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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>Poisson lognormal models for count data</title>
<meta charset="utf-8" />
<meta name="author" content="J. Chiquet, M. Mariadassou, S. Robin, B. Batardière + others MIA Paris-Saclay, AgroParisTech, INRAE, Sorbonne Universyty Last update 29 mars, 2023" />
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<body>
<textarea id="source">
class: center, middle, inverse, title-slide
.title[
# Poisson lognormal models for count data
]
.subtitle[
## Focus on some probabilistic factor models: Poisson LDA and Poisson PCA
]
.author[
### J. Chiquet, M. Mariadassou, S. Robin, B. Batardière + others<br /><br /> <small>MIA Paris-Saclay, AgroParisTech, INRAE, Sorbonne Universyty </small> <br /> <small>Last update 29 mars, 2023</small>
]
.date[
### <br/><a href="https://pln-team.github.io/PLNmodels" class="uri">https://pln-team.github.io/PLNmodels</a>
]
---
class: inverse, middle
# Outline
1. .large[**Framework** of multivariate Poisson lognormal models]
<br/>
2. .large[**PLN estimation** with variational inference]
<br/>
3. .large[**Focus** on factor analysis with PLN (LDA, PCA)]
<br/>
4. .large[**Illustration** on a scRNA dataset]
---
class: inverse, center, middle
# Multivariate Poisson lognormal models <br/> .small[Motivations, Framework]
<!-- STATISTICAL MODEL -->
---
# Generic form of data sets
Routinely gathered in ecology/microbiology/genomics
### Data tables
- .important[Abundances]: read counts of species/transcripts `\(j\)` in sample `\(i\)`
- .important[Covariates]: value of environmental variable `\(k\)` in sample `\(i\)`
- .important[Offsets]: sampling effort for species/transcripts `\(j\)` in sample `\(i\)`
### Need frameworks to model _dependencies between counts_
- understand **environmental effects** <br/>
`\(\rightsquigarrow\)` explanatory models (multivariate regression, classification)
- exhibit **patterns of diversity** <br/>
`\(\rightsquigarrow\)` summarize the information (clustering, dimension reduction)
- understand **between-species interactions** <br />
`\(\rightsquigarrow\)` 'network' inference (variable/covariance selection)
- correct for technical and **confounding effects** <br/>
`\(\rightsquigarrow\)` account for covariables and sampling effort
---
# Illustration in genomics
## scRNA data set
The dataset `scRNA` contains the counts of the 500 most varying transcripts in the mixtures of 5 cell lines in human liver (obtained with standard 10x scRNAseq Chromium protocol).
We subsample 500 random cells and the keep the 200 most varying genes
```r
data(scRNA); set.seed(1234)
train_set <- sample.int(nrow(scRNA), 500)
test_set <- setdiff(1:nrow(scRNA), train_set)
scRNA_train <- scRNA[train_set, ]
scRNA_test <- scRNA[test_set, ]
scRNA_train$counts <- scRNA_train$counts[, 1:200]
scRNA_test$counts <- scRNA_test$counts[, 1:200]
```
### Covariates
- `cell_line`: the cell line of the current row (among 5)
- `total_counts`: Total number reads for that cell
---
# Table of Counts
.pull-left[
Matrix view (log-transform + total-counts normalization)
]
.pull-right[
Histogram (log-transform + total-counts normalization)
<!-- -->
]
---
# Models for multivariate count data
### If we were in a Gaussian world...
The .important[general linear model] [MKB79] would be appropriate! For each sample `\(i = 1,\dots,n\)`,
`$$\underbrace{\mathbf{Y}_i}_{\text{abundances}} = \underbrace{\mathbf{x}_i^\top \mathbf{B}}_{\text{covariates}} + \underbrace{\mathbf{o}_i}_{\text{sampling effort}} + \boldsymbol\varepsilon_i, \quad \boldsymbol\varepsilon_i \sim \mathcal{N}(\mathbf{0}_p, \underbrace{\boldsymbol\Sigma}_{\text{between-species dependencies}})$$`
null covariance `\(\Leftrightarrow\)` independence `\(\rightsquigarrow\)` uncorrelated species/transcripts do not interact
.content-box-red[This model gives birth to Principal Component Analysis,
Discriminant Analysis, Gaussian Graphical Models, Gaussian Mixture models and many others `\(\dots\)`]
### With count data...
There is no generic model for multivariate counts
- Data transformation (log, `\(\sqrt{}\)`) : quick and dirty
- Non-Gaussian multivariate distributions [Ino+17]: do not scale to data dimension yet
- .important[Latent variable models]: interaction occur in a latent (unobserved) layer
---
# The Poisson Lognormal model (PLN)
The PLN model [AH89] is a .important[multivariate generalized linear model], where
- the counts `\(\mathbf{Y}_i\)` are the response variables
- the main effect is due to a linear combination of the covariates `\(\mathbf{x}_i\)`
- a vector of offsets `\(\mathbf{o}_i\)` can be specified for each sample.
