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Öğe An approach to the mean shift outlier model by Tikhonov regularization and conic programming(Ios Press, 2014) Taylan, Pakize; Yerlikaya-Oezkurt, Fatma; Weber, Gerhard-WilhelmIn statistical research, regression models based on data play a central role; one of these models is the linear regression model. However, this model may give misleading results when data contain outliers. The outliers in linear regression can be resolved in two stages: by using the Mean Shift Outlier Model (MSOM) and by providing a new solution for this model. First, we construct a Tikhonov regularization problem for the MSOM. Then, we treat this problem using convex optimization techniques, specifically conic quadratic programming, permitting the use of interior point methods. We present numerical examples, which reveal very good results, and we conclude with an outlook to future studies.Öğe C-LASSO estimator for generalized additive logistic regression based on B-Spline(Springer International Publishing, 2019) Taylan, Pakize; Weber, Gerhard Wilhelm; 0000-0003-0849-7771; 0000-0001-7204-8861[No abstract available]Öğe CG-lasso estimator for multivariate adaptive regression spline(Springer International Publishing Ag, 2019) Taylan, Pakize; Weber, Gerhard Wilhelm[Abstract Not Available]Öğe CMARS and GAM & CQP-Modern optimization methods applied to international credit default prediction(Elsevier, 2011) Alp, Ozge Sezgin; Buyukbebeci, Erkan; Cekic, Aysegul Iscanoglu; Ozkurt, Fatma Yerlikaya; Taylan, Pakize; Weber, Gerhard-WilhelmIn this paper, we apply newly developed methods called GAM & CQP and CMARS for country defaults. These are techniques refined by us using Conic Quadratic Programming. Moreover, we compare these new methods with common and regularly used classification tools, applied on 33 emerging markets' data in the period of 1980-2005. We conclude that GAM & CQP and CMARS provide an efficient alternative in predictions. The aim of this study is to develop a model for predicting the countries' default possibilities with the help of modern techniques of continuous optimization, especially conic quadratic programming. We want to show that the continuous optimization techniques used in data mining are also very successful in financial theory and application. By this paper we contribute to further benefits from model-based methods of applied mathematics in the financial sector. Herewith, we aim to help build up our nations. (C) 2010 Elsevier B.V. All rights reserved.Öğe CMARS: a new contribution to nonparametric regression with multivariate adaptive regression splines supported by continuous optimization(Taylor & Francis Ltd, 2012) Weber, Gerhard-Wilhelm; Batmaz, Inci; Koksal, Gulser; Taylan, Pakize; Yerlikaya-Ozkurt, FatmaRegression analysis is a widely used statistical method for modelling relationships between variables. Multivariate adaptive regression splines (MARS) especially is very useful for high-dimensional problems and fitting nonlinear multivariate functions. A special advantage of MARS lies in its ability to estimate contributions of some basis functions so that both additive and interactive effects of the predictors are allowed to determine the response variable. The MARS method consists of two parts: forward and backward algorithms. Through these algorithms, it seeks to achieve two objectives: a good fit to the data, but a simple model. In this article, we use a penalized residual sum of squares for MARS as a Tikhonov regularization problem, and treat this with continuous optimization technique, in particular, the framework of conic quadratic programming. We call this new approach to MARS as CMARS, and consider it as becoming an important complementary and model-based alternative to the backward stepwise algorithm. The performance of CMARS is also evaluated using different data sets with different features, and the results are discussed.Öğe Continuous optimization applied in MARS for modern applications in finance, science and technology(Vilnius Gediminas Technical Univ Press, Technika, 2008) Taylan, Pakize; Weber, Gerhard-Wilhelm; Yerlikaya, FatmaMultivariate adaptive regression spline (MARS) denotes a tool from statistics, important in classification and regression, with applicability in many areas of finance, science and technology. It is very useful in high dimensions and shows a great promise for fitting nonlinear multivariate functions. The MARS algorithm for estimating the model function consists of two subalgorithms. We propose not to use the second one (backward stepwise algorithm), but we construct a penalized residual sum of squares for a Tikhonov regularization problem which we treat using continuous optimization, considered to become a complementary technology and alternative to the backward stepwise algorithm. Especially, we employ conic quadratic programming (CQP), permitting the use of interior point methods.