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Data analysis is very important in many fields of science. The most preferred methods in data analysis is linear regression due to its simplicity to interpret and ease of application. One of the assumptions accepted while obtaining linear regression is that there is no correlation between the independent variables in the model which refers to absence of multicollinearity. As a result of multicollinearity, the variance of the parameter estimates will be high and this reduces the accuracy and reliability of the linear models. Shrinkage methods aim to handle the multicollinearity problem by minimizing the variance of the estimators in linear model. Ridge Regression, Lasso, and Elastic-Net methods are applied to different simulated data sets with different characteristics and also real world data sets. Based on performance results, the methods are compared according to multi-criteria decision making method named TOPSIS, and the order of preference is determined for each data set. [ABSTRACT FROM AUTHOR]

Recently, collection of huge amount of data and analysis of that much data have vital importance for human activities in many different application areas. Advanced statistical methods play crucial role for modeling of such data when the data contains outliers. Although there are number of outlier detection methods for revealing outlier observations in data, most of them may not be reasonable and appropriate for prediction purposes due to structural and requirements of modeling. In this study, density based clustering algorithm named Density-Based Spatial Clustering of Applications with Noise (DBSCAN) is considered in order to detect the location of outlier observations effectively with respect to form of the model for given data set. Based on obtained results, the Mean Shift Outlier Model (MSOM) is constructed as a robust linear model. This newly proposed computational approach based on DBSCAN uses power of data clustering and also minimize the impact of the outlier observations by MSOM. The numerical examples are also presented to reveal the performance of the proposed approach in this study. [ABSTRACT FROM AUTHOR]

Neuroscience 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. [ABSTRACT FROM AUTHOR]

A 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. [ABSTRACT FROM AUTHOR]

Most 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. [ABSTRACT FROM AUTHOR]

Regression 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. [ABSTRACT FROM AUTHOR]