Ab-initio prediction and reliability of protein structural genomics by PROPAINOR algorithm

We have formulated the ab-initio prediction of the 3D-structure of proteins as a probabilistic programming problem where the inter-residue 3D-distances are treated as random variables. Lower and upper bounds for these random variables and the corresponding probabilities are estimated by nonparametric statistical methods and knowledge-based heuristics. In this paper we focus on the probabilistic computation of the 3D-structure using these distance interval estimates. Validation of the predicted structures shows our method to be more accurate than other computational methods reported so far. Our method is also found to be computationally more efficient than other existing ab-initio structure prediction methods. Moreover, we provide a reliability index for the predicted structures too. Because of its computational simplicity and its applicability to any random sequence, our algorithm called PROPAINOR (PROtein structure Prediction by AI an Nonparametric Regression) has significant scope in computational protein structural genomics.     

Local Linear Estimation of Jump-Diffusion Models by Using Asymmetric Kernels

This article addresses the problem of nonparametric estimation of the first and second infinitesimal moments by using the local linear method of the underlying jump-diffusion models. The motivation behind the study is to use the asymmetric kernels instead of standard kernel smoothing. The basic idea relies on replacing the symmetric kernel by asymmetric kernel and provides a new way of obtaining the nonparametric estimation for jump-diffusion models. We prove that the estimators based on the local linear method for jump-diffusion models are consistent and asymptotically follow normal distribution under the condition of recurrence and stationarity.     

分式噪声驱动的一类随机偏微分方程的非参数估计}

研究了一类由分式噪声所驱动的随机偏微分方程的统计推断. 先构造了偏微分算子时间 相依系数的非参数估计量, 然后得到了该估计在最大值范数下的收敛率和渐近正态性. 该收敛率 由系数的平滑参数和分式噪声的Hurst参数共同决定.     

A more efficient algorithm for Convex Nonparametric Least Squares

Convex Nonparametric Least Squares (CNLSs) is a nonparametric regression method that does not require a priori specification of the functional form. The CNLS problem is solved by mathematical programming techniques; however, since the CNLS problem size grows quadratically as a function of the number of observations, standard quadratic programming (QP) and Nonlinear Programming (NLP) algorithms are inadequate for handling large samples, and the computational burdens become significant even for relatively small samples. This study proposes a generic algorithm that improves the computational performance in small samples and is able to solve problems that are currently unattainable. A Monte Carlo simulation is performed to evaluate the performance of six variants of the proposed algorithm. These experimental results indicate that the most effective variant can be identified given the sample size and the dimensionality. The computational benefits of the new algorithm are demonstrated by an empirical application that proved insurmountable for the standard QP and NLP algorithms.     

Fast and Stable Multiple Smoothing Parameter Selection in Smoothing Spline Analysis of Variance Models With Large Samples

The current parameterization and algorithm used to fit a smoothing spline analysis of variance (SSANOVA) model are computationally expensive, making a generalized additive model (GAM) the preferred method for multivariate smoothing. In this article, we propose an efficient reparameterization of the smoothing parameters in SSANOVA models, and a scalable algorithm for estimating multiple smoothing parameters in SSANOVAs. To validate our approach, we present two simulation studies comparing our reparameterization and algorithm to implementations of SSANOVAs and GAMs that are currently available in R. Our simulation results demonstrate that (a) our scalable SSANOVA algorithm outperforms the currently used SSANOVA algorithm, and (b) SSANOVAs can be a fast and reliable alternative to GAMs. We also provide an example with oceanographic data that demonstrates the practical advantage of our SSANOVA framework. Supplementary materials that are available online can be used to replicate the analyses in this article.     

