Question

linear regression 1.1 Localized Suppose we want to estimate localized linear regression by weighting the contribution of the data points by their distance to the query pointr,i.e. using the cost i=1 where (r)-2(is the inverse Euclidean distance between the training point and query (test) point Derive the modified normal equations for the above cost function E(r(a)). (Hint: first, rewrite the cost function in matrix/vector notation, using a diagonal matrix to represent the weights w).

Answer 1

Solution:-

f(a, b) = sigma^n_i = 1 (a + b_xi - y_i)^2 No to minimize f(a, b), we can make partial differential f(a, b)/partial differential a = 0 & partial differential/partial differential b f(a, b) = 0 sigma alpha (a + b_xi - y_i) = 0 or sigma (a + b_x_2 - y_1) = 0 or Za = sigma (y_i - b_xi) or na = sigma y_i - b sigma x_i or a = 1/n sigma y_i - b sigma x_i/n = y^bar - bx^bar similarly: partial differential/partial differential b f(a, b) = 0 or 2 sigma (a + b_xi - y_i) times I = 0 or sigma ax_i + sigma bx_i^2 - sigma y_1 times I = 0 or b = sigma y_i x_i - a sigma x_i/sigma x_2^2 or b sigma x_1^2 = sigma y_i x_i - (sigma y_i) (sigma y_i)/n + (sigma x_2)^2/eta b or b (eta sigma x_1^2 - (sigma Y-1)^2) = eta 2y_i y_i - (sigma y_i) (sigma y or b = as given (proved)