Cellwise-robust EM imputation for mixed data

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.

imputeCellEM(
  data,
  maxit_em = 100,
  eps_em = 0.005,
  gamma_init = 3,
  eps_init = 0.1,
  uncert = "conditional",
  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.

trace

logical; if TRUE, print progress information.

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.

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.

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.

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.

C. Raymaekers, P.J. Rousseeuw (2024) The cellwise minimum covariance determinant estimator. Journal of the American Statistical Association, 119(545), 576–588.

G. Zaccaria, L. Insolia, A. Farcomeni (2025) Robust model-based clustering with cellwise contamination. Technometrics, forthcoming.

M. Templ, A. Kowarik, P. Filzmoser (2011) Iterative stepwise regression imputation using standard and robust methods. Journal of Computational Statistics and 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()

Author

Matthias Templ

Examples

# \donttest{
data(sleep, package = "VIM")
result <- imputeCellEM(sleep)
head(result$data_imputed)
#>    BodyWgt BrainWgt     NonD    Dream Sleep      Span Gest Pred Exp Danger
#> 1 6654.000   5712.0 261.5162 34.24212   3.3  38.60000  645    3   5      3
#> 2    1.000      6.6   6.3000  2.00000   8.3   4.50000   42    3   1      3
#> 3    3.385     44.5 261.5160 34.24170  12.5  14.00000   60    1   1      1
#> 4    0.920      5.7 261.5161 34.24225  16.5 -63.57845   25    5   2      3
#> 5 2547.000   4603.0   2.1000  1.80000   3.9  69.00000  624    3   5      4
#> 6   10.550    179.5   9.1000  0.70000   9.8  27.00000  180    4   4      4

# Inspect estimated contamination rates
result$epsilon
#>  BodyWgt BrainWgt     NonD    Dream    Sleep     Span     Gest     Pred 
#>     0.49     0.49     0.49     0.49     0.49     0.49     0.49     0.49 
#>      Exp   Danger 
#>     0.49     0.49 

