Search results for: extended linear sigma model

Authors: Wen-Tsung Ho, Ku-Kuang Chang, Hsin-Yuan Chang

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In this study, we consider an economic lot-size batch-shipment scheduling problem (ELBSP) with extended basic period (EBP) and power-of-two (PoT) policies. In this problem, the supplier using a single facility to manufacture multiple products and equally sized batches are then delivered by the supplier to buyers over an infinite planning horizon. Further, the extended basic period (EBP) and power-of-two (PoT) policy are utilized. Relaxing the production schedule converts the ELBSP to an economic lot-size batch-shipment problem (ELBP) with EBP and PoT policies, and a nonlinear integer programming model of the ELBP is constructed. Using the replenishment cycle division and recursive tightening methods, optimal solutions are then solved separately for each product. The sum of these optimal solutions is the lower bound of the ELBSP. A proposed heuristic method with polynomial complexity is then applied to figure out the near-optimal solutions of the ELBSP. Numerical example is presented to confirm the efficacy of the proposed method.

Keywords: economic lot-size scheduling problem, extended basic period, replenishment cycle division, recursive tightening, power-of-two

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Authors: Ahmed Elsayed Ahmed, Ashraf Hafez, A. N. Ouda, Hossam Eldin Hussein Ahmed, Hala Mohamed ABD-Elkader

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Unmanned Aircraft Systems (UAS) are playing increasingly prominent roles in defense programs and defense strategies around the world. Technology advancements have enabled the development of it to do many excellent jobs as reconnaissance, surveillance, battle fighters, and communications relays. Simulating a small unmanned aerial vehicle (SUAV) dynamics and analyzing its behavior at the preflight stage is too important and more efficient. The first step in the UAV design is the mathematical modeling of the nonlinear equations of motion. In this paper, a survey with a standard method to obtain the full non-linear equations of motion is utilized,and then the linearization of the equations according to a steady state flight condition (trimming) is derived. This modeling technique is applied to an Ultrastick-25e fixed wing UAV to obtain the valued linear longitudinal and lateral models. At the end, the model is checked by matching between the behavior of the states of the non-linear UAV and the resulted linear model with doublet at the control surfaces.

Keywords: UAV, equations of motion, modeling, linearization

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Authors: Mahmoudi Noureddine, Bouregba Rachid

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In this paper the stress intensity factors of a slant-cracked plate of AISI 304 stainless steel, have been calculated using extended finite element method and finite element method (FEM) in ABAQUS software, the results were compared with theoretical values.

Keywords: stress intensity factors, extended finite element method, stainless steel, abaqus

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Authors: Fahmi Hassan, Noorizan Abdul Aziz, Yahaya Hassan, Hazlinda Abu Hassan

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Sepsis incidence in critical care settings is a major problem in health care. Extended antibiotic infusion is thought to be superior to traditional dosing especially when treating critically ill patients with sepsis. We compared clinical outcomes of critically ill patients with sepsis receiving 30-minute meropenem infusion and three-hour meropenem infusion. A retrospective case-control study was conducted among septic patients treated with meropenem infusion in ICUs of three hospitals. Patients included in the study received either extended or standard meropenem infusion as per the practice of individual settings. Outcomes and clinical data were retrospectively collected from the electronic databases and patients’ files. A total of 108 patients received extended meropenem infusion while another 117 patients received standard meropenem infusion. Patients receiving the extended meropenem infusion were found to have a significantly lower shorter length of hospital and ICU stay. It was also found that among those receiving extended meropenem infusion, 54.7% (64/117) had a reduction of SAPS II score, while only 44% (48/108) of patients receiving standard meropenem infusion had reduced scores. This study will strengthen the evidence in using extended meropenem infusion as a standard practice in critical care settings. As this is the first study of its kind done in Malaysia, it proves that prolonged meropenem infusion may be beneficial to critically ill patients with sepsis. However, randomized clinical trials with large sample size should be carried out in local settings in order to minimize other confounders that may influence with the result of the study.

