Ronaldo Dias

Density estimation via hybrid splines

The Hybrid Spline method (H-spline) is a method of density estimation which involves regression splines and smoothing splines methods. Using basis functions (B-splines), this method is much faster than Smoothing Spline Density Estimation approach (Gu, 1993). Simulations suggest that with more structured data (e.g., several modes) H-spline method estimates the modes as well as Logspline (Kooperberg and Stone, 1991). The H-spline algorithm is designed to compute a solution to the penalized likelihood problem. The smoothing parameter is updated jointly with the estimate via a cross-validation performance estimate, where the performance is measured by a proxy of the symmetrized Kullback-Leibler. The initial number of knots is determined automatically based on an estimate of the number of modes and the symmetry of the underlying density. The algorithm increases the number of knots by 1 until the symmetrized Kullback-Leibler distance, based on two consecutives estimates satisfies a condition which was determine...

Consistent variable selection for functional regression models

The dual problem of testing the predictive significance of a particular covariate, and identification of the set of relevant covariates is common in applied research and methodological investigations. To study this problem in the context of functional linear regression models with predictor variables observed over a grid and a scalar response, we consider basis expansions of the functional covariates and apply the likelihood ratio test. Based on p -values from testing each predictor, we propose a new variable selection method, which is consistent in selecting the relevant predictors from set of available predictors that is allowed to grow with the sample size n . Numerical simulations suggest that the proposed variable selection procedure outperforms existing methods found in the literature. A real dataset from weather stations in Japan is analyzed.

A Bayesian Approach to Hybrid Splines Non-Parametric Regression

A Bayesian approach is considered to estimate the number of basis functions and the smoothing parameter of the hybrid splines non-parametric regression procedure. The method used to obtain the estimate of the regression curve and its Bayesian confidence intervals is based on the reversible jump MCMC (Green 1995). Illustrations with simulated data are provided and show good performance of the proposed approach over the existing methods.

Sequential adaptive nonparametric regression via h-splines

The hybrid spline method (H‐spline) introduced by Dias (1994) is a hybrid method of curve estimation which combines ideas of regression spline and smoothing spline methods. In the context of nonparametric regression and by using basis functions (B‐splines), this method is much faster than smoothing spline methods (e.g. (Wahba, 1990)). The H‐spline algorithm is designed to compute a solution of the penalized least square problem, where the smoothing parameter is updated jointly with the number of basis functions in a performance‐oriented iteration. The algorithm increases the number of basis functions by one until the partial affinity between two consecutive estimates satisfies a constant determined empirically

A Hierarchical Model for Aggregated Functional Data

In many areas of science, one aims to estimate latent subpopulation mean curves based only on observations of aggregated population curves. By aggregated curves we mean linear combination of functional data that cannot be observed individually. We assume that several aggregated curves with linearly independent coefficients are available, and each aggregated curve is an independent partial realization of a Gaussian process with mean modeled through a weighted linear combination of the disaggregated curves. The mean of the Gaussian process is modeled using B-splines basis expansion methods. We propose a semiparametric, valid covariance function that is modeled as the product of a nonparametric variance function by a correlation function. The variance function is described as the square of a function that is expanded using B-splines basis functions. This results in a nonstationary covariance function and includes constant variance models as special cases. Inference is performed following the Bayesian paradigm allowing experts’ opinion, when available, to be accounted for. Moreover, it naturally provides the uncertainty associated with the parameters’ estimates and fitted values. We analyze artificial datasets and discuss how to choose among the different covariance models. We focus on two different real examples: a calibration problem for NIR spectroscopy data and an analysis of distribution of energy among different types of consumers. In the latter example, our proposed covariance function captures interesting features of the data. Further analysis of different artificial datasets, as well as computer code and data is available as supplementary material online.

Aggregated functional data model for near-infrared spectroscopy calibration and prediction

Calibration and prediction for NIR spectroscopy data are performed based on a functional interpretation of the Beer–Lambert formula. Considering that, for each chemical sample, the resulting spectrum is a continuous curve obtained as the summation of overlapped absorption spectra from each analyte plus a Gaussian error, we assume that each individual spectrum can be expanded as a linear combination of B-splines basis. Calibration is then performed using two procedures for estimating the individual analytes’ curves: basis smoothing and smoothing splines. Prediction is done by minimizing the square error of prediction. To assess the variance of the predicted values, we use a leave-one-out jackknife technique. Departures from the standard error models are discussed through a simulation study, in particular, how correlated errors impact on the calibration step and consequently on the analytes’ concentration prediction. Finally, the performance of our methodology is demonstrated through the analysis of two publicly available datasets.

Bayesian modeling of autoregressive partial linear models with scale mixture of normal errors

Normality and independence of error terms are typical assumptions for partial linear models. However, these assumptions may be unrealistic in many fields, such as economics, finance and biostatistics. In this paper, a Bayesian analysis for partial linear model with first-order autoregressive errors belonging to the class of the scale mixtures of normal distributions is studied in detail. The proposed model provides a useful generalization of the symmetrical linear regression model with independent errors, since the distribution of the error term covers both correlated and thick-tailed distributions, and has a convenient hierarchical representation allowing easy implementation of a Markov chain Monte Carlo scheme. In order to examine the robustness of the model against outlying and influential observations, a Bayesian case deletion influence diagnostics based on the Kullback–Leibler (K–L) divergence is presented. The proposed method is applied to monthly and daily returns of two Chilean companies.

Monte Carlo algorithm for trajectory optimization based on Markovian readings

This paper describes an efficient algorithm to find a smooth trajectory joining two points A and B with minimum length constrained to avoid fixed subsets. The basic assumption is that the locations of the obstacles are measured several times through a mechanism that corrects the sensors at each reading using the previous observation. The proposed algorithm is based on the penalized nonparametric method previously introduced that uses confidence ellipses as a fattening of the avoidance set. In this paper we obtain consistent estimates of the best trajectory using Monte Carlo construction of the confidence ellipse.

Approximations for non-symmetric truncated sequential and repeated significance tests: Approximations for non-symmetric truncated sequential and repeated significance tests

Let X1, X2…XN be a random sample from a N(μ, l) population. Using the B-value process {B(t); 0 ≤ t ≤ 1} introduced by Lan and Wittes (1988) one ran design a sequential test for H0 : μ = μ0 versus H1 : μ > μ0 for some functions γ1(t) > γ2(t): if B(t) > γ1(t) stop and reject H0 , if B(t) γ1(1). We suggest some approximations which are usually upper bounds for the significance level, power of the tests and lower bounds for the expected stopping time. Simulations show that the approximations get better as N → ∞ and/or μ → ∞. The exact values are difficult to obtain while the suggested approximations are very easy to compute in the case of linear and RST type boundaries and simulation results show that they are quite reasonable in several cases, even for moderate N.

Adaptive basis selection for functional data analysis via stochastic penalization

We propose an adaptive method of analyzing a collection of curves which can be, individually, modeled as a linear combination of spline basis functions. Through the introduction of latent Bernoulli variables, the number of basis functions, the variance of the error measurements and the coefficients of the expansion are determined. We provide a modification of the stochastic EM algorithm for which numerical results show that the estimates are very close to the true curve in the sense of L2 norm.