# 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)
#> --------------------------------------
#> cellEM: start of iteration 1
#>   log-likelihood: -3373.3739
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 1.703 ]
#> --------------------------------------
#> cellEM: start of iteration 2
#>   log-likelihood: -3517.4204
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 2.5461 ]
#>   parameter change: 0.749186
#> --------------------------------------
#> cellEM: start of iteration 3
#>   log-likelihood: -3355.8822
#>   epsilon range: [ 0.2149 , 0.49 ]
#>   gamma range: [ 1.01 , 2.962 ]
#>   parameter change: 0.756899
#> --------------------------------------
#> cellEM: start of iteration 4
#>   log-likelihood: -3408.1962
#>   epsilon range: [ 0.4668 , 0.49 ]
#>   gamma range: [ 1.01 , 3.7806 ]
#>   parameter change: 13.0363
#> --------------------------------------
#> cellEM: start of iteration 5
#>   log-likelihood: -3218.6519
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 6.0229 ]
#>   parameter change: 1.14561
#> --------------------------------------
#> cellEM: start of iteration 6
#>   log-likelihood: -3368.491
#>   epsilon range: [ 0.4836 , 0.49 ]
#>   gamma range: [ 1.01 , 68.1605 ]
#>   parameter change: 1.11707
#> --------------------------------------
#> cellEM: start of iteration 7
#>   log-likelihood: -3131.3632
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 99.4345 ]
#>   parameter change: 0.836874
#> --------------------------------------
#> cellEM: start of iteration 8
#>   log-likelihood: -3121.1872
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 82.6056 ]
#>   parameter change: 0.752232
#> --------------------------------------
#> cellEM: start of iteration 9
#>   log-likelihood: -3127.0606
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 85.9861 ]
#>   parameter change: 0.686364
#> --------------------------------------
#> cellEM: start of iteration 10
#>   log-likelihood: -2988.812
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 110.533 ]
#>   parameter change: 0.476713
#> --------------------------------------
#> cellEM: start of iteration 11
#>   log-likelihood: -2901.928
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 242.6742 ]
#>   parameter change: 0.801609
#> --------------------------------------
#> cellEM: start of iteration 12
#>   log-likelihood: -2723.5636
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 433.3493 ]
#>   parameter change: 0.893698
#> --------------------------------------
#> cellEM: start of iteration 13
#>   log-likelihood: -2715.4305
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 187.0453 ]
#>   parameter change: 89.1203
#> --------------------------------------
#> cellEM: start of iteration 14
#>   log-likelihood: -2794.0654
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 139.3917 ]
#>   parameter change: 24.9498
#> --------------------------------------
#> cellEM: start of iteration 15
#>   log-likelihood: -2804.1381
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 150.5891 ]
#>   parameter change: 0.999959
#> --------------------------------------
#> cellEM: start of iteration 16
#>   log-likelihood: -2730.7248
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 563.4647 ]
#>   parameter change: 21.5889
#> --------------------------------------
#> cellEM: start of iteration 17
#>   log-likelihood: -2840.0198
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 1209.8366 ]
#>   parameter change: 0.99999
#> --------------------------------------
#> cellEM: start of iteration 18
#>   log-likelihood: -2706.526
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 28.8037 ]
#>   parameter change: 3619.85
#> --------------------------------------
#> cellEM: start of iteration 19
#>   log-likelihood: -2686.6952
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 443.2427 ]
#>   parameter change: 0.999989
#> --------------------------------------
#> cellEM: start of iteration 20
#>   log-likelihood: -2793.5125
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 3044.1807 ]
#>   parameter change: 0.999952
#> --------------------------------------
#> cellEM: start of iteration 21
#>   log-likelihood: -2679.0941
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 732.261 ]
#>   parameter change: 26393.2
#> --------------------------------------
#> cellEM: start of iteration 22
#>   log-likelihood: -2798.6135
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 2.2221 , 3941.6867 ]
#>   parameter change: 0.999995
#> --------------------------------------
#> cellEM: start of iteration 23
#>   log-likelihood: -2618.8729
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.0955 , 461.6639 ]
#>   parameter change: 1308.38
#> --------------------------------------
#> cellEM: start of iteration 24
#>   log-likelihood: -2654.1438
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 787.6043 ]
#>   parameter change: 21.5953
#> --------------------------------------
#> cellEM: start of iteration 25
#>   log-likelihood: -2560.7619
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.1376 , 392.9904 ]
#>   parameter change: 0.999971
#> --------------------------------------
#> cellEM: start of iteration 26
#>   log-likelihood: -2666.0825
