`vignettes/PLNmixture.Rmd`

`PLNmixture.Rmd`

This vignette illustrates the standard use of the
`PLNmixture`

function and the methods accompanying the R6
Classes `PLNmixturefamily`

and
`PLNmixturefit`

.

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

The main function `PLNmixture`

integrates some features of
the **future** package to perform parallel computing: you
can set your plan to speed the fit by relying on 2 workers as
follows:

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.)

PLN-mixture for multivariate count data is a variant of the Poisson Lognormal model of Aitchison and Ho (1989) (see the PLN vignette as a reminder) which can be viewed as a PLN model with an additional mixture layer in the model: the latent observations found in the first layer are assumed to be drawn from a mixture of \(K\) multivariate Gaussian components. Each component \(k\) has a prior probability \(p(i \in k) = \pi_k\) such that \(\sum_k \pi_k = 1\). We denote by \(C_i\in \{1,\dots,K\}\) the multinomial variable \(\mathcal{M}(1,\boldsymbol{\pi} = (\pi_1,\dots,\pi_K))\) describing the component to which observation \(i\) belongs to. Introducing this additional layer, our PLN mixture model is as follows

\[ \begin{array}{rcl} \text{layer 2 (clustering)} & \mathbf{C}\_i \sim \mathcal{M}(1,\boldsymbol{\pi}) \\ \text{layer 1 (Gaussian)} & \mathbf{Z}\_i | \, \mathbf{C}\_i = k \sim \mathcal{N}({\boldsymbol\mu}^{(k)}, {\boldsymbol\Sigma}^{(k)}), \\ \text{observation space } & Y_{ij} \| Z_{ij} \quad \text{indep.} & \mathbf{Y}_i | \mathbf{Z}_i\sim\mathcal{P}\left(\exp\{\mathbf{Z}_i\}\right). \end{array} \]

Just like PLN, PLN-mixture generalizes to a formulation 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 \(i\) in each mixture component. The latent layer then reads

\[ \mathbf{Z}_i | \mathbf{C}_i = k \, \sim \mathcal{N}({\mathbf{o}_i + \mathbf{x}_i^{\top}{\boldsymbol\Theta} + \boldsymbol\mu}^{(k)},{\boldsymbol\Sigma}^{(k)}), \]

where \({\boldsymbol\Theta}\) is a \(d\times p\) matrix of regression parameters common to all the mixture components.

When using parametric mixture models like Gaussian mixture models, it is generally not recommended to have covariances matrices \({\boldsymbol\Sigma}^{(k)}\) with no special restriction, especially when dealing with a large number of variables. Indeed, the total number of parameters to estimate in such unrestricted model can become prohibitive.

To reduce the computational burden and avoid over-fitting the data,
two different, more constrained parametrizations of the covariance
matrices of each component are currently implemented in the
`PLNmodels`

package (on top of the general form of \(\Sigma_k\)):

The diagonal structure assumes that, given the group membership of a site, all variable abundances are independent. The spherical structure further assumes that all species have the same biological variability. In particular, in both parametrisations, all observed covariations are caused only by the group structure.

For readers familiar with the `mclust`

`R`

package (Fraley and Raftery 1999), which
implements Gaussian mixture models with many variants of covariance
matrices of each component, the spherical model corresponds to
`VII`

(spherical, unequal volume) and the diagonal model to
`VVI`

(diagonal, varying volume and shape). {Using
constrained forms of the covariance matrices enables} PLN-mixture to
{provide a clustering} even when the number of sites \(n\) remains of the same order, or smaller,
than the number of species \(p\).

Just like with all models fitted in PLNmodels, 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. In this case, it is not too difficult to show that PLN-mixture can be obtained by optimizing a collection of weighted standard PLN models.

In the package, the PLN-mixture model is adjusted with the function
`PLNmixture`

, which we review in this section. This function
adjusts the model for a series of value of \(k\) and provides a collection of objects
`PLNmixturefit`

stored in an object with class
`PLNmixturefamily`

.

The class `PLNmixturefit`

contains a collection of
components constituting the mixture, each of whom inherits from the
class `PLNfit`

, so we strongly recommend the reader to be
comfortable with `PLN`

and `PLNfit`

before using
`PLNmixture`

(see the PLN
vignette).

