Optimal multiple functional regression analysis using perturbation theory

Figures & data

Table 1. Multiple functional regression results for Data Set 1.

Data 1	x = [0 1 2 3 4 5 6 7] y = [0 1 5 9 14 25 37 49]	$\overline{x}$=[0.0000 0.1429 0.2857 0.4286 0.5714 0.7143 0.8571 1.0000] $\overline{y}$=[0.0000 0.0204 0.1020 0.1837 0.2857 0.5102 0.7551 1.0000]
K	Regression function	det(F)	ao …  … .aK	Σak	Sy/x
3	$a_0{\overline{x}}^2+a_1{\overline{x}}^3+a_2\overline{x}+a_3\sin \overline{x}$	2.88 10^−10	−0.6323	−12.0183	0.0184
			−11.7147
			82.4629
			−82.1343
2	$a_0{\overline{x}}^2+a_1{\overline{x}}^3+a_2\overline{x}$	0.0022	0.8047	1.0113	0.0212
			0.1582
			0.0484
1	$a_0{\overline{x}}^2+a_1{\overline{x}}^3$	0.0806	0.9426	1.0093	0.0196
			0.0667
0	$a_0{\overline{x}}^2$	1.9475	1.0017	1.0017	0.0189

Figure 1. Comparison of the normalized data (o) with the regression results for K = 3 (…), K = 2(-.), K = 1 (–), K = 0 (solid).

Table 2. Multiple functional regression results for Data Set 2.

Data 2	x = [ 0 1 2 3 4 5 6 7 8 9 10] y = [ 3 3.2 2.8 1 -0.9 -1.4 -0.1 1.7 2.3 1.1 -0.8]	$\overline{x}$=[0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0] $\overline{y}$=[ 0.9375 1.0000 0.8750 0.3125 -0.2813 -0.4375 -0.0313 0.5313 0.7187 0.3438 -0.2500]
K	Regression function	det(F)	ao …  … .aK	Σak	Sy/x
2	$a_0\frac{1}{1+10\overline{x}}+a_1\text{sin}\left(10\overline{x}\right)+a_2\text{cos}\left(10\overline{x}\right)$	41.2046	0.9370	1.5628	0.0046
			0.6206
			0.0052
1	$a_0\frac{1}{1+10\overline{x}}+a_1\text{sin}\left(10\overline{x}\right)$	7.4421	0.9398	1.5604	0.0059
			0.6206

Figure 2. Comparison of the normalized data (o) with the regression results for K = 2 (…), K = 1(solid).