Parametric Methods for Incomplete Big Data Sample Clauses

Parametric Methods for Incomplete Big Data. St¨xxxxx and Bu¨hlmann (2012) propose a likelihood approach to solve the so-called matrix completion problem, where the goal is to recover a matrix from an incomplete set of entries. They assume a multivariate normal model with p-dimensional covariance matrix and presented an EM algorithm that maximizes the l1-penalized observed log-likelihood. The method (MissGLasso) is found to be always better than KNNimpute in their simulation study. However, the E-step in the EM algorithm is rather complex and MissGLasso strongly depends on the MAR assumption and cannot handle data that are MNAR. In 2014, the authors develop a more efficient algorithm called MissPALasso (St¨xxxxx et al., 2014), focusing on improving MissGLasso in two aspects: inefficiency of the EM algorithm and non-sparseness of the regression coefficients. Song and Xxxxx (2004) provide a procedure for multiple imputation based on a common factor model to reduce the dimension of the parameters in a multivariate normal model. The assumed factor model has k underlying factors and can be described as Zi = α + Wiβ +si for i = 1, 2, ..., n, where α is a 1 ×p mean vector, Wi is a 1 ×k factor score vector, β is a k × p factor loading matrix, and si follows a multivariate normal distribution. With the complete-data likelihood, the parameters and factor scores W , as well as missing items can be simulated using a Xxxxx sampler, which is viewed as an application of data augmentation. In the presence of mixed continuous and binary data, Audigier et al. (2016) propose a new single imputation method based on a principal component method dedicated to mixed data: the factorial analysis for mixed data (FAMD). FAMD can reduce the di- mensionality of the data by providing a subspace that best represents the data, and account for the influence of the continuous and the categorial variables in the analysis as well. However, as a single imputation method, the proposed method does not account for the uncertainty brought by the imputation. Another parametric imputation method for high-dimensional mixed data is presented by He and Xxxxx (2014). It is a joint modeling approach, where latent variables are used to model the binary variables. They use the generalized parameter-expanded Metropolis-Hastings algorithm (Xxxxxxxxx et al., 2008) to sample the mixed covariance-correlation matrix for the joint distribution of continu- ous and latent variables associated with binary variables. The simulation study indicates that the mul...