Keywords: antibiotics, beta lactams, critical care, extended infusion, meropenem

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Authors: Sofia Stoica

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In this paper, we use historical data to predict the stock price of a tech company. To this end, we use a dataset consisting of the stock prices in the past five years of ten major tech companies – Adobe, Amazon, Apple, Facebook, Google, Microsoft, Netflix, Oracle, Salesforce, and Tesla. We experimented with a variety of models– a linear regressor model, K nearest Neighbors (KNN), a sequential neural network – and algorithms - Multiplicative Weight Update, and AdaBoost. We found that the sequential neural network performed the best, with a testing error of 0.18%. Interestingly, the linear model performed the second best with a testing error of 0.73%. These results show that using historical data is enough to obtain high accuracies, and a simple algorithm like linear regression has a performance similar to more sophisticated models while taking less time and resources to implement.

Keywords: finance, machine learning, opening price, stock market

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Authors: W. R. Li, J. K. Xia, R. Q. Peng, Z. Y. Guo, L. Jiang

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According to 3D magnetic circuit of the transverse flux PM linear machine, distribution law is presented, and analytical expression of axial end flux leakage is derived using numerical method. Maxwell stress tensor is used to solve detent force of mover. A 3D finite element model of the transverse flux PM machine is built to analyze the flux distribution and detent force. Experimental results of the prototype verified the validity of axial end flux leakage and detent force theoretical derivation, the research on axial end flux leakage and detent force provides a valuable reference to other types of linear machine.

Keywords: axial end flux leakage, detent force, flux distribution, transverse flux PM linear machine

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Authors: Xinhuang Wu, Yousef Sardahi

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A hybrid and optimal multi-loop control structure combining linear and nonlinear control algorithms are introduced in this paper to regulate the position of a quadcopter unmanned aerial vehicle (UAV) driven by four brushless DC motors. To this end, a nonlinear mathematical model of the UAV is derived and then linearized around one of its operating points. Using the nonlinear version of the model, a sliding mode control is used to derive the control laws of the motor thrust forces required to drive the UAV to a certain position. The linear model is used to design two controllers, XG-controller and YG-controller, responsible for calculating the required roll and pitch to maneuver the vehicle to the desired X and Y position. Three attitude controllers are designed to calculate the desired angular rates of rotors, assuming that the Euler angles are minimal. After that, a many-objective optimization problem involving 20 design parameters and ten objective functions is formulated and solved by HypE (Hypervolume estimation algorithm), one of the widely used many-objective optimization algorithms approaches. Both stability and performance constraints are imposed on the optimization problem. The optimization results in terms of Pareto sets and fronts are obtained and show that some of the design objectives are competing. That is, when one objective goes down, the other goes up. Also, Numerical simulations conducted on the nonlinear UAV model show that the proposed optimization method is quite effective.

Keywords: optimal control, many-objective optimization, sliding mode control, linear control, cascade controllers, UAV, drones

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Authors: Lin Wenli, Fu Linchun, Zhang Yi, Wu Ming

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To meet the demands of long-life and high-intelligence for satellites, the electrical power system should be provided with self-health condition evaluation capability. Any over-stress events in operations should be recorded. Based on Linear stresses accumulation damage theory, accumulative damage analysis was performed on thermal-mechanical-electrical united stresses for three components including the solar array, the batteries and the power conditioning unit. Then an overall health percentage evaluation model for satellite electrical power system was built. To obtain the accurate quantity for system health percentage, an automatic feedback closed-loop correction method for all coefficients in the evaluation model was present. The evaluation outputs could be referred as taking earlier fault-forecast and interventions for Ground Control Center or Satellites self.

Keywords: satellite electrical power system, health percentage, linear stresses accumulation damage, evaluation model

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Authors: Nur Aidya Hanum Aizam, Vikneswary Uvaraja

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The agenda of showing the scheduled time for performing certain tasks is known as timetabling. It widely used in many departments such as transportation, education, and production. Some difficulties arise to ensure all tasks happen in the time and place allocated. Therefore, many researchers invented various programming model to solve the scheduling problems from several fields. However, the studies in developing the general integer programming model for many timetabling problems are still questionable. Meanwhile, this thesis describe about creating a general model which solve different types of timetabling problems by considering the basic constraints. Initially, the common basic constraints from five different fields are selected and analyzed. A general basic integer programming model was created and then verified by using the medium set of data obtained randomly which is much similar to realistic data. The mathematical software, AIMMS with CPLEX as a solver has been used to solve the model. The model obtained is significant in solving many timetabling problems easily since it is modifiable to all types of scheduling problems which have same basic constraints.

