Question

3. {50 points} Our hypothesis function has the general form: hθ(x) = θ0 + θ1x. We give to hθ(x) values for θ0 and θ1 to approximate our output ’y’. In other words, we are trying to create a function called hθ(x) that is able to reliably map our input data (the x’s) to our output data-label (the y’s). Let us consider the
following training set: {x1, x2, x3, x4, x5}={(0,0), (1,1), (2,1), (3,2), (4,4)}.
(a) Plot all samples into a 2-dimensional Cartesian axes system
(b) {20 points} Make your initial guess about the values of the parameters θ0 and θ1. Give the corresponding hypothesis function and draw it on the Cartesian axes
(c) {10 points} Knowing the formula of the cost function, J(θ0, θ1) = 12m Pm
i=1(hθ(x(i)) − y(i))2, calculate
the error produced by your estimation
(d) {20 points} Calculate the error produced by hθ(x) having fixed the parameters θ0 = 0 and θ1 = 3

Answer 1

Solution : given hg(x) = 8 + 8, x given data is (x,.3.),0) (12.42) - (11) (23, Y; )= (2,12 (Rys. 94) = (3,2) Cas., ): (++ ) a) are shown in the lanterian system The points Lelow - 4 x (4,4) *(312) ()' (2,12 b) Initial guess loould be the equation ho(x) = x because three printa (0,0), (,1),(414) Satisfy alove equation ... 8:0, 0,= 1 The sthaight line is shown with dotted red . line in the Cartesian system above.

c) Cost function 4(0,0) = { 1,621)-4; 7 J& our estimation J(0,0) = (0.6)+(1-1+2-1)+(3-2)764-4) - ptr 2 d) 00:0, 0,33 then ho(x) = 32 ho(09:0, HOU= 3, ho(2)-6, ho(37=9, 1964)=12 is 4 (8,0)= (0-014 (3-19%+(6-1)7(9-2) + (12+4)* = 245 + 7 + 8? 142