Optimal Centers’ Allocation in Smoothing or Interpolating with Radial Basis Functions

This work was supported by FEDER/Junta de Andalucía-Consejería de Transformación Económica, Industria, Conocimiento y Universidades (Research Project A-FQM-76-UGR20, University of Granada) and by the Junta de Andalucía (Research Group FQM191).Function interpolation and approximation are classical problems of vital importance in many science/engineering areas and communities. In this paper, we propose a powerful methodology for the optimal placement of centers, when approximating or interpolating a curve or surface to a data set, using a base of functions of radial type. In fact, we chose a radial basis function under tension (RBFT), depending on a positive parameter, that also provides a convenient way to control the behavior of the corresponding interpolation or approximation method. We, therefore, propose a new technique, based on multi-objective genetic algorithms, to optimize both the number of centers of the base of radial functions and their optimal placement. To achieve this goal, we use a methodology based on an appropriate modification of a non-dominated genetic classification algorithm (of type NSGA-II). In our approach, the additional goal of maintaining the number of centers as small as possible was also taken into consideration. The good behavior and efficiency of the algorithm presented were tested using different experimental results, at least for functions of one independent variable.Junta de Andalucía-Consejería de Transformación Econímica, Industria, Conocimiento y Universidades A-FQM-76-UGR20Universidad de GranadaEuropean Regional Development FundJunta de Andalucía FQM19

Evolutionary computation of optimal knots allocation in smoothing methods by multivariate splines and radial function basis spaces

El objetivo de esta tesis doctoral ha sido el estudio, desarrollo y adaptación de una adecuada técnica de optimización multi-objetivo de tipo evolutivo (MOGA del inglés "Multi-Objective Genetic Algorithm") para el emplazamiento óptimo de los nodos (en el caso de bases de B-splines cúbicos o bicúbicos) o centros (cuando usamos funciones de base radiales) en sendos problemas de aproximación o de interpolación.Tesis Univ. Granada

Locally optimal designs for generalized linear models with applications to gamma models

Locally optimal designs for generalized linear models are derived at certain values of the regression parameters. In the present thesis analytic solutions for optimal designs are mostly developed. In particular situations numerical methods are employed. We restrict to D-, A- and Kiefer Phi kappa-optimality criteria. For general setup of the generalized linear model, by means of The General Equivalence Theorem, necessary and sufficient conditions in term of intensity values are obtained to characterize locally optimal designs. In this context, linear predictors with binary factors are assumed constituting first order models, models with interactions and models without intercept. Additionally, a particular approach is developed to identify locally D- or A-optimal design for the model with intercept from that for the model without intercept and vice versa. Gamma models with a power link function are considered constituting a particular class of generalized linear models. Relevant structures for the linear predictor are employed based on quantitative factors. The notions of locally essentially complete classes and locally complete classes of designs are introduced and such classes are established. On that basis locally D- and A-optimal designs are derived. In certain cases, the obtained results under generalized linear models with binary factors can be transferred to gamma models with quantitative factors. The explicit impact of the model parameters on the optimality of the designs is investigated. Furthermore, product type designs are derived for gamma models with product-type interactions. Moreover, gamma models having a linear predictor without intercept are considered. For a specific scenario sets of locally Phi kappa-optimal designs are developed. Further, by a suitable transformation between gamma models with and without intercept optimality results are transferred from one model to the other. Additionally with the aid of The General Equivalence Theorem optimality are characterized for multiple regression by a system of polynomial inequalities which can be solved analytically or by computer algebra. The robustness of the derived designs for gamma models with respect to misspecifications of the initial parameter values is examined by means of their local efficiencies. Optimal designs for multivariate generalized linear models are investigated. The components of the multivariate response might be combined with linear predictors via distinct link functions. We found that the locally optimal design for the univariate generalized linear models remains the same in the multivariate structure. In particular, product type designs are developed for the multivariate gamma model.Lokal optimale Versuchspläne für verallgemeinerte lineare Modelle werden für vorgegebene Werte der Regressionsparameter hergeleitet. In der vorliegenden Arbeit werden zumeist analytische Lösungen für optimale Versuchpläne entwickelt. In speziellen Situationen werden auch numerische Methoden verwendet. Wir beschränken unsere Untersuchungen auf das D- und A-Kriterium sowie Kiefers Phi kappa-Optimalitätskriterien. Im allgemeinen Rahmen der verallgemeinerten linearen Modelle werden mittels des allgemeinen Äquivalenzsatzes notwendige und hinreichende Bedingungen erhalten, die Intensitätswerte verwenden und lokal optimale Versuchspläne charakterisieren. In diesem Zusammenhang werden für lineare Prädiktoren mit binären Faktoren Modelle erster Ordnung, Modelle mit Wechselwirkungen und Modelle ohne Interzept (konstanten Term) betrachtet. Darüber hinaus wird eine spezielle Methode entwickelt, um lokal D- oder A-optimale Versuchspläne für ein Modell mit Interzept aus solchen für ein Modell ohne Interzept, und umgekehrt, zu konstruieren. Im Weiteren werden Gamma-Modelle mit einer Potenzfunktion als Link-Funktion (Power Link) betrachtet, die eine spezielle Klasse verallgemeinerter linearer Modelle bilden. Hierzu werden relevante Strukturen für lineare Prädiktoren verwendet, die auf quantitativen Faktoren basieren. Die Begriffe einer lokal wesentlich vollständigen Klasse und einer lokal vollständigen Klasse von Versuchsplänen werden eingeführt, und derartige Klassen werden für verallgemeinerte lineare Modelle mit binären Faktoren erhaltene Resultate. In geeigneten Fällen können die für verallgemeinerte lineare Modelle mit binären Faktoren erhaltene Resultate auf Gamma-Modelle mit quantitativen Faktoren übertragen werden. Zur Messung der Qualität wird der Einfluss der Modellparameter auf die Optimalität der Versuchspläne untersucht. Weiterhin werden Versuchspläne mit Produkt-Struktur als optimal für Gamma-Modelle mit produktartigen Wechselwirkungen identifiziert. Darüber hinaus werden auch Gamma-Modelle mit linearem Prädiktor ohne Interzept betrachtet. Für ein spezielles Szenario werden Mengen lokal Phi kappa-optimaler Versuchspläne gefunden. Durch eine geeignete Transformation werden Optimalitätsresultate für Gamma-Modelle mit Interzept auf Gamma-Modelle ohne Interzept, und umgekehrt, übertragen. Außerdem wird mit Hilfe des allgemeinen Äquivalenzsatzes die Optimalität für multiple Regression charakterisiert durch ein System polynomialer Ungleichungen, die analytisch oder mittels Computer Algebra gelöst werden können. Die Robustheit der hergeleiteten Versuchspläne für Gamma-Modelle bezüglich Fehlspezifikation der Parameter wird mittels ihrer lokalen Effizienzen überprüft. Schließlich werden optimale Versuchspläne für multivariate verallgemeinerte lineare Modelle untersucht. Dabei können die Komponenten der multivariaten Regressionsfunktion mit linearen Prädiktoren über verschiedene Link-Funktionen kombiniert werden. Es kann gezeigt werden dass der lokal optimale Versuchsplan für das univariate verallgemeinerte lineare Modell such für die multivariate Struktur optimal bleibt. Insbesondere werden Versuchspläne mit Produkt-Struktur für multivariate Gamma- Modelle entwickelt

