This paper introduces a novel Method for solving the generalized total Least Squares problem, an extension of the total Least Squares problem. The generalized total Least Squares problem emerges when solving overdetermined linear systems with the multiple right-hand sides $\mathbf{AX} \thickapprox \mathbf{B}$, where both the observation matrix $\mathbf{B}$ and the data matrix $\mathbf{A}$ contain errors. Our approach involves extending the Taylor series expansion to reformulate the generalized total Least Squares problem into a linear problem, allowing us to employ the tensor form of the generalized Least Squares algorithm for efficient computation. This technique streamlines the computational process and enhances solution accuracy. For a more detailed survey, we compare the proposed Method for solving the generalized total Least Squares problem with one of the matrix format Methods for the associated total Least Squares problem. Empirical results show that our Method significantly improves computational efficiency and solution precision. Additionally, we demonstrate its practical application in the context of image blurring.

This paper presents a new algorithm for the identification of a specific class of hybrid systems. Hybrid System identification is a challenging problem since it involves the estimation of discrete and continuous states simultaneously. Using the Method known as product of errors, this problem could be formulated such that, the identification of continuous state to be independent of discrete state estimation. We propose a new Iterative weighted Least Squares algorithm (IWLS) for the identification of switched auto regressive exogenous systems (SARX). In this Method, the parameters of only one subsystem are updated at each iteration while the parameters of the other subsystems are assumed known. In the Method, all four main parameters of hybrid systems, namely Subsystem degrees, Number of subsystems, Unknown parameters vector and switching signal are estimated. Simulation results shows that our proposed Method has a good performance in identifying the subsystems parameters and switching signal. Also, the superiority of our algorithm is shown by modeling of two SARX systems.

Determination of the diffusion coefficient on the base of solution of a linear inverse problem of the parameter estimation using the Least-square Method is presented in this research. For this propose a set of temperature measurements at a single sensor location inside the heat conducting body was considered. The corresponding direct problem was then solved by the application of the heat fundamental solution.

The main purpose of this article is to increase the efficiency of the Least Squares Method in numerical solution of ill-posed functional and physical equations. Determining the Least Squares of a given function in an arbitrary set is often an ill-posed problem. In this article, by defining artificial constraint and using Lagrange multipliers Method, the attempt is to turn n-dimensional Least Squares problems into (n-1) ones, in a way that the condition number of the corresponding system with(n-1) -dimensional problem will be low. At first, the new Method is introduced for2 and 3-term basis, then the presented Method is generalized for n-term basis. Finally, the numerical solution of some ill-posed problems like Fredholm integral equations of the first kind and singularly perturbed linear Fredholm integral equations of the second kind are approximated by chain Least Squares Method. Numerical comparisons indicate that the chain Least Squares Method yields accurate and stable approximations in many cases.

One of the most efficient statistical tools for modeling the relationship between a dependent variable and several independent variables is regression‎. ‎In practice‎, ‎observations relating to one or more variables‎, ‎or the relationship between variables‎, ‎may be vague or non-specific‎. ‎In such cases‎, ‎classic regression Methods will not have enough capability to model data‎, ‎and one of the alternative Methods is regression in a fuzzy environment‎. ‎The fuzzy logistic regression model provides a framework in the fuzzy environment to investigate the relationship between a binary response variable and a set of covariates‎. ‎The purpose of this paper is to attempt to develop a fuzzy model that is based on the idea of the possibility of success‎. ‎These possibilities are characterized {by several} linguistic phrases‎, ‎including low‎, ‎medium‎, ‎and high‎, ‎among others‎. ‎Next‎, ‎we {use a set of precise explanatory variable observations to model the logarithm transformation of‎ "‎possibilistic odds.‎" ‎We assume that the model's parameters are triangular fuzzy numbers.} We use the Least Squares Method in fuzzy linear regression to estimate the parameters of the provided model‎. ‎We compute three types of goodness-of-fit criteria to evaluate the model‎. ‎Ultimately‎, ‎we model suspected cases of Systemic Lupus Erythematosus (SLE) disease based on significant risk factors to identify the model's application‎. ‎We do this due to the widespread use of logistic regression in clinical studies and the prevalence of ambiguous observations in clinical diagnosis‎. ‎Furthermore‎, ‎to assess the prevalence of diabetes in the community‎, ‎we will collect a sample of plasma glucose levels‎, ‎measured two hours after a meal‎, ‎from each participant in a clinical survey‎. ‎The proposed model has the potential to rationally replace an ordinary model in modeling the clinically ambiguous condition‎, ‎according to the findings‎.

The concise review systematically summarises the state-of-the-art variants of Moving Least Squares (MLS) Method. MLS Method is a mathematical tool which could render cogent support in data interpolation, shape construction and formulation of meshfree schemes, particularly due to its flexibility to form complex arithmetic equation. However, the conventional MLS Method is suffering to deal with discontinuity of field variables. Varied strategies of overcoming such shortfall are discussed in current work. Although numerous MLS variants were proposed since the introduction of MLS Method in numerical/statistical analysis, there is no technical review made on how the Methods evolve. The current review is structured according to major strategies on how to improvise MLS Method: the modification of weight function, the manipulation of discrete norms, the inclusion of Iterative feature for residuals minimising and integration of these strategies for more robust computation. A wide range of advanced MLS variants have been compiled, summarised, and reappraised according to its underlying principle of improvement. In addition, inherent limitation of MLS Method and its possible strategy of improvement is discussed too in this article. The current work could render valuable reference to implement and develop advanced MLS schemes, whenever complexity of the specific scientific problems arose.