A Simple Method for Constructing Orthogonal Polynomials When the Independent Variable is Unequally Spaced

Biometrics ◽  
1959 ◽  
Vol 15 (2) ◽  
pp. 187 ◽  
Author(s):  
D. S. Robson
Keyword(s):  
Orthogonal Polynomials ◽  
Simple Method ◽  
Unequally Spaced ◽  
Independent Variable

scholarly journals On the Standard Errors in the Fitting of Polynomials to Unequally Spaced Observations

1953 ◽  
Vol 6 (2) ◽  
pp. 131 ◽  
Author(s):  
PG Guest
Keyword(s):  
Least Squares ◽  
Standard Errors ◽  
The Third ◽  
Unequally Spaced ◽  
Uniform Spacing ◽  
Independent Variable ◽  
Two Parameters ◽  
Second Degree

From the observed values of the independent variable two parameters are derived which specify the departure from uniform spacing. Expressions are obtained for the standard errors of the coefficients and fitted values in terms of these parameters, and numerical tables for the estimation of the errors are given. It is shown that the errors calculated in this way lie within a few per cent. of the exact least squares values for polynomials of the first and second degree, but when the polynomial is of the third degree the deviations may be much greater.

scholarly journals Logistic Regression: An Inferential Method for Identifying the Best Predictors

2019 ◽  
Vol 17 (2) ◽  
Author(s):  
Rand Wilcox
Keyword(s):  
Logistic Regression ◽  
Regression Model ◽  
Logistic Regression Model ◽  
Simple Method ◽  
Inferential Method ◽  
Independent Variable

When dealing with a logistic regression model, there is a simple method for estimating the strength of the association between the jth covariate and the dependent variable when all covariates are entered into the model. There is the issue of determining whether the jth independent variable has a stronger or weaker association than the kth independent variable. This note describes a method for dealing with this issue that was found to perform reasonably well in simulations.

An outline of the theory of the ‘Analysis of Variance’

1948 ◽  
Vol 7 (04) ◽  
pp. 235-252
Author(s):  
S. Vajda
Keyword(s):  
Regression Analysis ◽  
Orthogonal Polynomials ◽  
Statistical Method ◽  
Analysis Of Variance ◽  
Independent Variable ◽  
General Statistical ◽  
Special Case

It is intended in the following paragraphs to give an outline of a very general statistical method which originated in agricultural experimentation and which has since been used in many other fields, including that of mortality investigations. We will start with a special case, that of regression analysis involving only one independent variable. Well-known formulae will be developed with the aid of orthogonal polynomials and it will be shown how this tool can be used for generalizations. A new theorem referring to these polynomials, and its significance for the analysis of variance, will also be mentioned.

scholarly journals Form, Symmetry and Asymmetry of the Dental Arch: Orthogonal Analysis Revisited

2018 ◽  
Vol 16 (1) ◽  
pp. 3-8
Author(s):  
Toby Hughes ◽  
Lindsay Richards ◽  
Grant Townsend
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Theoretical Basis ◽  
Dental Arch ◽  
Recursive Procedure ◽  
Polynomial Curves ◽  
Unequally Spaced ◽  
Orthogonal Analysis ◽  
Shape Data ◽  
Arch Shape

There have been numerous attempts to quantify the shape of the dental arch mathematically, with orthogonal polynomial curves providing a robust and versatile method for quantifying variation in both shape and asymmetry. Lu (1966) first presented the theoretical basis for fitting orthogonal polynomials to arch shape data. Whilst theoretically sound, Lu’s original paper contained several arithmetic errors and a number of incorrect assumptions. In this paper we present corrections for these errors and extrapolate the theory to unequally-spaced arch shape data using a simple recursive procedure first developed by Robson (1959).

Failure to Make Orthogonal Polynomial Adjustments for Unequally Spaced Intervals Yields Inaccurate F Values in Trend Tests

1971 ◽  
Vol 33 (3_suppl) ◽  
pp. 1179-1183 ◽  
Author(s):  
Jon E. Roeckelein
Keyword(s):  
Orthogonal Polynomial ◽  
Linear Component ◽  
Major Result ◽  
Quadratic Regression ◽  
Trend Analyses ◽  
Quadratic Component ◽  
Unequally Spaced ◽  
Independent Variable ◽  
Made In

Trend tests on the same sets of data having unequally spaced intervals of the independent variable were conducted under conditions of adjustment and non-adjustment of the orthogonal coefficients for linear and quadratic regression. The major result was an inflated F value for the linear component and potentially spurious values for the quadratic component. It was recommended that authors explicitly cite procedures when adjustments of orthogonal coefficients have been made in trend analyses for unevenly spaced levels of the independent variable.

Reducing parabolic partial differential equations to canonical form

1994 ◽  
Vol 5 (2) ◽  
pp. 159-164 ◽  
Author(s):  
J. F. Harper
Keyword(s):  
Canonical Form ◽  
Parabolic Partial Differential Equation ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Simple Method ◽  
Information Theoretic ◽  
Second Derivatives ◽  
Special Cases ◽  
Black Scholes ◽  
Independent Variable

A simple method of reducing a parabolic partial differential equation to canonical form if it has only one term involving second derivatives is the following: find the general solution of the first-order equation obtained by ignoring that term and then seek a solution of the original equation which is a function of one more independent variable. Special cases of the method have been given before, but are not well known. Applications occur in fluid mechanics and the theory of finance, where the Black-Scholes equation yields to the method, and where the variable corresponding to time appears to run backwards, but there is an information-theoretic reason why it should.

Orthogonal polynomials satisfying fourth order differential equations

1981 ◽  
Vol 87 (3-4) ◽  
pp. 271-288 ◽  
Author(s):  
Allan M. Krall
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Jacobi Polynomials ◽  
Weight Functions ◽  
Classical Orthogonal Polynomials ◽  
Absolutely Continuous ◽  
Independent Variable ◽  
Fourth Order Differential Equation

SynopsisThese polynomials, which are intimately connected with the Legendre, Laguerre and Jacobi polynomials, are orthogonal with respect to Stieltjes weight functions which are absolutely continuous on (− 1, 1), (0, ∞) and (0, 1), respectively, but which have jumps at some of the intervals' ends. Each set satisfies a fourth order differential equation of the form Ly = λny, where the coefficients of the operator L depends only upon the independent variable. The polynomials also have other properties, which are usually associated with the classical orthogonal polynomials.

Sign in / Sign up

Export Citation Format

Share Document