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 317.4227 ]
#>   parameter change: 20.4909
#> --------------------------------------
#> cellEM: start of iteration 27
#>   log-likelihood: -2648.6721
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 71.9031 ]
#>   parameter change: 0.0903883
#> --------------------------------------
#> cellEM: start of iteration 28
#>   log-likelihood: -2604.5402
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 236.3047 ]
#>   parameter change: 0.999992
#> --------------------------------------
#> cellEM: start of iteration 29
#>   log-likelihood: -2669.0446
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 58.3517 ]
#>   parameter change: 38.8166
#> --------------------------------------
#> cellEM: start of iteration 30
#>   log-likelihood: -2686.6838
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 177.6893 ]
#>   parameter change: 0.566885
#> --------------------------------------
#> cellEM: start of iteration 31
#>   log-likelihood: -2687.3772
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 460.5568 ]
#>   parameter change: 0.0609497
#> --------------------------------------
#> cellEM: start of iteration 32
#>   log-likelihood: -2647.7021
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 109.9781 ]
#>   parameter change: 0.526468
#> --------------------------------------
#> cellEM: start of iteration 33
#>   log-likelihood: -2655.1597
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 143.6564 ]
#>   parameter change: 0.783629
#> --------------------------------------
#> cellEM: start of iteration 34
#>   log-likelihood: -2604.4498
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 2.7567 , 402.3008 ]
#>   parameter change: 0.905435
#> --------------------------------------
#> cellEM: start of iteration 35
#>   log-likelihood: -2662.2221
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.367 , 66.7333 ]
#>   parameter change: 2.83169
#> --------------------------------------
#> cellEM: start of iteration 36
#>   log-likelihood: -2802.2818
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 28.0276 , 30439.8972 ]
#>   parameter change: 1.04262
#> --------------------------------------
#> cellEM: start of iteration 37
#>   log-likelihood: -2772.7966
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 214.1785 , 7811.8365 ]
#>   parameter change: 0.0494632
#> --------------------------------------
#> cellEM: start of iteration 38
#>   log-likelihood: -2673.1201
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.1376 , 13277.9293 ]
#>   parameter change: 4308.48
#> --------------------------------------
#> cellEM: start of iteration 39
#>   log-likelihood: -2640.2015
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 33.4239 ]
#>   parameter change: 20.4909
#> --------------------------------------
#> cellEM: start of iteration 40
#>   log-likelihood: -2657.597
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 105.8003 ]
#>   parameter change: 0.101301
#> --------------------------------------
#> cellEM: start of iteration 41
#>   log-likelihood: -2581.1636
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 63.4833 ]
#>   parameter change: 0.968997
#> --------------------------------------
#> cellEM: start of iteration 42
#>   log-likelihood: -2637.1747
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 110.2806 ]
#>   parameter change: 3.05662
#> --------------------------------------
#> cellEM: start of iteration 43
#>   log-likelihood: -2810.0767
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 14.7137 , 9303.2774 ]
#>   parameter change: 0.999998
#> --------------------------------------
#> cellEM: start of iteration 44
#>   log-likelihood: -2809.1228
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 218.1305 , 42830.4413 ]
#>   parameter change: 0.884343
#> --------------------------------------
#> cellEM: start of iteration 45
#>   log-likelihood: -2778.1648
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 239.967 , 7581.194 ]
#>   parameter change: 14.0472
#> --------------------------------------
#> cellEM: start of iteration 46
#>   log-likelihood: -2673.2159
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.1376 , 13267.4064 ]
#>   parameter change: 5338.81
#> --------------------------------------
#> cellEM: start of iteration 47
#>   log-likelihood: -2640.2014
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 33.423 ]
#>   parameter change: 20.4909
#> --------------------------------------
#> cellEM: start of iteration 48
#>   log-likelihood: -2655.9748
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 105.734 ]
#>   parameter change: 0.101301
#> --------------------------------------
#> cellEM: start of iteration 49
#>   log-likelihood: -2564.7695
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 63.473 ]
#>   parameter change: 0.969024
#> --------------------------------------
#> cellEM: start of iteration 50
#>   log-likelihood: -2648.6084
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 110.2331 ]
#>   parameter change: 3.05502
#> --------------------------------------
#> cellEM: start of iteration 51
#>   log-likelihood: -2810.0771
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 14.7679 , 9250.3369 ]