Keywords: AIMMS mathematical software, integer linear programming, scheduling problems, timetabling

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Authors: Seyed Amin Vakili, Sahar Sadat Vakili, Seyed Ehsan Vakili, Nader Abdoli Yazdi

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Differential equations are of fundamental importance in engineering and applied mathematics, since many physical laws and relations appear mathematically in the form of such equations. The equilibrium state of structures consisting of one-dimensional elements can be described by an ordinary differential equation. The response of these kinds of structures under the loading, namely relationship between the displacement field and loading field, can be predicted by the solution of these differential equations and on satisfying the given boundary conditions. When the effect of change of geometry under loading is taken into account in modeling of equilibrium state, then these differential equations are partially integrable in quartered. They also exhibit instability characteristics when the structures are loaded compressively. The purpose of this paper is to represent the ability of the Modified Newmark Method in analyzing flexural-torsional instability of struts for both bifurcation and non-bifurcation structural systems. The results are shown to be very accurate with only a small number of iterations. The method is easily programmed, and has the advantages of simplicity and speeds of convergence and easily is extended to treat material and geometric nonlinearity including no prismatic members and linear and nonlinear spring restraints that would be encountered in frames. In this paper, these abilities of the method will be extended to the system of linear differential equations that govern strut flexural torsional stability.

Keywords: instability, torsion, flexural, buckling, modified newmark method stability

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Authors: Oliver Takawira, Thembi Hope

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This study will analyse the relationship between Bitcoin price movements and the Johannesburg stock exchange (JSE). The aim is to determine whether Bitcoin price movements affect the stock market performance. As crypto currencies continue to gain prominence as a safe asset during periods of economic distress, this raises the question of whether Bitcoin’s prosperity could affect investment in the stock market. To identify the existence of a short run and long run linear relationship, the study will apply the Autoregressive Distributed Lag Model (ARDL) bounds test and a Vector Error Correction Model (VECM) after testing the data for unit roots and cointegration using the Augmented Dicker Fuller (ADF) and Phillips-Perron (PP). The Non-Linear Auto Regressive Distributed Lag (NARDL) will then be used to check if there is a non-linear relationship between bitcoin prices and stock market prices.

Keywords: bitcoin, stock market, interest rates, ARDL

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Authors: Retius Chifurira

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Logistic regression model is the most used regression model to predict meteorological drought probabilities. When the dependent variable is extreme, the logistic model fails to adequately capture drought probabilities. In order to adequately predict drought probabilities, we use the generalized linear model (GLM) with the quantile function of the generalized extreme value distribution (GEVD) as the link function. The method maximum likelihood estimation is used to estimate the parameters of the generalized extreme value (GEV) regression model. We compare the performance of the logistic and the GEV regression models in predicting drought probabilities for Zimbabwe. The performance of the regression models are assessed using the goodness-of-fit tests, namely; relative root mean square error (RRMSE) and relative mean absolute error (RMAE). Results show that the GEV regression model performs better than the logistic model, thereby providing a good alternative candidate for predicting drought probabilities. This paper provides the first application of GLM derived from extreme value theory to predict drought probabilities for a drought-prone country such as Zimbabwe.