.content-box-red[
$$
\mathbf{Y}_i | \mathbf{Z}_i \sim \mathcal{P}\left(\exp\{\mathbf{Z}_i\}\right), \qquad \mathbf{Z}_i \sim \mathcal{N}({\mathbf{o}_i + \mathbf{x}_i^\top\mathbf{B}},\boldsymbol\Sigma), \\
$$
]
.pull-left[The unkwown parameters are
- `\(\mathbf{B}\)`, the regression parameters
- `\(\boldsymbol\Sigma\)`, the variance-covariance matrix
]
.pull-right[
Stacking all individuals together,
- `\(\mathbf{Y}\)` is the `\(n\times p\)` matrix of counts
- `\(\mathbf{X}\)` is the `\(n\times d\)` matrix of design
- `\(\mathbf{O}\)` is the `\(n\times p\)` matrix of offsets
]
### Properties: .small[.important[over-dispersion, arbitrary-signed covariances]]
- mean: `\(\mathbb{E}(Y_{ij}) = \exp \left( o_{ij} + \mathbf{x}_i^\top {\mathbf{B}}_{\cdot j} + \sigma_{jj}/2\right) > 0\)`
- variance: `\(\mathbb{V}(Y_{ij}) = \mathbb{E}(Y_{ij}) + \mathbb{E}(Y_{ij})^2 \left( e^{\sigma_{jj}} - 1 \right) > \mathbb{E}(Y_{ij})\)`
- covariance: `\(\mathrm{Cov}(Y_{ij}, Y_{ik}) = \mathbb{E}(Y_{ij}) \mathbb{E}(Y_{ik}) \left( e^{\sigma_{jk}} - 1 \right).\)`
---
class: inverse, center, middle
# Variational inference for standard PLN<br/> .small[Optimisation]
<!-- VARIATIONAL INFERENCE -->
---
# PLN Inference: general ingredients
Estimate `\(\theta = (\mathbf{B}, \boldsymbol\Sigma)\)`, predict the `\(\mathbf{Z}_i\)`, while the model marginal likelihood is
`$$p_\theta(\mathbf{Y}_i) = \int_{\mathbb{R}_p} \prod_{j=1}^p p_\theta(Y_{ij} | Z_{ij}) \, p_\theta(\mathbf{Z}_i) \mathrm{d}\mathbf{Z}_i$$`
### Expectation-Maximization
With `\(\mathcal{H}(p) = -\mathbb{E}_p(\log(p))\)` the entropy of `\(p\)`,
`$$\log p_\theta(\mathbf{Y}) = \mathbb{E}_{p_\theta(\mathbf{Z}\,|\,\mathbf{Y})} [\log p_\theta(\mathbf{Y}, \mathbf{Z})] + \mathcal{H}[p_\theta(\mathbf{Z}\,|\,\mathbf{Y})]$$`
EM requires to evaluate (some moments of) `\(p_\theta(\mathbf{Z} \,|\, \mathbf{Y})\)`, but there is no close form!
