Flexible Basis Function Research Articles

Functional joint models for chronic kidney disease in kidney transplant recipients.

This functional joint model paper is motivated by a chronic kidney disease study post kidney transplantation. The available kidney organ is a scarce resource because millions of end-stage renal patients are on the waiting list for kidney transplantation. The life of the transplanted kidney can be extended if the progression of the chronic kidney disease stage can be slowed, and so a major research question is how to extend the transplanted kidney life to maximize the usage of the scarce organ resource. The glomerular filtration rate is the best test to monitor the progression of the kidney function, and it is a continuous longitudinal outcome with repeated measures. The patient's survival status is characterized by time-to-event outcomes including kidney transplant failure, death with kidney function, and death without kidney function. Few studies have been carried out to simultaneously investigate these multiple clinical outcomes in chronic kidney disease stage patients based on a joint model. Therefore, this paper proposes a new functional joint model from this clinical chronic kidney disease study. The proposed joint models include a longitudinal sub-model with a flexible basis function for subject-level trajectories and a competing-risks sub-model for multiple time-to event outcomes. The different association structures can be accomplished through a time-dependent function of shared random effects from the longitudinal process or the whole longitudinal history in the competing-risks sub-model. The proposed joint model that utilizes basis function and competing-risks sub-model is an extension of the standard linear joint models. The application results from the proposed joint model can supply some useful clinical references for chronic kidney disease study post kidney transplantation.

An empirical comparison of two novel transformation models.

Continuous response variables are often transformed to meet modeling assumptions, but the choice of the transformation can be challenging. Two transformation models have recently been proposed: semiparametric cumulative probability models (CPMs) and parametric most likely transformation models (MLTs). Both approaches model the cumulative distribution function and require specifying a link function, which implicitly assumes that the responses follow a known distribution after some monotonic transformation. However, the two approaches estimate the transformation differently. With CPMs, an ordinal regression model is fit, which essentially treats each continuous response as a unique category and therefore nonparametrically estimates the transformation; CPMs are semiparametric linear transformation models. In contrast, with MLTs, the transformation is parameterized using flexible basis functions. Conditional expectations and quantiles are readily derived from both methods on the response variable's original scale. We compare the two methods with extensive simulations. We find that both methods generally have good performance with moderate and large sample sizes. MLTs slightly outperformed CPMs in small sample sizes under correct models. CPMs tended to be somewhat more robust to model misspecification and outcome rounding. Except in the simplest situations, both methods outperform basic transformation approaches commonly used in practice. We apply both methods to an HIV biomarker study.

Correlated-Gaussian approach to linear-chain states: Case of four α particles

We show that correlated Gaussians with good angular momentum and parity provide flexible basis functions for specific elongated shape. As its application we study linear-chain states of four-alpha particles in variation-after-projection calculations in which all the matrix elements are evaluated analytically. We find possible chain states for $J=0^+$, $2^+$, $4^+$ and perhaps $6^+$ with the bandhead energy being about 33 MeV from the ground state of $^{16}$O. No chain states with $J\geq 8$ are found. The nature of the rotational sequence of the chain states is clarified in contrast to a rigid-body rotation. The quadrupole deformation parameters estimated from the chain states increase from 0.59 to 1.07 for $2^+$ to $6^+$. This work suggests undeveloped fields for the correlated Gaussians beyond those problems which have hitherto been solved successfully.

Parametric Functional Principal Component Analysis

Functional principal component analysis (FPCA) is a popular approach in functional data analysis to explore major sources of variation in a sample of random curves. These major sources of variation are represented by functional principal components (FPCs). Most existing FPCA approaches use a set of flexible basis functions such as B-spline basis to represent the FPCs, and control the smoothness of the FPCs by adding roughness penalties. However, the flexible representations pose difficulties for users to understand and interpret the FPCs. In this article, we consider a variety of applications of FPCA and find that, in many situations, the shapes of top FPCs are simple enough to be approximated using simple parametric functions. We propose a parametric approach to estimate the top FPCs to enhance their interpretability for users. Our parametric approach can also circumvent the smoothing parameter selecting process in conventional nonparametric FPCA methods. In addition, our simulation study shows that the proposed parametric FPCA is more robust when outlier curves exist. The parametric FPCA method is demonstrated by analyzing several datasets from a variety of applications.