Öğe Enhancing classification modeling through feature selection and smoothness: A conic-fused lasso approach integrated with mean shift outlier modelling(Amer Inst Mathematical Sciences-Aims, 2024) Yerlikaya, Fatma Özkurt; Taylan, PakizeOutlier detection and variable selection are among main objectives of statistical analysis. In our study, we address the outlier problem for classification by using the Mean Shift Outlier Model (CLMSOM). Since the MSOM has more coefficients than the linear regression model, the complexity of the model MSOM is high. Therefore, we consider feature selection for MSOM by using fused Lasso (FLasso), which is beneficial and helpful in the cases where the number of explanatory variables or features is greater than the sample size. FLasso is penalizing both the coefficients and their successive differences by the L1-norm, and it allows sparsity for both of them, while Lasso only allows the coefficients by considering a nonsmooth optimization problem. In this study, we take into account Iterated Ridge approximation which enables us to use a smooth optimization for FLasso problem. Generated smooth optimization problem is solved by using one of continuous optimization techniques called Conic Quadratic Programming (CQP), which is enabling the utilization of interior point methods. The newly developed method is called Conic FLasso for classification by MSOM (CFLasso-CLMSOM) and is applied to real world data set to show its performance.Öğe Estimation in the partially nonlinear model by continuous optimization(Taylor and Francis Ltd., 2021) Özkurt, Fatma Yerlikaya; Taylan, Pakize; Tez, MüjganA useful model for data analysis is the partially nonlinear model where response variable is represented as the sum of a nonparametric and a parametric component. In this study, we propose a new procedure for estimating the parameters in the partially nonlinear models. Therefore, we consider penalized profile nonlinear least square problem where nonparametric components are expressed as a B-spline basis function, and then estimation problem is expressed in terms of conic quadratic programming which is a continuous optimization problem and solved interior point method. An application study is conducted to evaluate the performance of the proposed method by considering some well-known performance measures. The results are compared against parametric nonlinear model.Öğe Lineer olmayan ölçüm hatalı modellerde eğrilik ölçümleri(2018) Taylan, Pakize; Tez, MüjganÖlçüm hataları istatistiksel analizi kuvvetli olarak etkiler, çünkü gözlenebilir verileri oluşturan olasılık dağılımının, analizcinin çalıştığı model ile direkt olarak ilgili olan gözlenemeyen veriyi oluşturan olasılık dağılımından sapmasına neden olur. Lineer olmayan ölçüm hatalı modeller, gözlenemeyen doğru değerlerin bir fonksiyonel ilişkiyi sağladığı ve gözlenen değerlerin, doğru değerler ile ölçüm hatalarının toplamı olduğu kabul edilir. Bağımsız değişkendeki hata parametre tahmin yönteminin karmaşıklığını artırdığı gibi yüzeyin eğriliğini de etkileyebilir. Ölçüm hatasının yüzeyi ne kadar etkilediğini anlamak için modelin eğrilik ölçümü yapılır. Bu ölçümler, yapılan lineer yaklaşımın yeterliliği hakkında bilgi verir. Bu model için eğrilik incelemesi yapılırken, önce, gözlenemeyen doğru açıklayıcı değişken de modelin parametreleri ile birlikte eş zamanlı olarak tahmin edilir. Tahmin edilen doğru değerler ve 0 parametreleri kullanılarak modelin eğrilik incelemesi yapılır. Modelin ivme vektörleri teğet düzlemdeki ve teğet düzleme dik olan normal düzlemdeki bileşenlerine ayrılır. Bu ayrışım model için esas ve parametre etkileri eğriliğinin hesaplanmasında yarar sağlar. Parametre etkileri eğriliği, teğet düzlem üzerindeki parametre eğrilerinin düzgünsüzlüğünü ölçtüğünden parametrelemeye bağlıdır. Esas yada normal eğrilik fonksiyon için seçilen parametrelere bağlı olmayıp tepki fonksiyonuna ve tasarıma bağlıdır. Bu iki eğriliğin, verilerin ve parametrelerin skala değişimlerinden bağımsız olarak ölçülebilmesi için eğrilikler p = sVn faktörü ile çarpılır. Bu faktör ile çarpılmasının nedeni, 0 için çıkarsama bölgesi olan kürenin yarıçapı (Standart yarıçap) olmasıdır. Bütün bu hesaplamalar yapılırken eğrilik dizinlerinden faydalanılır. Şimdiye kadar yapılan bu tür çalışmalar, açıklayıcı değişken hatasız olduğu varsayımıyla klasik lineer olmayan modeller