#>   parameter change: 0.999998
#> --------------------------------------
#> cellEM: start of iteration 52
#>   log-likelihood: -2809.1226
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 219.1667 , 35098.265 ]
#>   parameter change: 0.885344
#> --------------------------------------
#> cellEM: start of iteration 53
#>   log-likelihood: -2778.1648
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 239.967 , 7457.7999 ]
#>   parameter change: 14.0471
#> --------------------------------------
#> cellEM: start of iteration 54
#>   log-likelihood: -2673.2669
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.1376 , 13262.577 ]
#>   parameter change: 5338.81
#> --------------------------------------
#> cellEM: start of iteration 55
#>   log-likelihood: -2640.2014
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 33.423 ]
#>   parameter change: 20.4909
#> --------------------------------------
#> cellEM: start of iteration 56
#>   log-likelihood: -2650.8499
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 105.7235 ]
#>   parameter change: 0.101301
#> --------------------------------------
#> cellEM: start of iteration 57
#>   log-likelihood: -2558.1975
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 63.4716 ]
#>   parameter change: 0.969028
#> --------------------------------------
#> cellEM: start of iteration 58
#>   log-likelihood: -2649.0562
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 110.227 ]
#>   parameter change: 3.05478
#> --------------------------------------
#> cellEM: start of iteration 59
#>   log-likelihood: -2810.0772
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 14.7759 , 9242.535 ]
#>   parameter change: 0.999998
#> --------------------------------------
#> cellEM: start of iteration 60
#>   log-likelihood: -2809.1225
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 219.3194 , 34678.5007 ]
#>   parameter change: 0.885492
#> --------------------------------------
#> cellEM: start of iteration 61
#>   log-likelihood: -2778.1648
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 239.967 , 7440.3836 ]
#>   parameter change: 14.0471
#> --------------------------------------
#> cellEM: start of iteration 62
#>   log-likelihood: -2673.2737
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.1376 , 13261.9606 ]
#>   parameter change: 5338.81
#> --------------------------------------
#> cellEM: start of iteration 63
#>   log-likelihood: -2640.2014
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 33.423 ]
#>   parameter change: 20.4909
#> --------------------------------------
#> cellEM: start of iteration 64
#>   log-likelihood: -2650.0311
#>   epsilon range: [ 0.49 , 0.49 ]
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#>   parameter change: 0.101301
#> --------------------------------------
#> cellEM: start of iteration 65
#>   log-likelihood: -2560.5102
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 63.4714 ]
#>   parameter change: 0.969028
#> --------------------------------------
#> cellEM: start of iteration 66
#>   log-likelihood: -2649.1273
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 110.2261 ]
#>   parameter change: 3.05475
#> --------------------------------------
#> cellEM: start of iteration 67
#>   log-likelihood: -2810.0772
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 14.7771 , 9241.3887 ]
#>   parameter change: 0.999998
#> --------------------------------------
#> cellEM: start of iteration 68
#>   log-likelihood: -2809.1225
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 219.3418 , 34597.1322 ]
#>   parameter change: 0.885513
#> --------------------------------------
#> cellEM: start of iteration 69
#>   log-likelihood: -2778.1648
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 239.967 , 7437.8418 ]
#>   parameter change: 14.0471
#> --------------------------------------
#> cellEM: start of iteration 70
#>   log-likelihood: -2673.2747
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.1376 , 13261.8734 ]
#>   parameter change: 5338.81
#> --------------------------------------
#> cellEM: start of iteration 71
#>   log-likelihood: -2640.2014
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 33.423 ]
#>   parameter change: 20.4909
#> --------------------------------------
#> cellEM: start of iteration 72
#>   log-likelihood: -2649.9102
#>   epsilon range: [ 0.49 , 0.49 ]
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#>   parameter change: 0.101301
#> --------------------------------------
#> cellEM: start of iteration 73
#>   log-likelihood: -2560.7924
#>   epsilon range: [ 0.49 , 0.49 ]
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#>   parameter change: 0.969028
#> --------------------------------------
#> cellEM: start of iteration 74
#>   log-likelihood: -2649.138
#>   epsilon range: [ 0.49 , 0.49 ]
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#>   parameter change: 3.05474
#> --------------------------------------
#> cellEM: start of iteration 75
#>   log-likelihood: -2810.0772
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 14.7773 , 9241.2203 ]
#>   parameter change: 0.999998
#> --------------------------------------
#> cellEM: start of iteration 76
#>   log-likelihood: -2809.1225
