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Welcome to the zinck Research Website!

This website documents the research and experiments related to the zinck package.

Description

zinck exploits a zero-inflated variant of the Latent Dirichlet Allocation (LDA) model to generate valid knockoffs that capture the key characteristics of microbiome data - mainly its compositional nature and high sparsity. It exhibits the properties of simultaneous variable selection and FDR control to identify microbial biomarkers. This package provides an implementation of zinck, which is trained either using the Automatic Differentiation Variational Inference (ADVI) algorithm or using a collapsed Gibbs sampler, facilitating variable selection for both continuous as well as binary outcomes.

The hierarchical structure of zinck

Before going into the structure of zinck, we define the following parameters:

  1. Number of biological samples: \(D\), number of microbial features: \(p\), number of latent clusters \(K\) and the sequencing depth of the \(d^{th}\) sample is given by \(N_d\), where \(d=1,\ldots,D\).

  2. \(\boldsymbol{\theta}_{d} = (\theta_{d1},\ldots,\theta_{dK})'\) is the vector of cluster mixing probabilities for the \(d^{th}\) biological sample, where \(d \in 1,2,\ldots,D\)

  3. \(\boldsymbol{\beta}_{k} = (\beta_{k1},\ldots,\beta_{k(p-1)})'\) and \(\boldsymbol{\beta}_{kp}=1-\sum_{i=1}^{p-1}\beta_{ki}\) is the vector of feature proportions for each cluster \(k=1,2,\ldots,K\).

  4. \(\mathbf{z}_{d}=\left(z_{d1},\ldots,z_{dN_{d}}\right)'\) is the vector of cluster assignments for the \(d^{th}\) biological sample. For instance, \(z_{dn}=k\) implies that the \(n^{th}\) sequencing read in the \(d^{th}\) sample belongs to the \(k^{th}\) cluster.

  5. \(\mathbf{w}_{d}=\left(w_{d1},\ldots,w_{dN_{d}}\right)'\) is the vector of the features drawn for each sequencing read for the \(d^{th}\) biological sample.

  6. \(\boldsymbol{\alpha}=(\alpha,\ldots,\alpha)^{K \times 1}\): symmetric hyperparameter of the Dirichlet prior of \(\boldsymbol{\theta}_{d}\).

  7. \(\boldsymbol{\pi}_{k}=(\pi_{k1},\ldots,\pi_{k(p-1)})'\): hyperparameter of the ZIGD distribution specifying the probability of being a structural zero for the \(j^{th}\) feature in the \(k^{th}\) subcommunity, where \(j=1,\ldots,p\) and \(k=1,\ldots,K\).

  8. \(\mathbf{a}=(a,\ldots,a)^{(p-1) \times 1}\) and \(\mathbf{b}=(b, \ldots, b)^{(p-1) \times 1}\) are the symmetric hyperparameter vectors on the ZIGD of \(\boldsymbol{\beta}_{k}\).

It is to be noted that ZIGD here refers to the zero-inflated Generalized Dirichlet distribution.

zinck is a probabilistic hierarchical model with the following specification:

\[ \begin{aligned} w_{dn}|z_{dn},\boldsymbol{\beta}_{z_{dn}} & \sim \text{Multinomial}(\boldsymbol{\beta}_{z_{dn}}) \\ \boldsymbol{\beta}_{z_{dn}}|\boldsymbol{\pi},\mathbf{a},\mathbf{b} & \sim \text{ZIGD}\left(\pi_{z_{dn}},a,b\right) \\ z_{dn}|\theta_{d} & \sim \text{Multinomial}(\theta_{d}) \\ \boldsymbol{\theta}_{d} & \sim \text{Dirichlet}(\boldsymbol{\alpha}) \end{aligned} \]

We denote the elements of the observed sample taxa matrix \(\mathbf{X}^{D \times p}\) by \((x_{dj})\) as the observed read count of the \(j^{th}\) taxon for the \(d^{th}\) subject: \[x_{dj}=\sum_{n=1}^{N_d} \mathbb{1}_{\{w_{dn}=j\}}\]

Generating knockoffs using zinck

We can exploit the knockoff generative model to learn the structure of the sample-feature count matrix \(\mathbf{X}\) and generate a valid knockoff copy, which is further used for FDR-controlled feature selection. We fit the augmented LDA model to the microbiome count data matrix \(\mathbf{X}\). The latent parameters of the model namely, \(\theta_{d}\) and \(\beta_{k}\) are learnt by approximating their joint posterior distribution via the Automatic Differentiation Variational Inference (ADVI) algorithm or by drawing MCMC samples using a Collapsed Gibbs sampler. We use the learnt parameters \(\tilde{\boldsymbol{\theta}}_d, \tilde{\boldsymbol{\beta}}_k\), for \(d=1,2,\ldots D\) and \(k=1,2,\ldots,K\) to generate a knockoff copy. For each sample \(d\), we first sample a cluster allocation \(z_{dn}\) from \(\text{Multinomial}(1,\tilde{\boldsymbol{\theta}}_d)\) and then we sample a feature \(w_{dn}\) from the selected cluster \(z_{dn}\), that is, \(w_{dn} \sim \text{Multinomial}(1,\tilde{\boldsymbol{\beta}}_{z_{dn}})\). Finally, we form the knockoff matrix \(\tilde{\mathbf{X}}^{D \times p} = \{\tilde{x}_{dj}\}\) by cumulating all the taxa read counts per subjects as illustrated previously.

Schematic Diagram for zinck

The figure below illustrates the hierarchical structure of the zinck model:

Nodes represent random variables, and edges indicate dependencies. The shaded node is observed, while the non-shaded nodes are latent. Replicated variables are indicated by plates: the outermost plate, labeled with \(D\), signifies multiple biological samples, each indexed by \(d\), and the inner plate, marked by \(N_d\), represents the sequencing reads within each sample. The plate on the right, labeled with \(K\), indicates replication across clusters.