A special nonlinear least-squares problem

AbstractIn this paper we consider the existence of the solution of a special nonlinear least-squares problem. We find the necessary conditions on the data, which insure the existence of the optimal parameters for the asymmetric S-function in the sense of the least squares

Reconstruction of the Earth\u27s Meridian Ellipse on the Basis of Experimental Measurements

Na bazi novih metoda matematičke optimizacije analizira se stari problem rekonstrukcije Zemljina rotacijskog elipsoida na bazi podataka mjerenja, koji je već početkom 18. stoljeća promatrao francuski matematičar i astronom markiz Pierre-Simon de Laplace. Posebno je analizirana, ilustrirana i modificirana efikasna metoda iz rada (Sabo i Scitovski 2009) za traženje optimalnih parametara najbolje l1 linearne aproksimacije. U znanstvenoj literaturi problem procjene najbolje l1 aproksimacije obično se povezuje s imenom hrvatskog znanstvenika J. R. Boškovića. Metode za traženje najbolje l1 aproksimacije u posljednje vrijeme dobivaju novi zamah u različitim tehničkim primjenama zbog svojstva robusnosti i mogućnosti implementacije u realnom vremenu na suvremenim računalima.On the basis of new mathematical optimization methods this paper analyzes an old problem of reconstructing the Earth\u27s ellipsoid of rotation on the basis of measurement data, which was considered by the French mathematician and astronomer marquis Pierre-Simon de Laplace as early as the beginning of the 18th century. An efficient method from the paper (Sabo and Scitovski 2009) for searching optimal parameters of the best least absolute deviations linear approximation is especially analyzed, illustrated and modified. In the scientific literature the best l1 approximation is usually associated with the name of Croatian scientist J. R. Bošković. The least absolute deviations methods have recently started to be extensively applied in different technical applications due to their property of robustness and the possibility of implementation in real time on modern computers

Reconstruction of the Earth\u27s Meridian Ellipse on the Basis of Experimental Measurements

Na bazi novih metoda matematičke optimizacije analizira se stari problem rekonstrukcije Zemljina rotacijskog elipsoida na bazi podataka mjerenja, koji je već početkom 18. stoljeća promatrao francuski matematičar i astronom markiz Pierre-Simon de Laplace. Posebno je analizirana, ilustrirana i modificirana efikasna metoda iz rada (Sabo i Scitovski 2009) za traženje optimalnih parametara najbolje l1 linearne aproksimacije. U znanstvenoj literaturi problem procjene najbolje l1 aproksimacije obično se povezuje s imenom hrvatskog znanstvenika J. R. Boškovića. Metode za traženje najbolje l1 aproksimacije u posljednje vrijeme dobivaju novi zamah u različitim tehničkim primjenama zbog svojstva robusnosti i mogućnosti implementacije u realnom vremenu na suvremenim računalima.On the basis of new mathematical optimization methods this paper analyzes an old problem of reconstructing the Earth\u27s ellipsoid of rotation on the basis of measurement data, which was considered by the French mathematician and astronomer marquis Pierre-Simon de Laplace as early as the beginning of the 18th century. An efficient method from the paper (Sabo and Scitovski 2009) for searching optimal parameters of the best least absolute deviations linear approximation is especially analyzed, illustrated and modified. In the scientific literature the best l1 approximation is usually associated with the name of Croatian scientist J. R. Bošković. The least absolute deviations methods have recently started to be extensively applied in different technical applications due to their property of robustness and the possibility of implementation in real time on modern computers

Parameter identification in the mathematical model of glucose and insulin tolerance test - the mathematical markers of diabetes

Glucose tolerance test (GTT) is standard diagnostic procedure that tests the efficiency of blood glucose-lowering hormones (insulin, incretins, leptin). Contrary, insulin tolerance test (ITT) is probing efficiency of blood glucose-rising hormones (glucagon, thyroxine, growth hormone, glucocorticoids, adrenalin, noradrenalin). These two hormone systems together maintain blood glucose levels in a narrow range. Various pathophysiological mechanisms give rise to a reversible condition - prediabetes which then progresses to an irreversible chronic disease - diabetes, both marked with deviation of blood glucose levels outside the set range. In diagnostic purpose, the patient is given glucose load, and blood glucose is measured right before and 2 hours after load. Measurements are more frequent after insulin injection (ITT) or if both tests are performed on experimental animals. In this paper we analyse the mathematical model for GTT and ITT. The obtained model function is an useful tool in describing the dynamics of blood glucose changes

