In a multiple LINEAR REGRESSION model, there are instances where one has to update the REGRESSION parameters. In such models as new data become available, by adding one row to the design matrix, the least-squares estimates for the parameters must be updated to reflect the impact of the new data. We will modify two existing methods of calculating REGRESSION coefficients in multiple LINEAR REGRESSION to make the computations more efficient. By resorting to an initial solution, we first employ the Sherman-Morrison formula to update the inverse of the transpose of the design matrix multiplied by the design matrix. We then modify the calculation of the product of the transpose of design matrix and the design matrix by the Cholesky decomposition method to solve the system. Finally, we compare these two modifications by several appropriate examples.

REGRESSION is a statistical technique used in finance, investment, and several other domains to assess the magnitude and precision of the association between a dependent variable (often represented as Y) and a set of other factors (referred to as independent variables). This work introduces a LINEAR programming approach for constructing REGRESSION models for Neutrosophic data. To achieve this objective, we use the least absolute deviation approach to transform the REGRESSION issue into a LINEAR programming problem. Ultimately, the efficacy of the suggested approach in resolving such problems has been shown via the presentation of a concrete illustration.

In the simulated binary crossover, offspring are generated from parents with a coefficient of variation and uses a probability distribution function for the coefficient and there is a LINEAR relationship between parents and offspring. Most existing methods of crossover operators generate offspring on the solution on the decision space during the search and so far, no suggestion has been proposed on making a REGRESSION model for generating the offspring on the objective space. In this paper, a Gaussian LINEAR REGRESSION crossover has been proposed. The idea is to apply LINEAR REGRESSION to model a relationship between parents and offspring in crossover operations through the Gaussian process. The reason for using this process is that the probability distribution of the simulated binary operator is based on the parent in the mating pool on decision space, while the probability distribution of the proposed method is on objective space in the mating pool. To optimize problems on the combinatorial sets, the proposed method is applied. The performance of the proposed algorithm was tested on Computational Expensive Optimization benchmark tests and indicates that the proposed operator is a competitive and promising approach.

Statistical REGRESSION analysis is a well-known method for formulating the relationship between the response variable (output) and some explanatory variables (input) using a set of observations based on the assumption of normal distributions. Fuzzy LINEAR REGRESSION is the most fundamental method in the field of fuzzy modeling in which the uncertain relationship between target and explanatory variables is estimated, and it has been effectively used repeatedly in a wide variety of real-world applications. In this article, we examine the fuzzy REGRESSION model with the coefficients of Neutrosophic fuzzy numbers. For this, we first write a generalization of the measure of the Diamond distance for these numbers, and then estimate the parameters of the model, which are Neutrosophic triangular fuzzy numbers, using the least square method. We show and finally by citing an example, we express the application of the presented model.

In this article, an approach for fitting a fuzzy LINEAR REGRESSION model based on support vectors is presentedwhen the response variable, model parameters and errors are considered as fuzzy numbers.In this method, the objective function is based on the sum of the absolute values ​​of the distances of the hypothetical points to the non-parallel border hyperplanes. The presented model has good robustness to the presence of outlier data. The proposed model has been compared with some other models based on three goodness of fit indices.

Introduction: In statistical analysis, the issues that statisticians focus on are identifying outlier observations. An outlier observation may occur due to a measurement error or a real value found in the study [2]. In either case, the detection of outlier observation is important because if this observation is obtained due to a measurement error, it can be ignored, but if this observation is a real value, then its identification can be useful in the future studies. Blessley et al. [3] identify outlier observations with the approach of removing each observation in each step and examining the effect of removing observation on coefficients estimation of LINEAR REGRESSION models. Other studies in this area were also carried out by Beckman and Cook [4], Brent and Louis [5] and Montgomery and Pack [2]. One way to identify outlier observations in REGRESSION models, is to measure the difference between the response variables and their expected values under fitted model. This identification in circular REGRESSION, is possible by using of a circular distance. According to the importance of identifying outlier observations in the LINEAR-circular REGRESSION model, the Difference of Means Circular Error statistic that was introduced by Abuzaid et al. [1] is applied for outlier detection in LINEAR-circular REGRESSION model. Material and methods: In this paper, the Difference of Means Circular Error statistics is applied for outlier detection in LINEAR-circular REGRESSION model and the cut-off points of this statistic are obtained by Monte Carlo simulations. In addition, the performance of this statistic is investigated with some simulation studies. Finally, this statistic is applied to identify outlier observations in speed and direction wind data set recorded at Mehrabad weather station in Tehran with parametric Bootstrap simulation method. Results and discussion: In this paper, we obtained the cut-off points of the DMCE statistics in a LINEAR-circular REGRESSION model using the Monte Carlo simulation method. These points were reduced to (k)n with the assumption of constant (n) k. Also, in simulation studies, the power of this statistic was obtained for large values of λ (contamination level) which was near one for various values of n and large values of k. DMCE statistic is applied to identify outlier observations in real data set. The performance of this statistic is desirable in detecting outlier observations in real data. Conclusion: The following conclusions were drawn from this research: Cut-off points of the DMCE statistics were reduced to (k) n with the assumption of constant (n ) k. In simulation studies, the power of DMCE statistic was near one for large values of λ (contamination level) and large values of k. The performance of DMCE statistic is desirable in detecting outlier observations in real data.