### Variational approximation [WJ08]
Use a proxy `\(q_\psi\)` of `\(p_\theta(\mathbf{Z}\,|\,\mathbf{Y})\)` minimizing a divergence in a class `\(\mathcal{Q}\)` .small[(e.g, Küllback-Leibler divergence)]
`$$q_\psi(\mathbf{Z})^\star \arg\min_{q\in\mathcal{Q}} D\left(q(\mathbf{Z}), p(\mathbf{Z} | \mathbf{Y})\right), \, \text{e.g.}, D(.,.) = KL(., .) = \mathbb{E}_{q_\psi}\left[\log \frac{q(z)}{p(z)}\right].$$`
---
# Inference: specific ingredients
Consider `\(\mathcal{Q}\)` the class of diagonal multivariate Gaussian distributions:
`$$\Big\{q: \, q(\mathbf{Z}) = \prod_i q_i(\mathbf{Z}_i), \, q_i(\mathbf{Z}_i) = \mathcal{N}\left(\mathbf{Z}_i; \mathbf{m}_i, \mathrm{diag}(\mathbf{s}_i \circ \mathbf{s}_i)\right), \boldsymbol\psi_i = (\mathbf{m}_i, \mathbf{s}_i) \in\mathbb{R}_p\times\mathbb{R}_p \Big\}$$`
and maximize the ELBO (Evidence Lower BOund)
`$$\begin{aligned}J(\theta, \psi) & = \log p_\theta(\mathbf{Y}) - KL[q_\psi (\mathbf{Z}) || p_\theta(\mathbf{Z} | \mathbf{Y})] \\ & = \mathbb{E}_{\psi} [\log p_\theta(\mathbf{Y}, \mathbf{Z})] + \mathcal{H}[q_\psi(\mathbf{Z})] \\ &
= \frac{1}{n} \sum_{i = 1}^n J_i(\theta, \psi_i),\end{aligned}$$`
where, letting `\(\mathbf{A}_i = \mathbb{E}_{q_i}[\exp(Z_i)] = \exp\left( \mathbf{o}_i + \mathbf{m}_i + \frac{1}{2}\mathbf{s}^2_i\right)\)`, we have
`$$\begin{aligned}
J_i(\theta, \psi_i) = &\mathbf{Y}_i^\intercal(\mathbf{o}_i + \mathbf{m}_i) - \left(\mathbf{A}_i - \frac{1}{2}\log(\mathbf{s}^2_i)\right) ^\intercal \mathbf{1}_p + \frac{1}{2} |\log|{\boldsymbol\Omega}| \\
& - \frac{1}{2}(\mathbf{m}_i - \boldsymbol{\Theta}\mathbf{x}_i)^\intercal \boldsymbol{\Omega} (\mathbf{m}_i - \boldsymbol{\Theta}\mathbf{x}_i) - \frac{1}{2} \mathrm{diag}(\boldsymbol\Omega)^\intercal\mathbf{s}^2_i + \mathrm{cst}
\end{aligned}$$`
---
# Resulting Variational EM
.important[Alternate] until convergence between
- VE step: optimize `\(\boldsymbol{\psi}\)` (can be written individually)
`$$\psi_i^{(h)} = \arg \max J_{i}(\theta^{(h)}, \psi_i) \left( = \arg\min_{q_i} KL[q_i(\mathbf{Z}_i) \,||\, p_{\theta^h}(\mathbf{Z}_i\,|\,\mathbf{Y}_i)] \right)$$`
- M step: optimize `\(\theta\)`
`$$\theta^{(h)} = \arg\max \frac{1}{n}\sum_{i=1}^{n}J_{Y_i}(\theta, \psi_i^{(h)})$$`
We end up with a `\(M\)`-estimator:
$$
`\begin{equation}
\hat{\theta}^{\text{ve}} = \arg\max_{\theta} \left( \frac{1}{n}\sum_{i=1}^{n} \sup_{\psi_i} J_i(\theta, \psi_i) \right) = \arg\max_{\theta} \underbrace{\left(\frac{1}{n}\sum_{i=1}^{n} \bar{J}_i(\theta) \right)}_{\bar{J}_n(\theta)}
\end{equation}`
$$
`\(\hat{\theta}^{\text{ve}}\)` is asymptotically unbiased. .important[Variance can be estimated with bootstrap/Jackknife.]
---
# Optimization of simple PLN models
### Property of the objective function
The ELBO `\(J(\theta, \psi)\)` is bi-concave, i.e.
- concave wrt `\(\psi = (\mathbf{M}, \mathbf{S})\)` for given `\(\theta\)`
- convace wrt `\(\theta = (\boldsymbol\Sigma, \mathbf{B})\)` for given `\(\psi\)`
but .important[not jointly concave] in general.
### M-step: analytical
`$$\hat{{\mathbf{B}}} = \left(\mathbf{X}^\top \mathbf{X}\right)^{-1} \mathbf{X} \mathbf{M}, \quad
\hat{{\boldsymbol\Sigma}} = \frac{1}{n} \left(\mathbf{M}-\mathbf{X}\hat{{\mathbf{B}}}\right)^\top \left(\mathbf{M}-\mathbf{X}\hat{\mathbf{B}}\right) + \frac{1}{n} \mathrm{diag}(\mathbf{1}^\intercal\mathbf{S}^2)$$`
### VE-step: gradient ascent
`$$\frac{\partial J(\psi)}{\partial \mathbf{M}} = \left(\mathbf{Y} - \mathbf{A} - (\mathbf{M} - \mathbf{X}{\mathbf{B}}) \mathbf{\Omega}\right), \qquad \frac{\partial J(\psi)}{\partial \mathbf{S}} = \frac{1}{\mathbf{S}} - \mathbf{S} \circ \mathbf{A} - \mathbf{S} \mathrm{D}_{\boldsymbol\Omega} .$$`
`\(\rightsquigarrow\)` Same routine for other PLN variants.
