Question

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.

Answer 1

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