We propose a nonparametric bivariate time-varying coefficient model for longitudinal measurements with the occurrence of a terminal event that is subject to right censoring. The time-varying coefficients capture the longitudinal trajectories of covariate effects along with both the followup time and the residual lifetime. The proposed model extends the parametric conditional approach given terminal event time in recent literature, and thus avoids potential model misspecification. We consider a kernel smoothing method for estimating regression coefficients in our model and use cross-validation for bandwidth selection, applying undersmoothing in the final analysis to eliminate the asymptotic bias of the kernel estimator. We show that the kernel estimates are asymptotically normal under mild regularity conditions, and provide an easily computable sandwich variance estimator. We conduct extensive simulations that show desirable performance of the proposed approach, and apply the method to analyzing the medical cost data for patients with end-stage renal disease.
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Machine learning techniques typically applied to dementia forecasting lack in their capabilities to jointly learn several tasks, handle time dependent heterogeneous data and missing values. In this paper, we propose a framework using the recently presented SSHIBA model for jointly learning different tasks on longitudinal data with missing values. The method uses Bayesian variational inference to impute missing values and combine information of several views. This way, we can combine different data-views from different time-points in a common latent space and learn the relations between each time-point while simultaneously modelling and predicting several output variables. We apply this model to predict together diagnosis, ventricle volume, and clinical scores in dementia. The results demonstrate that SSHIBA is capable of learning a good imputation of the missing values and outperforming the baselines while simultaneously predicting three different tasks.
Hawkes processes are point processes that model data where events occur in clusters through the self-exciting property of the intensity function. We consider a multivariate setting where multiple dimensions can influence each other with intensity function to allow for excitation and inhibition, both within and across dimensions. We discuss how such a model can be implemented and highlight challenges in the estimation procedure induced by a potentially negative intensity function. Furthermore, we introduce a new, stronger condition for stability that encompasses current approaches established in the literature. Finally, we examine the total number of offsprings to reparametrise the model and subsequently use Normal and sparsity-inducing priors in a Bayesian estimation procedure on simulated data.
Moment methods are an important means of density estimation, but they are generally strongly dependent on the choice of feasible functions, which severely affects the performance. We propose a non-classical parameterization for density estimation using the sample moments, which does not require the choice of such functions. The parameterization is induced by the Kullback-Leibler distance, and the solution of it, which is proved to exist and be unique subject to simple prior that does not depend on data, can be obtained by convex optimization. Simulation results show the performance of the proposed estimator in estimating multi-modal densities which are mixtures of different types of functions.
The widespread availability of high-dimensional biological data has made the simultaneous screening of many biological characteristics a central problem in computational biology and allied sciences. While the dimensionality of such datasets continues to grow, so too does the complexity of biomarker identification from exposure patterns in health studies measuring baseline confounders; moreover, doing so while avoiding model misspecification remains an issue only partially addressed. Efficient estimators capable of incorporating flexible, data adaptive regression techniques in estimating relevant components of the data-generating distribution provide an avenue for avoiding model misspecification; however, in the context of high-dimensional problems that require the simultaneous estimation of numerous parameters, standard variance estimators have proven unstable, resulting in unreliable Type-I error control even under standard multiple testing corrections. We present a general approach for applying empirical Bayes shrinkage to variance estimators of a family of efficient, asymptotically linear estimators of population intervention causal effects. Our generalization of shrinkage-based variance estimators increases inferential stability in high-dimensional settings, facilitating the application of these estimators for deriving nonparametric variable importance measures in high-dimensional biological datasets with modest sample sizes. The result is a data adaptive approach for robustly uncovering stable causal associations in high-dimensional data in studies with limited samples. Our generalized variance estimator is evaluated against alternative variance estimators in numerical experiments. Identification of biomarkers with the proposed methodology is demonstrated in an analysis of high-dimensional DNA methylation data from an observational study on the epigenetic effects of tobacco smoking.