---
# Implementations
#### Medium scale problems (R/C++ package)
- **algorithm**: conservative convex separable approximations [Sva02]
- **implementation**: `NLopt` nonlinear-optimization library [Joh11] <br/>
`\(\rightsquigarrow\)` Up to thousands of sites ( `\(n \approx 1000s\)` ), hundreds of species ( `\(p\approx 100s\)` )
#### Large scale problems (Python/Pytorch module)
- **algorithm**: `Rprop` (gradient sign + adaptive variable-specific update) [RB93]
- **implementation**: `torch` with GPU auto-differentiation [FL22; Pas+17] <br/>
`\(\rightsquigarrow\)` Up to `\(n \approx 100,000\)` and `\(p\approx 10,000s\)`
<div class="figure" style="text-align: center">
<img src="figs/final_n=10000,p=2000.png" alt="n = 10,000, p = 2,000, d = 2 (running time: 1 min 40s)" width="30%" />
<p class="caption">n = 10,000, p = 2,000, d = 2 (running time: 1 min 40s)</p>
</div>
---
# Availability
### Help and documentation
- github group <https://github.com/pln-team>
- PLNmodels website <https://pln-team.github.io/PLNmodels>
### R/C++ Package `PLNmodels`
Last stable release on CRAN, development version available on GitHub).
```r
install.packages("PLNmodels")
remotes::install_github("PLN-team/PLNmodels@dev")
```
```r
library(PLNmodels)
packageVersion("PLNmodels")
```
```
## [1] '1.0.1'
```
### Python module `pyPLNmodels`
A PyTorch implementation is available in pyPI [https://pypi.org/project/pyPLNmodels/](https://pypi.org/project/pyPLNmodels/)
---
# PLN with offsets and covariates
- Cell-line effect is in the regression coefficient (groupwise or common mean)
- Spurious effect regarding the interactions between genes (full or diagonal covariance).
## Offset: .small[modeling sampling effort]
The predefined offset uses the total sum of reads.
```r
M01_scRNA <- PLN(counts ~ 1 + offset(log(total_counts)), scRNA_train)
M02_scRNA <- PLN(counts ~ 1 + offset(log(total_counts)), scRNA_train,
control = PLN_param(covariance = "diagonal"))
```
## Covariates: .small[cell-line effect ('ANOVA'-like) ]
The `cell_line` is a natural candidate for explaining a large of the variance.
```r
M11_scRNA <- PLN(counts ~ 0 + cell_line + offset(log(total_counts)), scRNA_train)
M12_scRNA <- PLN(counts ~ 0 + cell_line + offset(log(total_counts)), scRNA_train,
control = PLN_param(covariance = "diagonal"))
```
---
# PLN with offsets and covariates (2)
There is a clear gain in introducing the cell_line covariate in the model:
```r
rbind(M01 = M01_scRNA$criteria, M02 = M02_scRNA$criteria,
M11 = M11_scRNA$criteria, M12 = M12_scRNA$criteria) %>%
knitr::kable(format = "html")
```
<table>
<thead>
<tr>
<th style="text-align:left;"> </th>
<th style="text-align:right;"> nb_param </th>
<th style="text-align:right;"> loglik </th>
<th style="text-align:right;"> BIC </th>
<th style="text-align:right;"> ICL </th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;"> M01 </td>
<td style="text-align:right;"> 20300 </td>
<td style="text-align:right;"> -203318.4 </td>
<td style="text-align:right;"> -266396.7 </td>
<td style="text-align:right;"> -294420.7 </td>
</tr>
<tr>
<td style="text-align:left;"> M02 </td>
<td style="text-align:right;"> 400 </td>
<td style="text-align:right;"> -258421.9 </td>
<td style="text-align:right;"> -259664.8 </td>
<td style="text-align:right;"> -342792.9 </td>
</tr>
<tr>
<td style="text-align:left;"> M11 </td>
<td style="text-align:right;"> 21100 </td>
<td style="text-align:right;"> -199111.2 </td>
<td style="text-align:right;"> -264675.3 </td>
<td style="text-align:right;"> -289661.4 </td>
</tr>
<tr>
<td style="text-align:left;"> M12 </td>
<td style="text-align:right;"> 1200 </td>
<td style="text-align:right;"> -216457.5 </td>
<td style="text-align:right;"> -220186.3 </td>
<td style="text-align:right;"> -275057.6 </td>
</tr>
</tbody>
</table>
---
# PLN with offsets and covariates (3)
Looking at the coefficients `\(\mathbf{B}\)` associated with `cell_line` bring additional insights:
<img src="slides_files/figure-html/scRNA matrix plot-1.png" style="display: block; margin: auto;" />
---
# Simple torch example in `R`
```r
data("oaks")
system.time(myPLN_torch <-
PLN(Abundance ~ 1 + offset(log(Offset)),
data = oaks, control = PLN_param(backend = "torch", trace = 0)))
```
```
## user system elapsed
## 3.378 0.438 3.067
```
```r
system.time(myPLN_nlopt <-
PLN(Abundance ~ 1 + offset(log(Offset)),
data = oaks, control = PLN_param(backend = "nlopt", trace = 0)))
```
```
## user system elapsed
## 7.466 0.613 6.900
```
```r
myPLN_torch$loglik
```
```
## [1] -32224.15
```
```r
myPLN_nlopt$loglik
```
```
## [1] -32097.67
```
---
class: inverse, center, middle
# Probabilistic factor models in PLN<br/> .small[Poisson LDA and Poisson PCA]
---
# Natural extensions of PLN
### Various tasks of multivariate analysis
- .important[Dimension Reduction]: rank constraint matrix `\(\boldsymbol\Sigma\)`.