An Algorithm for Nonlinear,Nonparametric Model Choice and Prediction

We introduce an algorithm which, in the context of nonlinear regression on vector-valued explanatory variables, aims to choose those combinations of vector components that provide best prediction. The algorithm is constructed specifically so that it devotes attention to components that might be of relatively little predictive value by themselves, and so might be ignored by more conventional methodology for model choice, but which, in combination with other difficult-to-find components, can be particularly beneficial for prediction. The design of the algorithm is also motivated by a desire to choose vector components that become redundant once appropriate combinations of other, more relevant components are selected. Our theoretical arguments show these goals are met in the sense that, with probability converging to 1 as sample size increases, the algorithm correctly determines a small, fixed number of variables on which the regression mean, g say, depends, even if dimension diverges to infinity much faster than n. Moreover, the estimated regression mean based on those variables approximates g with an error that, to first order, equals the error which would arise if we were told in advance the correct variables. In this sense, the estimator achieves oracle performance. Our numerical work indicates that the algorithm is suitable for very high dimensional problems, where it keeps computational labor in check by using a novel sequential argument, and also for more conventional prediction problems, where dimension is relatively low.     

Empirical Likelihood Confidence Bands in Density Estimation

Abstract

Empirical likelihood methods are developed for constructing confidence bands in problems of nonparametric density estimation. These techniques have an advantage over more conventional methods in that the shape of the bands is determined solely by the data. We show how to construct an empirical likelihood functional, rather than a function, and contour it to produce the confidence bands. Analogs of Wilks's theorem are established in this infinite-parameter setting and may be used to select the appropriate contour. An alternative calibration, based on the bootstrap, is also suggested. Large-sample theory is developed to show that the bands have asymptotically correct coverage, and a numerical example is presented to demonstrate the technique. Comparisons are made with the use of bootstrap replications to choose both the shape and size of the bands.     

Hypothesis Tests of Convergence in Markov Chain Monte Carlo

Abstract

Deciding when a Markov chain has reached its stationary distribution is a major problem in applications of Markov Chain Monte Carlo methods. Many methods have been proposed ranging from simple graphical methods to complicated numerical methods. Most such methods require a lot of user interaction with the chain which can be very tedious and time-consuming for a slowly mixing chain. This article describes a system to reduce the burden on the user in assessing convergence. The method uses simple nonparametric hypothesis testing techniques to examine the output of several independent chains and so determines whether there is any evidence against the hypothesis of convergence. We illustrate the proposed method on some examples from the literature.     

Using Spatial Statistics to Select Model Complexity

Testing for nonindependence among the residuals from a regression or time series model is a common approach to evaluating the adequacy of a fitted model. This idea underlies the familiar Durbin–Watson statistic, and previous works illustrate how the spatial autocorrelation among residuals can be used to test a candidate linear model. We propose here that a version of Moran's I statistic for spatial autocorrelation, applied to residuals from a fitted model, is a practical general tool for selecting model complexity under the assumption of iid additive errors. The “space” is defined by the independent variables, and the presence of significant spatial autocorrelation in residuals is evidence that a more complex model is needed to capture all of the structure in the data. An advantage of this approach is its generality, which results from the fact that no properties of the fitted model are used other than consistency. The problem of smoothing parameter selection in nonparametric regression is used to illustrate the performance of model selection based on residual spatial autocorrelation (RSA). In simulation trials comparing RSA with established selection criteria based on minimizing mean square prediction error, smooths selected by RSA exhibit fewer spurious features such as minima and maxima. In some cases, at higher noise levels, RSA smooths achieved a lower average mean square error than smooths selected by GCV. We also briefly describe a possible modification of the method for non-iid errors having short-range correlations, for example, time-series errors or spatial data. Some other potential applications are suggested, including variable selection in regression models.     

Bayesian Methods for Wavelet Series in Single-Index Models

Single-index models have found applications in econometrics and biometrics, where multidimensional regression models are often encountered. This article proposes a nonparametric estimation approach that combines wavelet methods for nonequispaced designs with Bayesian models. We consider a wavelet series expansion of the unknown regression function and set prior distributions for the wavelet coefficients and the other model parameters. To ensure model identifiability, the direction parameter is represented via its polar coordinates. We employ ad hoc hierarchical mixture priors that perform shrinkage on wavelet coefficients and use Markov chain Monte Carlo methods for a posteriori inference. We investigate an independence-type Metropolis-Hastings algorithm to produce samples for the direction parameter. Our method leads to simultaneous estimates of the link function and of the index parameters. We present results on both simulated and real data, where we look at comparisons with other methods.