Keywords: generalized extreme value distribution, general linear model, mean annual rainfall, meteorological drought probabilities

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Authors: Asraful Islam

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Extended dependency graph (EDG) is a state-of-the-art isomorphic graph to represent normal logic programs (NLPs) that can characterize the consistency of NLPs by graph analysis. To construct the vertices and arcs of an EDG, additional renaming atoms and rules besides those the given program provides are used, resulting in higher space complexity compared to the corresponding traditional dependency graph (TDG). In this article, we propose an extended labeled dependency graph (ELDG) to represent an NLP that shares an equal number of nodes and arcs with TDG and prove that it is isomorphic to the domain program. The number of nodes and arcs used in the underlying dependency graphs are formulated to compare the space complexity. Results show that ELDG uses less memory to store nodes, arcs, and cycles compared to EDG. To exhibit the desirability of ELDG, firstly, the stable models of the kernel form of NLP are characterized by the admissible coloring of ELDG; secondly, a relation of the stable models of a kernel program with the handles of the minimal, odd cycles appearing in the corresponding ELDG has been established; thirdly, to our best knowledge, for the first time an inverse transformation from a dependency graph to the representing NLP w.r.t. ELDG has been defined that enables transferring analytical results from the graph to the program straightforwardly.

Keywords: normal logic program, isomorphism of graph, extended labelled dependency graph, inverse graph transforma-tion, graph colouring

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Authors: Pitsanu Tongkhow, Pichet Jiraprasertwong

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This research investigates risk factors for defective products in autoparts factories. Under a Bayesian framework, a generalized linear mixed model (GLMM) in which the dependent variable, the number of defective products, has a Poisson distribution is adopted. Its performance is compared with the Poisson GLM under a Bayesian framework. The factors considered are production process, machines, and workers. The products coded RT50 are observed. The study found that the Poisson GLMM is more appropriate than the Poisson GLM. For the production Process factor, the highest risk of producing defective products is Process 1, for the Machine factor, the highest risk is Machine 5, and for the Worker factor, the highest risk is Worker 6.

Keywords: defective autoparts products, Bayesian framework, generalized linear mixed model (GLMM), risk factors

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Authors: Fuziyah Ishak, Siti Norazura Ahmad

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Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations.

Keywords: accuracy, extended trapezoidal method, numerical solution, Volterra integro-differential equations

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Authors: Lavinia B. Dulla

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The exploration study of the fuzzy regression approach attempts to present that fuzzy regression can be used as a possible alternative to classical regression. It likewise seeks to assess the differences and characteristics of simple linear regression and fuzzy regression using the width of prediction interval, mean absolute deviation, and variance of residuals. Based on the simple linear regression model, the fuzzy regression approach is worth considering as an alternative to simple linear regression when the sample size is between 10 and 20. As the sample size increases, the fuzzy regression approach is not applicable to use since the assumption regarding large sample size is already operating within the framework of simple linear regression. Nonetheless, it can be suggested for a practical alternative when decisions often have to be made on the basis of small data.

Keywords: fuzzy regression approach, minimum fuzziness criterion, interval regression, prediction interval

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Authors: Fariba Azizi, Firoozeh Haghighi, Viliam Makis

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In this paper, we propose a method to model the relationship between failure time and degradation for a simple step stress test where underlying degradation path is linear and different causes of failure are possible. It is assumed that the intensity function depends only on the degradation value. No assumptions are made about the distribution of the failure times. A simple step-stress test is used to shorten failure time of products and a tampered failure rate (TFR) model is proposed to describe the effect of the changing stress on the intensities. We assume that some of the products that fail during the test have a cause of failure that is only known to belong to a certain subset of all possible failures. This case is known as masking. In the presence of masking, the maximum likelihood estimates (MLEs) of the model parameters are obtained through an expectation-maximization (EM) algorithm by treating the causes of failure as missing values. The effect of incomplete information on the estimation of parameters is studied through a Monte-Carlo simulation. Finally, a real example is analyzed to illustrate the application of the proposed methods.

Keywords: cause of failure, linear degradation path, reliability function, expectation-maximization algorithm, intensity, masked data

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Authors: Tamou Nasser, Ahmed Essadki, Ali Boukhriss

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This paper presents the control of doubly fed induction generator (DFIG) used in the wind energy conversion systems. Maximum power point tracking (MPPT) strategy is used to extract the maximum of power during the conversion and taking care that the system does not exceed the operating limits. This is done by acting on the pitch angle to control the orientation of the turbine's blades. Having regard to its robustness and performance, active disturbance rejection control (ADRC) based on the extended state observer (ESO) is employed to achieve the control of both rotor and grid side converters. Simulations are carried out using MATLAB simulink.