`$$\mathbf{Z}_i \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma = \mathbf{C}\mathbf{C}^\top), \quad \mathbf{C} \in \mathcal{M}_{pk} \text{ with orthogonal columns}.$$`
- .important[Classification]: maximize separation between groups with means
`$$\mathbf{Z}_i \sim \mathcal{N}(\sum_k {\boldsymbol\mu}_k \mathbf{1}_{\{i\in k\}}, \boldsymbol\Sigma), \quad \text{for known memberships}.$$`
- .important[Clustering]: mixture model in the latent space
`$$\mathbf{Z}_i \mid i \in k \sim \mathcal{N}(\boldsymbol\mu_k, \boldsymbol\Sigma_k), \quad \text{for unknown memberships}.$$`
- .important[Network inference]: sparsity constraint on inverse covariance.
`$$\mathbf{Z}_i \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma = \boldsymbol\Omega^{-1}), \quad \|\boldsymbol\Omega \|_1 < c.$$`
- .important[Variable selection]: sparsity constraint on regression coefficients
`$$\mathbf{Z}_i \sim \mathcal{N}(\mathbf{x}_i^\top\mathbf{B}, \boldsymbol\Sigma), \quad \|\mathbf{B} \|_1 < c.$$`
---
# Natural extensions of PLN
### Tasks seen in Factor Analysis
- .important[Dimension Reduction]: rank constraint matrix `\(\boldsymbol\Sigma\)`. .important[PCA]
`$$\mathbf{Z}_i \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma = \mathbf{C}\mathbf{C}^\top), \quad \mathbf{C} \in \mathcal{M}_{pk} \text{ with orthogonal columns}.$$`
- .important[Classification]: maximize separation between groups with means .important[LDA]
`$$\mathbf{Z}_i \sim \mathcal{N}(\sum_k {\boldsymbol\mu}_k \mathbf{1}_{\{i\in k\}}, \boldsymbol\Sigma), \quad \text{for known memberships}.$$`
---
# Background: Gaussian LDA
### Model
Assume the samples are distributed in `\(K\)` groups and note
- `\(\mathbf{G}\)` the group membership matrix
- `\(\mathbf{U} = [{\boldsymbol\mu}_1, \dots, {\boldsymbol\mu}_K]\)` the matrix of group-specific means
The model is
`$$\mathbf{Z}_i = \mathcal{N}(\mathbf{G}_i^\top \mathbf{U}, {\boldsymbol\Sigma})$$`
`\(\rightsquigarrow\)` Aim of LDA: Find the linear combination(s) `\(\mathbf{Z} u\)` maximizing separation between groups
### Solution
Find the first eigenvectors of `\(\mathbf{\Sigma}_w^{-1} \mathbf{\Sigma}_b\)` where
- `\(\mathbf{\Sigma}_w\)` is the _within_-group variance matrix, i.e. the unbiased estimated of `\({\boldsymbol\Sigma}\)`:
- `\(\mathbf{\Sigma}_b\)` is the _between_-group variance matrix, estimated from `\(\mathbf{U}\)`.
