Academic literature on the topic 'Newton-Raphson Iterative Proportional Fitting Algorithm'

1. A procedure based on compartmental modeling called the "constrained inverse computation" was developed for estimating the electrotonic structure of neurons. With the constrained inverse computation, a set of N electrotonic parameters are estimated iteratively with use of a Newton-Raphson algorithm given values of N parameters that can be measured or estimated from experimental data. 2. The constrained inverse computation is illustrated by several applications to the basic example of a neuron represented as one cylinder coupled to a soma. The number of unknown parameters estimated was different (ranging from 2 to 6) when different sets of constraints were chosen. The unknowns were chosen from the following: dendritic membrane resistivity Rmd, soma membrane resistivity Rms, intracellular resistivity Ri, membrane capacity Cm, dendritic membrane area AD, soma membrane area As, electrotonic length L, and resistivity-free length, rfl (rfl = 2l/d1/2 where l and d are length and diameter of the cylinder). The values of the unknown parameters were estimated from the values of an equal number of known parameters, which were chosen from the following: the time constants and coefficients of a voltage transient tau 0, tau 1, ..., C0, C1, ..., voltage-clamp time constants tau vc1, tau vc2, ..., and input resistance RN. Note that initially, morphological data were treated as unknown, rather than known. 3. When complete morphology was not known, parameters from voltage and current transients, combined with the input resistance were not sufficient to completely specify the electrotonic structure of the neuron. For a neuron represented as a cylinder coupled to a soma, there were an infinite number of combinations of Rmd, Rms, Ri, Cm, AS, AD, and L that could be fitted to the same voltage and current transients and input resistance. 4. One reason for the nonuniqueness when complete morphology was not specified is that the Ri estimate is intrinsically bound to the morphology. Ri enters the inverse computation only in the calculation of the electrotonic length of a compartment. The electrotonic length of a compartment is l[4 Ri/(dRmd)]1/2, where l and d are the length and diameter of the compartment. Without complete morphology, the inverse computation cannot distinguish between a change in d or l and a change in Ri. Even when morphology is known, the accuracy of the Ri estimate obtained by any fitting procedure is affected by systematic errors in length and diameter measurements (i.e., tissue shrinkage); the Ri estimate is inversely proportional to the length measurement and proportional to the square root of the diameter measurement.(ABSTRACT TRUNCATED AT 400 WORDS)

AbstractThis paper presents a combined electro-thermal model to serve the aim of accurate output power prediction of photovoltaic systems, based on the concept of the thermal energy balance. The electrical sub-model is built based on fitting a surface to the current-voltage curves collected under wide range temperatures and irradiances. For this purpose, the current-voltage characteristic curves are reproduced using two different methods. The thermal sub-model considers all the effective heat transfer mechanisms to estimate the photovoltaic module junction temperature. The Newton-Raphson iterative method is used as a solving algorithm to calculate the photovoltaic junction temperature. The collected results prove the applicability of the model under a wide range of environmental conditions.

The creation and maintenance of complex forest structures has become an important forestry objective. Complex forest structures, often expressed in multimodal shapes of tree size/diameter (DBH) distributions, are challenging to model. Mixture probability density functions of two- or three-component gamma, log-normal, and Weibull mixture models offer a solution and can additionally provide insights into forest dynamics. Model parameters can be efficiently estimated with the maximum likelihood (ML) approach using iterative methods such as the Newton-Raphson (NR) algorithm. However, the NR algorithm is sensitive to the choice of initial values and does not always converge. As an alternative, we explored the use of the iterative expectation-maximization (EM) algorithm for estimating parameters of the aforementioned mixture models because it always converges to ML estimators. Since forestry data frequently occur both in grouped (classified) and ungrouped (raw) forms, the EM algorithm was applied to explore the goodness-of-fit of the gamma, log-normal, and Weibull mixture distributions in three sample plots that exhibited irregular, multimodal, highly skewed, and heavy-tailed DBH distributions where some size classes were empty. The EM-based goodness-of-fit was further compared against a nonparametric kernel-based density estimation (NK) model and the recently popularized gamma-shaped mixture (GSM) models using the ungrouped data. In this example application, the EM algorithm provided well-fitting two- or three-component mixture models for all three model families. The number of components of the best-fitting models differed among the three sample plots (but not among model families) and the mixture models of the log-normal and gamma families provided a better fit than the Weibull distribution for grouped and ungrouped data. For ungrouped data, both log-normal and gamma mixture distributions outperformed the GSM model and, with the exception of the multimodal diameter distribution, also the NK model. The EM algorithm appears to be a promising tool for modeling complex forest structures.