We fit a collection of \(K=5\) models with one iteration of forward smoothing of the log-likelihood as follows:

```
mixture_models <- PLNmixture(
Abundance ~ 1 + offset(log(Offset)),
data = trichoptera,
clusters = 1:5,
control_main = list(smoothing = "forward", iterates = 1)
)
```

```
##
## Initialization...
##
## Adjusting 5 PLN mixture models.
## number of cluster = 1
number of cluster = 2
number of cluster = 3
number of cluster = 4
number of cluster = 5
##
## Smoothing PLN mixture models.
## Going forward ++++
## Post-treatments
## DONE!
```

Note the use of the `formula`

object to specify the model,
similar to the one used in the function `PLN`

.

`PLNmixturefamily`

The `mixture_models`

variable is an `R6`

object
with class `PLNmixturefamily`

, which comes with a couple of
methods. The most basic is the `show/print`

method, which
outputs a brief summary of the estimation process:

`mixture_models`

```
## --------------------------------------------------------
## COLLECTION OF 5 POISSON LOGNORMAL MODELS
## --------------------------------------------------------
## Task: Mixture Model
## ========================================================
## - Number of clusters considered: from 1 to 5
## - Best model (regarding BIC): cluster = 2
## - Best model (regarding ICL): cluster = 5
```

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

param | nb_param | loglik | BIC | ICL |
---|---|---|---|---|

1 | 18 | -1158.503 | -1193.530 | -2152.619 |

2 | 37 | -1104.242 | -1176.241 | -1981.670 |

3 | 56 | -1070.276 | -1179.247 | -1887.299 |

4 | 75 | -1041.149 | -1187.092 | -1790.786 |

5 | 94 | -1019.064 | -1201.980 | -1417.825 |

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

field:

param | nb_param | objective | convergence | outer_iterations | |
---|---|---|---|---|---|

out | 1 | 18 | 1158.503345 | 0.000000 | 2.000000 |

elt | 2 | 37 | 1104.241949 | 0.000001 | 2.000000 |

elt.1 | 3 | 56 | 1070.276423 | 0.000001 | 2.000000 |

elt.2 | 4 | 75 | 1041.148778 | 0.000000 | 2.000000 |

elt.3 | 5 | 94 | 1019.064399 | 0.000002 | 2.000000 |

A visual representation of the optimization can be obtained be representing the objective function

`mixture_models$plot_objective()`

Comprehensive information about `PLNmixturefamily`

is
available via `?PLNmixturefamily`

.

The `plot`

method of `PLNmixturefamily`

displays evolution of the criteria mentioned above, and is a good
starting point for model selection:

`plot(mixture_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(mixture_models, reverse = TRUE)`

From those plots, we can see that the best model in terms of BIC is
obtained for a number of clusters of 2. We may extract the corresponding
model with the method `getBestModel()`

. A model with a
specific number of clusters can also be extracted with the
`getModel()`

method:

```
myMix_BIC <- getBestModel(mixture_models, "BIC")
myMix_2 <- getModel(mixture_models, 2)
```

`PLNmixturefit`

Object `myMix_BIC`

is an `R6Class`

object with
class `PLNmixturefit`

which in turns has a couple of methods.
A good place to start is the `show/print`

method:

`myMix_BIC`

```
## Poisson Lognormal mixture model with 2 components and spherical covariances.
## * Useful fields
## $posteriorProb, $memberships, $mixtureParam, $group_means
## $model_par, $latent, $latent_pos, $optim_par
## $loglik, $BIC, $ICL, $loglik_vec, $nb_param, $criteria
## $component[[i]] (a PLNfit with associated methods and fields)
## * Useful S3 methods
## print(), coef(), sigma(), fitted(), predict()
```

The user can easily access several fields of the
`PLNmixturefit`

object using active binding or
`S3`

methods:

- the vector of group memberships:

`myMix_BIC$memberships`

```
## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 2 2 2 1 1 1 1 1 1 1
## [39] 1 1 1 1 1 1 1 1 1 1 2
```

- the group proportions:

`myMix_BIC$mixtureParam`

`## [1] 0.8163265 0.1836735`

- the posterior probabilities (often close to the boundaries \(\{0,1\}\)):

1 | 0 |

1 | 0 |

1 | 0 |

1 | 0 |

1 | 0 |

1 | 0 |

- a list of \(K\) \(p \times p\) covariance matrices \(\hat{\boldsymbol{\Sigma}}\) (here spherical variances):

```
## [[1]]
## Che Hyc Hym Hys Psy Aga Glo Ath
## 0.7177328 0.7177328 0.7177328 0.7177328 0.7177328 0.7177328 0.7177328 0.7177328
## Cea Ced Set All Han Hfo Hsp Hve
## 0.7177328 0.7177328 0.7177328 0.7177328 0.7177328 0.7177328 0.7177328 0.7177328
## Sta
## 0.7177328
##
## [[2]]
## Che Hyc Hym Hys Psy Aga Glo Ath
## 0.9607135 0.9607135 0.9607135 0.9607135 0.9607135 0.9607135 0.9607135 0.9607135
## Cea Ced Set All Han Hfo Hsp Hve
## 0.9607135 0.9607135 0.9607135 0.9607135 0.9607135 0.9607135 0.9607135 0.9607135
## Sta
## 0.9607135
```

