Iterative Nonlinear Least-squares Research Articles

The pituitary gland secretes in bursts: appraising the nature of glandular secretory impulses by simultaneous multiple-parameter deconvolution of plasma hormone concentrations.

To investigate patterns of endogenous hormone release, we have proposed a biophysical model in which measured hormone concentrations at any given instant reflect the operation of a suitable cumulation function (secretory input) convolved with an appropriate elimination mechanism (metabolic clearance). The cumulation function underlying a macroscopic hormone secretory burst can be represented by a random (Gaussian) distribution of instantaneous molecular secretory rates, which are centered with some finite and determinable standard deviation about a particular moment in time. The hormone elimination mechanism is described by a mono- or biexponential clearance function. The resultant convolution integral is solved by iterative nonlinear least-squares parameter estimation, in which all plasma hormone concentrations and their variances are considered simultaneously. Experiments with human endocrine time series revealed that the spontaneous secretory patterns of any of multiple distinct anterior pituitary hormones (luteinizing hormone, follicle-stimulating hormone, growth hormone, prolactin, thyrotropin, and adrenocorticotropic hormone) can be described effectively by this parsimonious model. In addition, endogenous hormone disappearance rates determined by deconvolution agreed well with those reported earlier that were determined after exogenous hormone injections. Moreover, this model predicted that durations of underlying secretory impulses are extremely brief; i.e., the standard deviations of the Gaussian distributions of instantaneous secretory rates range from 4.5 min (luteinizing hormone) to 16 min (growth hormone) compared to plasma hormone concentration peaks of 90-140 min in duration. Accordingly, we conclude that observed physiological patterns of fluctuating plasma hormone concentrations can be accounted for by distinct, highly delimited, random bursts of hormone release separated by intervals of secretory quiescence.

A calculator program for individualizing factor VIII dosage.

We developed a calculator program to individualize Factor VIII dosage on the basis of the concentration-time data obtained after a single test-dose. The program is designed for the Hewlett-Packard 41-CV calculator. The calculation procedure is the 1977 version of the Sawchuk and Zaske method. The program performs model parameter estimation through a nonlinear iterative least-squares technique (the modified Gauss-Newton method). We tested our program in three patients with classic hemophilia who required multiple-dose treatment with Factor VIII. In each patient, individual kinetic parameters were estimated from the serial plasma levels measured after the test-dose. The predicted concentration-time curve resulting from all the administered doses was calculated based on the estimated kinetic parameters. Good agreement between predicted and measured levels during the multiple-dose regimen was observed in all patients.

Calculation of bioconcentration factors from kinetic data by non-linear iterative least-squares regression analysis using a programmable minicalculator

A program to fit theoretically modelled uptake and clearance periods of the bioconcentration of chemicals to experimental data using a programmable pocket calculator with thermal printer is reported. The bioconcentration factor (BCF) is calculated from the ratio of uptake and clearance rate constants. Because of the impossibility to linearize the kinetic equations an iterative nonlinear GAUSS-NEWTON least-squares fit has been applied for the pocket calculator programm ACCUTI-59, which calculates from a set of start parameters the best values for uptake and clearance rate constants, BCF, and the standard errors as well as the sum of squared deviations. A listing of the program is given.

Infrared intensities of pregn-1,4-diene-3,20-dione derivatives

The IR absorption intensities of the νCO and νCC bands of some pregn-1,4-diene-3,20-dione derivatives have been determined by means of a Fourier-transform infrared spectrometer. The integrated intensities were calculated by fitting Gauss-Cauchy sum functions to the experimental spectra using an iterative non-linear least-squares method and numerical integration. The intensities of the A-ring vibrations of pregn-4-ene-3,20-dione and pregn-1,4-diene-3,20-dione derivatives were compared, and the νCO band intensities of different 21-acylated derivatives were investigated.

Impairment of diazepam metabolism by low-dose estrogen-containing oral-contraceptive steroids.