In functionally graded saturated poroelastic circular plates with immovable simply supported and clamped rims, the axisymmetric nonlinear bending under transverse thermo-mechanical loading has been parametrically studied and compared with the axisymmetric postbuckling and nonlinear bending under thermal loading. Based on the classical plate theory, Love–Kirchhoff hypotheses and Sander’s assumptions, the general coupled nonlinear radial and transverse equilibrium equations, central continuity, symmetry and boundary conditions has been derived in ordinary and state-spatial forms. The corresponding difference equations have been achieved by using the generalized differential quadrature method. The equations have been assembled and numerically solved by using the Newton–Raphson iterative algorithm. The effects of the mechanical and thermal loads, pore distribution type, porosity parameter, Skempton’s coefficient, and thickness and boundary condition type on the behavior of the deflection, whether caused by thermo-mechanical bending, thermal postbuckling, or thermal bending, have been investigated in detail. From the parametric study, a novel quantity determining bending behavior has been found. The axisymmetric themo-mechanical nonlinear bending deflection is inversely and nonlinearly proportional to thermal load when the quantity is greater than a critical value and is nonlinearly proportional to thermal load when the quantity is less than a critical value. It was verified that the plate behavior complies with the general rules known for FG saturated poroelastic circular plates and with those known for metal–ceramic functionally graded circular plates whose governing equations are mathematically analogous to those of the current research.

Abstract This paper considers the problem of semi-parametric proportional hazards model fitting where observed survival times contain event times and also interval, left and right censoring times. Although this is not a new topic, many existing methods suffer from poor computational performance. In this paper, we adopt a more versatile penalized likelihood method to estimate the baseline hazard and the regression coefficients simultaneously. The baseline hazard is approximated using basis functions such as M-splines. A penalty is introduced to regularize the baseline hazard estimate and also to ease dependence of the estimates on the knots of the basis functions. We propose a Newton–MI (multiplicative iterative) algorithm to fit this model. We also present novel asymptotic properties of our estimates, allowing for the possibility that some parameters of the approximate baseline hazard may lie on the parameter space boundary. Comparisons of our method against other similar approaches are made through an intensive simulation study. Results demonstrate that our method is very stable and encounters virtually no numerical issues. A real data application involving melanoma recurrence is presented and an R package ‘survivalMPL’ implementing the method is available on R CRAN.

