Orthogonal Linear Regression

In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable. ^1

Essentially generate a model which approximates a set of given points $a=\{(x_i,y_i)\in R|i=1,...,k\}$ we calculate $$Y_i={\beta }_0+{\beta }_1{x}_i+{ϵ}_i$$

if we assume ${ϵ}_i=0$ we get the formula for orthogonal regression $$Y_i={\beta }_0+{\beta }_1{x}_i$$ which can be calculated using the following Python implementation

import numpy as np
from matplotlib import pyplot as plt


def calc_line(a: np.array) -> np.array:
    """
    Calculate 2D orthogonal regression line
    A_i(i|y_i) for all i = 1, ..., k

    :param a: Array of y-values
    :returns: x-values of orthogonal regression line for the k given y-values
    """
    size = a.size
    u1 = np.linspace(1, size, size)
    u2 = np.ones([size, ])
    right = np.array([a.T.dot(u1), a.T.dot(u2)])
    left = np.array([[u1.T.dot(u1), u1.T.dot(u2)], [u1.T.dot(u2), u2.T.dot(u2)]])
    beta_1, beta_0 = np.linalg.solve(left, right)
    return beta_0 + u1 * beta_1