#>   epsilon range: [ 0.49 , 0.49 ]
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#>   parameter change: 0.885517
#> --------------------------------------
#> cellEM: start of iteration 77
#>   log-likelihood: -2778.1648
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 239.967 , 7437.4687 ]
#>   parameter change: 14.0471
#> --------------------------------------
#> cellEM: start of iteration 78
#>   log-likelihood: -2673.2748
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.1376 , 13261.86 ]
#>   parameter change: 5338.81
#> --------------------------------------
#> cellEM: start of iteration 79
#>   log-likelihood: -2640.2014
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 33.423 ]
#>   parameter change: 20.4909
#> --------------------------------------
#> cellEM: start of iteration 80
#>   log-likelihood: -2649.8924
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 105.7216 ]
#>   parameter change: 0.101301
#> --------------------------------------
#> cellEM: start of iteration 81
#>   log-likelihood: -2560.8312
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 63.4713 ]
#>   parameter change: 0.969028
#> --------------------------------------
#> cellEM: start of iteration 82
#>   log-likelihood: -2649.1397
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 110.226 ]
#>   parameter change: 3.05474
#> --------------------------------------
#> cellEM: start of iteration 83
#>   log-likelihood: -2810.0772
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 14.7773 , 9241.1956 ]
#>   parameter change: 0.999998
#> --------------------------------------
#> cellEM: start of iteration 84
#>   log-likelihood: -2809.1225
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 219.3456 , 34582.7976 ]
#>   parameter change: 0.885517
#> --------------------------------------
#> cellEM: start of iteration 85
#>   log-likelihood: -2778.1648
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 239.967 , 7437.4138 ]
#>   parameter change: 14.0471
#> --------------------------------------
#> cellEM: start of iteration 86
#>   log-likelihood: -2673.2749
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.1376 , 13261.8572 ]
#>   parameter change: 5338.81
#> --------------------------------------
#> cellEM: start of iteration 87
#>   log-likelihood: -2640.2014
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 33.423 ]
#>   parameter change: 20.4909
#> --------------------------------------
#> cellEM: start of iteration 88
#>   log-likelihood: -2649.8896
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 105.7216 ]
#>   parameter change: 0.101301
#> --------------------------------------
#> cellEM: start of iteration 89
#>   log-likelihood: -2560.8369
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 63.4713 ]
#>   parameter change: 0.969028
#> --------------------------------------
#> cellEM: start of iteration 90
#>   log-likelihood: -2649.14
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 110.226 ]
#>   parameter change: 3.05474
#> --------------------------------------
#> cellEM: start of iteration 91
#>   log-likelihood: -2810.0772
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 14.7773 , 9241.1919 ]
#>   parameter change: 0.999998
#> --------------------------------------
#> cellEM: start of iteration 92
#>   log-likelihood: -2809.1225
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 219.3457 , 34582.5264 ]
#>   parameter change: 0.885517
#> --------------------------------------
#> cellEM: start of iteration 93
#>   log-likelihood: -2778.1648
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 239.967 , 7437.4056 ]
#>   parameter change: 14.0471
#> --------------------------------------
#> cellEM: start of iteration 94
#>   log-likelihood: -2673.2749
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.1376 , 13261.8563 ]
#>   parameter change: 5338.81
#> --------------------------------------
#> cellEM: start of iteration 95
#>   log-likelihood: -2640.2014
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 33.423 ]
#>   parameter change: 20.4909
#> --------------------------------------
#> cellEM: start of iteration 96
#>   log-likelihood: -2649.8891
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 105.7216 ]
#>   parameter change: 0.101301
#> --------------------------------------
#> cellEM: start of iteration 97
#>   log-likelihood: -2560.8377
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 63.4713 ]
#>   parameter change: 0.969028
#> --------------------------------------
#> cellEM: start of iteration 98
#>   log-likelihood: -2649.1401
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 1.01 , 110.226 ]
#>   parameter change: 3.05474
#> --------------------------------------
#> cellEM: start of iteration 99
#>   log-likelihood: -2810.0772
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 14.7773 , 9241.1914 ]
#>   parameter change: 0.999998
#> --------------------------------------
#> cellEM: start of iteration 100
#>   log-likelihood: -2809.1225
#>   epsilon range: [ 0.49 , 0.49 ]
#>   gamma range: [ 219.3457 , 34582.4873 ]
#>   parameter change: 0.885517
#> cellEM did not converge after 100 iterations.

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