Missing data is a common problem in clinical data collection, which causes difficulty in the statistical analysis of such data. To overcome problems caused by incomplete data, we propose a new imputation method called projective resampling imputation mean estimation (PRIME), which can also address ``the curse of dimensionality" problem in imputation with less information loss. We use various sample sizes, missing-data rates, covariate correlations, and noise levels in simulation studies, and all results show that PRIME outperformes other methods such as iterative least-squares estimation (ILSE), maximum likelihood (ML), and complete-case analysis (CC). Moreover, we conduct a study of influential factors in cardiac surgery-associated acute kidney injury (CSA-AKI), which show that our method performs better than the other models. Finally, we prove that PRIME has a consistent property under some regular conditions.
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The Chebyshev or $\ell_{\infty}$ estimator is an unconventional alternative to the ordinary least squares in solving linear regressions. It is defined as the minimizer of the $\ell_{\infty}$ objective function \begin{align*} \hat{\boldsymbol{\beta}} := \arg\min_{\boldsymbol{\beta}} \|\boldsymbol{Y} - \mathbf{X}\boldsymbol{\beta}\|_{\infty}. \end{align*} The asymptotic distribution of the Chebyshev estimator under fixed number of covariates were recently studied (Knight, 2020), yet finite sample guarantees and generalizations to high-dimensional settings remain open. In this paper, we develop non-asymptotic upper bounds on the estimation error $\|\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}^*\|_2$ for a Chebyshev estimator $\hat{\boldsymbol{\beta}}$, in a regression setting with uniformly distributed noise $\varepsilon_i\sim U([-a,a])$ where $a$ is either known or unknown. With relatively mild assumptions on the (random) design matrix $\mathbf{X}$, we can bound the error rate by $\frac{C_p}{n}$ with high probability, for some constant $C_p$ depending on the dimension $p$ and the law of the design. Furthermore, we illustrate that there exist designs for which the Chebyshev estimator is (nearly) minimax optimal. In addition we show that "Chebyshev's LASSO" has advantages over the regular LASSO in high dimensional situations, provided that the noise is uniform. Specifically, we argue that it achieves a much faster rate of estimation under certain assumptions on the growth rate of the sparsity level and the ambient dimension with respect to the sample size.
Archetypal analysis is an unsupervised learning method for exploratory data analysis. One major challenge that limits the applicability of archetypal analysis in practice is the inherent computational complexity of the existing algorithms. In this paper, we provide a novel approximation approach to partially address this issue. Utilizing probabilistic ideas from high-dimensional geometry, we introduce two preprocessing techniques to reduce the dimension and representation cardinality of the data, respectively. We prove that, provided the data is approximately embedded in a low-dimensional linear subspace and the convex hull of the corresponding representations is well approximated by a polytope with a few vertices, our method can effectively reduce the scaling of archetypal analysis. Moreover, the solution of the reduced problem is near-optimal in terms of prediction errors. Our approach can be combined with other acceleration techniques to further mitigate the intrinsic complexity of archetypal analysis. We demonstrate the usefulness of our results by applying our method to summarize several moderately large-scale datasets.
We propose an Extended Hybrid High-Order scheme for the Poisson problem with solution possessing weak singularities. Some general assumptions are stated on the nature of this singularity and the remaining part of the solution. The method is formulated by enriching the local polynomial spaces with appropriate singular functions. Via a detailed error analysis, the method is shown to converge optimally in both discrete and continuous energy norms. Some tests are conducted in two dimensions for singularities arising from irregular geometries in the domain. The numerical simulations illustrate the established error estimates, and show the method to be a significant improvement over a standard Hybrid High-Order method.
In this paper, we study the weight spectrum of linear codes with \emph{super-linear} field size and use the probabilistic method to show that for nearly all such codes, the corresponding weight spectrum is very close to that of a maximum distance separable (MDS) code.
