Univariate Case Research Articles

Bayesian classification of ripening stages of tomato fruit using acoustic impact and colorimeter sensor data

In this work, the concept of data fusion is applied to nondestructive testing data for classification of fresh intact tomatoes based on their ripening stages. A Bayesian classifier considering a multivariate, three-class problem was incorporated for data fusion. Probability of error was estimated numerically for univariate and multivariate cases based on Bhattacharyya distance. Numerical results showed that multi-sensorial data fusion reduces the classification error considerably. The Bayesian classifier was tested on data of tomato fruits taken by the following nondestructive tests: colorimeter and acoustic impact. Results of Bayesian classifier agree with numerical estimations showing an 11% classification error in the multivariate (multi-sensor) case compared with a 48% obtained by the univariate case (single sensor).

Modified Taylor reproducing formulas and \emph{h-p} clouds

We study two different approximations of a multivariate function f by operators of the form Σ N i=1 τ r [f,x i ]Wi(x)Wi(x),where {W i } is an m-reproducing partition of unity and T [f, x i ](x) are modified Taylor polynomials of degree r expanded at x i . The first approximation was introduced by Xuli (2003) in the univariate case and generalized for convex domains by Guessab et al. (2005). The second one was introduced by Duarte (1995) and proved in the univariate case. In this paper, we first relax the Guessab's convexity assumption and we prove Duarte's reproduction formula in the multivariate case. Then, we introduce two related reproducing quasi-interpolation operators in Sobolev spaces. A weighted error estimate and Jackson's type inequalities for h-p cloud function spaces are obtained. Last, numerical examples are analyzed to show the approximative power of the method.

On the Finite Sample Behavior of Fixed Bandwidth Bickel–Rosenblatt Test for Univariate and Multivariate Uniformity

The Bickel–Rosenblatt (BR) goodness-of-fit test with fixed bandwidth was introduced by Fan in 1998. Although its asymptotic properties have been studied by several authors, little is known about its finite sample performance. Restricting our attention to the test of uniformity in the d-unit cube for d ≥ 1, we present in this article a description of the finite sample behavior of the BR test as a function of the bandwidth h. For d = 1 our analysis is based not only on empirical power results but also on the Bahadur's concept of efficiency. The numerical evaluation of the Bahadur local slopes of the BR test statistic for different values of h for a set of Legendre and trigonometric alternatives give us some additional insight about the role played by the smoothing parameter in the detection of departures from the null hypothesis. For d > 1 we develop a Monte-Carlo study based on a set of meta-type uniforme alternative distributions and a rule-of-thumb for the practical choice of the bandwidth is proposed. For both univariate and multivariate cases, comparisons with existing uniformity tests are presented. The BR test reveals an overall good comparative performance, being clearly superior to the considered competiting tests for bivariate data.

Accurate and precise 2D–3D registration based on X-ray intensity

This paper addresses the problem of estimating the 3D rigid pose of an object from its digitized X-ray projection. We considered the cases of homogeneous (CAD models) and inhomogeneous (attenuation map obtained from computed tomography) X-ray attenuation in an optimization framework based on a mutual information similarity measure. Convergence of object pose recovery is highly precise and obtained with sub-millimeter accuracy for both screen-film and digital radiographs by three major enhancements: (i) special care is given to the model of Parzen distribution used in the mutual information estimator (data pre-sphering in the bivariate case and bandwidth estimation in the univariate case); (ii) a quasi-global optimization scheme based on a modified version of stochastic clustering is used in conjunction with an object mesh resampling stage to reduce variance of the final pose estimator; (iii) nonlinear response to the radiograph is also estimated for screen-film radiographs.

Comments on 'Fourier Series Method for Measurement of Multivariate Volatilities' by P. Malliavin and M. E. Mancino

Malliavin and Mancino (2002) suggest a promising method for measuring asset volatilities, which was later used by many researchers. However, there are some mistakes in proof in original paper. Considering univariate case and making some further simplifying assumptions I show how one can give a correct proof. Most likely, there are no principle difficulties to modify this proof for the general case that was initially considered by Malliavin and Mancino.

Robust estimation of multivariate regression model

This paper studies robust estimation of multivariate regression model using kernel weighted local linear regression. A robust estimation procedure is proposed for estimating the regression function and its partial derivatives. The proposed estimators are jointly asymptotically normal and attain nonparametric optimal convergence rate. One-step approximations to the robust estimators are introduced to reduce computational burden. The one-step local M-estimators are shown to achieve the same efficiency as the fully iterative local M-estimators as long as the initial estimators are good enough. The proposed estimators inherit the excellent edge-effect behavior of the local polynomial methods in the univariate case and at the same time overcome the disadvantages of the local least-squares based smoothers. Simulations are conducted to demonstrate the performance of the proposed estimators. Real data sets are analyzed to illustrate the practical utility of the proposed methodology.