Keywords: active disturbance rejection control, extended state observer, doubly fed induction generator, maximum power point tracking

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Authors: Intan Maisarah Abd Rahim, Herlina Abdul Rahim, Rashidah Ghazali

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Diabetes is a medical condition that can lead to various diseases such as stroke, heart disease, blindness and obesity. In clinical practice, the concern of the diabetic patients towards the blood glucose examination is rather alarming as some of the individual describing it as something painful with pinprick and pinch. As for some patient with high level of glucose level, pricking the fingers multiple times a day with the conventional glucose meter for close monitoring can be tiresome, time consuming and painful. With these concerns, several non-invasive techniques were used by researchers in measuring the glucose level in blood, including ultrasonic sensor implementation, multisensory systems, absorbance of transmittance, bio-impedance, voltage intensity, and thermography. This paper is discussing the application of the near-infrared (NIR) spectroscopy as a non-invasive method in measuring the glucose level and the implementation of the linear system identification model in predicting the output data for the NIR measurement. In this study, the wavelengths considered are at the 1450 nm and 1950 nm. Both of these wavelengths showed the most reliable information on the glucose presence in blood. Then, the linear Autoregressive Moving Average Exogenous model (ARMAX) model with both un-regularized and regularized methods was implemented in predicting the output result for the NIR measurement in order to investigate the practicality of the linear system in this study. However, the result showed only 50.11% accuracy obtained from the system which is far from the satisfying results that should be obtained.

Keywords: diabetes, glucose level, linear, near-infrared, non-invasive, prediction system

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Authors: Sergio Haram Sarmiento, Arcady Ponosov

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Based on appropriate multivariate statistical methodology, we suggest a generic framework for efficient parameter estimation for ordinary differential equations and the corresponding nonlinear models. In this framework classical linear regression strategies is refined into a nonlinear regression by a locally linear modelling technique (known as metamodelling). The approach identifies those latent variables of the given model that accumulate most information about it among all approximations of the same dimension. The method is applied to several benchmark problems, in particular, to the so-called ”power-law systems”, being non-linear differential equations typically used in Biochemical System Theory.

Keywords: principal component analysis, generalized law of mass action, parameter estimation, metamodels

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Authors: Mohamed Hassan Abdullahi

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Decomposition is used as a synthesis tool in several physical systems. It can also be used for tearing and restructuring, which is large-scale system analysis. On the other hand, the commutativity of series-connected systems has fascinated the interest of researchers, and its advantages have been emphasized in the literature. The presentation looks into the necessary conditions for decomposing any third-order discrete-time linear time-varying system into a commutative pair of first- and second-order systems. Additional requirements are derived in the case of nonzero initial conditions. MATLAB simulations are used to verify the findings. The work is unique and is being published for the first time. It is critical from the standpoints of synthesis and/or design. Because many design techniques in engineering systems rely on tearing and reconstruction, this is the process of putting together simple components to create a finished product. Furthermore, it is demonstrated that regarding sensitivity to initial conditions, some combinations may be better than others. The results of this work can be extended for the decomposition of fourth-order discrete-time linear time-varying systems into lower-order commutative pairs, as two second-order commutative subsystems or one first-order and one third-order commutative subsystems.

Keywords: commutativity, decomposition, discrete time-varying systems, systems

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Authors: Farhad Ghasemi

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Along with transforming the international system into a complex and chaotic system, the fundamental question arises: how can deterrence be reconstructed conceptually and theoretically in this system model? The deterrence system is much more complex today than it was seven decades ago. This article suggests that the perception of deterrence as a linear system is a fundamental mistake because it does not consider the new dynamics of the international system, including network power dynamics. The author aims to improve this point by focusing on complexity and chaos theories, especially their nonlinearity and cascading failure principles. This article proposes that the perception of deterrence as a linear system is a fundamental mistake, as the new dynamics of the surrounding international system do not take into account. The author recognizes deterrence as a nonlinear system and introduces it as a concept in strategic studies.

