Question

In: Statistics and Probability

Find the function of the form f(t) = c0 + c1sin(t) + c2cos(t) + c3sin(2t) +...

Find the function of the form

f(t) = c0 + c1sin(t) + c2cos(t) + c3sin(2t) + c4cos(2t)

that best fits the data points (0,0), (0.5,0.5), (1,1), (1.5,1.5), (2,2), (2.5,2.5), (3,3), using least squares. Sketch the solution, together with the function g(t) = t.

Solutions

Expert Solution

t sin t cos t sin (2t) cos(2t) y
0 0 1 0 1 0
0.5 0.479426 0.877583 0.841471 0.540302 0.5
1 0.841471 0.540302 0.909297 -0.41615 1
1.5 0.997495 0.070737 0.14112 -0.98999 1.5
2 0.909297 -0.41615 -0.7568 -0.65364 2
2.5 0.598472 -0.80114 -0.95892 0.283662 2.5
3 0.14112 -0.98999 -0.27942 0.96017 3

using Excel

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.999896184
R Square 0.999792378
Adjusted R Square 0.999377134
Standard Error 0.026956942
Observations 7
ANOVA
df SS MS F Significance F
Regression 4 6.998546647 1.749636662 2407.723582 0.000415201
Residual 2 0.001453353 0.000726677
Total 6 7
Coefficients Standard Error t Stat P-value Lower 95%
Intercept 1.5 0.075852952 19.77510382 0.00254742 1.173631089
sin t 0.108974218 0.121544186 0.896581088 0.464559036 -0.413988204
cos t -1.536691216 0.022576322 -68.06649975 0.000215771 -1.63382929
sin (2t) 0.302691969 0.024514511 12.34746117 0.006495271 0.197214543
cos(2t) 0.043147694 0.057848625 0.745872425 0.533494831 -0.205754848

y^= 1.5 + 0.108974 sin t -1.53669 cos t + 0.3026919 sin 2t + 0.043147 cos 2t

t sin t cos t sin (2t) cos(2t) y y^
0 0 1 0 1 0 0.006456
0.5 0.479426 0.877583 0.841471 0.540302 0.5 0.481691
1 0.841471 0.540302 0.909297 -0.41615 1 1.018702
1.5 0.997495 0.070737 0.14112 -0.98999 1.5 1.5
2 0.909297 -0.41615 -0.7568 -0.65364 2 1.981298
2.5 0.598472 -0.80114 -0.95892 0.283662 2.5 2.518309
3 0.14112 -0.98999 -0.27942 0.96017 3 2.993544


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