---
# Poisson lognormal LDA (1)
Similar to normal PLN with
- `\(\mathbf{X} \rightarrow (\mathbf{X}, \mathbf{G})\)`
- `\(\mathbf{B} \rightarrow (\mathbf{B}, \mathbf{U})\)`
### Inference
- Use .important[variational inference] to estimate `\((\mathbf{B}, \mathbf{U})\)`, `\(\mathbf{\Sigma}\)` and `\(\mathbf{Z}_i\)`
- Compute `\(\hat{\mathbf{\Sigma}}_b\)` as
`$$\hat{\mathbf{\Sigma}}_b = \frac1{K-1} \sum_k n_k (\hat{{\boldsymbol\mu}}_k - \hat{{\boldsymbol\mu}}_\bullet) (\hat{{\boldsymbol\mu}}_k - \hat{{\boldsymbol\mu}}_\bullet)^\intercal$$`
- Compute first `\(K-1\)` eigenvectors of `\(\hat{{\boldsymbol\Sigma}}^{-1} \hat{\mathbf{\Sigma}}_b = \mathbf{P}^\top \Lambda \mathbf{P}\)`
---
# Poisson lognormal LDA (2)
### Graphical representation
Mimick Gaussian LDA:
- Center the estimated latent positions `\(\tilde{\mathbf{Z}}\)`
- Compute the estimated coordinates along the discriminant axes
`$$\tilde{\mathbf{Z}}^{LDA} = \tilde{\mathbf{Z}} \mathbf{P} \Lambda^{1/2}$$`
### Prediction
For each group `\(k\)`
- Assume that the new sample `\(\mathbf{Y}_{\text{new}}\)` comes from group `\(k\)`
- Compute (variational) _likelihood_ `\(p_k = \mathbb{P}(\mathbf{Y}_{\text{new}} | \hat{{\boldsymbol\Sigma}}, \hat{\mathbf{\Sigma}_b}, \hat{{\boldsymbol\mu}}_k)\)`
- Compute posterior probability `\(\pi_k \propto \frac{n_k p_k}{n}\)`
`\(\rightsquigarrow\)` Assign to group with highest `\(\pi_k\)`
---
# Discriminant Analysis (scRNA, 1)
Use the `cell-line` variable for grouping (`grouping` is a factor of group to be considered)
```r
myLDA_cell_line <-
PLNLDA(counts ~ 1 + offset(log(total_counts)), grouping = cell_line,
data = scRNA_train)
```
.pull-left[
<!-- -->
]
.pull-right[
<!-- -->
]
---
# Discriminant Analysis (scRNA, 2)
Consider now a diagonal covariance.
```r
myLDA_cell_line_diag <-
PLNLDA(counts ~ 1 + offset(log(total_counts)), grouping = cell_line,
data = scRNA_train, control = PLN_param(covariance = "diagonal"))
```
.pull-left[
<!-- -->
]
.pull-right[
<!-- -->
]
---
# Discriminant Analysis (scRNA, 3)
We can prediction cell-line of some new data: let us try either diagonal or fully parametrized covariance.
```r
pred_cell_line <- predict(myLDA_cell_line ,
newdata = scRNA_test, type = "response")
pred_cell_line_diag <- predict(myLDA_cell_line_diag,
newdata = scRNA_test, type = "response")
```
The ARI on the test set is impressive.footnote[The problem should be easy though...]
```r
aricode::ARI(pred_cell_line, scRNA_test$cell_line)
```
```
## [1] 0.9770576
```
```r
aricode::ARI(pred_cell_line_diag, scRNA_test$cell_line)
```
```
## [1] 0.9683799
```
---
# Discriminant Analysis (scRNA, 4)
Let us explore the discriminant groups:
```r
heatmap(exp(myLDA_cell_line$group_means))
```
<img src="slides_files/figure-html/LDA-heatmap-factors-1.png" style="display: block; margin: auto;" />
Indeed, some groups of gene caracterize well the cell-lines.
---
# Poisson Lognormal PCA
### Model
`$$\begin{array}{rcl}
\mathbf{Z}_i & \sim^\text{iid} \mathcal{N}_p({\boldsymbol 0}_p, {\boldsymbol\Sigma}), & \text{rank}({\boldsymbol\Sigma}) = k \ll p \\
\mathbf{Y}_i \,|\, \mathbf{Z}_i & \sim \mathcal{P}(\exp\{\mathbf{O}_i + \mathbf{X}_i \mathbf{B} + \mathbf{Z}_i\})
\end{array}$$`
Recall that: `\(\text{rank}({\boldsymbol\Sigma}) = q \quad \Leftrightarrow \quad \exists \mathbf{C} (p \times q): \Sigma = \mathbf{C} \mathbf{C}^\intercal\)`.
