Addendum to fitting polynomials to main effects in multi factor experiments: A note concerning polynomial coefficients when all factors are quantitative

When all factors are quantitative, cell means may be expressed as a polynomial function of products of powers of the associated quantitative classification variables. Existence and uniqueness of this polynomial is established for unbalanced data with unequal spacing for no missing cells. The relationship between the coefficients of this polynomial and the main effects polynomials are derived for main effects which are arbitrary weighted averages of the cell means.     

Bayesian curve estimation by polynomial of random order

A Bayesian method of estimating an unknown regression curve by a polynomial of random order is proposed. A joint distribution is assigned over both the set of possible orders of the polynomial and the polynomial coefficients. Reversible jumps Markov chain Monte Carlo (MCMC) (Green, Biometrika 82 (1995) 711-32), are used to compute required posteriors. The methodology is extended to polynomials of random order with discontinuities and to piecewise polynomials of random order to handle wiggly curves. The effectiveness of the methodology is illustrated with a number of examples.     

An algebraic and Bayesian technology for object recognition

This paper presents a new robust, low computational cost technology for recognizing free-form objects in three-dimensional (3D) range data, or, in two dimensional (2D) curve data in the image plane. Objects are represented by implicit polynomials (i.e. 3D algebraic surfaces or 2D algebraic curves) of degree greater than two, and are recognized by computing and matching vectors of their algebraic invariants (which are functions of their coefficients that are invariant to translations, rotations and general linear transformations). Such polynomials of the fourth degree can represent objects considerably more complicated than quadrics and super-quadrics, and can realize object recognition at significantly lower computational cost. Unfortunately, the coefficients of high degree implicit polynomials are highly sensitive to small changes in the data to which the polynomials are fit, thus often making recognition based on these polynomial coefficients or their invariants unreliable. We take two approaches to the problem: one involves restricting the polynomials to those which represent bounded curves and surfaces, and the other approach is to use Bayesian recognizers. The Bayesian recognizers are amazingly stable and reliable, even when the polynomials have unbounded zero sets and very large coefficient variability. The Bayesian recognizers are a unique interplay of algebraic functions and statistical methods. In this paper, we present these recognizers and show that they work effectively, even when data are missing along a large portion of an object boundary due, for example, to partial occlusion.     

An algebraic and Bayesian technology for object recognition

This paper presents a new robust, low computational cost technology for recognizing free-form objects in three-dimensional (3D) range data, or, in two dimensional (2D) curve data in the image plane. Objects are represented by implicit polynomials (i.e. 3D algebraic surfaces or 2D algebraic curves) of degree greater than two, and are recognized by computing and matching vectors of their algebraic invariants (which are functions of their coefficients that are invariant to translations, rotations and general linear transformations). Such polynomials of the fourth degree can represent objects considerably more complicated than quadrics and super-quadrics, and can realize object recognition at significantly lower computational cost. Unfortunately, the coefficients of high degree implicit polynomials are highly sensitive to small changes in the data to which the polynomials are fit, thus often making recognition based on these polynomial coefficients or their invariants unreliable. We take two approaches to the problem: one involves restricting the polynomials to those which represent bounded curves and surfaces, and the other approach is to use Bayesian recognizers. The Bayesian recognizers are amazingly stable and reliable, even when the polynomials have unbounded zero sets and very large coefficient variability. The Bayesian recognizers are a unique interplay of algebraic functions and statistical methods. In this paper, we present these recognizers and show that they work effectively, even when data are missing along a large portion of an object boundary due, for example, to partial occlusion.     

The exact proability that the roots of quadratic,cubic, and quartic equations are all real if the equation coefficients are random

Suppose that the coefficients of a polynomial equation are Independent random variables defined on subsets of real numbers, The purpose of this paper is to find the exact probability that all roots of a random polynomial equation are real. Since a polynomial equation of degree higher than four with arbitrary coefficients cannot be solved algrebraically, this paper will consider quadratic, cubic and quartic equations only. The general results are obtained in each case, Also, a number of special cases are furnished.     

A unified treatment for the asymptotic normality of the coefficients of polynomials related to orthogonal ones

Suitable transformations on the orthogonal polynomials lead to polynomials with nonnegative coefficients. In this work, the asymptotic normality for the nonnegative coefficients of these polynomials is derived based on the nature of the weight function of the orthogonal polynomials. In particular, orthogonal polynomial cases from both classical and semi-classical systems are included as well as the singular behaved Pollaczek polynomial case.     

