A New Algorithm for Generalization of Least Square Method for Straight Line Regression in Cartesian System for Fully Correlated Both Coordinates

This work presents the estimation of parameters and uncertainty of straight regression line using the least square method when both coordinates in Cartesian System XOY are affected by errors and fully correlated. The maximalization of the likelihood multivariate Gaussian function as equivalent of minimalization of objection function is derived for common vector variable Z included X and  Y vectors random variable – vectors of coordinates of measurement points. In this way  all kind  of possible correlations are taken into account – within the metrological literature, there exist some works taking them into account. The core of the presentation is the mathematical manipulation based on linear algebra matrixes and vectors  with elements of functional analysis. Any novelty claimed in this field will be carefully demonstrated. The problem is reduced to the determination of numerical one-dimensional characteristic e.g. objection function as function of slope a of straight regression line. The algorithm allows to determine numerically  the covariance matrix  and  the coverage corridor  for straight line regression. The implementation of the numerical method is applied in script running in MATLAB environment. Finally, the comparison of results  based on the previous published tests are carried out