### Estimation
Variational inference
$$\text{maximize } J({\boldsymbol\beta}, {\boldsymbol\psi}) $$
`\(\rightsquigarrow\)` Still bi-concave in `\({\boldsymbol\beta} = (\mathbf{C}, \mathbf{B})\)` and `\({\boldsymbol\psi} = (\mathbf{M}, \mathbf{S})\)`
---
# Model selection and Visualization
### Number of components/rank `\(k\)` needs to be chosen.
`\(\log p_{\hat{\boldsymbol\beta}}(\mathbf{Y})\)` intractable: use variational "likelihood" `\(J(\hat{\boldsymbol\beta}, \hat{\boldsymbol\psi})\)`
- BIC `\(\rightsquigarrow\)` `\(\text{vBIC}_k = J(\hat{\boldsymbol\beta}, \tilde{p}) - \frac12 p (d + k) \log(n)\)`
- ICL `\(\rightsquigarrow\)` `\(\text{vICL}_k = \text{vBIC}_k - \mathcal{H}(\tilde{p})\)`
$$
\hat{k} = \arg\max_k \text{vBIC}_k
\qquad \text{or} \qquad
\hat{k} = \arg\max_k \text{vICL}_k
$$
### Visualization
- Gaussian PCA: Optimal subspaces nested when `\(q\)` increases.
- PLN-pPCA: Non-nested subspaces.
For the selected dimension dimension `\(\hat{k}\)`:
- Compute the estimated latent positions `\(\mathbb{E}_q(\mathbf{Z}_i) = \mathbf{M} \hat{\mathbf{C}}^\top\)`
- Perform PCA on the `\(\mathbf{M} \hat{\mathbf{C}}^\top\)`
- Display result in any dimension `\(k \leq \hat{k}\)`
---
# PCA: Goodness of fit
.important[pPCA:] Cumulated sum of the eigenvalues = \% of variance preserved on the first `\(q\)` components.
### PLN-pPCA
Deviance based criterion.
- Compute `\(\tilde{\mathbf{Z}}^{(k)} = \mathbf{O} + \mathbf{X} \hat{\mathbf{B}}^\top + \mathbf{M}^{(k)} \left(\hat{\mathbf{C}}^{(k)}\right)^\top\)`
- Take `\(\lambda_{ij}^{(k)} = \exp\left(\tilde{Z}_{ij}^{(k)}\right)\)`
- Define `\(\lambda_{ij}^{\min} = \exp( \tilde{Z}_{ij}^0)\)` and `\(\lambda_{ij}^{\max} = Y_{ij}\)`
- Compute the Poisson log-likelihood `\(\ell_k = \log \mathbb{P}(\mathbf{Y}; \lambda^{(k)})\)`
### Pseudo R²
`$$R_k^2 = \frac{\ell_k - \ell_{\min}}{\ell_{\max} - \ell_{\min}}$$`
---
# A PCA analysis of the scRNA data set (1)
```r
PCA_scRNA <- PLNPCA(counts ~ 1 + offset(log(total_counts)), data = scRNA_train,
ranks = c(1, 2, seq(5, 40, 5)))
```
### Model Selection
<img src="slides_files/figure-html/PCA offset vizu cell-line-1.png" style="display: block; margin: auto;" />
---
# A PCA analysis of the scRNA data set (2)
### Biplot
Individual + Factor map - 40 most contributing genes
<small>
<img src="slides_files/figure-html/PCA offset vizu tree-1.png" style="display: block; margin: auto;" />
</small>
---
class: inverse, center, middle
# On-going works
---
# Optimisation, Algorithms
With .important[Bastien Batardière, Joon Kwon, Julien Stoehr]
## Exact Maximization
Direct likelihood optim (SGD with Important Sampling)
## Optimization
Optimisation guarantees coupling adaptive SGD + variance reduction
---
# PLN LDA: Ongoing Extension
With .important[Nicolas Jouvin]
### Quadratic Discriminant Analysis
- Relax assumption of common covariance between groups
- Tests Linear vs Quadratic Discriminant
### Regularized Discriminant Analysis
- Ridge-like regularized Covariance
- Sparse Covariance
- Block-wise Covariance
`\(\rightsquigarrow\)` better assess the structure of the sub-population
---
# PLN PCA: Ongoing Extension
With .important[Nicolas Jouvin]
## Mixture of PLN PCA
- Assume several sub-population in the latent space
- Each sub-population has is represented by a different linear combinasion of the initial variables
- In the PLN setup, several variational approximation are possible
