imputeCellEM: Cellwise-robust EM imputation for mixed data

In VIM: Visualization and Imputation of Missing Values

Cellwise-robust EM imputation for mixed data

Description

EM algorithm with latent contamination indicators that jointly estimates clean distribution parameters and identifies cellwise outliers. Each continuous cell has a posterior probability of being clean vs contaminated. The clean distribution is modeled as multivariate normal for continuous variables, with categorical variables handled via conditional multinomial logistic regression.

Usage

imputeCellEM(
  data,
  maxit_em = 100,
  eps_em = 0.005,
  gamma_init = 3,
  eps_init = 0.1,
  uncert = "conditional",
  conditioning = "weighted",
  trust_min = 0.5,
  trace = FALSE
)

Arguments

data	data.frame with missing values (mixed continuous + categorical).
maxit_em	maximum EM iterations (default: 100).
eps_em	convergence tolerance on the relative change in estimated parameters (mu, Sigma). Default: 5e-3.
gamma_init	initial scale inflation factor for the contamination distribution. Contaminated cells are modeled as having variance (gamma * sigma)^2 with gamma > 1. Default: 3.
eps_init	initial contamination probability per variable (default: 0.1). Must be in (0, 0.5).
uncert	imputation uncertainty method: "conditional" (default) draws from the conditional normal distribution, or "pmm" uses predictive mean matching.
conditioning	how observed cells enter the E-step conditional moments. "weighted" (default) multiplies each conditioning deviation x_{ik} - \mu_k by its current cell weight w_{ik} – the posterior-expected clean deviation, so likely contaminated cells are shrunk towards the mean before they enter the conditional. "trust" conditions only on trusted cells: deviations with w_{ik} < trust_min are set to zero (hard threshold). "unweighted" uses the raw deviations (all observed cells fully trusted, as in early versions).
trust_min	trust threshold in (0, 1) used when conditioning = "trust" (default: 0.5). Ignored otherwise.
trace	logical; if TRUE, print progress information.

Details

The algorithm proceeds as follows:

1. Initialization. Missing values are filled by initialise (medians for continuous, modes for categorical). Initial location and scale are estimated robustly (median and MAD). Cell weights are initialized to 1.

2. E-step (for each continuous variable j, each observation i):

   • Compute the conditional mean and variance of x_{ij} given the other continuous variables, using the current \mu and \Sigma.

   • For observed cells: compute the posterior probability that the cell is clean vs contaminated, yielding cell weight w_{ij}.

   • For missing cells: impute from the conditional distribution (with optional PMM).

3. M-step:

   • Update \mu: cell-weighted mean.

   • Update \Sigma: pairwise cell-weighted covariance using w_{ij} \cdot w_{ik} (not row-level min).

   • Update contamination rates: \varepsilon_j = 1 - \mathrm{mean}(w_{ij}) over observed cells.

   • Update contamination scale \gamma_j from weighted variance of contaminated cells.

   • For categorical variables: fit weighted multinomial logistic with row weights derived from continuous cell weights.

4. Convergence. Check relative change in estimated parameters (\mu, \Sigma); stop when below eps_em or after maxit_em iterations. The observed-data log-likelihood (sum of per-variable conditional mixture log-likelihoods) is tracked for diagnostics.

This method differs from cellGMM (Zaccaria et al., 2025) in using a single clean component rather than a mixture of clean clusters, and in supporting mixed continuous + categorical data.

The pseudo_loglik component is a composite (pseudo) log-likelihood: the sum of per-variable conditional mixture log-likelihoods, not the proper observed-data joint log-likelihood. It is useful for monitoring convergence but should not be compared across models or used for model selection criteria such as AIC/BIC.

This implementation is an ECM (Expectation Conditional Maximization) variant rather than a pure EM algorithm, because the conditional variance in the E-step is computed using the updated Sigma from the current M-step rather than the Sigma from the previous iteration. As a result, strict log-likelihood monotonicity is not guaranteed, but is observed empirically in practice.

Value

A list with components:

data_imputed	the imputed data.frame
cellweights	n x p matrix of posterior clean probabilities. Continuous observed cells have values in [0, 1]; missing cells and categorical columns have weight 1.
mu	estimated clean location vector (continuous variables only)
Sigma	estimated clean covariance matrix (continuous variables only)
epsilon	named numeric vector of estimated per-variable contamination rates (continuous variables only)
converged	logical indicating convergence
iterations	number of EM iterations performed
pseudo_loglik	numeric vector of composite (pseudo) log-likelihood values, one per iteration (computed after each M-step). This is a sum of per-variable conditional mixture log-likelihoods, not the proper observed-data joint log-likelihood.

Note

Model uncertainty via bootstrap (Rubin's combining rules for multiple imputation) is not yet implemented. The current version provides single imputation with stochastic uncertainty (PMM or residual draw). For valid multiple imputation, call the function repeatedly with different seeds and combine using Rubin's rules.

Author(s)

Matthias Templ

References

A.P. Dempster, N.M. Laird, D.B. Rubin (1977) Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society: Series B, 39(1), 1–38.

C.F.J. Wu (1983) On the Convergence Properties of the EM Algorithm. The Annals of Statistics, 11(1), 95–103.

J. Raymaekers, P.J. Rousseeuw (2024) The cellwise minimum covariance determinant estimator. Journal of the American Statistical Association, 119(548), 2610–2621.

G. Zaccaria, L.A. Garcia-Escudero, F. Greselin, A. Mayo-Iscar (2025) Cellwise outlier detection in heterogeneous populations. Technometrics, 67(4), 643–654.

M. Templ, A. Kowarik, P. Filzmoser (2011) Iterative stepwise regression imputation using standard and robust methods. Computational Statistics & Data Analysis, Vol. 55, pp. 2793–2806.

See Also

imputeCellIRMI, imputeCellM, initialise, irmi

Other imputation methods: hotdeck(), impPCA(), imputeCellIRMI(), imputeCellM(), imputeCellMCD(), imputeCellwise(), imputeRobust(), imputeRobustChain(), irmi(), kNN(), matchImpute(), medianSamp(), rangerImpute(), regressionImp(), sampleCat(), vimmi, vimpute(), xgboostImpute()

Examples

data(sleep, package = "VIM")
result <- imputeCellEM(sleep)
head(result$data_imputed)

# Inspect estimated contamination rates
result$epsilon

# Cell weight matrix (1 = clean, low = likely contaminated)
image(result$cellweights, main = "Cell weights")

# Log-likelihood trace
plot(result$pseudo_loglik, type = "b", xlab = "Iteration",
     ylab = "Pseudo log-likelihood")

# With predictive mean matching for imputation
result2 <- imputeCellEM(sleep, uncert = "pmm", trace = TRUE)

# Mixed data example
data(testdata)
result3 <- imputeCellEM(testdata$wna)

VIM documentation built on Sept. 2, 2026, 5:07 p.m.