Keywords: complexity, international system, deterrence, linear deterrence, nonlinear deterrence

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Authors: Edward J. Kansa, Pavel Holoborodko

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Continuously differentiable radial basis functions (RBFs) are meshfree, converge faster as the dimensionality increases, and is theoretically spectrally convergent. When implemented on current single and double precision computers, such RBFs can suffer from ill-conditioning because the systems of equations needed to be solved to find the expansion coefficients are full. However, the Advanpix extended precision software package allows computer mathematics to resemble asymptotically ideal Platonic mathematics. Additionally, full systems with extended precision execute faster graphical processors units and field-programmable gate arrays because no branching is needed. Sparse equation systems are fast for iterative solvers in a very limited number of cases.

Keywords: partial differential equations, Meshfree radial basis functions, , no restrictions on spatial dimensions, Extended arithmetic precision.

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Authors: Suleiman Obeidat, Nabeel Mandahawi

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In this work, the packaging process for a move is improved. The customers of this move need their household stuff to be moved from their current house to the new one with minimum damage, in an organized manner, on time and with the minimum cost. Our goal was to improve the process between 10% and 20% time efficiency, 90% reduction in damaged parts and an acceptable improvement in the cost of the total move process. The expected ROI was 833%. Many improvement techniques have been used in terms of the way the boxes are prepared, their preparation cost, packing the goods, labeling them and moving them to a place for moving out. DMAIC technique is used in this work: SIPOC diagram, value stream map of “As Is” process, Root Cause Analysis, Maps of “Future State” and “Ideal State” and an Improvement Plan. A value of ROI=624% is obtained which is lower than the expected value of 833%. The work explains the techniques of improvement and the deficiencies in the old process.

Keywords: packaging, lean tools, six sigma, DMAIC methodology, SIPOC

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Authors: R. Vigneswaran, S. Thilakanathan

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Variable time-stepping algorithms for solving dynamical systems performed poorly for long time computations which pass close to a fixed point. To overcome this difficulty, several authors considered phase space error controls for numerical simulation of dynamical systems. In one generalized phase space error control, a step-size selection scheme was proposed, which allows this error control to be incorporated into the standard adaptive algorithm as an extra constraint at negligible extra computational cost. For this generalized error control, it was already analyzed the forward Euler method applied to the linear system whose coefficient matrix has real and negative eigenvalues. In this paper, this result was extended to the linear system whose coefficient matrix has complex eigenvalues with negative real parts. Some theoretical results were obtained and numerical experiments were carried out to support the theoretical results.

Keywords: adaptivity, fixed point, long time simulations, stability, linear system

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Authors: Haci Mehmet Guzey, Levent Acar

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In this paper, a novel decentralized controller is developed for linear systems with nonlinear unknown interconnections. A model linear decoupled system is assigned for each system. By using the difference actual and model state dynamics, the problem is formulated as inverse problem. Then, the interconnected dynamics are approximated by using Galerkin’s expansion method for inverse problems. Two different sets of orthogonal basis functions are utilized to approximate the interconnected dynamics. Approximated interconnections are utilized in the controller to cancel the interconnections and decouple the systems. Subsequently, the interconnected systems behave as a collection of decoupled systems.

Keywords: decentralized control, inverse problems, large scale systems, nonlinear interconnections, basis functions, system identification

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Authors: Sadia Arshad, Ayesha Sohail, Sana Javed, Khadija Maqbool, Salma Kanwal

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The dynamical study of the carriers play an essential role in the evolution and global transmission of infectious diseases and will be discussed in this study. To make this approach novel, we will consider the fractional order model which is generalization of integer order derivative to an arbitrary number. Since the integration involved is non local therefore this property of fractional operator is very useful to study epidemic model for infectious diseases. An extended numerical method (ODE solver) is implemented on the model equations and we will present the simulations of the model for different values of fractional order to study the effect of carriers on transmission dynamics. Global dynamics of fractional model are established by using the reproduction number.