## Functional PCA
PhD of Barbara Bricout (S. Robin, S. Donnet)
Features are time points
---
# Temporal/Spatial dependencies
With .important[Stéphane Robin, Mahendra Mariadassou]
## 'In-row' dependency structure
Consider, e.g., a multivariate Gaussian auto-regressive process on the latent variable `\(\mathbf{Z}\)`
`$$\mathbf{Z}_1 \sim \mathcal{N}(0, {\boldsymbol\Sigma}), \quad \mathbf{Z}_t = \mathbf{A} \mathbf{Z}_{t-1} + {\boldsymbol\varepsilon}_t \quad
({\boldsymbol\varepsilon})_t \text{ iid } \sim \mathcal{N}(0, \Delta^{-1})$$`
The counts are described by `\((\mathbf{Y}_t) \in \mathbb{N}^p\)` with time `\(t\geq 0\)`
`$$(\mathbf{Y}_t) \text{ indep. } \mathbf{Z}_t: \quad \mathbf{Y}_t \sim \mathcal{P}\left(\exp\left\{ \mathbf{x}_t^\top \mathbf{B} + \mathbf{Z}_t\right\}\right)$$`
`\(\rightsquigarrow\)` Camille Mondon's MsC
## Inference
Based on Kalman Filter in the latent layer + variational approximation
---
# A zero-inflated PLN
With .important[Bastien Batardière, Mahendra Mariadassou]
### Motivations
- account for a large amount of zero, i.e. with single-cell data,
- try to separate "true" zeros from "technical"/dropouts
### The Model
Use two latent vectors `\(\mathbf{W}_i\)` and `\(\mathbf{Z}_i\)` to model excess of zeroes and dependence structure
`$$\begin{aligned}
\mathbf{Z}_i & \sim \mathcal{N}({\mathbf{o}_i + \mathbf{x}_i^\top\mathbf{B}},\boldsymbol\Sigma) \\
W_{ij} & \sim \mathcal{B}(\text{logit}^{-1}(\mathbf{x}_i^\top{\mathbf{B}}_j^0)) \\
Y_{ij} \, | \, W_{ij}, Z_{ij} & \sim W_{ij}\delta_0 + (1-W_{ij}) \mathcal{P}\left(\exp\{Z_{ij}\}\right), \\
\end{aligned}$$`
The unkwown parameters are
- `\(\mathbf{B}\)`, the regression parameters (from the PLN component)
- `\(\mathbf{B}^0\)`, the regression parameters (from the Bernoulli component)
- `\(\boldsymbol\Sigma\)`, the variance-covariance matrix
`\(\rightsquigarrow\)` ZI-PLN is a mixture of PLN and Bernoulli distribution with shared covariates.
---
# ZI-PLN Inference
### Variational approximation 1.
`\begin{equation*}
p(\mathbf{Z}_i, \mathbf{W}_i \mathbf{Y}_i) \approx q_{\psi}(\mathbf{Z}_i, \mathbf{W}_i) \approx q_{\psi_1}(\mathbf{Z}_i) q_{\psi_2}(\mathbf{W}_i)
\end{equation*}`
with
`\begin{equation*}
q_{\psi_1}(\mathbf{Z}_i) = \mathcal{N}(\mathbf{Z}_i; \mathbf{m}_{i}, \mathrm{diag}(\mathbf{s}_{i} \circ \mathbf{s}_{i})), \qquad q_{\psi_2}(\mathbf{W}_i) = \otimes_{j=1}^p \mathcal{B}(W_{ij}, \pi_{ij})
\end{equation*}`
.important[Tested, works _partially_] Too rough variational approximation
### Variational approximation 1.
`\begin{equation*}
p(\mathbf{Z}_i, \mathbf{W}_i \mathbf{Y}_i) \approx q_{\psi}(\mathbf{Z}_i, \mathbf{W}_i) \approx q_{\psi_1}(\mathbf{Z}_i|\mathbf{W}_i) q_{\psi_2}(\mathbf{W}_i)
\end{equation*}`
`\begin{equation*}
q_{\psi_1}(\mathbf{Z}_i | \mathbf{W}_i) = \text{ mixture of Gaussians}, \quad q_{\psi_2}(\mathbf{W}_i) = \otimes_{j=1}^p \mathcal{B}(W_{ij}, \pi_{ij})
\end{equation*}`
---
# Conclusion
## Summary
- PLN = generic model for multivariate count data analysis
- Flexible modeling of the covariance structure, allows for covariates
- Efficient V-EM algorithm
## Advertisement
[https://computo.sfds.asso.fr](https://computo.sfds.asso.fr), a journal promoting reproducible research in ML and stat.
---
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</textarea>
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