Least squares adaptive polynomials

A concept of adaptive least squares polynomials is introduced for modelling time series data. A recursion algorithm for updating coefficients of the adaptive polynomial (of a fixed degree) is derived. This concept assumes that the weights w are such that i) the importance of the data values, in terms of their weights, relative to each other stays fixed, and that ii) they satisfy the update property, i.e., the polynomial does not change if a new data value is a polynomial extrapolate. Closed form results are provided for exponential weights as a special case as they are shown to possess the update property when used with polynomials.

The concept of adaptive polynomials is similar to the linear adaptive prediction provided by the Kalman filter or the Least Mean Square algorithm of Widrow and Hoff. They can be useful in interpolating, tracking and analyzing nonstationary data.     

Estimating Mixing Densities in Exponential Family Models for Discrete Variables

This paper is concerned with estimating a mixing density g using a random sample from the mixture distribution f(x)=∫f x | θ)g(θ)dθ where f(· | θ) is a known discrete exponen tial family of density functions. Recently two techniques for estimating g have been proposed. The first uses Fourier analysis and the method of kernels and the second uses orthogonal polynomials. It is known that the first technique is capable of yielding estimators that achieve (or almost achieve) the minimax convergence rate. We show that this is true for the technique based on orthogonal polynomials as well. The practical implementation of these estimators is also addressed. Computer experiments indicate that the kernel estimators give somewhat disappoint ing finite sample results. However, the orthogonal polynomial estimators appear to do much better. To improve on the finite sample performance of the orthogonal polynomial estimators, a way of estimating the optimal truncation parameter is proposed. The resultant estimators retain the convergence rates of the previous estimators and a Monte Carlo finite sample study reveals that they perform well relative to the ones based on the optimal truncation parameter.     

Growth curve models with restrictions on random parameters

In the classical growth curve setting, individuals are repeatedly measured over time on an outcome of interest. The objective of statistical modeling is to fit some function of time, generally a polynomial, that describes the outcome's behavior. The polynomial coefficients are assumed drawn from a multivariate normal mixing distribution. At times, it may be known that each individual's polynomial must follow a restricted form. When the polynomial coefficients lie close to the restriction boundary, or the outcome is subject to substantial measurement error, or relatively few observations per individual are recorded, it can be advantageous to incorporate known restrictions. This paper introduces a class of models where the polynomial coefficients are assumed drawn from a restricted multivariate normal whose support is confined to a theoretically permissible region. The model can handle a variety of restrictions on the space of random parameters. The restricted support ensures that each individual's random polynomial is theoretically plausible. Estimation, posterior calculations, and comparisons with the unrestricted approach are provided.     

Extended Analysis of At Least Partially Ordered Multi‐factor ANOVA

For multifactor experimental designs in which the levels of at least one of the factors are ordered we show how to construct components that provide a deep nonparametric scrutiny of the data. The components assess generalised correlations and the resulting tests include and extend the Page and umbrella tests. Application of the tests described is straightforward. Orthonormal polynomials on the ANOVA responses and the factors are required and the formulae needed are given subsequently. These depend on the moments of the responses and of each factor and are easily calculated. Products of at least two of these orthonormal polynomials are then used as inputs into standard ANOVA routines. For example, using the first order orthonormal polynomial on factor A and the second order orthonormal polynomial on the ANOVA response will assess if, with increasing levels of factor A there is an umbrella response with either an increase and then a decrease or a decrease and then an increase.     

Inequalities Among Some Measures of Location

Inequalities involving some sample means and order statistics are established. An upper bound of the absolute difference between the sample mean and median is also derived. Interesting inequalities among the sample mean and the median are obtained for cases when all the observations have the same sign. Some other algebraic inequalities are derived by taking expected values of the sample results and then applying them to some continuous distributions. It is also proved that the mean of a non-negative continuous random variable is at least as large as p times 100(1 ? p)th percentile.     

Optimal Prediction in Stochastic Regression Models with Application to the Analysis of Repeated Surveys

Several results relating to the optimal prediction of regression coefficients and random variables under a general linear model with stochastic coefficients are presented. These results are then applied to the analysis of repeated sample surveys over time. In particular, if the finite population can be modelled by a superpopulation model, a fully efficient method for the analysis of repeated surveys is proposed.     

