Study of random process theory aids digital data processing

Study of techniques for all random process technology, including stationary, nonstationary, and Gaussian bivariate, aids digital data processing. It presents material on digital filtering, correlation function, optimal spectral smoothing, deterministic data processing, and nonstationary spectrum and correlation analyses

A Key Note on Performance of Smoothing Parameterizations in Kernel Density Estimation

The univariate kernel density estimator requires one smoothing parameter while the bivariate and other higher dimensional kernel density estimators demand more than one smoothing parameter depending on the form of smoothing parameterizations used. The smoothing parameters of the higher dimensional kernels are presented in a matrix called the smoothing matrix. The two forms of parameterizations frequently used in higher dimensional kernel estimators are diagonal or constrained parameterization and full or unconstrained parameterization. While the full parameterization has no restrictions, the diagonal has some form of restrictions. The study investigates the performance of smoothing parameterizations of bivariate kernel estimator using asymptotic mean integrated squared error as error criterion function. The results show that in retention of statistical properties of data and production of smaller values of asymptotic mean integrated squared error as tabulated, the full smoothing parameterization outperforms its diagonal counterpart.Keywords: Smoothing Matrix, Kernel Estimator, Integrated Variance, Integrated Squared Bias, Asymptotic Mean Integration Squared Error (AMISE).

Generalized time-frequency coherency for assessing neural interactions in electrophysiological recordings

Time-frequency coherence has been widely used to quantify statistical dependencies in bivariate data and has proven to be vital for the study of neural interactions in electrophysiological recordings. Conventional methods establish time-frequency coherence by smoothing the cross and power spectra using identical smoothing procedures. Smoothing entails a trade-off between time-frequency resolution and statistical consistency and is critical for detecting instantaneous coherence in single-trial data. Here, we propose a generalized method to estimate time-frequency coherency by using different smoothing procedures for the cross spectra versus power spectra. This novel method has an improved trade-off between time resolution and statistical consistency compared to conventional methods, as verified by two simulated data sets. The methods are then applied to single-trial surface encephalography recorded from human subjects for comparative purposes. Our approach extracted robust alpha- and gamma-band synchronization over the visual cortex that was not detected by conventional methods, demonstrating the efficacy of this method

An algorithm for surface smoothing with rational splines

Discussed is an algorithm for smoothing surfaces with spline functions containing tension parameters. The bivariate spline functions used are tensor products of univariate rational-spline functions. A distinct tension parameter corresponds to each rectangular strip defined by a pair of consecutive spline knots along either axis. Equations are derived for writing the bivariate rational spline in terms of functions and derivatives at the knots. Estimates of these values are obtained via weighted least squares subject to continuity constraints at the knots. The algorithm is illustrated on a set of terrain elevation data

A Statistical Method for Estimating Luminosity Functions using Truncated Data

The observational limitations of astronomical surveys lead to significant statistical inference challenges. One such challenge is the estimation of luminosity functions given redshift $z$ and absolute magnitude $M$ measurements from an irregularly truncated sample of objects. This is a bivariate density estimation problem; we develop here a statistically rigorous method which (1) does not assume a strict parametric form for the bivariate density; (2) does not assume independence between redshift and absolute magnitude (and hence allows evolution of the luminosity function with redshift); (3) does not require dividing the data into arbitrary bins; and (4) naturally incorporates a varying selection function. We accomplish this by decomposing the bivariate density into nonparametric and parametric portions. There is a simple way of estimating the integrated mean squared error of the estimator; smoothing parameters are selected to minimize this quantity. Results are presented from the analysis of a sample of quasars.Comment: 30 pages, 9 figures, Accepted for publication in Ap

Single-trial multiwavelet coherence in application to neurophysiological time series

A method of single-trial coherence analysis is presented, through the application of continuous muldwavelets. Multiwavelets allow the construction of spectra and bivariate statistics such as coherence within single trials. Spectral estimates are made consistent through optimal time-frequency localization and smoothing. The use of multiwavelets is considered along with an alternative single-trial method prevalent in the literature, with the focus being on statistical, interpretive and computational aspects. The multiwavelet approach is shown to possess many desirable properties, including optimal conditioning, statistical descriptions and computational efficiency. The methods. are then applied to bivariate surrogate and neurophysiological data for calibration and comparative study. Neurophysiological data were recorded intracellularly from two spinal motoneurones innervating the posterior,biceps muscle during fictive locomotion in the decerebrated cat

Exploring local dependence

This paper discusses two graphical methods for the investigation of local association of two continuous random variables. Often, scalar dependence measures, such as correlation, cannot reflect the complex dependence structure of two variables. However, dependence graphs have the potential to assess a far richer class of bivariate dependence structures. The two graphical methods discussed in this article are the chi-plot and the local dependence map. After the introduction of these methods they are applied to different data sets. These data sets contain simulated data and daily stock return series. With these examples the application possibilities of the two local dependence graphs are shown.Association, bivariate distribution, chi-plot, copula, correlation, kernel smoothing, local dependence, permutation test

Smooth plug-in inverse estimators in the current status continuous mark model

We consider the problem of estimating the joint distribution function of the event time and a continuous mark variable when the event time is subject to interval censoring case 1 and the continuous mark variable is only observed in case the event occurred before the time of inspection. The nonparametric maximum likelihood estimator in this model is known to be inconsistent. We study two alternative smooth estimators, based on the explicit (inverse) expression of the distribution function of interest in terms of the density of the observable vector. We derive the pointwise asymptotic distribution of both estimators.Comment: 29 pages, 12 figure