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Missing Not At Random (MNAR) values lead to significant biases in the data, since the probability of missingness depends on the unobserved values.They are not ignorable in the sense that they often require defining a model for the missing data mechanism, which makes inference or imputation tasks more complex. Furthermore, this implies a strong textit{a priori} on the parametric form of the distribution.However, some works have obtained guarantees on the estimation of parameters in the presence of MNAR data, without specifying the distribution of missing data citep{mohan2018estimation, tang2003analysis}. This is very useful in practice, but is limited to simple cases such as self-masked MNAR values in data generated according to linear regression models.We continue this line of research, but extend it to a more general MNAR mechanism, in a more general model of the probabilistic principal component analysis (PPCA), textit{i.e.}, a low-rank model with random effects. We prove identifiability of the PPCA parameters. We then propose an estimation of the loading coefficients and a data imputation method. They are based on estimators of means, variances and covariances of missing variables, for which consistency is discussed. These estimators have the great advantage of being calculated using only the observed data, leveraging the underlying low-rank structure of the data. We illustrate the relevance of the method with numerical experiments on synthetic data and also on real data collected from a medical register.
Missing values challenge data analysis because many supervised and unsupervised learning methods cannot be applied directly to incomplete data. Matrix completion based on low-rank assumptions are very powerful solution for dealing with missing values
We introduce uncertainty regions to perform inference on partial correlations when data are missing not at random. These uncertainty regions are shown to have a desired asymptotic coverage. Their finite sample performance is illustrated via simulations and real data example.
Fan et al. [$mathit{Annals}$ $mathit{of}$ $mathit{Statistics}$ $textbf{47}$(6) (2019) 3009-3031] proposed a distributed principal component analysis (PCA) algorithm to significantly reduce the communication cost between multiple servers. In this pape
Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered for example as movement trajectories on the surface of the earth, are an important special case. We consider an intrinsic
Let $X$ be a mean zero Gaussian random vector in a separable Hilbert space ${mathbb H}$ with covariance operator $Sigma:={mathbb E}(Xotimes X).$ Let $Sigma=sum_{rgeq 1}mu_r P_r$ be the spectral decomposition of $Sigma$ with distinct eigenvalues $mu_1