A Note Concerning Professor Jolicoeur’s Comments

1975 ◽  
Vol 32 (8) ◽  
pp. 1494-1498 ◽  
Author(s):  
W. E. Ricker
Keyword(s):  
Functional Relationship ◽  
Regression Line ◽  
Major Axis ◽  
Bivariate Normal Distribution ◽  
General Sense ◽  
Bivariate Normal ◽  
Sum Of Products ◽  
Data Points ◽  
Objective Meaning ◽  
Standard Major Axis

The ordinary major axis of a bivariate normal distribution is unsuitable for describing the functional relationship (in Lindley’s general sense) between two naturally variable quantities, because it is not invariant with scale and because qualitatively unlike quantities are subtracted in computing it. The bivariate structural relationship is unsuitable because one of the parameters required, λ, cannot be estimated from naturally variable data; in fact λ has no objective meaning for such data. The standard major axis (or GM regression line) has neither of these defects, and it is the line that minimizes the sum of products of the absolute vertical and horizontal deviations of data points from itself.

XXIX.—On a Problem in Correlated Errors

1948 ◽  
Vol 62 (3) ◽  
pp. 273-277
Author(s):  
A. C. Aitken
Keyword(s):  
Bivariate Normal Distribution ◽  
Mean Value ◽  
Sampling Variance ◽  
Correlated Errors ◽  
General Application ◽  
Normal Equations ◽  
Bivariate Normal ◽  
Sum Of Products ◽  
The Mean ◽  
Made In

The problem with which this paper is concerned arose in the discussion of a series of chronometric observations, but it is of more general application, and is capable of wide extension. Pairs of readings (xi yi) were taken at times ti, i = 1, 2, …, n. These readings were known to be affected by respective errors (ξi ηi) from sources different but possessing some common part. It was important to have an estimate of the consequent correlation and to assess its precision. The assumptions made in the particular experiment were that x and y were both linear in t, representable by x = a0 + a1t, y = b0 + b1t, and that the distributions of error in x and y were normal. The parameters a0 and a1, b0 and b1 were therefore obtained from two separate sets of normal equations, and the unknown correlation was then estimated from the sum of products of corresponding residuals ui, vi, one from each set. In the corresponding situation in n samples (xi,yi) from a bivariate normal distribution the mean value of is (n − 1) ρσ1σ2, where σ12, σ22 are the variances of x and y and ρσ1σ2 is their product moment. One might therefore anticipate, by analogy, that in the present case the mean value of Σuivi would be (n − 2)ρσ1σ2. So indeed it proves to be, and the sampling variance of Σuivi conforms likewise with standard results; but it is desirable, by an extension of the problem, both to see why this is so and to take notice of cases where the analogy fails to hold.

A note on the generalization of bivariate normal quadrant probabilities to bivariate elliptical distributions

2018 ◽  
Author(s):  
Oscar Lorenzo Olvera Astivia
Keyword(s):  
Normal Distribution ◽  
Bivariate Normal Distribution ◽  
Elliptical Distributions ◽  
Bivariate Normal ◽  
Geometric Argument

I present a geometric argument to show that the quadrant probability for the bivariate normal distribution can be generalized to the case of all elliptical distributions.

Estimation of response slopes in respiratory control

1984 ◽  
Vol 56 (2) ◽  
pp. 536-539 ◽  
Author(s):  
D. L. Sherrill ◽  
G. D. Swanson
Keyword(s):  
Statistical Approach ◽  
Ventilatory Response ◽  
Respiratory Control ◽  
Major Axis ◽  
Bivariate Normal Distribution ◽  
Directional Statistics ◽  
Control Behavior ◽  
Slope Estimate ◽  
Partial Pressure Of Co2 ◽  
Reduced Major Axis

The ventilatory response to changes in alveolar (arterial) CO2 is widely used as an index of respiratory control behavior. Methods for estimating these response slopes should incorporate the possibility that there may be errors in both the independent (partial pressure of CO2) and dependent (ventilation) variables. In a recent paper Daubenspeck and Ogden (J. Appl. Physiol. Respirat. Environ. Exercise Physiol. 45:823–829, 1978) have suggested problems inherent in the traditional technique of reduced major axis and have suggested a more contemporary technique of directional statistics. We have previously analyzed both techniques and developed a method to overcome the problems of reduced major axis and problems inherent in the use of directional statistics. Under the assumption of a bivariate normal distribution, we demonstrate that our slope estimate is similar to the maximum likelihood estimate proposed by Mardia et al. (J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 54: 309–313, 1983) for this problem. In addition, we demonstrate a bootstrap statistical approach when the distributions are not normally distributed. These concepts are illustrated using O2-CO2 interaction data.

scholarly journals PSIII-5 Comparison of Bivariate Machine Learning and Linear Model for Genomic Prediction with Different Heritability, QTL and SNP Panel Scenarios

2020 ◽  
Vol 98 (Supplement_4) ◽  
pp. 230-231
Author(s):  
Sunday O Peters ◽  
Mahmut Sinecan ◽  
Kadir Kizilkaya ◽  
Milt Thomas
Keyword(s):  
Neural Network ◽  
Artificial Neural Network ◽  
Normal Distribution ◽  
Genomic Prediction ◽  
Network Models ◽  
Bivariate Normal Distribution ◽  
Snp Markers ◽  
Neural Network Models ◽  
Bivariate Normal ◽  
Artificial Neural

Abstract This simulation study used actual SNP genotypes on the first chromosome of Brangus beef cattle to simulate 0.50 genetically correlated two traits with heritabilities of 0.25 and 0.50 determined either by 50, 100, 250 or 500 QTL and then aimed to compare the accuracies of genomic prediction from bivariate linear and artificial neural network with 1 to 10 neurons models based on G genomic relationship matrix. QTL effects of 50, 100, 250 and 500 SNPs from the 3361 SNPs of 719 animals were sampled from a bivariate normal distribution. In each QTL scenario, the breeding values (Σgijβj) of animal i for two traits were generated by using genotype (gij) of animal i at QTL j and the effects (βj) of QTL j from a bivariate normal distribution. Phenotypic values of animal i for traits were generated by adding residuals from a bivariate normal distribution to the breeding values of animal i. Genomic predictions for traits were carried out by bivariate Feed Forward MultiLayer Perceptron ANN-1–10 neurons and linear (GBLUP) models. Three sets of SNP panels were used for genomic prediction: only QTL genotypes (Panel1), all SNP markers, including the QTL (Panel2), and all SNP markers, excluding the QTL (Panel3). Correlations from 10-fold cross validation for traits were used to assess predictive ability of bivariate linear (GBLUP) and artificial neural network models based on 4 QTL scenarios with 3 Panels of SNP panels. Table 1 shows that the trait with high heritability (0.50) resulted in higher correlation than the trait with low heritability (0.25) in bivariate linear (GBLUP) and artificial neural network models. However, bivariate linear (GBLUP) model produced higher correlation than bivariate neural network. Panel1 performed the best correlations for all QTL scenarios, then Panel2 including QTL and SNP markers resulted in better prediction than Panel3.

Tables of the Integral of the Elliptical Bivariate Normal Distribution over Offset Circles.

1962 ◽  
Vol 16 (77) ◽  
pp. 116
Author(s):  
Author's Summary ◽  
G. W. Rosenthal ◽  
J. J. Rodden
Keyword(s):  
Normal Distribution ◽  
Bivariate Normal Distribution ◽  
Bivariate Normal
Sign in / Sign up

Export Citation Format

Share Document