One application of least squares is to find a function that best fits a set of data. For example, given a set of t values and the corresponding f(t) values, we might want to find the coefficients of f(t) = a0+a1t+· · ·+antn . However, we might not know in advance the degree of the polynomial that best fits the data. We can try polynomials of different degrees and compare the residuals to see which polynomial best fits the data.

i. use least squares to fit it to a linear model, f(t) = a0 + a1t

ii. calculates the norm of the residual, || \vec{b} ? A\vec{x}||2, for the linear model

iii. uses least squares to fit it to a quadratic model, f(t) = a0 + a1t + a2t2

iv. calculate the norm of the residual, || \vec{b} ? A\vec{x} ||2, for the quadratic model

v. plot the original points, that is, t versus f(t)

data set 1 for producing functions of the form

f(t) = a0 + a1t + a2t2 + ...

t1 = [6.4 3.8 8.1 5.3 3.5 9.4 8.8 5.5 6.2 5.9 2.1 3.0 4.7 2.3 8.4 1.9 ...
2.3 1.7 2.3 4.4 3.1 9.2 4.3 1.8 9.0 9.8 4.4 1.1 2.6 4.1 5.9 2.6 ...
6.0 7.1 2.2 1.2 3.0 3.2 4.2 5.1 0.9 2.6 8.0 0.3 9.3 7.3 4.9 5.8 ...
2.4 4.6 9.6 5.5 5.2 2.3 4.9 6.2 6.8 4.0 3.7 9.9 0.4 8.9 9.1 8.0 ...
1.0 2.6 3.4 6.8 1.4 7.2 1.1 6.5 4.9 7.8 7.2 9.0 8.9 3.3 7.0 2.0 ...
0.3 7.4 5.0 4.8 9.0 6.1 6.2 8.6 8.1 5.8 1.8 2.4 8.9 0.3 4.9 1.7 ...
9.8 7.1 5.0 4.7]'

ft1 = [89.08 79.12 80.28 87.32 76.00 67.28 76.12 90.00 87.12 88.88 ...
56.68 66.00 84.32 59.92 77.88 51.28 59.92 45.72 57.92 84.28 ...
67.28 71.72 83.52 47.52 74.00 65.92 80.28 36.08 64.48 77.88 ...
88.88 64.48 87.00 87.68 58.32 36.12 68.00 72.52 78.72 86.48 ...
33.88 60.48 83.00 20.32 68.52 86.92 87.48 86.72 59.48 85.68 ...
64.68 88.00 88.92 55.92 85.48 91.12 86.52 79.00 78.12 60.48 ...
20.68 77.08 70.88 83.00 36.00 60.48 72.88 90.52 40.08 87.32 ...
38.08 87.00 85.48 86.32 85.32 74.00 77.08 69.72 88.00 55.00 ...
16.32 86.48 88.00 82.92 74.00 91.08 87.12 78.08 84.28 86.72 ...
49.52 61.48 73.08 18.32 87.48 45.72 63.92 89.68 84.00 84.32]'

data set 2 for producing functions of the form
f(t) = a0 + a1t + a2t2 + ...

t2 = [8.5 5.6 9.3 7.0 5.8 8.2 8.8 9.9 0.0 8.7 6.1 9.9 5.3 4.8 8.0 2.3 ...
5.0 9.0 5.7 8.5 7.4 5.9 2.5 6.7 0.8 6.3 6.6 7.3 8.9 9.8 7.7 5.8 ...
9.3 5.8 0.2 1.2 8.6 4.8 8.4 2.1 5.5 6.3 0.3 6.1 3.6 0.5 4.9 1.9 ...
1.2 2.1 1.5 1.9 0.4 6.4 2.8 5.4 7.0 5.0 5.4 4.5 1.2 4.9 8.5 8.7 ...
2.7 2.1 5.6 6.4 4.2 2.1 9.5 0.8 1.1 1.4 1.7 6.2 5.7 0.5 9.3 7.3 ...
7.4 0.6 8.6 9.3 9.8 8.6 7.9 5.1 1.8 4.0 1.3 0.3 9.4 3.0 3.0 3.3 ...
4.7 6.5 0.3 8.4]'

ft2 = [-21.5 -5.0 -27.5 -14.0 -6.0 -22.0 -23.0 -26.5 19.0 -22.5 -7.5 ...
-30.5 -5.5 -1.0 -21.0 9.5 -2.0 -26.0 -7.5 -19.5 -18.0 -8.5 ...
10.5 -14.5 17.0 -8.5 -14.0 -15.5 -21.5 -30.0 -17.5 -6.0 -27.5 ...
-8.0 22.0 13.0 -22.0 -1.0 -23.0 10.5 -4.5 -12.5 19.5 -7.5 1.0 ...
18.5 -1.5 9.5 15.0 12.5 11.5 11.5 21.0 -13.0 7.0 -4.0 -16.0 ...
-4.0 -4.0 -3.5 15.0 -1.5 -23.5 -22.5 9.5 8.5 -7.0 -9.0 -2.0 ...
10.5 -24.5 15.0 15.5 16.0 10.5 -10.0 -5.5 16.5 -25.5 -13.5 ...
-18.0 18.0 -20.0 -27.5 -28.0 -20.0 -20.5 -4.5 14.0 -1.0 14.5 ...
21.5 -28.0 6.0 8.0 2.5 -2.5 -9.5 17.5 -21.0]'