Application of MARS for the Construction of Nonparametric Models

2000 Mathematics Subject Classification: 62G08, 62P30.This paper presents the main features of the relatively new statistical technique called Multivariate Adaptive Regression Splines (MARS) and the corresponding software product. The MARS method is designed for statistical analysis of data, when standard parametric modeling by multiple regression or logistic regression methods is not applicable. A case study from the area of laser technology, especially for modeling of UV Cu+ Ne-CuBr laser is performed. The obtained results are in a good agreement with practical issues. It is shown that the constructed nonparametric MARS models can be applied in estimation and prediction of current and future experiments in order to improve the output laser power.This paper is partially supported by projects VU-MI-205, NSF of the Bulgarian Ministry of Education, Youth and Science and RS09–FMI–013, ISM-4 of NPD, Plovdiv University “Paisii Hilendarski”

Mathematical Modeling and Simulation of Radial Temperature Profile of Strontium Bromide Lasers

For metal and metal halide vapor lasers excited by high frequency pulsed discharge, the thermal effect mainly caused by the radial temperature distribution is of considerable importance for stable laser operation and improvement of laser output characteristics. A short survey of the obtained analytical and numerical-analytical mathematical models of the temperature profile in a high-powered He-SrBr2 laser is presented. The models are described by the steady-state heat conduction equation with mixed type nonlinear boundary conditions for the arbitrary form of the volume power density. A complete model of radial heat flow between the two tubes is established for precise calculating the inner wall temperature. The models are applied for simulating temperature profiles for newly designed laser. The author’s software prototype LasSim is used for carrying out the mathematical models and simulations

Using the Business Process Execution Language for Managing Scientific Processes

This paper describes the use of the Business Process Execution Language for Web Services (BPEL4WS/BPEL) for managing scientific workflows. This work is result of our attempt to adopt Service Oriented Architecture in order to perform Web services – based simulation of metal vapor lasers. Scientific workflows can be more demanding in their requirements than business processes. In the context of addressing these requirements, the features of the BPEL4WS specification are discussed, which is widely regarded as the de-facto standard for orchestrating Web services for business workflows. A typical use case of calculation the electric field potential and intensity distributions is discussed as an example of building a BPEL process to perform distributed simulation constructed by loosely-coupled services

Modeling of Output Characteristics of a UV Cu+ Ne-CuBr Laser

This paper examines experiment data for a Ne-CuBr UV copper ion laser excited by longitudinal pulsed discharge emitting in multiline regime. The flexible multivariate adaptive regression splines (MARSs) method has been used to develop nonparametric regression models describing the laser output power and service life of the devices. The models have been constructed as explicit functions of 9 basic input laser characteristics. The obtained models account for local nonlinearities of the relationships within the various multivariate subregions. The built best MARS models account for over 98% of data. The models are used to estimate the investigated output laser characteristics of existing UV lasers. The capabilities for using the models in predicting existing and future experiments have been demonstrated. Specific analyses have been presented comparing the models with actual experiments. The obtained results are applicable for guiding and planning the engineering experiment. The modeling methodology can be applied for a wide range of similar lasers and laser devices

Application of the Classification and Regression Trees for Modeling the Laser Output Power of a Copper Bromide Vapor Laser

This study examines the available experiment data for a copper bromide vapor laser (CuBr laser), emitting in the visible spectrum at 2 wavelengths—510.6 and 578.2 nm. Laser output power is estimated based on 10 independent input parameters. The CART method is used to build a binary regression tree of solutions with respect to output power. In the case of a linear model, an approximation of 98% has been achieved and 99% for the model of interactions between predictors up to the the second order with an relative error under 5%. The resulting CART tree takes into account which input quantities influence the formation of classification groups and in what manner. This makes it possible to estimate which ones are significant from an engineering point of view for the development and operation of the considered type of lasers, thus assisting in the design and improvement of laser technology

Modeling of the Radial Heat Flow and Cooling Processes in a Deep Ultraviolet Cu

An improved theoretical model of the gas temperature profile in the cross-section of an ultraviolet copper ion excited copper bromide laser is developed. The model is based on the solution of the one-dimensional heat conduction equation subject to special nonlinear boundary conditions, describing the heat interaction between the laser tube and its surroundings. It takes into account the nonuniform distribution of the volume power density along with the radius of the laser tube. The problem is reduced to the boundary value problem of the first kind. An explicit solution of this model is obtained. The model is applied for the evaluation of the gas temperature profiles of the laser in the conditions of free and forced air-cooling. Comparison with other simple models assumed constant volume power density is made. In particular, a simple expression for calculating the average gas temperature is found

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Special Issue “Statistical Data Modeling and Machine Learning with Applications II”

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New Recursive Representations for the Favard Constants with Application to Multiple Singular Integrals and Summation of Series

There are obtained integral form and recurrence representations for some Fourier series and connected with them Favard constants. The method is based on preliminary integration of Fourier series which permits to establish general recursion formulas for Favard constants. This gives the opportunity for effective summation of infinite series and calculation of some classes of multiple singular integrals by the Favard constants

Assessment of Students’ Achievements and Competencies in Mathematics Using CART and CART Ensembles and Bagging with Combined Model Improvement by MARS

The aim of this study is to evaluate students’ achievements in mathematics using three machine learning regression methods: classification and regression trees (CART), CART ensembles and bagging (CART-EB) and multivariate adaptive regression splines (MARS). A novel ensemble methodology is proposed based on the combination of CART and CART-EB models in a new ensemble to regress the actual data using MARS. Results of a final exam test, control and home assignments, and other learning activities to assess students’ knowledge and competencies in applied mathematics are examined. The exam test combines problems on elements of mathematical analysis, statistics and a small practical project. The project is the new competence-oriented element, which requires students to formulate problems themselves, to choose different solutions and to use or not use specialized software. Initially, empirical data are statistically modeled using six CART and six CART-EB competing models. The models achieve a goodness-of-fit up to 96% to actual data. The impact of the examined factors on the students’ success at the final exam is determined. Using the best of these models and proposed novel ensemble procedure, final MARS models are built that outperform the other models for predicting the achievements of students in applied mathematics