üzerineydi. Biz bu çalışmada açıklayıcı değişken ölçüm hatalı olursa eğrilik probleminin nasıl etkileneceği noktasından hareketle, ölçüm hatalı lineer olmayan modeller üzerine araştırma yaptık. Bu araştırmada kestirim probleminde yüzeyin eğriliklerinin nasıl etkilendiğini diferansiyel geometri yaklaşımıyla gösterdik. Literatürde rastlanmayan bu çalışmayı, lineer olmayan özel bir model üzerinde hem ölçüm hatalı hem de ölçüm hatasız açıklayıcı değişken için istatistiksel çıkarsamalar yaparak parametre etkileri eğriliği ve esas eğriliği karşılaştırdık. Sonuçta ölçüm hatalı parametre etkileri eğriliği ve esas eğriliğin büyük olduğu, dolayısıyla da değişkendeki hatanın parametre tahmini üzerinde etkisinin yanı sıra yüzey eğriliği üzerine de etkisinin olduğunu gördük.Öğe Lineer olmayan ölçüm hatalı modellerde eğrilik ölçümleri(Dicle Üniversitesi, Fen Bilimleri Enstitüsü, 1999) Taylan, Pakize; Tez, MüjganÖlçüm hataları istatistiksel analizi kuvvetli olarak etkiler, çünkü gözlenebilir verileri oluşturan olasılık dağılımının, analizcinin çalıştığı model ile direkt olarak ilgili olan gözlenemeyen veriyi oluşturan olasılık dağılımından sapmasına neden olur. Lineer olmayan ölçüm hatalı modeller, gözlenemeyen doğru değerlerin bir fonksiyonel ilişkiyi sağladığı ve gözlenen değerlerin, doğru değerler ile ölçüm hatalarının toplamı olduğu kabul edilir. Bağımsız değişkendeki hata parametre tahmin yönteminin karmaşıklığını artırdığı gibi yüzeyin eğriliğini de etkileyebilir. Ölçüm hatasının yüzeyi ne kadar etkilediğini anlamak için modelin eğrilik ölçümü yapılır. Bu ölçümler, yapılan lineer yaklaşımın yeterliliği hakkında bilgi verir. Bu model için eğrilik incelemesi yapılırken, önce, gözlenemeyen doğru açıklayıcı değişken de modelin parametreleri ile birlikte eş zamanlı olarak tahmin edilir. Tahmin edilen doğru değerler ve 0 parametreleri kullanılarak modelin eğrilik incelemesi yapılır. Modelin ivme vektörleri teğet düzlemdeki ve teğet düzleme dik olan normal düzlemdeki bileşenlerine ayrılır. Bu ayrışım model için esas ve parametre etkileri eğriliğinin hesaplanmasında yarar sağlar. Parametre etkileri eğriliği, teğet düzlem üzerindeki parametre eğrilerinin düzgünsüzlüğünü ölçtüğünden parametrelemeye bağlıdır. Esas yada normal eğrilik fonksiyon için seçilen parametrelere bağlı olmayıp tepki fonksiyonuna ve tasarıma bağlıdır. Bu iki eğriliğin, verilerin ve parametrelerin skala değişimlerinden bağımsız olarak ölçülebilmesi için eğrilikler p = sVn faktörü ile çarpılır. Bu faktör ile çarpılmasının nedeni, 0 için çıkarsama bölgesi olan kürenin yarıçapı (Standart yarıçap) olmasıdır. Bütün bu hesaplamalar yapılırken eğrilik dizinlerinden faydalanılır. Şimdiye kadar yapılan bu tür çalışmalar, açıklayıcı değişken hatasız olduğu varsayımıyla klasik lineer olmayan modeller üzerineydi. Biz bu çalışmada açıklayıcı değişken ölçüm hatalı olursa eğrilik probleminin nasıl etkileneceği noktasından hareketle, ölçüm hatalı lineer olmayan modeller üzerine araştırma yaptık. Bu araştırmada kestirim probleminde yüzeyin eğriliklerinin nasıl etkilendiğini diferansiyel geometri yaklaşımıyla gösterdik. Literatürde rastlanmayan bu çalışmayı, lineer olmayan özel bir model üzerinde hem ölçüm hatalı hem de ölçüm hatasız açıklayıcı değişken için istatistiksel çıkarsamalar yaparak parametre etkileri eğriliği ve esas eğriliği karşılaştırdık. Sonuçta ölçüm hatalı parametre etkileri eğriliği ve esas eğriliğin büyük olduğu, dolayısıyla da değişkendeki hatanın parametre tahmini üzerinde etkisinin yanı sıra yüzey eğriliği üzerine de etkisinin olduğunu gördük.Öğe MATHEMATICAL AND DATA MINING CONTRIBUTIONS TO DYNAMICS AND OPTIMIZATION OF GENE-ENVIRONMENT NETWORKS(Nova Science Publishers, Inc, 2010) Weber, Gerhard Wilhelm; Taylan, Pakize; Akteke-Ozturk, Basak; Ugur, OmurThis paper further introduces continuous optimization into the fields of computational biology and environmental protection which belong to the most challenging and emerging areas of science. It refines earlier ones of our models on gene-environment patterns by the use of optimization theory. We emphasize that it bases on and presents work done in [61, 66]. Furthermore, our paper tries to detect and overcome some structural frontiers of our methods applied to the recently introduced gene-environment networks. Based on the experimental data, we investigate the ordinary differential equations having nonlinearities on the right-hand side and a generalized treatment of the absolute shift term which represents the environmental effects. The genetic process is studied by a time-discretization, in particular, Runge-Kutta type discretization. The possibility of detecting stability and