Given n data points (x1, y1), ... (Xn, Yn), the linear least- squares fit is the linear function f(x) = mx +b that minimizes the sum of the squares (Figure E (m, b) = Ev; - f(x}))² j=l Show that the minimum value of E occurs for m and b satisfying the two equations Σ+ bn= Σy j=l Ex3+b£xj = £ m j=l j=l j=1 (Xµ, Yn) y= mx + b (X2, y2) (x1, y1) FIGURE The linear least-squares fit minimizes the sum of the squares of the vertical distances from the data points to the line.

44/ For the set of data ( x, y) X y 4 8 8 7 7 8 12 Find the Residual value of y – ŷ for each of these points and find the Least Square

Find the best (in the sense of least squares over (-1,1)) second degree polynomial approx- imation to f(x) = x³. %3D

Find the least squares fit solution for paramters a & b of function: y = a/(x^2+b)  Express the result for a & b in terms of the data (xi, yi), i=1,...,n