The longterm use of low-dose estrogen containing oral contraceptives (OCs) and its impairment of diazepam clearance is reported. 8 healthy women, ranging in age from 52 to 72 years, participated in the study. All of the women had been taking low-dose estrogen containing OCs for more than 3 months. 1 of the subjects was a cigarette smoker. 8 healthy controls (all nonsmokers) who were not using OCs also participated. They ranged in age from 27-31 years. Diazepam (10 mg) was given by intravenous infusion over 15-30 seconds. Venous blood samples were drawn into heparinized tubes before the infusion, at the end of the infusion, at 5, 15, 30, and 45 minutes, at 1, 1.5, 2, 2.5, 3, 4, 6, 8, 12, and 24 hours, and daily for 7 days after the infusion. Plasma concentrations of diazepam after intravenous infusion were analyzed by weighted iterative nonlinear least-squares regression techniques. The volume of distribution of diazepam was not significantly different between the groups, but the apparent elimination half-life of diazepam was significantly longer and the total metabolic clearance significantly less in the OC users than in the control group. The differences were not confounded by variations in protein binding. The mean diazepam free fraction was identical in the 2 groups. The clinical result of the decrease in diazepam clearance reported here would be increased steady-state plasma diazepam concentration after longterm use at a given daily dose, with the potential for increasing the clinical effects of diazepam in this population.

High-resolution Fourier spectrometry of 14N 2 and 15N 2 infrared emission spectrum: Extensive analysis of the W3Δ u - B3Π g system

The infrared spectrum of the nitrogen molecule, excited in a microwave discharge, has been recorded in high resolution by Fourier spectrometry in the range 2500–15 000 cm −1. The precision of the measurements is estimated to be about 0.003 cm −1. We have analyzed 19 bands of the W 3Δ u - B 3Π g system of 14N 2, with 0 ≤ v′ ≤ 7, and three bands of 15N 2 lying between 2500 and 5900 cm −1. The molecular constants of the 3Δ u and B 3Π g states have been determined by direct approach using an iterative nonlinear least-squares procedure. It has proved convenient to describe the levels of W 3Δ u state in a case a basis although in fact they approximate those of Hund's case b. Derived values of equilibrium constants of W 3Δ u are, in cm −1: T e = 8875.3347 (with origin taken in A, 3Σ u + v = 0 level); ω e = 1506.4859; ω e χ e = 12.5469; B e = 1.4702537; α B = 0.0170389; D e = 0.55958 × 10 −5. RKR potential energy curves for the two states are constructed, and the Franck-Condon factors calculated for the W- B system.

A two-stage identification scheme for the determination of the parameters of a model of left heart and systemic circulation.

An identification scheme is developed for the determination of several parameters of a modified "Windkessel" model of the systemic arterial system for an individual patient undergoing cardiac catheterization. The scheme utilizes a modification of the Prony method [10], [11] as a "starter method" to determine good nominal values for the model parameters being varied. These values then serve as input to a well-known iterative nonlinear least-squares identification method (Marquardt method [14]) which then converges rapidly to frmal values of the parameters. Solution of the model equations with these parameter values yields the best fit in a least-squares sense of model-generated and observed aortic and brachial artery pressures. This two stage or sequential Prony-Marquardt technique represents an extension of our previous work associated with the analysis of multiexponential decay curves [18], and is applied here to the identification of parameters associated with the humam arterial system. When coupled with a method of determining the contractile mechanics of the left ventricle (eg., the ventricular elastance concept [l]-[5]), this identification scheme permits a functional characterization of the hemodynamic properties of the left ventricle and its systemic load, for an individual subject.

High resolution Fourier spectrometry of 14N 2 infrared emission spectrum : Analysis of the (2-0) band of the W3Δ u → B3Π g system

The rotational analysis of the (2-0) band of the W 3Δ u → B 3Π g system of 14N 2 was performed from high resolution Fourier records. The experimental wavenumbers were reduced to molecular parameters by using an iterative nonlinear least-squares procedure. The values obtained for the v = 2 level of the W 3Δ u state are (cm −1) T=12 563.5132 ∗, A=5.7091, ϵ=0.4502, B=1.427647, D=5.648×10 −6, p=0.0115 (For the T value, energy referred to v = 0, J = 0 of A 3Σ u +.)

