Mixture-Based Muscle Synergy Analysis (MFA & MPCA)

Yusuke Matsui (Nagoya University)

1. Background

Muscle synergy analysis is often applied to reduce high-dimensional electromyographic (EMG) signals into a smaller set of “synergies” or activation patterns. Two mixture-based methods for identifying group-level synergies in heterogeneous populations are:

Mixture Factor Analysis (MFA): Assumes each cluster’s covariance is decomposed by a Factor Analysis model with diagonal noise.
Mixture PCA (MPCA): Assumes each cluster is modeled by a probabilistic PCA approach with isotropic noise.

Both approaches can handle heterogeneous data, grouping subjects (or trials) into different synergy “clusters,” each with its own synergy loadings.

Compared to a single global factor/PCA decomposition, mixture-based approaches can uncover fundamentally different synergy patterns in subpopulations (e.g. half the subjects might use synergy pattern A, the other half synergy pattern B)

2. Method Overview and Effective Use Cases

2.1 Mixture Factor Analysis (MFA)

Cluster membership Each subject i belongs to one of K clusters with probability pi_k.
Factor model per cluster Within cluster k, each subject’s data X_i (dimension time x M) follows a low-dimensional factor model: X_i(t) = mu_k + Lambda_k * Z_i(t) + epsilon_{i,t}, where mu_k is the mean vector, Lambda_k is (M x r) factor loadings (synergy matrix), and epsilon_{i,t} ~ N(0, Psi_k) with Psi_k diagonal.

2.2 Mixture PCA (MPCA)

Similarly, we can define a mixture of probabilistic PCA (PPCA) models:

Cluster membership Each subject i belongs to one of K clusters with probability pi_k.
PCA model per cluster Data X_i(t) is modeled with X_i(t) = mu_k + W_k * Z_i(t) + epsilon_{i,t}, where W_k is (M x r). The noise is isotropic sigma^2_k * I.

Use cases:

Heterogeneous populations E.g. subsets of participants with distinct synergy structures.
Large datasets Enough participants/trials to fit multiple clusters.
Exploratory approach You suspect more than one synergy pattern set in the population.

3. Data (Input) Format

All the provided functions assume:

list_of_data: A list of length N, where each element is a (T_i x M) matrix:
- N: number of subjects/trials.
- T_i: time points for subject i.
- M: number of EMG channels (muscles).

Example:

N <- 10
list_of_data <- vector("list", N)
for (i in seq_len(N)) {
  T_i <- 100
  M   <- 8
  X_i <- matrix(rnorm(T_i * M), nrow=T_i, ncol=M)
  list_of_data[[i]] <- X_i
}

4. Single Mixture Model Estimation

4.1 Mixture Factor Analysis (MFA)

Use mfa_em_fit() to fit an MFA model for fixed K (clusters) and r (factors):

library(synergyMixR)

# Suppose we want K=2 clusters, r=3 factors
fit_mfa <- mfa_em_fit(
  list_of_data = list_of_data,
  K = 2,
  r = 3,
  max_iter = 50,
  nIterFA  = 5,
  tol      = 1e-3,
  n_init   = 1,
  use_kmeans_init = TRUE
)

This returns a fitted MFA object with elements like $z (cluster assignments), $Lambda, $mu, $Psi, etc.

4.2 Mixture PCA (MPCA)

Use mixture_pca_em_fit() similarly:

fit_mpca <- mixture_pca_em_fit(
  list_of_data = list_of_data,
  K = 2,
  r = 3,
  max_iter = 50,
  nIterPCA = 20,   # sub-iterations for the PCA updates
  tol = 1e-3,
  method = "EM",   # or "closed_form"
  n_init = 1,
  use_kmeans_init = TRUE
)

This returns an MPCA object, typically with $z, $W or $P/$D, $mu, $sigma2, $logLik, etc.

5. Parameter Meanings and Typical Settings

K (clusters): e.g., 1 to 5 for an exploratory synergy analysis.
r (synergy/factor dimension): typically 1-5 for muscle synergy contexts.
max_iter: outer EM iteration limit, e.g. 50-100.
nIterFA (MFA) / nIterPCA (MPCA): sub-iterations for each cluster’s parameter updates.
tol: convergence tolerance, typically ~1e-3.
n_init: number of random initializations. More helps avoid local minima.
use_kmeans_init: if TRUE, tries a k-means-based assignment to seed EM.

6. Visualizing the Fitted Model

6.1 MFA: Synergy Loadings & Factor Scores

After mfa_em_fit(), you can visualize synergy loadings:

# Heatmap of factor loadings
p_loadings <- plot_cluster_synergy_loadings_mfa(fit_mfa, plot_type="heatmap")
print(p_loadings)

# Factor scores (time evolution) per subject, faceted by cluster & factor
fit_mfa <- compute_factor_scores_mfa(fit_mfa, list_of_data)
p_scores <- plot_all_factor_scores_mfa(fit_mfa, list_of_data, overlay_subjects=TRUE)
print(p_scores)

6.2 MPCA: Loadings & Factor Scores

Similar calls for MPCA:

# Heatmap of loadings W
p_loadings_mpca <- plot_cluster_synergy_loadings_mpca(fit_mpca, plot_type="heatmap")
print(p_loadings_mpca)

# Factor scores
fit_mpca <- compute_factor_scores_mpca(fit_mpca, list_of_data)
p_scores_mpca <- plot_all_factor_scores_mpca(fit_mpca, list_of_data, overlay_subjects=TRUE)
print(p_scores_mpca)

7. Optimal Parameter Search (K, r)

If you do not know the best (K, r) in advance, you can search over a grid:

7.1 MFA Grid Search

search_res_mfa <- select_optimal_K_r_mfa(
  list_of_data,
  Kvec = 1:5,
  rvec = 1:5,
  max_iter = 50,
  nIterFA  = 5,
  tol = 1e-3,
  n_init = 1,
  use_kmeans_init = TRUE
)

search_res_mfa$summary         # data.frame with K, r, logLik, BIC
search_res_mfa$best_model_info # best (K, r)

7.2 MPCA Grid Search

search_res_mpca <- select_optimal_K_r_mpca(
  list_of_data,
  Kvec = 1:5,
  rvec = 1:5,
  max_iter = 50,
  nIterPCA = 20,
  tol = 1e-3,
  method = "EM",
  n_init = 1,
  use_kmeans_init = TRUE
)

search_res_mpca$summary
search_res_mpca$best_model_info

Both select_optimal_K_r_mfa() and select_optimal_K_r_mpca() compute a BIC for each combination, returning the best by BIC.

8. Refining the Chosen Model

Once you identify the best (K, r), you can:

Extract the best model:

best_fit_mfa <- search_res_mfa$best_model_info$model
best_fit_mpca <- search_res_mpca$best_model_info$model

Optionally re-run with more initializations, stricter tolerances, or greater iteration limits.

9. Closing Remarks

Interpretation: Always check whether synergy loadings make physiological sense. Mixed-subject synergy analysis can reveal distinct synergy “clusters” that might be lost in a single global decomposition.
Computation: Large (K, r) grids and big data may require parallelization or focusing on smaller parameter ranges.
Rotation: Varimax or similar methods can improve interpretability by making synergy loadings sparser or more distinct.

Using MFA or MPCA in a mixture setting can substantially enhance your ability to discover and interpret distinct synergy patterns in heterogeneous EMG datasets.

Mixture-Based Muscle Synergy Analysis (MFA & MPCA)

Yusuke Matsui

2026-08-14