instability regions is being shown by a utilization of the combinatorial algorithm of Brayton and Tong which is based on the orbits of polyhedra. The time-continuous and discrete systems can be represented by means of matrices allowing biological implications, they encode and are motivated by our gene-environment networks. A specific contribution of this paper consists in a careful but rigorous integration of the environment into modeling and dynamics, and in further new sights. Relations to parameter estimation within modeling, especially, by using optimization, are indicated, and future research is addressed, especially towards the use of stochastic differential equations. This practically motivated and theoretically elaborated work is devoted for a contribution to better health care, progress in medicine, a better education and more healthy living conditions recommended.Öğe Mathematical contributions to dynamics and optimization of gene-environment networks(Taylor & Francis Ltd, 2008) Weber, Gerhard-Wilhelm; Tezel, Aysun; Taylan, Pakize; Soyler, Alper; Cetin, MehmetThis article contributes to a further introduction of continuous optimization in the field of computational biology which is one of the most challenging and emerging areas of science, in addition to foundations presented and the state-of-the-art displayed in [C.A. Floudas and P.M. Pardalos, eds., Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches, Kluwer Academic Publishers, Boston, 2000]. Based on a summary of earlier works by the coauthors and their colleagues, it refines the model on gene-environment patterns by a problem from generalized semi-infinite programming (GSIP), and characterizes the condition of its structural stability. Furthermore, our paper tries to detect and understand structural frontiers of our methods applied to the recently introduced gene-environment networks and tries to overcome them. Computational biology is interdisciplinary, but it also looks for its mathematical foundations. From data got by DNA microarray experiments, non-linear ordinary differential equations are extracted by the optimization of least-squares errors; then we derive corresponding time-discretized dynamical systems. Using a combinatorial algorithm with polyhedra sequences we can detect the regions of parametric stability, contributing to a testing the goodness of data fitting of the model. To represent and interpret the dynamics, certain matrices, genetic networks and, more generally, gene-environment networks serve. Here, we consider n genes in possible dependence with m special environmental factors and a cumulative one. These networks are subject of discrete mathematical questions, but very large structures, such that we need to simplify them. This is undertaken in a careful optimization with constraints, aiming at a balanced connectedness, incorporates any type of a priori knowledge or request and should be done carefully enough to be robust against disturbation by the environment. In this way, we take into account attacks on the network, knockout phenomena and catastrophies, but also changes in lifestyle and effects of education as far as they can approximately be quantified. We characterize the structural stability of the GSIP problem against perturbations like changes between data series or due to outliers. We give explanations on the numerics and the use of splines. This study is an attempt to demonstrate some beauty and applicabilty of continuous optimization which might together one day give a support in health care, food engineering, biomedicine and -technology, including elements of bioenergy and biomaterials.Öğe Model Choice in Mixtural Experiments(TÜİK, 2004) Urgan, Nurkut Nuray; Taylan, PakizeExperiments with mixtures are considered in which the response to a mixture depends on the proportions of the components present, but not on the total amount of the mixture. The purpose of it is to obtain the best mixture which does not increase the cost and does not reduce the quality of the product by considering the physical chemical and economical properties of the different proportions of the mixture. This causes some restrictions over the components. Generally the set of the constraint made by fixing the total amount of the mixture and changing the total of the components to be q is the number of comnponents and xi is the proportion of ith component in the mixture. Scheffé's and Cox’s mixture models are the most used ones for the experiments with mixtures. In this study, Scheffé's and Cox's mixture models were compared and this applied to a chemical experiment.