Multilevel Stochastic Optimization for Imputation in Massive Medical Data Records

TL;DR

This work tackles the challenge of imputing missing numerical data in massive medical records by introducing a multilevel Kriging/BLUP framework. It reformulates the BLUP problem into a well-conditioned multilevel system using projections, enabling an exact solution with dramatically reduced computational burden. On HCUP/NIS datasets, the method delivers superior accuracy and robustness, outperforming traditional imputation approaches (PMM, PPD, BEM, DA) and discriminative deep learning, with substantial error reductions and speedups. The approach makes Kriging practical for large-scale imputation and suggests avenues for multiple imputation and extension to categorical data in future work.

Abstract

It has long been a recognized problem that many datasets contain significant levels of missing numerical data. A potentially critical predicate for application of machine learning methods to datasets involves addressing this problem. However, this is a challenging task. In this paper, we apply a recently developed multi-level stochastic optimization approach to the problem of imputation in massive medical records. The approach is based on computational applied mathematics techniques and is highly accurate. In particular, for the Best Linear Unbiased Predictor (BLUP) this multi-level formulation is exact, and is significantly faster and more numerically stable. This permits practical application of Kriging methods to data imputation problems for massive datasets. We test this approach on data from the National Inpatient Sample (NIS) data records, Healthcare Cost and Utilization Project (HCUP), Agency for Healthcare Research and Quality. Numerical results show that the multi-level method significantly outperforms current approaches and is numerically robust. It has superior accuracy as compared with methods recommended in the recent report from HCUP. Benchmark tests show up to 75% reductions in error. Furthermore, the results are also superior to recent state of the art methods such as discriminative deep learning.

Figures (4)

• Figure 1: Multilevel Kriging/BLUP flowchart. Training: The data is split into the predictors $\mathbf{x}_T$ and the observation data $\mathbf{y}_T$ corresponding to the variable that will be imputed. Multilevel: The multilevel operators are constructed $(\mathbf{L},\mathbf{W})$. Estimation: The coefficients $\hat{\boldsymbol{\theta}}$ of the covariance coefficients are estimated. BLUP (Imputation): Given the multilevel operators $(\mathbf{L},\mathbf{W})$, the covariance coefficients $\hat{\boldsymbol{\theta}}$ and the predictors for the missing values $\mathbf{x}_0$ imputing the missing variable.
• Figure 2: Prediction error comparison ( totchg) for the 2013 NIS Dataset with respect to the number of data points $N$, where 90% are used for training and 10% for validation.
• Figure 3: (a) Population histogram statistical comparison of kNN-R, kNN, GLS and Kriging with respect to the validation data set for 90,000 training and 10,000 validation datasets ($N = 100,000$). Notice that Kriging more faithfully reproduces the population statistics of the validation total charge data set. This is the advantage of the unbiased constrained in the stochastic optimization. Note that PMM, PPD, BEM and DA methods also give similar results to kNN-R. (b) Total charge ( totchg) imputation statistical errors comparisons. Kriging provides the best imputation performance for all error measures. (c) Length of stay ( los) imputation statistical errors comparisons. For los, in general Kriging performs well.
• Figure 4: Performance comparison for imputation of the total charge variable among Kriging/BLUP, KNN-Reg, KNN, GLS and DDL for different training/validation proportions of the data. On the horizontal axis we have the percentage proportion for the training dataset. The vertical axis corresponds to the rMSE, MAPE and mean lnQ metrics. As observed for all the metrics rMSE, MAPE and mean lnQ the Kriging/BLUP method produces in almost all cases the best results.

Theorems & Definitions (11)

• Remark
• Theorem 4.1
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• Theorem 4.2
• proof
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