Algorithm Principles in MOFSR

Zaoqu Liu

2026-01-29

Overview

MOFSR implements 15 state-of-the-art multi-omics clustering algorithms. This vignette provides theoretical background and mathematical foundations for each method.

Algorithm Classification

Paradigm	Algorithms	Key Idea
Network Fusion	SNF, wSNF, CIMLR, NEMO	Construct similarity networks and fuse at network level
Matrix Factorization	IntNMF, LRAcluster	Joint non-negative or low-rank decomposition
Factor Analysis	MOFA, MCIA	Latent factor extraction across views
Canonical Correlation	RGCCA, SGCCA, CPCA	Maximize correlations between views
Bayesian	iClusterBayes, BCC	Probabilistic joint modeling
Ensemble	PINSPlus, Late Fusion	Combine view-specific clusterings

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Network-Based Methods

Similarity Network Fusion (SNF)

Reference: Wang et al., Nature Methods, 2014

SNF constructs patient similarity networks for each omics type and iteratively fuses them into a unified network.

Mathematical Framework:

1. Affinity Matrix Construction: For data matrix $X^{(v)}$ of view $v$:

$$W^{(v)}_{ij} = \frac{\exp(-d_{ij}^2/\mu\epsilon_{ij})}{\sum_{k \in N_i} \exp(-d_{ik}^2/\mu\epsilon_{ik})}$$

where $d_{ij}$ is the Euclidean distance between samples $i$ and $j$, $N_i$ is the set of $K$ nearest neighbors, and $\epsilon_{ij} = \frac{\text{mean}(d(i,N_i)) + \text{mean}(d(j,N_j)) + d_{ij}}{3}$.

2. Network Normalization: Construct transition probability matrix:

$$P^{(v)} = D^{-1}W^{(v)}$$

3. Iterative Fusion: Update networks until convergence:

$$P^{(v)}_{t+1} = S^{(v)} \times \frac{\sum_{u \neq v} P^{(u)}_t}{|V|-1} \times (S^{(v)})^T$$

where $S^{(v)}$ is the normalized kernel matrix.

library(MOFSR)
set.seed(42)

# Generate sample data
n <- 50
data_list <- list(
  omics1 = matrix(rnorm(n * 100), 100, n),
  omics2 = matrix(rnorm(n * 80), 80, n)
)
colnames(data_list$omics1) <- colnames(data_list$omics2) <- paste0("S", 1:n)

# Step 1: Compute affinity matrices
W1 <- snf_affinity_matrix(data_list$omics1, K = 10)
W2 <- snf_affinity_matrix(data_list$omics2, K = 10)

# Step 2: Fuse networks
W_fused <- snf_fuse(list(W1, W2), K = 10, t = 20)

# Step 3: Spectral clustering
clusters <- spectral_clustering(W_fused, n_clusters = 3)
table(clusters)

CIMLR (Cancer Integration via Multi-kernel Learning)

Reference: Ramazzotti et al., Nature Communications, 2018

CIMLR extends SNF with multiple kernel learning to automatically learn optimal kernel combinations.

Key Innovation: Instead of a single Gaussian kernel, CIMLR uses:

$$K = \sum_{l=1}^{L} \beta_l K_l$$

where $K_l$ are Gaussian kernels with different bandwidths and $\beta_l$ are learned weights.

# Run CIMLR
result_cimlr <- run_cimlr(data_list, n_clusters = 3, n_cores = 1)
head(result_cimlr)

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Matrix Factorization Methods

Integrative Non-negative Matrix Factorization (IntNMF)

Reference: Chalise & Fridley, PLoS ONE, 2017

IntNMF simultaneously factorizes multiple non-negative data matrices while sharing a common factor matrix.

Objective Function:

$$\min_{W,H^{(v)}} \sum_{v=1}^{V} \lambda_v \|X^{(v)} - WH^{(v)}\|_F^2$$

subject to $W \geq 0$, $H^{(v)} \geq 0$

where: - $X^{(v)} \in \mathbb{R}^{n \times p_v}$: Data matrix for view $v$ - $W \in \mathbb{R}^{n \times K}$: Shared sample factor matrix - $H^{(v)} \in \mathbb{R}^{K \times p_v}$: View-specific feature factor matrix

Update Rules (multiplicative):

$$W_{ik} \leftarrow W_{ik} \frac{\sum_v \lambda_v (X^{(v)}H^{(v)T})_{ik}}{\sum_v \lambda_v (WH^{(v)}H^{(v)T})_{ik}}$$

$$H^{(v)}_{kj} \leftarrow H^{(v)}_{kj} \frac{(W^TX^{(v)})_{kj}}{(W^TWH^{(v)})_{kj}}$$

# IntNMF requires non-negative data
pos_data <- lapply(data_list, function(x) abs(x))

# Run IntNMF
result_intnmf <- intnmf_cluster(pos_data, n_clusters = 3, max_iter = 100)
table(result_intnmf$cluster)

LRAcluster (Low-Rank Approximation)

Reference: Wu et al., Cancer Informatics, 2015

LRAcluster uses low-rank approximation with different noise models for different data types.

Framework: For count data (e.g., RNA-seq), assume Poisson distribution:

$$X_{ij} \sim \text{Poisson}(\mu_{ij})$$$$\log(\mu) = UV^T$$

where $U$ and $V$ are low-rank factors.

