TABLE 2.

Performance of principal component analysis (PCA) of hybrid functional and vector data (HFV-PCA) and Ramsay and Silverman’s method (RS-PCA) in terms of $\mathrm{RSE}\left({\widehat{\lambda }}_m\right)$, $\mathrm{RSE}\left({\widehat{\xi }}_m\right)$ and $\mathrm{ASE}\left({\widehat{\rho }}_m\right)$, m = 1,2,3, averaged across 500 simulated datasets of size n = 200 (with standard deviations in parentheses) for each setting (I/II) and functional data type (SIR/DIR/DR). “Ratio” denotes the RSE and ASE ratios of HFV-PCA to RS-PCA

		Setting I			Setting II
Type	R(A)SE	HFV-PCA	RS-PCA	Ratio	HFV-PCA	RS-PCA	Ratio
SIR	$\mathrm{RSE}\left({\widehat{\lambda }}_1\right)$	0.025 (0.026)	0.134 (0.245)	0.189	0.019 (0.022)	1.207 (6.743)	0.016
	$\mathrm{RSE}\left({\widehat{\lambda }}_2\right)$	0.050 (0.041)	0.493 (0.527)	0.102	0.048 (0.045)	1.168 (1.183)	0.041
	$\mathrm{RSE}\left({\widehat{\lambda }}_3\right)$	0.192 (0.057)	1.152 (0.939)	0.167	0.138 (0.066)	3.642 (1.964)	0.038
	$\mathrm{RSE}\left({\widehat{\xi }}_1\right)$	0.024 (0.020)	0.148 (0.213)	0.161	0.021 (0.018)	0.584 (0.381)	0.036
	$\mathrm{RSE}\left({\widehat{\xi }}_2\right)$	0.045 (0.027)	0.339 (0.307)	0.132	0.052 (0.036)	0.761 (0.501)	0.068
	$\mathrm{RSE}\left({\widehat{\xi }}_3\right)$	0.255 (0.068)	1.579 (0.418)	0.162	0.165 (0.094)	1.783 (0.181)	0.092
	$\mathrm{ASE}\left({\widehat{\rho }}_1\right)$	0.047 (0.012)	0.427 (0.333)	0.111	0.059 (0.012)	1.193 (1.008)	0.049
	$\mathrm{ASE}\left({\widehat{\rho }}_2\right)$	0.043 (0.015)	0.484 (0.278)	0.088	0.052 (0.016)	0.856 (0.415)	0.060
	$\mathrm{ASE}\left({\widehat{\rho }}_3\right)$	0.026 (0.007)	0.634 (0.172)	0.041	0.030 (0.010)	0.886 (0.148)	0.034
DIR	$\mathrm{RSE}\left({\widehat{\lambda }}_1\right)$	0.010 (0.013)	0.011 (0.016)	0.903	0.015 (0.011)	0.011 (0.014)	1.336
	$\mathrm{RSE}\left({\widehat{\lambda }}_2\right)$	0.012 (0.015)	0.013 (0.018)	0.931	0.016 (0.017)	0.017 (0.023)	0.946
	$\mathrm{RSE}\left({\widehat{\lambda }}_3\right)$	0.009 (0.014)	0.013 (0.022)	0.700	0.011 (0.021)	0.021 (0.028)	0.499
	$\mathrm{RSE}\left({\widehat{\xi }}_1\right)$	0.015 (0.013)	0.017 (0.016)	0.889	0.091 (0.027)	0.091 (0.027)	0.396
	$\mathrm{RSE}\left({\widehat{\xi }}_2\right)$	0.047 (0.028)	0.031 (0.025)	1.514	0.037 (0.041)	0.037 (0.041)	1.292
	$\mathrm{RSE}\left({\widehat{\xi }}_3\right)$	0.060 (0.030)	0.040 (0.027)	1.500	0.265 (0.151)	0.265 (0.151)	0.265
	$\mathrm{ASE}\left({\widehat{\rho }}_1\right)$	0.024 (0.009)	0.030 (0.012)	0.796	0.033 (0.011)	0.051 (0.033)	0.655
	$\mathrm{ASE}\left({\widehat{\rho }}_2\right)$	0.032 (0.014)	0.035 (0.017)	0.918	0.037 (0.015)	0.053 (0.026)	0.703
	$\mathrm{ASE}\left({\widehat{\rho }}_3\right)$	0.028 (0.011)	0.029 (0.010)	0.938	0.036 (0.014)	0.047 (0.047)	0.772
DR	$\mathrm{RSE}\left({\widehat{\lambda }}_1\right)$	0.010 (0.014)	0.010 (0.014)	0.999	0.010 (0.015)	0.012 (0.014)	0.867
	$\mathrm{RSE}\left({\widehat{\lambda }}_2\right)$	0.010 (0.014)	0.010 (0.014)	1.012	0.011 (0.014)	0.010 (0.014)	1.015
	$\mathrm{RSE}\left({\widehat{\lambda }}_3\right)$	0.010 (0.015)	0.010 (0.015)	1.004	0.010 (0.014)	0.028 (0.024)	0.352
	$\mathrm{RSE}\left({\widehat{\xi }}_1\right)$	0.014 (0.014)	0.015 (0.015)	0.950	0.020 (0.014)	0.078 (0.018)	0.251
	$\mathrm{RSE}\left({\widehat{\xi }}_2\right)$	0.029 (0.025)	0.027 (0.024)	1.098	0.034 (0.023)	0.025 (0.021)	1.343
	$\mathrm{RSE}\left({\widehat{\xi }}_3\right)$	0.030 (0.025)	0.031 (0.026)	0.983	0.041 (0.025)	0.192 (0.033)	0.212
	$\mathrm{ASE}\left({\widehat{\rho }}_1\right)$	0.013 (0.009)	0.014 (0.010)	0.972	0.014 (0.009)	0.021 (0.012)	0.664
	$\mathrm{ASE}\left({\widehat{\rho }}_2\right)$	0.019 (0.015)	0.019 (0.015)	0.999	0.019 (0.014)	0.019 (0.016)	0.995
	$\mathrm{ASE}\left({\widehat{\rho }}_3\right)$	0.014 (0.010)	0.014 (0.010)	0.999	0.014 (0.009)	0.019 (0.008)	0.747

Abbreviations: ASE, average squared error; DR, dense and regular; DIR, dense and irregular; RSE, relative squared error; SIR, sparse and irregular.