximated) \(R^2\) which is displayed for each value of \(q\) (see (Chiquet, Mariadassou, and Robin 2018) for details on its computation).

From this plot, we can see that the best model in terms of BIC or ICL is obtained for a rank \(q=4\) or \(q=3\). We may extract the corresponding model with the method `getBestModel("ICL")`. A model with a specific rank can be extracted with the `getModel()` method:

    myPCA_ICL <- getBestModel(PCA_models, "ICL")
    myPCA_BIC <- getModel(PCA_models, 3) # getBestModel(PCA_models, "BIC")  is equivalent here

#### Structure of `PLNPCAfit`

Objects `myPCA_ICL` and `myPCA_BIC` are `R6Class` objects of class `PLNPCAfit` which in turns own a couple of methods, some inherited from `PLNfit` and some others specific, mostly for visualization purposes. The `plot` method provides individual maps and correlation circles as in usual PCA. If an additional classification exists for the observations – which is the case here with the available classification of the trapping nights – , it can be passed as an argument to the function.1

    plot(myPCA_ICL, ind_cols = trichoptera$Group)

Among other fields and methods (see `?PLNPCAfit` for a comprehensive view), the most interesting for the end-user in the context of PCA are

- the regression coefficient matrix

    coef(myPCA_ICL) %>% head() %>% knitr::kable()

| (Intercept)
--- | ---
Che | -7.5001934
Hyc | -8.1851526
Hym | -3.0413520
Hys | -6.9625049
Psy | -0.5249485
Aga | -3.8102276

- the estimated covariance matrix \(\boldsymbol\Sigma\) with fixed rank

    sigma(myPCA_ICL) %>% corrplot(is.corr = FALSE)

- the rotation matrix (in the latent space)

    myPCA_ICL$rotation %>% head() %>% knitr::kable()

| PC1 | PC2 | PC3
--- | --- | --- | ---
Che | -0.2098340 | 0.3280969 | -0.0322464
Hyc | -0.4346332 | -0.2488820 | 0.2115507
Hym | -0.1284943 | 0.1577914 | 0.2973014
Hys | -0.4261529 | 0.3796084 | 0.2965003
Psy | 0.0488866 | 0.0187237 | -0.0692717
Aga | 0.0576168 | 0.2683950 | 0.2081702

- the principal components values (or scores)

    myPCA_ICL$scores %>% head() %>% knitr::kable()

PC1 | PC2 | PC3
--- | --- | ---
-1.786121 | -0.5451126 | 0.7217146
3.559240 | 0.9127011 | 2.1018535
7.466732 | -1.3686051 | 0.5159175
6.367155 | -1.4132232 | -1.7538134
4.613973 | 0.0006481 | -1.0402287
3.983468 | 0.8103992 | 0.4699022

`PLNPCAfit` also inherits from the methods of `PLNfit` (see the appropriate vignette). Most are recalled via the show method:

    myPCA_ICL

    ## Poisson Lognormal with rank constrained for PCA (rank = 3)
    ## ==================================================================
    ##  nb_param   loglik      BIC      ICL
    ##        65 -637.742 -764.226 -824.153
    ## ==================================================================
    ## * 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 PCA
    ##     $percent_var, $corr_circle, $scores, $rotation, $eig, $var, $ind
    ## * Additional S3 methods for PCA
    ##     plot.PLNPCAfit()

### Additional visualization

We provide simple plotting functions but a wealth of plotting utilities are available for factorial analyses results. The following bindings allow you to use widely popular tools to make your own plots: `$eig`, `$var` and `$ind`.

    ## All summaries associated to the individuals
    str(myPCA_ICL$ind)

    ## List of 4
    ##  $ coord  : num [1:49, 1:3] -1.79 3.56 7.47 6.37 4.61 ...
    ##   ..- attr(*, "dimnames")=List of 2
    ##   .. ..$ : chr [1:49] "1" "2" "3" "4" ...
    ##   .. ..$ : chr [1:3] "Dim.1" "Dim.2" "Dim.3"
    ##  $ cos2   : num [1:49, 1:3] 0.796 0.707 0.963 0.889 0.952 ...
    ##   ..- attr(*, "dimnames")=List of 2
    ##   .. ..$ : chr [1:49] "1" "2" "3" "4" ...
    ##   .. ..$ : chr [1:3] "Dim.1" "Dim.2" "Dim.3"
    ##  $ contrib: num [1:49, 1:3] 0.642 2.548 11.213 8.154 4.282 ...
    ##   ..- attr(*, "dimnames")=List of 2
    ##   .. ..$ : chr [1:49] "1" "2" "3" "4" ...
    ##   .. ..$ : chr [1:3] "Dim.1" "Dim.2" "Dim.3"
    ##  $ dist   : Named num [1:49] 2 4.23 7.61 6.75 4.73 ...
    ##   ..- attr(*, "names")= chr [1:49] "1" "2" "3" "4" ...