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Factor Analysis Methods

Multi-Omics Factor Analysis (MOFA)

MOFSR implements a simplified variational inference approach inspired by MOFA.

Generative Model:

$$X^{(v)} = W^{(v)}Z + \epsilon^{(v)}$$

where: - $Z \in \mathbb{R}^{K \times n}$: Latent factors - $W^{(v)} \in \mathbb{R}^{p_v \times K}$: View-specific loadings - $\epsilon^{(v)}$: Gaussian noise

Prior Structure (ARD - Automatic Relevance Determination):

$$W^{(v)}_{jk} \sim \mathcal{N}(0, 1/\alpha^{(v)}_k)$$$$\alpha^{(v)}_k \sim \text{Gamma}(a_0, b_0)$$

This allows automatic selection of relevant factors per view.

# Run factor analysis
result_mofa <- multi_view_factor_analysis(data_list, n_factors = 5, max_iter = 100)

# Examine variance explained
cat("Variance explained per factor:\n")
print(round(result_mofa$variance_explained, 3))

Multiple Co-Inertia Analysis (MCIA)

Reference: Meng et al., BMC Bioinformatics, 2014

MCIA extends co-inertia analysis to multiple tables, finding common structures across omics.

Objective: Maximize the sum of squared covariances between the global scores and individual table scores:

$$\max_{\mathbf{f}} \sum_{v=1}^{V} \text{Cov}^2(\mathbf{f}, X^{(v)}\mathbf{q}^{(v)})$$

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Canonical Correlation Methods

RGCCA (Regularized Generalized CCA)

Reference: Tenenhaus & Tenenhaus, Psychometrika, 2011

RGCCA finds latent components that maximize the sum of correlations between connected blocks.

Optimization Problem:

$$\max_{a^{(1)}, \ldots, a^{(V)}} \sum_{j < k} c_{jk} g(\text{Cov}(X^{(j)}a^{(j)}, X^{(k)}a^{(k)}))$$

subject to $(1-\tau_j)\text{Var}(X^{(j)}a^{(j)}) + \tau_j\|a^{(j)}\|^2 = 1$

where $c_{jk}$ defines the design matrix (which blocks are connected) and $\tau_j$ is the regularization parameter.

# Run RGCCA
result_rgcca <- run_rgcca(data_list, n_components = 3, n_clusters = 3)
head(result_rgcca)

SGCCA (Sparse GCCA)

SGCCA adds L1 penalty for feature selection:

$$\|a^{(j)}\|_1 \leq s_j$$

This induces sparsity in the loadings, selecting relevant features.

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Bayesian Methods

iClusterBayes

Reference: Mo et al., Biostatistics, 2018

iClusterBayes uses a Bayesian framework with Gibbs sampling.

Model:

$$X^{(v)} = W^{(v)}Z + \epsilon^{(v)}$$

with priors: - $W^{(v)}_{jk} \sim \mathcal{N}(0, \sigma^2_w)$ - $Z_{ki} \sim \mathcal{N}(0, 1)$ - $\sigma^{-2}_v \sim \text{Gamma}(a, b)$

Bayesian Consensus Clustering (BCC)

Reference: Lock & Dunson, Bioinformatics, 2013

BCC models the relationship between local (view-specific) and global clusters using a Dirichlet-Multinomial framework.

Generative Process:

1. Global cluster: $c_i \sim \text{Multinomial}(\pi)$
2. Local cluster: $c^{(v)}_i | c_i \sim \text{Multinomial}(\theta_{c_i}^{(v)})$

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Ensemble Methods

PINSPlus (Perturbation Clustering)

Reference: Nguyen et al., Genome Research, 2017

PINSPlus uses perturbation-based subsampling to identify stable clusters.

Algorithm:

1. Subsample data with noise perturbation
2. Cluster each subsample
3. Build connectivity matrix
4. Final clustering from connectivity matrix

# Run PINSPlus
result_pins <- run_pinsplus(data_list, n_clusters = 3, n_reps = 20)
head(result_pins)

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Algorithm Selection Guide

Scenario	Recommended Algorithms
High-dimensional, sparse data	CIMLR, SGCCA
Missing data	MOFA, iClusterBayes
Large sample size	SNF, IntNMF
Need feature selection	CIMLR, SGCCA
Interpretable factors	MOFA, MCIA
Robust to noise	PINSPlus, BCC

References

1. Wang B, et al. (2014). Similarity network fusion for aggregating data types on a genomic scale. Nature Methods, 11(3):333-337.
2. Ramazzotti D, et al. (2018). Multi-omic tumor data reveal diversity of molecular mechanisms underlying survival. Nature Communications, 9(1):4453.
3. Mo Q, et al. (2018). A fully Bayesian latent variable model for integrative clustering analysis of multi-type omics data. Biostatistics, 19(1):71-86.
4. Chalise P, Fridley BL (2017). Integrative clustering of multi-level ’omic data based on non-negative matrix factorization algorithm. PLoS ONE, 12(5):e0176278.
5. Tenenhaus A, Tenenhaus M (2011). Regularized generalized canonical correlation analysis. Psychometrika, 76(2):257-284.

Session Info

sessionInfo()