A comparative study of the K-means algorithm and the normal mixture model for clustering: Univariate case

This paper gives a comparative study of the K-means algorithm and the mixture model (MM) method for clustering normal data. The EM algorithm is used to compute the maximum likelihood estimators (MLEs) of the parameters of the MM model. These parameters include mixing proportions, which may be thought of as the prior probabilities of different clusters; the maximum posterior (Bayes) rule is used for clustering. Hence, asymptotically the MM method approaches the Bayes rule for known parameters, which is optimal in terms of minimizing the expected misclassification rate (EMCR). The paper gives a thorough analytic comparison of the two methods for the univariate case under both homoscedasticity and heteroscedasticity. Simulation results are given to compare the two methods for a range of sample sizes. The comparison, which is limited to two clusters, shows that the MM method has substantially lower EMCR particularly when the mixing proportions are unbalanced. The two methods have asymptotically the same EMCR under homoscedasticity (resp., heteroscedasticity) when the mixing proportions of the two clusters are equal (resp., unequal), but for small samples the MM method sometimes performs slightly worse because of the errors in estimating unknown parameters.

A Polynomial Method for Temporal Disaggregation of Multivariate Time Series

A problem often encountered in the preparation of economic and business time series is that much of current data is only available in aggregate form. For example, we rarely see daily or weekly production data, but only monthly or quarterly figures. There has been considerable research done on the disaggregation of univariate time series. However, in the multivariate case, we find only little research (e.g., Di Fonzo, 1990; Rossi, 1982, etc.). An example of temporal disaggregation in the multivariate case is the problem of estimating quarterly consumption series for some categories of consumer goods using yearly observations of each category and quarterly values of total consumer's expenditure. This article considers the problem of temporal disaggregation in the multivariate case. The problem considered here is the estimation of M (> 1) high-frequency time series using the relevant low-frequency data. The M estimated high-frequency time series must fulfill both temporal and contemporaneous aggregation constraints. An econometric computer package G, developed by Almon (1988), provides an univariate polynomial method to convert annual series to quarterly figures by interpolation. In this article, we propose a procedure which extend the polynomial method (Almon, 1988) to the multivariate case. The proposed method is used in the case of temporal disaggregation of multivariate time series when related series are not available. Previous work in the multivariate case have dealt with the same problem but only when related series are available. The first section of the article is devoted to the formulation of the problem in the univariate and multivariate case, while the second section gives the details of the method we propose. The method is illustrated through the example of deriving quarterly US Gross Domestic Product by major type of product from the annual series.

Performance‐based seismic bridge design for damage and loss limit states

AbstractRecent efforts in the earthquake engineering community have focused largely on developing a performance‐based earthquake engineering assessment methodology. However, equally important is applying such a methodology to the design of structures. Probabilistic performance‐based seismic design enables design and acceptance criteria to be specified by a continuum of limit states and incorporates uncertainty in all the sources of data. This paper introduces a non‐iterative method, based on the Pacific Earthquake Engineering Research Center framework, for design of bridge structures in terms of single or multiple physical design parameters, such as column height and diameter. Design parameter solutions are obtained from explicit consideration of uncertainty in the hazard, demand, damage, and loss to the structure by way of specifying a level of confidence in the resulting design under different performance objectives. Similar to other performance‐based methodologies, performance objectives consist of a performance level and a hazard level. Design equations are derived for performance levels defined in terms of bridge damage and loss. Examples are presented of both univariate and bivariate design parameter cases for typical reinforced concrete highway overpass bridges in California. Copyright © 2007 John Wiley & Sons, Ltd.

Extension of a Two-Stage Conditionally Unbiased Estimator of the Selected Population to the Bivariate Normal Case

Cohen and Sackrowitz (1989) derive a uniformly minimum variance conditionally unbiased estimator (UMVCUE) for the mean of a selected population for the univariate normal case with variance known and unknown as well as for the gamma case in a two-stage design. We extend this methodology to the bivariate normal case where the covariance matrix is assumed to be known. The population with the largest sample mean of the first dimension is selected for additional observations in a second stage. The goal of the analysis is to find an unbiased estimate of the mean of the second dimension with all of the data.