Keywords: Fractional differential equation, Numerical simulations, epidemic model, transmission dynamics

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Authors: Hartomo Soewardi, Bachtiar Jouhari

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Ineffectiveness and inefficiency of the building permit process in Indonesia still becomes a major problems for people to apply. Long time of service, the complicated administration process, and an expensive fees are a process that causing a dissatisfaction and discomfort for applicant. Therefore, it is critical to improve the quality of service of building permit system. Objectives of this research is to develop a better process of the system to improve productivity and quality service. Lean six sigma concept by using DMAIC procedures was used to analyze the existing system. Moreover, improvement of the system was conducted by using the Axiomatic Design method. Verification test was done to test the hypothesis of the proposed system design. Result of this research shows that proposed system can produce increasing 61.8% of efficiency on service time, and more effective and easier.

Keywords: axiomatic design, bbuilding permit system, DMAIC, Lean Six Sigma

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Authors: Riadh Zorgati, Thomas Triboulet

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In quite diverse application areas such as astronomy, medical imaging, geophysics or nondestructive evaluation, many problems related to calibration, fitting or estimation of a large number of input parameters of a model from a small amount of output noisy data, can be cast as inverse problems. Due to noisy data corruption, insufficient data and model errors, most inverse problems are ill-posed in a Hadamard sense, i.e. existence, uniqueness and stability of the solution are not guaranteed. A wide class of inverse problems in physics relates to the Fredholm equation of the first kind. The ill-posedness of such inverse problem results, after discretization, in a very ill-conditioned linear system of equations, the condition number of the associated matrix can typically range from 109 to 1018. This condition number plays the role of an amplifier of uncertainties on data during inversion and then, renders the inverse problem difficult to handle numerically. Similar problems appear in other areas such as numerical optimization when using interior points algorithms for solving linear programs leads to face ill-conditioned systems of linear equations. Devising efficient solution approaches for such system of equations is therefore of great practical interest. Efficient iterative algorithms are proposed for solving a system of linear equations. The approach is based on a preconditioning of the initial matrix of the system with an approximation of a generalized inverse leading to a stochastic preconditioned matrix. This approach, valid for non-negative matrices, is first extended to hermitian, semi-definite positive matrices and then generalized to any complex rectangular matrices. The main results obtained are as follows: 1) We are able to build a generalized inverse of any complex rectangular matrix which satisfies the convergence condition requested in iterative algorithms for solving a system of linear equations. This completes the (short) list of generalized inverse having this property, after Kaczmarz and Cimmino matrices. Theoretical results on both the characterization of the type of generalized inverse obtained and the convergence are derived. 2) Thanks to its properties, this matrix can be efficiently used in different solving schemes as Richardson-Tanabe or preconditioned conjugate gradients. 3) By using Lp norms, we propose generalized Kaczmarz’s type matrices. We also show how Cimmino's matrix can be considered as a particular case consisting in choosing the Euclidian norm in an asymmetrical structure. 4) Regarding numerical results obtained on some pathological well-known test-cases (Hilbert, Nakasaka, …), some of the proposed algorithms are empirically shown to be more efficient on ill-conditioned problems and more robust to error propagation than the known classical techniques we have tested (Gauss, Moore-Penrose inverse, minimum residue, conjugate gradients, Kaczmarz, Cimmino). We end on a very early prospective application of our approach based on stochastic matrices aiming at computing some parameters (such as the extreme values, the mean, the variance, …) of the solution of a linear system prior to its resolution. Such an approach, if it were to be efficient, would be a source of information on the solution of a system of linear equations.

Keywords: conditioning, generalized inverse, linear system, norms, stochastic matrix

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Authors: Puttharapong Sakulwaropas, Uraiwan Jaroengeratikun

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Casualty insurance business, the optimal premium pricing and adequate cost for an insurance company are important in risk management. Normally, the insurance pure premium can be determined by multiplying the claim frequency with the claim cost. The aim of this research was to study in the application of generalized linear models to select the risk factor for model of claim frequency and claim cost for estimating a pure premium. In this study, the data set was the claim of comprehensive motor insurance, which was provided by one of the insurance company in Thailand. The results of this study found that the risk factors significantly related to pure premium at the 0.05 level consisted of no claim bonus (NCB) and used of the car (Car code).

Keywords: generalized linear models, risk factor, pure premium, regression model

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