Il presente lavoro di tesi illustra un'applicazione dei modelli grafici per il l’analisi del credit scoring comportamentale o behavioural scoring. Quest'ultimo e' definito come: ‘the systems and models that allow lenders to make better decisions in managing existing clients by forcasting their future performance’, secondo Thomas (1999). La classe di modelli grafici presa in considerazione e’ quella dei modelli garfici a catena. Sono dei modelli statistici multivariati che consetono di modellizzare in modo appropriato le relazioni tra le variabili che descrivono il comporatemento dei titoloari della carta. Dato che sono basati su un'espansione log-lineare della funzione di densità delle variabili consentono di rappresentare anche graficamente associazioni orientate, inerenti sottoinsiemi di variabili. Consentono, inoltre, di individuare la struttura che rappresenti in modo più parsimonioso possibile tali relazioni e modellare simultaneamente più di una variabile risposta. Sono utili quando esiste un ordinamento anche parziale tra le variabili che permette di suddividerle in meramente esogene, gruppi d’intermedie tra loro concatenate e di risposta. Nei modelli grafici la struttura d’indipendenza delle variabili viene rappresentata visivamente attraverso un grafo. Nel grafo le variabili sono rappresentate da nodi legati da archi i quali mostrano le dipendenze in probabilità tra le variabili. La mancanza di un arco implica che due nodi sono indipendenti dati gli altri nodi. Tali modelli risultano particolarmente utili per la teoria che li accomuna con i sistemi esperti, per cui una volta selezionato il modello è possibile interrogare il sistema esperto per modellare la distribuzione di probabilità congiunta e marginale delle variabili. Nel primo capitolo vengono presentati i principali modelli statistici adottati nel credit scoring. Il secondo capitolo prende in considerazione le variabili categoriche. Le informazioni sui titolari di carta di credito sono, infatti, compendiate in tabelle di contingenza. Si introducono le nozioni d’indipendenza tra due variabili e di indipendenza condizionata tra più di due variabili. Si elencano alcune misure d’associazione tra variabili, in particolare, si introducono i rapporti di odds che costituiscono la base per la costruzione dei modelli multivariati utilizzati. Nel terzo capitolo vengono illustrati i modelli log-lineari e logistici che appartengono alla famiglia dei modelli lineari generalizzati. Essendo metodi multivariati consentono di studiare l’associazione tra le variabili considerandole simultaneamente. In particolare viene descritta una speciale parametrizzazione log-lineare che permette di tener conto della scala ordinale con cui sono misurate alcune delle variabili categoriche utilizzate. Questa è anche utile per trovare la migliore categorizzazione delle variabili continue. Si richiamano, inoltre, i risultati relativi alla stima di massima verosimiglianza dei parametri dei modelli, accennando anche agli algoritmi numerici iterativi necessari per la risoluzione delle equazioni di verosimiglianza rispetto ai parametri incogniti. Si fa riferimento al test del rapporto di verosimiglianza per valutare la bontà di adattamento del modello ai dati. Il capitolo quarto introduce alla teoria dei grafi, esponendone i concetti principali ed evidenziando alcune proprietà che consentono la rappresentazione visiva del modello mediante il grafo, mettendone in luce i vantaggi interpretativi. In tale capitolo si accenna anche al problema derivante dalla sparsità della tabella di contingenza, quando le dimensioni sono elevate. Vengono pertanto descritti alcuni metodi adottati per far fronte a tale problema ponendo l’accento sulle definizioni di collassabilità. Il quinto capitolo illustra un’applicazione dei metodi descritti su un campione composto da circa sessantamila titolari di carta di credito revolving, rilasciata da una delle maggiori società finanziarie italiane operanti nel settore. Le variabili prese in esame sono quelle descriventi le caratteristiche socioeconomiche del titolare della carta, desumibili dal modulo che il cliente compila alla richiesta di finanziamento e lo stato del conto del cliente in due periodi successivi. Ogni mese, infatti, i clienti vengono classificati dalla società in: ‘attivi’, ‘inattivi’ o ‘dormienti’ a seconda di come si presenta il saldo del conto. Lo scopo del lavoro è stato quello di ricercare indipendenze condizionate tra le variabili in particolare rispetto alle due variabili obbiettivo e definire il profilo di coloro che utilizzano maggiormente la carta. Le conclusioni riguardanti le analisi effettuate al capitolo quinto sono riportate nell’ultima sezione. L’appendice descrive alcuni dei principali programmi relativi ai software statistici utilizzati per le elaborazioni.
In this thesis work the use of graphical models is proposed to the analysis of credit scoring. In particular the applied application is related to the behavioural scoring which is defined by Thomas (1999) as ‘the systems and models that allow lenders to make better decisions in managing existing clients by forecasting their future performance’. The multivariate statistical models, named chain graph models, proposed for the application allow us to model in a proper way the relation between the variables describing the behaviour of the holders of the credit card. The proposed models are named chain graph models. They are based on a log-linear expansion of the density function of the variables. They allow to: depict oriented association between subset of variables; to detect the structure which accounts for a parsimonious description of the relations between variables; to model simultaneously more than one response variable. They are useful in particular when there is a partial ordering between variables such that they can be divided into exogenous, intermediate and responses. In the graphical models the independence structure is represented by a graph. The variables are represented by nodes, joint by edges showing the dependence in probability among variables. The missing edge means that two nodes are independent given the other nodes. Such class of models is very useful for the theory which combines them with the expert systems. In fact, once the model has been selected, it is possible to link it to the expert system to model the joint and marginal probability of the variables. The first chapter introduces the most used statistical models for the credit scoring analysis. The second chapter introduces the categorical variables. The information related to the credit card holder are stored in a contingency table. It illustrates also the notion of independence between two variables and conditional independence among more than two variables. The odds ratio is introduced as a measure of association between two variables. It is the base of the model formulation. The third chapter introduces the log-linear and logistic models belonging to the family of generalized linear models. They are multivariate methods allowing to study the association between variables considering them simultaneously. A log-linear parameterization is described in details. Its advantage is also that it allow us to take into account of the ordinal scale on which the categorical variables are measured. This is also useful to find the better categorization of the continuous variables. The results related to the maximum likelihood estimation of the model parameters are mentioned as well as the numerical iterative algorithm which are used to solve the likelihood equations with respect to the unknown parameters. The score test is illustrated to evaluate the goodness of fit of the model to the data. Chapter 4 introduces some main concepts of the graph theory in connection with their properties which allow us to depict the model through the graph, showing the interpretative advantages. The sparsity of the contingency table is also mentioned, when there are many cells. The collapsibility conditions are considered as well. Finally, Chapter 5 illustrates the application of the proposed methodology on a sample composed by 70000 revolving credit card holders. The data are released by a one of biggest Italian financial society working in this sector. The variables are the socioeconomic characteristics of the credit card holder, taken form the form filled by the customer when asking for the credit. Every months the society refines the classification of the customers in active, inactive or asleep according to the balance. The application of the proposed method was devoted to find the existing conditional independences between variables related to the two responses which are the balance of the account at two subsequent dates and therefore to define the profiles of most frequently users of the revolving credit card. The chapter ends with some conclusive remarks. The appendix of the chapter reports the code of the used statistical softwares.