Abstract:
We study a nonlinear regression problem of fitting a circle (or a circular arc) to scattered data. We prove that under any standard assumptions on the statistical distribution of errors that are commonly adopted in the literature, the estimates of the circle center and radius have infinite moments. We also discuss methodological implications of this fact.

Abstract:
The error between appropriately smooth functions and their radial basis function interpolants, as the interpolation points fill out a bounded domain in R^d, is a well studied artifact. In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basis function -- the native space. The native space contains functions possessing a certain amount of smoothness. This paper establishes error estimates when the function being interpolated is conspicuously rough.

Abstract:
In this paper we investigate the approximation properties of kernel interpolants on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on $\R^d$, such as radial basis functions (RBFs), to a smooth, compact embedded submanifold $\M\subset \R^d$. For restricted kernels having finite smoothness, we provide a complete characterization of the native space on $\M$. After this and some preliminary setup, we present Sobolev-type error estimates for the interpolation problem. Numerical results verifying the theory are also presented for a one-dimensional curve embedded in $\R^3$ and a two-dimensional torus.

Abstract:
estimates of the repeatability and heritability of 19 measures of performance in jersey cows were obtained using an animal model with a relationship matrix and a derivative-free restricted maximum likelihood algorithm. the data consisted of 935 records for 374 cows by 69 sires over the period 1969-1987. the estimates were similar to those obtained by ordinary least squares methods reported for the same data set and in other studies, but had smaller error variances. a likelihood ratio test showed agreement between these heritability estimates and those in the literature. the heritability estimates of milk, fat, protein, lactose-mineral, solids-not-fat, and total solids yields were about 0.25; for the corresponding percentages, and for the protein to fat and solids-not-fat to fat ratios, the estimates were 0.50. heritability estimates were 0.10 or less for the time from parturition to first breeding and for three measures of somatic cell counts. these estimates of heritability in a dairy cattle population in a subtropical environment were not different from those of populations in temperate climates.

Abstract:
Estimates of the repeatability and heritability of 19 measures of performance in Jersey cows were obtained using an animal model with a relationship matrix and a derivative-free restricted maximum likelihood algorithm. The data consisted of 935 records for 374 cows by 69 sires over the period 1969-1987. The estimates were similar to those obtained by ordinary least squares methods reported for the same data set and in other studies, but had smaller error variances. A likelihood ratio test showed agreement between these heritability estimates and those in the literature. The heritability estimates of milk, fat, protein, lactose-mineral, solids-not-fat, and total solids yields were about 0.25; for the corresponding percentages, and for the protein to fat and solids-not-fat to fat ratios, the estimates were 0.50. Heritability estimates were 0.10 or less for the time from parturition to first breeding and for three measures of somatic cell counts. These estimates of heritability in a dairy cattle population in a subtropical environment were not different from those of populations in temperate climates.

Abstract:
The breeding records of 179 Red Sindhi cattle maintained at the Livestock Experiment Station Hab Choki, Balochistan, during the years 1982-97 were used in the present study. An effort was made to estimate the repeatability for some productive and reproductive traits by using the Mixed model least squares and maximum likelihood procedures. The repeatability estimates for milk yield, lactation length, dry period, service period, gestation period and calving interval were 0.361 ± 0.05, 0.03 ± 0.05, 0.029 ± 0.06, 0.17± 0.06, 0.94 ± 0.55 and 0.167 ± 0.06, respectively in model 1, whereas in model 2 estimates for milk yield, lactation length, dry period, Service period, gestation period and calving interval were 0.354 ± 0.05. 0.041 ± 0.05, 0.040 ± 0.06, 0.173 ± 0.06, 0.98 ± 0.056 and 0.165 ± 0.06, respectively. The repeatability estimates are used in determining the amount of culling that can be safely done on the basis of records. A high estimate of repeatability provides enough evidence for selection or rejection of the individual based on single record. Whereas a low estimate of repeatability justifies accumulation of further records. A moderate estimate of repeatability (0.361) for lactation milk yield obtained in the present study indicates that the cows can be selected for milk yield on the basis of relatively lesser records in the present herd.

Abstract:
The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function on a uniform grid to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe the application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.

Abstract:
This study discusses the problem of modeling inherently positive data sets using Modified Quadratic Shepard interpolation method. There are number of scientific and business domains where data is inherently positive. For example numbers of people, mass, volume and percentage mass concentration are meaningless when negative. This interplant generates negative values while modeling such data that may cause ambiguity. We present a scaling technique that constrains the interpolant to produce non-negative graph through scattered positive data sets.

Abstract:
An important problem in applications is the approximation of a function $f$ from a finite set of randomly scattered data $f(x_j)$. A common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials $\{e^{2\pi i kx}\}$. This leads to fast numerical algorithms, but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data, an assumption that is rarely satisfied in practice. To overcome this drawback we impose Neumann boundary conditions on the data. This implies the use of cosine polynomials $\cos (\pi kx)$ as basis functions. We show that scattered data approximation using cosine polynomials leads to a least squares problem involving certain Toeplitz+Hankel matrices. We derive estimates on the condition number of these matrices. Unlike other Toeplitz+Hankel matrices, the Toeplitz+Hankel matrices arising in our context cannot be diagonalized by the discrete cosine transform, but they still allow a fast matrix-vector multiplication via DCT which gives rise to fast conjugate gradient type algorithms. We show how the results can be generalized to higher dimensions. Finally we demonstrate the performance of the proposed method by applying it to a two-dimensional geophysical scattered data problem.

Abstract:
this study aimed to estimate the repeatability coefficient and determine the minimum number of samples required for effective selection for yield of custard apple. twenty progenies were evaluated in randomized blocks, five replications and four plants per plot. the fruits were collected, counted and weighed every two days of the year. estimates of the repeatability coefficients were obtained by the methods of analysis of variance - anova and principal components - pc. the estimates from the repeatability analysis of biennial data are higher than those based on individual years. the estimates of the pc method were accurate even in the first harvest, unlike anova. four biennia were sufficient to ensure effective progeny selection of custard apple.