The Modified Gauss-Newton Method for the Fitting of Non-Linear Regression Functions by Least Squares
TL;DR: The experimental scientist is frequently faced with the task of determining the functional relation between a response, y and a number of inputs, x, x2,. * *, x, with the help of empirical data.
Abstract: The experimental scientist is frequently faced with the task of determining the functional relation between a 'response', y and a number of 'inputs', x , x2, . * * , x, with the help of empirical data. Often the mathematical form of the functional relation is assumed to be known and is written in the form of a 'regression function',
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TL;DR: In this article, the authors discuss a class of biased linear estimators employing generalized inverses and establish a unifying perspective on nonlinear estimation from nonorthogonal data.
Abstract: A principal objective of this paper is to discuss a class of biased linear estimators employing generalized inverses. A second objective is to establish a unifying perspective. The paper exhibits theoretical properties shared by generalized inverse estimators, ridge estimators, and corresponding nonlinear estimation procedures. From this perspective it becomes clear why all these methods work so well in practical estimation from nonorthogonal data.
1,828 citations
Cites methods from "The Modified Gauss-Newton Method fo..."
...Procedures [2, 9, 18] that use a simple fraction of the computed correction, viz....
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TL;DR: HR and HR X BP, both easily measured hemodynamic variables, are good predictors of MVO, during exercise in normotensive patients with ischemic heart disease, and including variables reflecting the contractile state of the heart and ventricular volume may further improve the predictability.
Abstract: In order to evaluate hemodynamic predictors of myocardial oxygen consumption (MVO2), 27 normotensive men with angina pectoris were studied at rest and during a steady state at sympton-tolerated maximal exercise (STME). Myocardial blood flow (MBF) was measured by the nitrous oxide method using gas chromatography. MBF increased by 71% from a resting value of 57.4 +/- 10.2 to 98.3 +/- 15.6 ml/100 g LV/min (P less than 0.001) during STME while MVO2 increased by 81% from a resting value of 6.7 +/- 1.3 to 12.1 +/- 2.8 ml O2/100 g LV/min (P less than 0.001). MVO2 correlated well with heart rate (HR) (r = 0.79), with HR x blood pressure (BP) (r = 0.83), and, adding end-diastolic pressure and peak LV dp/dt as independent variables, slightly improved this correlation (r = .86). Including the ejection period (tension-time index) did not improve the correlation (r = 0.80). Thus, HR and HR x BP, both easily measured hemodynamic variables, are good predictors of MVO2 during exercise in normotensive patients with ischemic heart disease. Including variables reflecting the contractile state of the heart and ventricular volume may further improve the predictability.
825 citations
TL;DR: 'Several theoretical formulations have recently been proposed to explain the shape of peripheral indicator dilution curves, but these have found limited application and have not been subjected to extensive experimental verification using a large number of normal and abnormal curves.
Abstract: • Several theoretical formulations have recently been proposed to explain the shape of peripheral indicator dilution curves." One such mathematical approach utilizes convolution integrals,' '*• "*• °~ in which concentration of the indicator is handled as an unknown and unspecified function of time, C(t). The widespread use of this method, however, will be limited because the required mathematical manipulations, when applied to curves obtained in vivo, involve intricate and lengthy calculations. This type of analysis would be facilitated if the indicator concentration could be conveniently expressed as a specific functon of time.' If such a suitable function were available, it would also be possible to characterize more accurately normal and abnormal indicator dilution curves and perhaps to gain insight into some of the factors determining the shape of the curves. In addition, the availability of such a function would allow more efficient processing of experimental curves by high-speed computers. A number of theoretical and empirical mathematical expressions for C(t) have been suggested.' "•' These have found limited application and have not been subjected to extensive experimental verification using a large number of normal and abnormal curves. One expression for C(t), proposed by Evans in
606 citations
TL;DR: In this article, the ordinal logistic regression model that McCullagh called the proportional odds model is extended to models that allow non-proportional odds for a subset of the explanatory variables.
Abstract: SUMMARY The ordinal logistic regression model that McCullagh calls the proportional odds model is extended to models that allow non-proportional odds for a subset of the explanatory variables The maximum likelihood method is used for estimation of parameters of general and restricted partial proportional odds models as well as for the derivation of Wald, Rao score and likelihood ratio tests These tests assess association without assuming proportional odds and test proportional odds against various alternatives Simulation results compare the score test for proportional odds with tests suggested by Koch, Amara and Singer that are based on a series of binary logistic models
601 citations
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TL;DR: In this article, the problem of least square problems with non-linear normal equations is solved by an extension of the standard method which insures improvement of the initial solution, which can also be considered an extension to Newton's method.
Abstract: The standard method for solving least squares problems which lead to non-linear normal equations depends upon a reduction of the residuals to linear form by first order Taylor approximations taken about an initial or trial solution for the parameters.2 If the usual least squares procedure, performed with these linear approximations, yields new values for the parameters which are not sufficiently close to the initial values, the neglect of second and higher order terms may invalidate the process, and may actually give rise to a larger value of the sum of the squares of the residuals than that corresponding to the initial solution. This failure of the standard method to improve the initial solution has received some notice in statistical applications of least squares3 and has been encountered rather frequently in connection with certain engineering applications involving the approximate representation of one function by another. The purpose of this article is to show how the problem may be solved by an extension of the standard method which insures improvement of the initial solution.4 The process can also be used for solving non-linear simultaneous equations, in which case it may be considered an extension of Newton's method. Let the function to be approximated be h{x, y, z, • • • ), and let the approximating function be H{oc, y, z, • • ■ ; a, j3, y, ■ • ■ ), where a, /3, 7, • ■ ■ are the unknown parameters. Then the residuals at the points, yit zit • • • ), i = 1, 2, ■ • • , n, are
11,253 citations
TL;DR: The work described in this article is the result of a study extending over the past few years by a chemist and a statistician, which has come about mainly in answer to problems of determining optimum conditions in chemical investigations, but they believe that the methods will be of value in other fields where experimentation is sequential and the error fairly small.
Abstract: The work described is the result of a study extending over the past few years by a chemist and a statistician. Development has come about mainly in answer to problems of determining optimum conditions in chemical investigations, but we believe that the methods will be of value in other fields where experimentation is sequential and the error fairly small.
4,359 citations
1,045 citations