New techniques for approximate realisation

The problem of approximate realisation is described and various methods are discussed. A new method is then given which directly identifies the system poles from pulse response data by finding the local minima of a real function of a complex variable. These estimates of the poles are then refined using a new iterative nonlinear least-squares algorithm. Finally, these methods are applied to a ‘seismic wavelet’, and are shown to give good qualitative and quantitative information on the system being modelled.

Program for optimal computation of EPR spin-hamiltonian parameters for ions in tetragonal symmetry

EPR studies have been carried out on Mn(II)-doped single crystals of [Zn(H2O)6]PtCl6, [Mg(H2O)6]PtCl6 and [Cd(H2O)6]PtCl6 from room temperature to 77 K at X-band frequencies (⋍9.5 GHz). MN(II) ions substituting the divalent metals exhibit a unique magnetic behavior with the z-axes directed along the c-axes of the crystals. A non-linear iterative least-squares fitting procedure for the analysis of EPR spectra is presented. In this method calculation of the spin-Hamiltonian parameters can be carried out for any orientation of the external magnetic field with respect to the crystal symmetry axis.

Integral Approach to Nonisothermal Estimation of Activation Energies

A simple experimental procedure is described, enabling estimates of the activation energy and preexponential factor for drug decomposition to be obtained from a single experiment. A computer program was written which bypasses the problem of slope determination and the consequent lack of final errors associated with graphical methods. It uses numerical integration and an iterative nonlinear least-squares treatment to ensure reliable and consistent parameter estimates and provides information on parameter uncertainties. The method is quite sensitive to deviations from the mathematical model and has been illustrated with the decomposition of riboflavin in alkaline solution.

An Alternative Econometric Approach to the Permanent Income Hypothesis: An International Comparison: A Comment

In their recent paper in this Review (1971) Singh and Drost's major aim has been to upon the of those research workers who have tried to estimate the aggregate consumption function based on the Permanent Income HIypothesis (PIH) suggested by Friedman (1957). In order to improve the workmanship in this field they suggest two iterative methods for the estimation of two alternative formulations of tne PIH. In this note we demonstrate first that in the case of their 'Method I' their suggested nonlinear iterative method will, in fact, result in Ordinary Least Squares (OLS) estimates, and hence the use of any iterative method will be unnecessary. Second, we show that the iterative method suggested for the second formulation of the PIH is also unnecessary and the required estimate can be obtained by solving a quadratic equation. Furthermore, their estimate of the marginal propensity to coinsume in the latter case will not be consistent unless the variances. of the transitory components of consumption and income are assumed to be equal to each other. In thei,r paper Singh and Drost (S-D) suggest an iterative procedure based on Wold's proposal to estimate k and A in the following relation.' Ct = k (1-xA) Yt + ACt-1 + Wt (1) which can also be written. as Ct= alYt + a2Ct1 + Wt (2) with a, = k(l-A), a2 = X where Wt's are assumed to be independently distributed with mean zero and a constant variance. S-D's argument for requiring an iterative procedure to obtain least squares estimates for k and A is that . . it would not be possible to separate out k from the estimate of k (1 -A) for the reason of identification. But since there is. a unique one-to-one relation between the parameter space (ai,a2) and (k,A), given that A # 1, such an argument is not valid and the us.e of Nonlinear Iterative Least Squares (NILES) to compute the least squares estimates of A and k will result in exactly the same estimates as the OLS estimates given by

Regression aspects of canonical correlation

The first canonical correlation coefficient is the maximum value of the correlation coefficient between a linear expression in one set of variables and a linear expression in another set of variables. Other stationary values of the correlation coefficients between the linear expressions are also called canonical correlations. The standard approach to the canonical correlations which goes back on Hotelling’s classical paper [5] is that of searching directly for the stationary values of the correlations, but this leads to a matrix equation of which the order is the total number of variables involved. But before his main paper on the subject, Hotelling in a short paper gave a regression treatment of the problem, which leads to a matrix equation of the order equal to that of the number of the variables in one of the sets [4]. Conveniently then the smallest set is used. The disadvantage with this approach is that it treats the two sets of variables in an unsymmetrical way. Sometimes however it may be an advantage, namely if only one set is stochastic [I, p. 2961. Recently Wold [7] has suggested estimation by nonlinear iterative least squares (NILES) procedure, later on called nonlinear iterative partial least squares (NIPALS) methods [8]. The parameters are arranged in groups and new proxies are obtained for each group successively during each iterative step. In this way the NIPALS methods differ from the classical approaches, where the iteration is performed simultaneously for all parameters. The NIPALS procedure for canonical correlation as given by Wold [7] makes use of least squares regres-