This paper is concerned with the asymptotic distribution of the largest eigenvalues for some nonlinear random matrix ensemble stemming from the study of neural networks. More precisely we consider $M= \frac{1}{m} YY^\top$ with $Y=f(WX)$ where $W$ and $X$ are random rectangular matrices with i.i.d. centered entries. This models the data covariance matrix or the Conjugate Kernel of a single layered random Feed-Forward Neural Network. The function $f$ is applied entrywise and can be seen as the activation function of the neural network. We show that the largest eigenvalue has the same limit (in probability) as that of some well-known linear random matrix ensembles. In particular, we relate the asymptotic limit of the largest eigenvalue for the nonlinear model to that of an information-plus-noise random matrix, establishing a possible phase transition depending on the function $f$ and the distribution of $W$ and $X$. This may be of interest for applications to machine learning.
Due to a high spatial angle resolution and low circuit cost of massive hybrid analog and digital (HAD) multiple-input multiple-output (MIMO), it is viewed as a key technology for future wireless networks. Combining a massive HAD-MIMO with direction of arrinal (DOA) will provide a high-precision even ultra-high-precision DOA measurement performance approaching the fully-digital (FD) MIMO. However, phase ambiguity is a challenge issue for a massive HAD-MIMO DOA estimation. In this paper, we review three aspects: detection, estimation, and Cramer-Rao lower bound (CRLB) with low-resolution ADCs at receiver. First, a multi-layer-neural-network (MLNN) detector is proposed to infer the existence of passive emitters. Then, a two-layer HAD (TLHAD) MIMO structure is proposed to eliminate phase ambiguity using only one-snapshot. Simulation results show that the proposed MLNN detector is much better than both the existing generalized likelihood ratio test (GRLT) and the ratio of maximum eigen-value (Max-EV) to minimum eigen-value (R-MaxEV-MinEV) in terms of detection probability. Additionally, the proposed TLHAD structure can achieve the corresponding CRLB using single snapshot.
In this work we present a novel bulk-surface virtual element method (BSVEM) for the numerical approximation of elliptic bulk-surface partial differential equations (BSPDEs) in three space dimensions. The BSVEM is based on the discretisation of the bulk domain into polyhedral elements with arbitrarily many faces. The polyhedral approximation of the bulk induces a polygonal approximation of the surface. Firstly, we present a geometric error analysis of bulk-surface polyhedral meshes independent of the numerical method. Then, we show that BSVEM has optimal second-order convergence in space, provided the exact solution is $H^{2+3/4}$ in the bulk and $H^2$ on the surface, where the additional $\frac{3}{4}$ is due to the combined effect of surface curvature and polyhedral elements close to the boundary. We show that general polyhedra can be exploited to reduce the computational time of the matrix assembly. To demonstrate optimal convergence results, a numerical example is presented on the unit sphere.
The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As non-parametric model, it offers a very flexible way of modeling time-varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non-negative for all covariate values in their domain. We describe an efficient algorithm to find the maximum likelihood estimator. The method is contrasted with the ordinary least squares approach in a simulation study, and the method is illustrated on a realistic data set.
We propose a new method for multivariate response regression and covariance estimation when elements of the response vector are of mixed types, for example some continuous and some discrete. Our method is based on a model which assumes the observable mixed-type response vector is connected to a latent multivariate normal response linear regression through a link function. We explore the properties of this model and show its parameters are identifiable under reasonable conditions. We impose no parametric restrictions on the covariance of the latent normal other than positive definiteness, thereby avoiding assumptions about unobservable variables which can be difficult to verify in practice. To accommodate this generality, we propose a novel algorithm for approximate maximum likelihood estimation that works "off-the-shelf" with many different combinations of response types, and which scales well in the dimension of the response vector. Our method typically gives better predictions and parameter estimates than fitting separate models for the different response types and allows for approximate likelihood ratio testing of relevant hypotheses such as independence of responses. The usefulness of the proposed method is illustrated in simulations; and one biomedical and one genomic data example.
Existing high-dimensional statistical methods are largely established for analyzing individual-level data. In this work, we study estimation and inference for high-dimensional linear models where we only observe "proxy data", which include the marginal statistics and sample covariance matrix that are computed based on different sets of individuals. We develop a rate optimal method for estimation and inference for the regression coefficient vector and its linear functionals based on the proxy data. Moreover, we show the intrinsic limitations in the proxy-data based inference: the minimax optimal rate for estimation is slower than that in the conventional case where individual data are observed; the power for testing and multiple testing does not go to one as the signal strength goes to infinity. These interesting findings are illustrated through simulation studies and an analysis of a dataset concerning the genetic associations of hindlimb muscle weight in a mouse population.
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