Agnese Panzera

Nonparametric Regression for Spherical Data

We develop nonparametric smoothing for regression when both the predictor and the response variables are defined on a sphere of whatever dimension. A local polynomial fitting approach is pursued, which retains all the advantages in terms of rate optimality, interpretability, and ease of implementation widely observed in the standard setting. Our estimates have a multi-output nature, meaning that each coordinate is separately estimated, within a scheme of a regression with a linear response. The main properties include linearity and rotational equivariance. This research has been motivated by the fact that very few models describe this kind of regression. Such current methods are surely not widely employable since they have a parametric nature, and also require the same dimensionality for prediction and response spaces, along with nonrandom design. Our approach does not suffer these limitations. Real-data case studies and simulation experiments are used to illustrate the effectiveness of the method.

Non‐Parametric Smoothing and Prediction for Nonlinear Circular Time Series

Not much research has been done in the field of circular time-series analysis. We propose a non-parametric theory for smoothing and prediction in the time domain for circular time-series data. Our model is based on local constant and local linear fitting estimates of a minimizer of an angular risk function. Both asymptotic arguments and empirical examples are used to describe the accuracy of our methods.

Non-parametric regression for compositional data

Regression for compositional data has been considered only from a parametric point of view. We introduce local constant and local linear smoothing for this problem, and treat the cases when the response, the predictor or both of them are compositions. To this end, we introduce suitable series expansions of the regression function at a point, along with a class of simplicial kernels. Our methods are formulated according to the Aitchison geometry of the simplex and then, using some relevant properties of the isometric log-ratio transformation, are developed following the principle of ‘working on coordinates’. Asymptotic properties and real-data case studies show the effectiveness of the methods.

Validating protein structure using kernel density estimates

Measuring the quality of determined protein structures is a very important problem in bioinformatics. Kernel density estimation is a well-known nonparametric method which is often used for exploratory data analysis. Recent advances, which have extended previous linear methods to multi-dimensional circular data, give a sound basis for the analysis of conformational angles of protein backbones, which lie on the torus. By using an energy test, which is based on interpoint distances, we initially investigate the dependence of the angles on the amino acid type. Then, by computing tail probabilities which are based on amino-acid conditional density estimates, a method is proposed which permits inference on a test set of data. This can be used, for example, to validate protein structures, choose between possible protein predictions and highlight unusual residue angles.

Smooth estimation of circular cumulative distribution functions and quantiles

Smooth nonparametric estimators based on a kernel method are proposed for cumulative distribution functions (CDFs) and quantiles of circular data. A sound motivation for this is that although for euclidean data similar estimators have been widely studied, for circular data nothing similar seems to exist; albeit, remarkably, in the circular-setting local methods are implemented more easily because of the absence of boundaries on the circle. The only alternative to our method seems to be the empirical CDF, that does not take into account circularity of data when the estimate is near the cut-point, as our local method naturally does. The definition of circular CDF is different from its euclidean counterpart in many respects, and this will give rise to estimators exhibiting some ‘unusual’ features such as, for example, global efficiency measures containing a location parameter and a covariance term. Simulations along with real data case studies illustrate the findings.

Nonparametric Rotations for Sphere-Sphere Regression

ABSTRACT Regression of data represented as points on a hypersphere has traditionally been treated using parametric families of transformations that include the simple rigid rotation as an important, special case. On the other hand, nonparametric methods have generally focused on modeling a scalar response through a spherical predictor by representing the regression function as a polynomial, leading to component-wise estimation of a spherical response. We propose a very flexible, simple regression model where for each location of the manifold a specific rotation matrix is to be estimated. To make this approach tractable, we assume continuity of the regression function that, in turn, allows for approximations of rotation matrices based on a series expansion. It is seen that the nonrigidity of our technique motivates an iterative estimation within a Newton–Raphson learning scheme, which exhibits bias reduction properties. Extensions to general shape matching are also outlined. Both simulations and real data are used to illustrate the results. Supplementary materials for this article are available online.

Clustering Upper Level Units in Multilevel Models for Ordinal Data

We consider an explorative method for unsupervised clustering of upper level units in a two-level hierarchical setting. The idea lies in applying a density-based clustering algorithm to the predicted random effects obtained from a multilevel cumulative logit model. We illustrate the proposed approach throughout the analysis of data from European Social Survey about political trust in European countries.

A note on nonparametric estimation of circular conditional densities

ABSTRACT The conditional density offers the most informative summary of the relationship between explanatory and response variables. We need to estimate it in place of the simple conditional mean when its shape is not well-behaved. A motivation for estimating conditional densities, specific to the circular setting, lies in the fact that a natural alternative of it, like quantile regression, could be considered problematic because circular quantiles are not rotationally equivariant. We treat conditional density estimation as a local polynomial fitting problem as proposed by Fan et al. [Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika. 1996;83:189–206] in the Euclidean setting, and discuss a class of estimators in the cases when the conditioning variable is either circular or linear. Asymptotic properties for some members of the proposed class are derived. The effectiveness of the methods for finite sample sizes is illustrated by simulation experiments and an example using real data.

Practical performance of local likelihood for circular density estimation

ABSTRACT Local likelihood has been mainly developed from an asymptotic point of view, with little attention to finite sample size issues. The present paper provides simulation evidence of how likelihood density estimation practically performs from two points of view. First, we explore the impact of the normalization step of the final estimate, second we show the effectiveness of higher order fits in identifying modes present in the population when small sample sizes are available. We refer to circular data, nevertheless it is easily seen that our findings straightforwardly extend to the Euclidean setting, where they appear to be somehow new.

Nonparametric estimating equations for circular probability density functions and their derivatives

We propose estimating equations whose unknown parameters are the values taken by a circular density and its derivatives at a point. Specifically, we solve equations which relate local versions of population trigonometric moments with their sample counterparts. Major advantages of our approach are: higher order bias without asymptotic variance inflation, closed form for the estimators, and absence of numerical tasks. We also investigate situations where the observed data are dependent. Theoretical results along with simulation experiments are provided.