On a mixture vector autoregressive model

AbstractThe authors show how to extend univariate mixture autoregressive models to a multivariate time series context. Similar to the univariate case, the multivariate model consists of a mixture of stationary or nonstationary autoregressive components. The authors give the first and second order stationarity conditions for a multivariate case up to order 2. They also derive the second order stationarity condition for the univariate mixture model up to arbitrary order. They describe an EM algorithm for estimation, as well as a diagnostic checking procedure. They study the performance of their method via simulations and include a real application.

Relating crop yield to topographic attributes using Spatial Analysis Neural Networks and regression

Land-surface topographic attributes can be useful for estimating stable spatial patterns of crop yield caused by spatial variability in soils and water availability. We present spatial analyses of grain yield for three fields of dryland winter wheat in northeastern Colorado using topographic attributes as explanatory variables. Topographic attributes including elevation, slope, aspect, curvature, specific contributing area, and wetness index were computed from a 10-m digital elevation model. A Spatial Analysis Neural Network (SANN) algorithm was used for joint spatial interpolation and yield prediction from the topographic attributes. SANN prediction errors were compared with the results of Multiple Linear Regression (MLR). SANN and MLR were assessed in terms of bias and relative root mean squared error (rRMSE) using validation data. SANN out-performed MLR in multivariate estimation, but not for the univariate cases. The greatest advantage of SANN was seen using four or more topographic attributes, whereas MLR showed diminishing efficiency with more than three explanatory variables. Prediction/interpolation errors within a given field were reduced substantially by using the spatial coordinates (latitude and longitude) in tandem with topographic attributes. The rRMSE value reached a minimum of 0.44 (model efficiency, E = 0.80) for interpolation with SANN on the West field. Using only topographic attributes as input, the minimum rRMSE values were 0.59 ( E = 0.65) for SANN with 5 variables and 0.72 ( E = 0.48) for MLR with 4 or 5 explanatory variables. Thus, this study demonstrated the utility of SANN with topographic attributes that contain implicit soil and water information for estimating spatial patterns of dryland crop yield.

Univariate and multivariate versions of the negative binomial-inverse Gaussian distributions with applications

In this paper we propose a new compound negative binomial distribution by mixing the p negative binomial parameter with an inverse Gaussian distribution and where we consider the reparameterization p = exp ( − λ ) . This new formulation provides a tractable model with attractive properties which make it suitable for application not only in the insurance setting but also in other fields where overdispersion is observed. Basic properties of the new distribution are studied. A recurrence for the probabilities of the new distribution and an integral equation for the probability density function of the compound version, when the claim severities are absolutely continuous, are derived. A multivariate version of the new distribution is proposed. For this multivariate version, we provide marginal distributions, the means vector, the covariance matrix and a simple formula for computing multivariate probabilities. Estimation methods are discussed. Finally, examples of application for both univariate and bivariate cases are given.

Multi-Lag Term Structure Models with Stochastic Risk Premia

The purpose of this paper is to propose discrete-time term structure models where the historical dynamics of the factor (xt) is given, in the univariate case, by a Gaussian AR(p) process, and, in the multivariate case, by a Gaussian n-dimensional VAR(p) process. The factor (xt) is considered as a latent or an observable variable and, in the second case, (xt) is given by the short rate (in the scalar setting) or by a vector of several yields (in the multivariate setting). We consider an exponential-affine stochastic discount factor (SDF) with a stochastic factor risk correction coefficient defined, at time t, as an affine function of Xt = (xt, . . . , xt?p+1)0 and, consequently, the yield-to-maturity formula at time t is an affine function of the p most recent lagged values of xt+1. We study the Gaussian AR(p) and the Gaussian VAR(p) Factor-Based Term Structure Models. We investigate, under the risk-neutral and the S-forward probability, the Moving Average (or discrete-time Heath, Jarrow and Morton) representation of the yield and short-term forward rate processes. This representation gives the possibility to exactly replicate the currently-observed yield curve. We also study the problem of matching the theoretical and currently-observed market term structure by means of the Extended AR(p) approach.

The multivariate Faà di Bruno formula and multivariate Taylor expansions with explicit integral remainder term

AbstractThe Faà di Bruno formulæ for higher-order derivatives of a composite function are important in analysis for a variety of applications. There is a substantial literature on the univariate case, but despite significant applications the multivariate case has until recently received limited study. We present a succinct result which is a natural generalization of the univariate version. The derivation makes use of an explicit integralform of the remainder term for multivariate Taylor expansions.