36. Influence of data processing pipelines on EEG-fMRI results in patients with epilepsy

Simultaneous EEG-fMRI is increasingly used for the noninvasive pre-surgical evaluation of epileptic patients to localize the epileptogenic zone. In this retrospective study of EEG-fMRI data in patients with pharmacoresistant epilepsy, we compared a wide range of data processing strategies using validation with resection masks after successful epilepsy surgery. The aims of this study were to find how various data processing strategies influence EEG-fMRI results, and to identify the best approach for data processing. Thirteen subjects (9 F, 4 M) with pharmacoresistant epilepsy and good outcome after epilepsy surgery were included in the study. Simultaneous EEG-fMRI data (1.5 T scanner) was acquired before the surgery (300 scans per session, TR = 3 s). The position of interictal discharges (IED) was marked and used as onsets for event-related regressors in statistical model. SPM8 software was used for data processing. In total, 240 statistical analyses were calculated for each subject comprising all possible combinations of the used variants of preprocessing and GLM settings. The resection mask was created individually for each patient using clinical MR images acquired 3 months after the surgery. Several parameters (e.g. sensitivity, cosine criterion) were calculated for each dataset and processing pipeline to evaluate the concordance between spike-informed EEG-fMRI results and the resection mask. Multivariate statistical analysis was performed in SPSS software. We found that the preprocessing type (mainly basic pipeline vs. correction for cardiac artifact) does not affect the results. The study revealed two main findings. The first is the optimal processing pipeline – only canonical HRF as a basis function, IED stimulation time series shifted 2 s earlier than positions from EEG description, and massive filtering of artifact (24 movement regressors, signals from white matter and CSF, and global signal). The second finding is related to a more general understanding of the influence of various processing options on results. The superiority of canonical HRF over more flexible basis functions is probably due to our concordance measure, which is based on a single epileptic focus represented with a resection mask, and some type of similarity between the activation map and the mask. The finding of earlier BOLD responses is in concordance with the predominantly prespiking character of the BOLD response presented in previous studies.

Inferring continuous dynamic social influence and personal preference for temporal behavior prediction

It is always attractive and challenging to explore the intricate behavior data and uncover people's motivations, preference and habits, which can greatly benefit many tasks including link prediction, item recommendation, etc. Traditional work usually studies people's behaviors without time information in a static or discrete manner, assuming the underlying factors stay invariant in a long period. However, we believe people's behaviors are dynamic, and the contributing factors including the social influence and personal preference for behaviors are varying continuously over time. Such continuous dynamics convey important knowledge about people's behavior patterns; ignoring them would lead to inaccurate models. In this work, we address the continuous dynamic modeling of temporal behaviors. To model the fully continuous temporal dynamics of behaviors and the underlying factors, we propose the DP-Space, a dynamic preference probability space, which can capture their smooth variation in various shapes over time with flexible basis functions. Upon that we propose a generative dynamic behavior model, ConTyor, which considers the temporal item-adoption behaviors as joint effect of dynamic social influence and varying personal preference over continuous time. We also develop effective inference methods for ConTyor and present its applications. We conduct a comprehensive experimental study using real-world datasets to evaluate the effectiveness of our model and the temporal modeling. Results verify that ConTyor outperforms existing state-of-the-art static and temporal models in behavior predictions. Moreover, in our detailed study on temporal modeling, we show that temporal modeling is superior to static approaches and modeling over continuous time is further better than that over discrete time. We also demonstrate that the ancient behavior data can still become important and beneficial if modeled well.