- the regression coefficient matrix and other model of parameters (results not shown here, redundant with other fields)

```
coef(myMix_BIC, 'main') # equivalent to myMix_BIC$model_par$Theta
coef(myMix_BIC, 'mixture') # equivalent to myMix_BIC$model_par$Pi, myMix_BIC$mixtureParam
coef(myMix_BIC, 'means') # equivalent to myMix_BIC$model_par$Mu, myMix_BIC$group_means
coef(myMix_BIC, 'covariance') # equivalent to myMix_BIC$model_par$Sigma, sigma(myMix_BIC)
```

- the \(p \times K\) matrix of group means \(\mathbf{M}\)

group_1 | group_2 | |
---|---|---|

Che | -7.16 | -18.02 |

Hyc | -8.33 | -7.71 |

Hym | -2.68 | -4.87 |

Hys | -6.59 | -8.40 |

Psy | -0.49 | -0.82 |

Aga | -3.72 | -6.97 |

In turn, each component of a `PLNmixturefit`

is a
`PLNfit`

object (see the corresponding vignette)

`myMix_BIC$components[[1]]`

```
## A multivariate Poisson Lognormal fit with spherical covariance model.
## ==================================================================
## nb_param loglik BIC ICL
## 18 -828.202 -863.228 -1564.048
## ==================================================================
## * 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()
```

The `PLNmixturefit`

class also benefits from two important
methods: `plot`

and `predict`

.

`plot`

method
We can visualize the clustered latent position by performing a PCA on the latent layer:

`plot(myMix_BIC, "pca")`

We can also plot the data matrix with samples reordered by clusters to check whether it exhibits strong pattern or not. The limits between clusters are highlighted by grey lines.

`plot(myMix_BIC, "matrix")`

`predict`

method
For PLNmixture, the goal of `predict`

is to predict the
membership based on observed newly *species counts*.

By default, the `predict`

use the argument
`type = "posterior"`

to output the matrix of posterior
probabilities \(\hat{\pi}_k\)

```
predicted.class <- predict(myMix_BIC, newdata = trichoptera)
## equivalent to
## predicted.class <- predict(myMIX_BIC, newdata = trichoptera, type = "posterior")
predicted.class %>% head() %>% knitr::kable(digits = 2)
```

1 | 0 |

1 | 0 |

1 | 0 |

1 | 0 |

1 | 0 |

1 | 0 |

Setting `type = "response"`

, we can predict the most
likely cluster \(\hat{k} = \arg\max_{k =
1\dots K} \{ \hat{\pi_k}\}\) instead:

```
predicted.class <- predict(myMix_BIC, newdata = trichoptera,
prior = myMix_BIC$posteriorProb, type = "response")
predicted.class
```

```
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2
## 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
## 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2
## Levels: 1 2
```

We can assess that the predictions are quite similar to the real
group (*this is not a proper validation of the method as we used data
set for both model fitting and prediction and are thus at risk of
overfitting*).

Finally, we can get the coordinates of the new data on the same graph
at the original ones with `type = "position"`

. This is done
by averaging the latent positions \(\hat{\mathbf{Z}}_i + \boldsymbol{\mu}_k\)
(found when the sample is assumed to come from group \(k\)) and weighting them with the \(\hat{\pi}_k\). Some samples, have
compositions that put them very far from their group mean.

```
predicted.position <- predict(myMix_BIC, newdata = trichoptera,
prior = myMix_BIC$posteriorProb, type = "position")
prcomp(predicted.position) %>%
factoextra::fviz_pca_ind(col.ind = predicted.class)
```

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

`future::plan("sequential")`

Aitchison, J., and C. H. Ho. 1989. “The Multivariate Poisson-Log
Normal Distribution.” *Biometrika* 76 (4): 643–53.

Fraley, Chris, and Adrian E. Raftery. 1999. “MCLUST: Software for
Model-Based Cluster Analysis.” *Journal of Classification*
16 (2): 297–306. https://doi.org/10.1007/s003579900058.