SHORT-TERM AND LONG-TERM WATER LEVEL PREDICTION AT ONE RIVER MEASUREMENT LOCATION

Global hydrological cycles mainly depend on climate changes whose occurrence is predominantly triggered by solar and terrestrial influence, and the knowledge of the high water regime is widely applied in hydrology. Regular monitoring and studying of river water level behavior is important from several perspectives. On the basis of the given data, by using modifications of general approaches known from literature, especially from investigation in hydrology, the problem of long- and short-term water level forecast at one river measurement location is considered in the paper. Long-term forecasting is considered as the problem of investigating the periodicity of water level behavior by using linear-trigonometric regression and short-term forecasting is based on the modification of the nearest neighbor method. The proposed methods are tested on data referring to the Drava River level by Donji Miholjac, Croatia, in the period between the beginning of 1900 and the end of 2012

Least-squares fitting Gompertz curve

AbstractIn this paper we consider the least-squares (LS) fitting of the Gompertz curve to the given nonconstant data (pi,ti,yi), i=1,…,m, m⩾3. We give necessary and sufficient conditions which guarantee the existence of the LS estimate, suggest a choice of a good initial approximation and give some numerical examples

A fast and efficient method for solving the multiple line detection problem

In this paper, we consider the multiple line detection problem on the basis of a data points set coming from a number of lines not known in advance. A new and efficient method is proposed, which is based upon center-based clustering, and it solves this problem quickly and precisely. The method has been tested on 100 randomly generated data sets. In comparison to the incremental algorithm, the method gives significantly better results. Also, in order to identify a partition with the most appropriate number of clusters, a new index has been proposed for the case of a cluster whose lines are cluster-centers. The index can also be generalized for other geometrical objects

Matematičko modeliranje ekonomskih pojava koje teže zasićenju

U ekonomskim istraživanjima često je potrebno neku pojavu opisati funkcijom čija se vrijednost asimptotski približava razini zasićenja. U radu se promatra klasa tzv. S-funkcija, koje služe za kratkoročne i dugoročne prognoze. Poseban naglasak dat je na analizu klasa funkcija, koje ovisno o parametru, mogu imati pozitivnu ili negativnu asimetriju ili se mogu preobraziti u poznatu logističku funkciju. Procjenjivanje parametara ovakvih funkcija je složen nelinearni problem najmanjih kvadrata, koji ne mora uvijek imati rješenje. Ekonomska interpretacija svojstava tih funkcija je izuzetno značajna za strategijsko upravljanje ekonomskim sistemima

On a global optimization problem

U radu se promatra sljedeći problem globalne optimizacije [argminlimits_{ainmathbb{R}^n}F(a),quad F(a)=intlimits_0^{+infty}e^{-x}left(1+a_1x+cdots+a_nx^nright)^2dx.] Pokazano je da ovaj problem ima jedinstveno rješenje, koje se može odrediti rješavanjem odgovarajućeg problema najmanjih kvadrata ili kao specijalni slučaj jednog općenitijeg problema najbolje aproksimacije u unitarnom vektorskom prostoru. U drugom slučaju primijenjeni su Laguerrovi ortogonalni polinomi. Rješavanje problema ilustrirano je s nekoliko numeričkih primjera.We consider the following global optimization problem [argminlimits_{ainmathbb{R}^n}F(a),quad F(a)=intlimits_0^{+infty}e^{-x}left(1+a_1x+cdots+a_nx^nright)^2dx. ] It is shown that {this problem has a unique solution, which can be determined by solving the corresponding least squares problem or as a special case of a general best approximation problem} in a unitary vector space. In the latter case, Laguerre polynomials are applied. The problem solving is illustrated by several numerical examples

ANALYSIS OF THE CHICK GROWTH CHARACTERISTICS BY MEANS OF ASYMETRIC S-FUNCTION

U radu su istraživane fenotipske značajke rasta Arbor Acres pilića do 6. tjedna odnosno 10. tjedna. U istraživanja je uključeno po 30 pilića ženskog odnosno muškog spola. U modeliranju krivulje rasta upotrebljena je asimetrična S-funkcija. Pomoću navedene funkcije utvrđene su karakteristične faze rasta i točka infleksije. Istraživanje je pokazalo da je asimetrična S-funkcija prikladna i za prognozu životne mase pilića do 10. tjedna.This article deals with the analysis of phenotypic growth characteristics of the Arbor Acres chicks till the 6th, i. e. 10th week of age. The study included 30 female and male chicks. An asymmetric S function was applied as a growth curve. The characteristics of the growth stages and the point of inflection were determined. The study showed that the asymmetric S-function was adequate for forecasting the weight, i. e. live mass of the chicks till the 10th week of age