The random coefficients model $Y_i={\beta_0}_i+{\beta_1}_i {X_1}_i+{\beta_2}_i {X_2}_i+\ldots+{\beta_d}_i {X_d}_i$, with $\mathbf{X}_i$, $Y_i$, $\mathbf{\beta}_i$ i.i.d, and $\mathbf{\beta}_i$ independent of $X_i$ is often used to capture unobserved heterogeneity in a population. We propose a quasi-maximum likelihood method to estimate the joint density distribution of the random coefficient model. This method implicitly involves the inversion of the Radon transformation in order to reconstruct the joint distribution, and hence is an inverse problem. Nonparametric estimation for the joint density of $\mathbf{\beta}_i=({\beta_0}_i,\ldots, {\beta_d}_i)$ based on kernel methods or Fourier inversion have been proposed in recent years. Most of these methods assume a heavy tailed design density $f_\mathbf{X}$. To add stability to the solution, we apply regularization methods. We analyze the convergence of the method without assuming heavy tails for $f_\mathbf{X}$ and illustrate performance by applying the method on simulated and real data. To add stability to the solution, we apply a Tikhonov-type regularization method.
Latent class growth analysis is a popular approach to identify underlying subpopulations. Several implementations, such as LCGA (Mplus), Proc Traj (SAS) and lcmm (R) are specially designed for this purpose. Motivated by data collection of psychological instruments over time in a large North American cancer centre, we compare these implementations using various simulated Edmonton Symptom Assessment System revised (ESAS-r) scores, an ordinal outcome from 0 to 10, as well as the real data consisting of more than 20,000 patients. We found that Mplus and lcmm lead to high correct classification rate, but Proc Traj over estimated the number of classes and failed to converge. While Mplus is computationally faster than lcmm, it does not allow more than 10 levels. We therefore suggest first analyzing data on the ordinal scale using lcmm. If computational time becomes an issue, then one can group the scores into categories and implement them in Mplus.
We propose straightforward nonparametric estimators for the mean and the covariance functions of functional data. Our setup covers a wide range of practical situations. The random trajectories are, not necessarily differentiable, have unknown regularity, and are measured with error at discrete design points. The measurement error could be heteroscedastic. The design points could be either randomly drawn or common for all curves. The definition of our nonparametric estimators depends on the local regularity of the stochastic process generating the functional data. We first propose a simple estimator of this local regularity which takes strength from the replication and regularization features of functional data. Next, we use the "smoothing first, then estimate" approach for the mean and the covariance functions. The new nonparametric estimators achieve optimal rates of convergence. They can be applied with both sparsely or densely sampled curves, are easy to calculate and to update, and perform well in simulations. Simulations built upon a real data example on household power consumption illustrate the effectiveness of the new approach.
In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower dimensional manifold, thus the high dimensionality problem can be overcome by learning the lower dimensionality behavior. However, in real life applications, data is often very noisy. In this work, we propose a method to approximate $\mathcal{M}$ a $d$-dimensional $C^{m+1}$ smooth submanifold of $\mathbb{R}^n$ ($d \ll n$) based upon noisy scattered data points (i.e., a data cloud). We assume that the data points are located "near" the lower dimensional manifold and suggest a non-linear moving least-squares projection on an approximating $d$-dimensional manifold. Under some mild assumptions, the resulting approximant is shown to be infinitely smooth and of high approximation order (i.e., $O(h^{m+1})$, where $h$ is the fill distance and $m$ is the degree of the local polynomial approximation). The method presented here assumes no analytic knowledge of the approximated manifold and the approximation algorithm is linear in the large dimension $n$. Furthermore, the approximating manifold can serve as a framework to perform operations directly on the high dimensional data in a computationally efficient manner. This way, the preparatory step of dimension reduction, which induces distortions to the data, can be avoided altogether.
Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.
This study considers the 3D human pose estimation problem in a single RGB image by proposing a conditional random field (CRF) model over 2D poses, in which the 3D pose is obtained as a byproduct of the inference process. The unary term of the proposed CRF model is defined based on a powerful heat-map regression network, which has been proposed for 2D human pose estimation. This study also presents a regression network for lifting the 2D pose to 3D pose and proposes the prior term based on the consistency between the estimated 3D pose and the 2D pose. To obtain the approximate solution of the proposed CRF model, the N-best strategy is adopted. The proposed inference algorithm can be viewed as sequential processes of bottom-up generation of 2D and 3D pose proposals from the input 2D image based on deep networks and top-down verification of such proposals by checking their consistencies. To evaluate the proposed method, we use two large-scale datasets: Human3.6M and HumanEva. Experimental results show that the proposed method achieves the state-of-the-art 3D human pose estimation performance.
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