The multiscale coarse-graining method. IX. A general method for construction of three body coarse-grained force fields

The multiscale coarse-graining (MS-CG) method is a method for constructing a coarse-grained (CG) model of a system using data obtained from molecular dynamics simulations of the corresponding atomically detailed model. The formal statistical mechanical derivation of the method shows that the potential energy function extracted from an MS-CG calculation is a variational approximation for the true potential of mean force of the CG sites, one that becomes exact in the limit that a complete basis set is used in the variational calculation if enough data are obtained from the atomistic simulations. Most applications of the MS-CG method have employed a representation for the nonbonded part of the CG potential that is a sum of all possible pair interactions. This approach, despite being quite successful for some CG models, is inadequate for some others. Here we propose a systematic method for including three body terms as well as two body terms in the nonbonded part of the CG potential energy. The current method is more general than a previous version presented in a recent paper of this series [L. Larini, L. Lu, and G. A. Voth, J. Chem. Phys. 132, 164107 (2010)], in the sense that it does not make any restrictive choices for the functional form of the three body potential. We use hierarchical multiresolution functions that are similar to wavelets to develop very flexible basis function expansions with both two and three body basis functions. The variational problem is solved by a numerical technique that is capable of automatically selecting an appropriate subset of basis functions from a large initial set. We apply the method to two very different coarse-grained models: a solvent free model of a two component solution made of identical Lennard-Jones particles and a one site model of SPC/E water where a site is placed at the center of mass of each water molecule. These calculations show that the inclusion of three body terms in the nonbonded CG potential can lead to significant improvement in the accuracy of CG potentials and hence of CG simulations.

Compact and flexible basis functions for quantum Monte Carlo calculations

Molecular calculations in quantum Monte Carlo frequently employ a mixed basis consisting of contracted and primitive Gaussian functions. While standard basis sets of varying size and accuracy are available in the literature, we demonstrate that reoptimizing the primitive function exponents within quantum Monte Carlo yields more compact basis sets for a given accuracy. Particularly large gains are achieved for highly excited states. For calculations using nondiverging pseudopotentials, we introduce Gauss-Slater basis functions that behave as Gaussians at short distances and Slaters at long distances. These basis functions further improve the energy and fluctuations of the local energy for a given basis size. Gains achieved by exponent optimization and Gauss-Slater basis use are exemplified by calculations for the ground state of carbon, the lowest lying excited states of carbon with (5)S(o), (3)P(o), (1)D(o), and (3)F(o) symmetries, carbon dimer, and naphthalene. Basis-size reduction enables quantum Monte Carlo treatment of larger molecules at high accuracy.

Gaussian finite-element mixed-basis method for electronic structure calculations

As a fully flexible basis function, the finite element (FE) function was introduced for the representation of the wave function in the molecular electronic state calculation within the density functional theory. The most serious disadvantage, which hindered the extensive application of the FE basis function, was the excessive computational cost for the representation of the core electron. In order to reduce this deficiency, the Gaussian-FE mixed basis method was proposed. The cooperation with Gaussian basis functions was utilized for representing the steeply varying part of the electronic wave function around the nuclei. Based on the results of the calculations for a set of small molecules, it was confirmed that the augmentation of the FE basis function using the Gaussian basis function allowed us to adopt a coarse uniform grid, and reduce the dependency on the relative configuration of FE nodal positions and atomic positions. These features were desirable to carry out the all-electron calculations of polyatomic molecules in the three-dimensional Cartesian coordinates while conserving the flexibility of the FE basis function.

Correlated two-electron wave functions of any symmetry.

Using a procedure originally due to Hylleraas, a convenient expansion in coupled spherical harmonics which terminates in a very small number of terms is applied to the treatment of fully correlated two-electron wave functions of any symmetry (total angular momentum, parity, and spin). Coupled equations satisfied by these wave functions are derived which are well adapted to computation and which we discretize on a numerical lattice utilizing the basis-spline collocation method. Use of this method which relies on very flexible basis functions is intended to facilitate subsequently the consideration of time-dependent rearrangments such as autoionization, photoionization, and electron-impact excitation and ionization. Here, we describe the underlying theoretical and computational methods concerning our treatment of the two-electron problem, the lattice discretization, and partial eigensolution by damped relaxation. Results of explicit calculations are given regarding the ground state and two low-lying singly excited states of helium.