An Alternative Econometric Approach to the Permanent Income Hypothesis: An International Comparison

PpTHE theory of aggregative consumption function has undergone several developments since Kuznets' [27] saving-income estimates for the United States showed disagreement with Keynes' [ 2 1 ] fundamental psychological rule. Noteworthy among these developments are the Relative Income Hypothesis [10] the Life-Cycle Hypothesis [33] and the Permanent Income Hypothesis [ 13 ]. Although these different hypotheses are rather complementary yet the Permanent Income Hypothesis (PIH) has been more widely studied than the others. According to the PIH, measured income and consumption are composed of two components permanent and transitory. Under the assumption of zero correlation between transitory and permanent and also between the two transitory components, postulates that permanent consumption constitutes a constant proportion of permanent income. The basic concepts of permanent income and consumption, although theoretically attractive, involved considerable difficulties in regard to their measurability. This stimulated a number of debates and empirical researches. Some corroborated the PIH, while others showed a strong disagreement over some of its major assumptions. A close investigation of these debates and researches suggests that Friedman's own proposal to use Cagan's [7] distributed lag scheme, as one of the alternatives for constructing an estimate of permanent income in time series studies, has perhaps invited more troubles than actually solved. Clearly, does not allow for a separation of the permanent and transitory components from the observed data.' It is not surprising then that Laumas [28] and Choudhury [8] can obtain significant marginal propensities to consume transitory income for Canada and India respectively, and that Walters [41] can show a relationship between transitory and permanent income. A cursory look at such studies may sometimes shatter the faith in the strict version of the PIH but if one studies them more carefully one feels inclined to speak with Ovid it often happens that the material is better than the workmanship. The purpose of the present paper is two-fold: first, to improve upon the workmanship and second, to investigate whether the PIH is applicable to economies with different structures. The improvement in the 'workmanship' lies in the application of Nonlinear Iterative Least Squares (NILES) procedure 2 to estimate the coefficient of proportionality when both permanent income and consumption are unknown. The advantages of this method over the commonly used Ordinary Least Squares procedure are that overcomes the problem of identification in the consumption function arising from the nonlinearity of parameters and that yields individual estimates of the parameters with least squares properties. In addition to this, is computationally more convenient than other nonlinear procedures. This method has been applied on the one hand to the existing approach in which the estimate of permanent income has been defined in terms of Cagan's distributed lag scheme, and on the other hand to an alternative approach in which considers the PIH on the lines of the classical two-variables linear model where both variables are subject to errors of observation. In neither case do we violate any of Friedman's assumptions. The advantage of the second approach lies in the fact that the estimation of permanent income does not require any extraneous informa* The authors would like to express their gratitude to Professors A. R. Dobell, S. J. Turnovsky and T. A. Wilson for constructive criticism and encouragement, and to the referee for helpful comments on an earlier draft of this paper. Needless to say, the responsibility for any errors lies with the authors alone. 'Singh [38], pp. 3-6. 2For the NILES procedure see Wold [43], pp. 433-434 and 438-440.

Discussion of Calculation of Power Function with Special Reference to Respiratory Metabolism in Fish

The rationale for employing a nonlinear iterative least-squares technique for fitting the well-known power function to oxygen consumption–body weight data is set forth. Twenty-six sets of routine or standard metabolism data from six authors were used to demonstrate the relative merits of two methods of calculating parameter values for the power function. The conclusion was reached that if accuracy in predicting oxygen consumption over a wide range of values of body weight is desired, an iterative curve fitting method may be superior to the much used technique of performing a linear regression on logarithmically transformed data.