    ## Coordinates of the individuals in the principal plane
    head(myPCA_ICL$ind$coord)

    ##       Dim.1         Dim.2      Dim.3
    ## 1 -1.786121 -0.5451125902  0.7217146
    ## 2  3.559240  0.9127011127  2.1018535
    ## 3  7.466732 -1.3686051158  0.5159175
    ## 4  6.367155 -1.4132231837 -1.7538134
    ## 5  4.613973  0.0006481148 -1.0402287
    ## 6  3.983468  0.8103992452  0.4699022

You can also use high level functions from the factoextra package to extract relevant informations

    ## Eigenvalues
    factoextra::get_eig(myPCA_ICL)

    ##       eigenvalue variance.percent cumulative.variance.percent
    ## Dim.1   497.2111         38.66810                    38.66810
    ## Dim.2   315.2595         24.51772                    63.18582
    ## Dim.3   220.6377         17.15899                    80.34481

    ## Variables
    factoextra::get_pca_var(myPCA_ICL)

    ## Principal Component Analysis Results for variables
    ##  ===================================================
    ##   Name       Description
    ## 1 "$coord"   "Coordinates for the variables"
    ## 2 "$cor"     "Correlations between variables and dimensions"
    ## 3 "$cos2"    "Cos2 for the variables"
    ## 4 "$contrib" "contributions of the variables"

    ## Individuals
    factoextra::get_pca_ind(myPCA_ICL)

    ## Principal Component Analysis Results for individuals
    ##  ===================================================
    ##   Name       Description
    ## 1 "$coord"   "Coordinates for the individuals"
    ## 2 "$cos2"    "Cos2 for the individuals"
    ## 3 "$contrib" "contributions of the individuals"

And some of the very nice plotting methods such as biplots, correlation circles and scatter plots of the scores.

    factoextra::fviz_pca_biplot(myPCA_ICL)

    factoextra::fviz_pca_var(myPCA_ICL)

    factoextra::fviz_pca_ind(myPCA_ICL)

### A model accounting for meteorological covariates

A contribution of PLN-PCA is to let the possibility to taking into account some covariates in the parameter space. Such a strategy often completely changes the interpretation of PCA. Indeed, the covariates are often responsible for some strong structure in the data. The effect of the covariates should be removed since they are often quite obvious for the analyst and may hide some more important and subtle effects.

In the case at hand, the covariates corresponds to the meteorological variables. Let us try to introduce some of them in our model, for instance, the temperature, the wind and the cloudiness. This can be done thanks to the model formula:

    PCA_models_cov <-
      PLNPCA(
        Abundance ~ 1 + offset(log(Offset)) + Temperature + Wind + Cloudiness,
        data  = trichoptera,
        ranks = 1:4
      )

    ##
    ##  Initialization...
    ##
    ##  Adjusting 4 PLN models for PCA analysis.
    ##   Rank approximation = 1
         Rank approximation = 4
         Rank approximation = 2
         Rank approximation = 3
    ##  Post-treatments
    ##  DONE!

Again, the best model is obtained for \(q=3\) classes.

    plot(PCA_models_cov)

    myPCA_cov <- getBestModel(PCA_models_cov, "ICL")

Suppose that we want to have a closer look to the first two axes. This can be done thanks to the plot method:

    gridExtra::grid.arrange(
      plot(myPCA_cov, map = "individual", ind_cols = trichoptera$Group, plot = FALSE),
      plot(myPCA_cov, map = "variable", plot = FALSE),
      ncol = 2
    )

We can check that the fitted value of the counts – even with this low-rank covariance matrix – are close to the observed ones:

    data.frame(
      fitted   = as.vector(fitted(myPCA_cov)),
      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

## References

Aitchison, J., and C. H. Ho. 1989. “The Multivariate Poisson-Log Normal Distribution.” Biometrika 76 (4): 643–53.
Chiquet, Julien, Mahendra Mariadassou, and Stéphane Robin. 2018. “Variational Inference for Probabilistic Poisson PCA.” The Annals of Applied Statistics 12: 2674–98. https://doi.org/10.1214/18%2DAOAS1177.
Johnson, Steven G. 2011. The NLopt Nonlinear-Optimization Package. https://nlopt.readthedocs.io/en/latest/.
Svanberg, Krister. 2002. “A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations.” SIAM Journal on Optimization 12 (2): 555–73.
Tipping, M. E, and C. M Bishop. 1999. “Probabilistic Principal Component Analysis.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 61 (3): 611–22.

1. With our PLN-PCA (and any pPCA model for count data, where successive models are not nested), it is important to performed the model selection of \(q\) prior to visualization, since the model with rank \(q=3\) is not nested in the model with rank \(q=4\). Hence, percentage of variance must be interpreted with care: it sums to 100% but must be put in perspective with the model \(R^2\), giving an approximation of the total percentage of variance explained with the current model.↩︎