Question

a. As discussed in class, a coefficient matrix Z constructed with basis functions can be used to solve for the unknown coefficients a; of a polynomial function. Use this technique to determine the coefficients do and a; for a linear fit (y = ao tax) to the following data: x 0 1 2 3 y 2.43 9.62 13.55 20.12 Hint: Use basis functions 1 and x to create your coefficient matrix Z. Solve for the vector a using the equation: Z Za = Z'y. Show your work for finding the inverse using the formula: A-1 = jarac", where C is the cofactor matrix of A. The coefficient of determination r can be calculated with the following equations: - St where S is the sum of the squares of the residuals given by: S= =1(); – 7)?, and S, is the sum of squares: sy = -(y -ỹ)”. b. Calculate r2 for your linear fit from part (a) and discuss the implications of this value.

Answer 1

29 ہر 0 205 | 9 ،93 9,62 ۹,62 2 ا216 .3 8( 8ه با ہا ۹۷9 , | 0 . 5 3 . 9 561 15 و 0 . 098 بار - 2 28366 = 2866 690 = 2 | 13 , 25 | 10 | 30 و60 | 8o012 = 5 . 142 - 9 . 08 3 6 = 2

х = 6 - ). 5 , Я = 9512. — 1 ч. а) = xxЯ — л муу - — Q = а1. og — чx ). 5 x 11. 93 ену — чx (15) - ge - а, - 91.08 – 68. 58 = 8 5 - - - 19 — 9 - а. - у - а, у = ||. Ч3 – 5. TX) 15 - g. RX = 5.

So ai = 5.7) and (ao = 2.88 Now Ir2 = St-S2 St = I ligge Se = ² yi- } i) ² | J = Gota, x = 2.88 + 5.7 x o = 2,88 J = aota, la V = 2.88 +5.7 X 1 = 8.58 go = ao ta, xz V = 2.88 +5.7x2 = 14.28 = dot a, xyl = 2.88 +5.783 - 1998 St= 164.2866 Sr= 118366 164.2866- 1.8366 = 164.2866 1 ² = -0.9888=82