Analytic solutions for locally optimal designs for gamma models having linear predictors without intercept

The gamma model is a generalized linear model for gamma-distributed outcomes. The model is widely applied in psychology, ecology or medicine. Recently, Gaffke et al. (J Stat Plan Inference 203:199–214, 2019) established a complete class and an essentially complete class of designs for gamma models to obtain locally optimal designs in particular when the linear predictor includes an intercept term. In this paper we extend this approach to gamma models having linear predictors without intercept. For a specific scenario sets of locally D- and A-optimal designs are established. It turns out that the optimality problem can be transformed to one under gamma models with intercept leading to a reduction in the dimension of the experimental region. On that basis optimality results can be transferred from one model to the other and vice versa. Additionally by means of the general equivalence theorem optimality can be characterized formultiple regression by a system of polynomial inequalitieswhich can be solved analytically or by computer algebra. Thus necessary and sufficient conditions can be obtained on the parameter values for the local D-optimality of specific designs. The robustness of the derived designs with respect to misspecification of the initial parameter values is examined by means of their local D-efficiencies.Projekt DEAL 202

تقييم استخدام المياه العادمة المعالجة على الخواص النيميائية للتربة وانتاجية المحاصيل في قطاع غزة

The increasing demand for water in the arid and semi-arid regions has resulted in the emergence of wastewater application for agriculture. Using treated wastewater as marginal quality water in agriculture is a justified practice, yet care should be taken to minimize adverse environmental impacts and to prevent soil deterioration. The objective of this research is to investigate the impact of using treated wastewater for irrigation on soil chemical properties and plant productivity. An reuse pilot study was carried out in Al-Zaitoun agricultural farm in the Gaza Strip from May to September2011. Acomparison was carried out between the soil properties in two experimental plots; one was irrigated with the effluent from Gaza Wastewater Treatment Plant over a period of four months, and the other was irrigated with fresh water in the same period. The irrigation water was applied by drip irrigation system, and the crop used was sorghum. The fresh water samples were obtained from the local well in the farm, and treated wastewater samples were obtained from the wastewater collection basin and analyzed for TSS, EC, pH, TDS, Ca, Mg, Na, SAR, CO3, Cl, NO3, TKN, TP, K, PO4, and selected heavy metals "Cd, Co, Cu, Fe, Mn, Ni, Pb, Zn" in addition to BOD, COD and the number of fecal and total coliforms before irrigation. Composite soil samples were taken from depth of 0-30 cm in both plots and analyzed for the main chemical parameters. The results indicated that the level of TDS, Na, Cl, TSS, Zn and Fe were higher in the effluent than the fresh water; it was above the recommended Palestinian standard for dry fodder irrigated by treated wastewater. Also, irrigation with wastewater lead to significant increase in O.M, CEC, K, TP, Ca, Mg, Na, and Cl in soil than irrigation with fresh water. In addition, the increases of Zn, Fe, Mn, and Pb in soil and sorghum plant irrigated with treated wastewater were significant in comparison with the plants irrigated with fresh water. Further, treated wastewater increased the plants height, and grain weight of sorghum

Equivariance and invariance for optimal designs in generalized linear models exemplified by a class of gamma models

The main intention of the present work is to outline the concept of equivariance and invariance in the design of experiments for generalized linear models and to demonstrate its usefulness. In contrast with linear models, pairs of transformations have to be employed for generalized linear models. These transformations act simultaneously on the experimental settings and on the location parameters in the linear component. Then, the concept of equivariance provides a tool to transfer locally optimal designs from one experimental region to another when the nominal values of the parameters are changed accordingly. The stronger concept of invariance requires a whole group of equivariant transformations. It can be used to characterize optimal designs which reflect the symmetries resulting from the group actions. The general concepts are illustrated by models with gamma distributed response and a canonical link. There, for a given transformation of the experimental settings, the transformation of the parameters is not unique and may be chosen to be nonlinear in order to fully exploit the model structure. In this case, we can derive invariant maximin efficient designs for the Dand the IMSE-criterion.Projekt DEAL 202