Öğe A new approach to multivariate adaptive regression splines by using Tikhonov regularization and continuous optimization(Springer, 2010) Taylan, Pakize; Weber, Gerhard-Wilhelm; Ozkurt, Fatma YerlikayaThis paper introduces a model-based approach to the important data mining tool Multivariate adaptive regression splines (MARS), which has originally been organized in a more model-free way. Indeed, MARS denotes a modern methodology from statistical learning which is important in both classification and regression, with an increasing number of applications in many areas of science, economy and technology. It is very useful for high-dimensional problems and shows a great promise for fitting nonlinear multivariate functions. The MARS algorithm for estimating the model function consists of two algorithms, these are the forward and the backward stepwise algorithm. In our paper, we propose not to use the backward stepwise algorithm. Instead, we construct a penalized residual sum of squares for MARS as a Tikhonov regularization problem which is also known as ridge regression. We treat this problem using continuous optimization techniques which we consider to become an important complementary technology and model-based alternative to the concept of the backward stepwise algorithm. In particular, we apply the elegant framework of conic quadratic programming. This is an area of convex optimization which is very well-structured, herewith, resembling linear programming and, hence, permitting the use of powerful interior point methods. Based on these theoretical and algorithmical studies, this paper also contains an application to diabetes data. We evaluate and compare the performance of the established MARS and our new CMARS in classifying diabetic persons, where CMARS turns out to be very competitive and promising.Öğe New computational methods for classification problems in the existence of outliers based on conic quadratic optimization(Taylor and Francis Inc., 2020) Özkurt, Fatma Yerlikaya; Taylan, PakizeMost of the statistical research involves classification which is a procedure utilized to establish prediction models to set apart and classify new observations in the dataset from every fields of science, technology, and economics. However, these models may give misclassification results when dataset contains outliers (extreme data points). Therefore, we dealt with outliers in classification problem: firstly, by combining robustness of mean-shift outlier model and then stability of Tikhonov regularization based on continuous optimization method called Conic Quadratic Programming. These new methodologies are performed on classification dataset within the existence of outliers, and the results are compared with parametric model by using well-known performance measures.Öğe A new outlier detection method based on convex optimization: application to diagnosis of Parkinson's disease(Taylor & Francis Ltd, 2021) Taylan, Pakize; Yerlikaya-Ozkurt, Fatma; Bilgic Ucak, Burcu; Weber, Gerhard-WilhelmNeuroscience is a combination of different scientific disciplines which investigate the nervous system for understanding of the biological basis. Recently, applications to the diagnosis of neurodegenerative diseases like Parkinson's disease have become very promising by considering different statistical regression models. However, well-known statistical regression models may give misleading results for the diagnosis of the neurodegenerative diseases when experimental data contain outlier observations that lie an abnormal distance from the other observation. The main achievements of this study consist of a novel mathematics-supported approach beside statistical regression models to identify and treat the outlier observations without direct elimination for a great and emerging challenge in humankind, such as neurodegenerative diseases. By this approach, a new method named as CMTMSOM is proposed with the contributions of the powerful convex and continuous optimization techniques referred to as conic quadratic programing. This method, based on the mean-shift outlier regression model, is developed by combining robustness of M-estimation and stability of Tikhonov regularization. We apply our method and other parametric models on Parkinson telemonitoring dataset which is a real-world dataset in Neuroscience. Then, we compare these methods by using well-known method-free performance measures. The results indicate that the CMTMSOM method performs better than current parametric models.