Algorithm Principles in MOFSR

Algorithm Classification

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)

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.

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:

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

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.

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: \(c^{(v)}_i | c_i \sim \text{Multinomial}(\theta_{c_i}^{(v)})\)

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)