Question

4. Given five data points: (x, y, z) = (0,0,3), (0,1,2), (1,0,3), (1,1,5), (1, 2, 6), use the normal equation method to find the 3D plane of the form z = do + 01x + a2y that best fits the given data (in least squares sense). You may use MATLAB to verify your solution.

Answer 1

Let z = ata at ay be the required. plane that would be a best fit to the geven data We use method of least squares to find the equation (1). The normal expuation for the least squares plane are E zao + an Exit azy, C Eziz aon tarni tay TENT Ez Qori + QE2ye 11 Žai Ye = a Žyot a Laiyo ta, Zy? REI LA We prepare a table to find Exi, EY, ZZ, Ex? Ey ² Eriyu, Enzi Z entaming the columns X Y Z 0 2 2² xil xiza Zilr 0 0 3 0 0 0 0 0 0 or mor m o - --OO Nuoto - 1 0 3 1 0 2 0 3 0 1511 25 15 15 2 to 1 4 36 2 6 12 Addung 34 19 3 683 31 Substituting Zxe = 3, Zyn = 4, Ex? - 3, Exi%c3 E219, 224=14, ²4,2=6 in normal equations we get Ez 9 e 19 19 = 500 +31+ 402 9 14 = 300 +30 + 3 Q2Y - - 42 +39 + baz

system of equations in (9) by using We solve the matrix method. systeme) can be written as matrix fhm 15 3 41 PRO7 P 197 8 3a el 14 14 ; 6L azJ [19] We perform a sequence of row operations only to reduce to coefficient matrix into a triangular matrix, then back substitution will give the slutions ao, ai, az 5R-3R 5 3 7 197 5 R-4R 15-4 15-12 -16-57 -> LO 1542 30-16023 L95-76 1534150075 197 Lo3 14 La Liq NOR- R215 3 4 7rao7 197 36 313 Lo 0 25 Laz) [15 A₂ = 1 from Roz we get 250 = 25 sai +39=13 = = 13-392 from R₂ weget substituting aq=1, we get a = 13-3110 - 10 From R 500+ 39, +42,= 19 + 2 = 19-391-402 Subali keting anh, 9 = 20 =(19-31" 3-4 (1) 2 Substituting ao=2, are 5 and a = 1 in equation ( 12 weget z = 2 + 5x + y) is the required plane obtained by least squaros method 6 55

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