Recurrence relations of coefficients of the generalized hypergeometric function in multivariate analysis

James(1960) defined the zonal polynomials and used it to represent the joint distributions of latent roots of VVisfiart matrix. The zonal polviionnals played an important role to define the generalized hypergeometric function of symmetric matrix argument Since then, many density functions and moments based on Wishart matrix have been expressed in terms of the generalized hy¬pergeometric Function. The purpose of this paper is to get the recurrence relations for the coefficients of it. In Section 1 we derive a partial differen¬tial equations having the generalized hypergeometric function as the unique solution. Then we ubtain the recurrence relations until order 7 in Section 2.     

A monte carlo comparison of tests of parallelism of the ponses over time of two populations vith non-uniform observation times

Repeated measures data collected at random observation times are quite common in clinical studies and are often difficult to analyze. A Monte Carlo comparison of four analysis procedures with respect to significance level and power is presented. The basic procedures compared are successive difference analyses and three procedures using the data as summarized in the estimated quadratic polynomial regression coefficients for each subject. These three procedures are (1) Hotelling's T-square, (2) Multivariate Multisample Rank Sum Test (MMRST) and (3) Multivariate Multisample Median Test (MMMT).

For the variety of dispersion structures, sample sizes and treatement groups simulated the MMRST and successive difference analysis were the most satisfactory.     

A Simulation Comparison of Approximate Tests for Fixed Effects in Random Coefficients Growth Curve Models

Often, the response variables on sampling units are observed repeatedly over time. The sampling units may come from different populations, such as treatment groups. This setting is routinely modeled by a random coefficients growth curve model, and the techniques of general linear mixed models are applied to address the primary research aim. An alternative approach is to reduce each subject’s data to summary measures, such as within-subject averages or regression coefficients. One may then test for equality of means of the summary measures (or functions of them) among treatment groups. Here, we compare by simulation the performance characteristics of three approximate tests based on summary measures and one based on the full data, focusing mainly on accuracy of p-values. We find that performances of these procedures can be quite different for small samples in several different configurations of parameter values. The summary-measures approach performed at least as well as the full-data mixed models approach.     

Symmetric Sheffer sequences and their applications to lattice path counting

A sequence of Sheffer polynomials is symmetric, if the value of the nth degree polynomial at any natural number m agrees with the mth degree polynomial at n. While the above property sounds rather esoteric, symmetric Sheffer sequences frequently describe the elegant results of standard lattice path enumeration. We characterize all symmetric Sheffer sequences, and explain their role from the initial value problem point of view. Applications occur in the enumeration of nonintersecting weighted lattice paths, and the discussion of certain correlated random walks.     

New Measures of Spread and a Simpler Formula for the Normal Distribution

Spread can be measured, with some advantages, by measures based on squared pairwise differences instead of measures based on squared differences from the mean. The measures are equal to multiples of the various versions of the standard deviation. Their advantages are that the measure of spread does not depend on a previously defined measure of location, that the spread of a sample and of a population are both square roots of simple averages and are both intuitively reasonable, and that the formula for the normal density is simplified. Computation is not significantly increased.     

E-optimal designs for polynomial regression without intercept

We give all E-optimal designs for the mean parameter vector in polynomial regression of degree d without intercept in one real variable. The derivation is based on interplays between E-optimal design problems in the present and in certain heteroscedastic polynomial setups with intercept. Thereby the optimal supports are found to be related to the alternation points of the Chebyshev polynomials of the first kind, but the structure of optimal designs essentially depends on the regression degree being odd or even. In the former case the E-optimal designs are precisely the (infinitely many) scalar optimal designs, where the scalar parameter system refers to the Chebyshev coefficients, while for even d there is exactly one optimal design. In both cases explicit formulae for the corresponding optimal weights are obtained. Remarks on extending the results to E-optimality for subparameters of the mean vector (in heteroscdastic setups) are also given.     

A Short-Cut Derivation for the Solution of Autoregressive Models from Sharp Algebraic Arguments

This article uses algebraic arguments to cast light on the solution of vector autoregressive models in the presence of unit roots. First, the linear case and then the multi-lag specification are investigated. Clear-cut representations of the model solutions are obtained, closed-form expressions of the coefficient matrices are provided, and integration features and cointegration mechanisms for stationarity recovery are elucidated.     

敏感性问题随机化回答模型的改进

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