Öğe On curvature measurements of the nonlinear errors in variable models by application study(Taru Publications, 2018) Taylan, Pakize; Uysal, Ersin; Tez, MujganRelative curvature measurements are of great importance from a practical point of view since it determines the validity of the linearized approximation used in estimation problems for nonlinear regression models. But, these measurements can be negatively affected when an explanatory variable contains a measurement error as well as response variables and can prevent accurate inferences. In our study, we considered the curvature measurement of nonlinear errors in variable models to investigate adequacy of the linear approximation in case the explanatory variables are subjected to measurement error and how the parameter estimation problem is affected by this error, using the geometric concepts such as parameter-effects and intrinsic curvatures of the model function. Then, for the two cases of the explanatory variable, curvature calculations and statistical inferences were made on the chemical model called Michaelis-Menten, in which the rate of reaction against a substrate concentration is measured, by using different data sets.Öğe On foundations of estimation for nonparametric regression with continuous optimization(IGI Global, 2019) Taylan, PakizeThe aim of parametric regression models like linear regression and nonlinear regression are to produce a reasonable relationship between response and independent variables based on the assumption of linearity and predetermined nonlinearity in the regression parameters by finite set of parameters. Nonparametric regression techniques are widely-used statistical techniques, and they not only relax the assumption of linearity in the regression parameters, but they also do not need a predetermined functional form as nonlinearity for the relationship between response and independent variables. It is capable of handling higher dimensional problem and sizes of sample than regression that considers parametric models because the data should provide both the model building and the model estimates. For this purpose, firstly, PRSS problems for MARS, ADMs, and CR will be constructed. Secondly, the solution of the generated problems will be obtained with CQP, one of the famous methods of convex optimization, and these solutions will be called CMARS, CADMs, and CKR, respectively.Öğe ON FOUNDATIONS OF PARAMETER ESTIMATION FOR GENERALIZED PARTIAL LINEAR MODELS WITH B-SPLINES AND CONTINUOUS OPTIMIZATION(Amer Inst Physics, 2010) Taylan, Pakize; Weber, Gerhard-Wilhelm; Liu, LianGeneralized linear models are widely-used statistical techniques. As an extension, generalized partial linear models utilize semiparametric methods and augment the usual parametric terms by a single nonparametric component of a continuous covariate. In this paper, after a short introduction, we present our model in the generalized additive context with a focus on penalized maximum likelihood and on the penalized iteratively reweighted least squares (P-IRLS) problem based on B-splines which is attractive for nonparametric components. Then, we approach solving the P-IRLS problem using continuous optimization techniques. They become an important complementary technology and alternative to the penalty methods with the flexibility of choosing the penalty parameter adaptively. In particular, we model and treat the constrained P-IRLS problem by the elegant framework of conic quadratic programming. This paper is of a more theoretical nature and a preparation of real-world applications in future.Öğe On the foundations of parameter estimation for generalized partial linear models with B-splines and continuous optimization(Pergamon-Elsevier Science Ltd, 2010) Taylan, Pakize; Weber, Gerhard-Wilhelm; Liu, Lian; Yerlikaya-Ozkurt, FatmaGeneralized linear models are widely used in statistical techniques. As an extension, generalized partial linear models utilize semiparametric methods and augment the usual parametric terms with a single nonparametric component of a continuous covariate. In this paper, after a short introduction, we present our model in the generalized additive context with a focus on the penalized maximum likelihood and the penalized iteratively reweighted least squares (P-IRLS) problem based on B-splines, which is attractive for nonparametric components. Then, we approach solving the P-IRLS problem using continuous optimization techniques. They have come to constitute an important complementary approach, alternative to the penalty methods, with flexibility for choosing the penalty parameter adaptively. In particular, we model and treat the constrained P-IRIS problem by using the elegant framework of conic quadratic programming. The method is illustrated using a small numerical example. (C) 2010 Elsevier Ltd. All rights reserved.