On Large and Superlarge Deviations of Sums of Independent Random Vectors Under Cramér's Condition. II

The present paper is a continuation of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 51 (2007), pp. 227–255]. In that paper we studied, in the univariate case, the asymptotics of the probabilities that a sum of independent identically distributed random variables will hit a half-interval $[x,x+\Delta)$ in the zone of superlarge deviations when the relative (scaled) deviations ${\alpha=x/n}$ unboundedly increase together with the number of summands n and, at the same time, remain in the analyticity domain of the large deviations rate function for the summands. In the multivariate case, the first part of the paper presented sufficient conditions which ensure that integrolocal and local theorems of the same universal type as in the large and normal deviations zones will also hold in the superlarge deviations zone. The second part of the paper deals with the same problems for three classes on the most wide-spread univariate distributions, for which one can obtain simple sufficient conditions, enabling one to find the asymptotics of the desired probabilities, as $x/n\to \infty$, in the above-mentioned universal form. These are the classes of the so-called exponentially and “superexponentially" fast decaying regular distributions. For them, we also establish limit theorems for the Cramér transforms with parameter values close to the “critical" one. Moreover, we obtain asymptotic expansion for the large deviations rate function.

Novel bivariate moment-closure approximations

Nonlinear stochastic models are typically intractable to analytic solutions and hence, moment-closure schemes are used to provide approximations to these models. Existing closure approximations are often unable to describe transient aspects caused by extinction behaviour in a stochastic process. Recent work has tackled this problem in the univariate case. In this study, we address this problem by introducing novel bivariate moment-closure methods based on mixture distributions. Novel closure approximations are developed, based on the beta-binomial, zero-modified distributions and the log-Normal, designed to capture the behaviour of the stochastic SIS model with varying population size, around the threshold between persistence and extinction of disease. The idea of conditional dependence between variables of interest underlies these mixture approximations. In the first approximation, we assume that the distribution of infectives ( I) conditional on population size ( N) is governed by the beta-binomial and for the second form, we assume that I is governed by zero-modified beta-binomial distribution where in either case N follows a log-Normal distribution. We analyse the impact of coupling and inter-dependency between population variables on the behaviour of the approximations developed. Thus, the approximations are applied in two situations in the case of the SIS model where: (1) the death rate is independent of disease status; and (2) the death rate is disease-dependent. Comparison with simulation shows that these mixture approximations are able to predict disease extinction behaviour and describe transient aspects of the process.

Transfer-function realization for multipliers of the Arveson space

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers for the reproducing kernel Hilbert space H ( k d ) on the unit ball B d ⊂ C d , where k d is the positive kernel k d ( λ , ζ ) = 1 / ( 1 − 〈 λ , ζ 〉 ) on B d . We study this space from the point of view of realization theory and functional models of de Branges–Rovnyak type. We highlight features which depart from the classical univariate case: coisometric realizations have only partial uniqueness properties, the nonuniqueness can be described explicitly, and this description assumes a particularly concrete form in the functional-model context.

코퓰러과 극단치이론을 이용한 위험척도의 추정 및 성과분석

금융위험의 측정 및 관리를 위한 도구로서 분포의 꼬리 부분과 관련한 위험척도로 VaR가 현재 널리 활용되고 있다. 특히 VaR의 정확한 추정을 위해 정규분포를 가정한 기존의 방법보다는 극단치이론을 이용한 방법이 최근 관심을 끌고 있다. 지금까지 극단치이론을 이용한VaR의 추정에 관한 연구는 대부분 단변량의 경우에 대해 이루어졌다. 본 논문에서는 코퓰러를 극단치이론에 결부시켜 다변량 극단치분포를 모형화하여 포트폴리오 위험측정을 다루고 있다. 특히 본 연구에서는 포트폴리오 위험 척도로 VaR와 더불어 ES에 대한 추정 방법도 함께 논의하였다. 포트폴리오 위험측정을 위한 방법으로 본 논문에서 논의한 코퓰러-극단치이론에 의한 접근방법이 기존의 분산-공분산 방법보다 상대적으로 우수한지를 실증자료에 대한 사후검증을 통해 살펴보았다. VaR, a tail-related risk measure is now widely used as a tool for a measurement and a management of financial risks. For more accurate measurement of VaR, recently we are particularly concerned about the approach based on extreme value theory rather than the traditional method based on the assumption of normal distribution. However, many studies about the approaches using extreme value theory was done only for the univariate case. In this paper, we discuss portfolio risk measurements with modelling multivariate extreme value distributions by combining copulas and extreme value theory. We also discuss the estimation of ES together with VaR as portfolio risk measures. Finally, we investigate the relative superiority of EVT-copula approach than variance-covariance method through the back-testing of an empirical data.

Simple formulas for certain polynomials

Simple (hitherto unknown) representations are derived for the generalized Hermite, Laguerre and Jacobi polynomials. For simplicity, the univariate case is considered. The same results hold for the corresponding multivariate and matrix–variate polynomials.