Stability of principal components under normal and non-normal parent populations and different covariance structures scenarios

Figures & data

Figure 1. Estimated densities of angular displacement (degrees) between the population and sample eigenvectors as a function of covariance matrix pattern, covariance matrix parameter (ρ) and sample size (n), for both normal and non-normal parent populations. Shaded area correspond to cosines between 0.95 and 1 (stable solutions) (a) First eigenvector (b) Second eigenvector.

Table 1. Simulation parameters.

Description	Parameter	Values
Population		Normal/Non-normal
Number of variables	p	3, 6, 10, 15, 20, 25
Correlation	ρ	0.1, 0.3, 0.5, 0.7, 0.9
Type of covariance matrix	$\mathbf{\Sigma }$	CS, AR(1), TOEPLITZ
Sample size	n	30, 50, 100, 150, 200, 500

Figure 2. QQplots comparing the empirical distribution of the angle between the population and sample eigenvectors under normal and non-normal parent populations as a function of covariance matrix pattern, covariance matrix parameter (ρ) and sample size (n). Shaded area correspond to cosines between 0.95 and 1 (stable solutions). (a) First eigenvector (b) Second eigenvector.

Figure 3. Estimated densities of the global angular displacement as a function of covariance matrix pattern, covariance matrix parameter (ρ) and sample size (n), for both normal and non-normal parent populations. Dashed vertical lines correspond to perfect stability. (a) First 2 principal components ($G_2$) (b) All principal components ($G_p$).

Figure 4. ROC curves. (a) First principal component (b) Second principal component.

Table 2. Stability diagnosis summary (Equation (8(8) $\frac{se({\ell }_k)}{\mathrm{\Delta }{\ell }_k}=\frac{{\ell }_k(2/n{)}^{1/2}}{({\ell }_k-{\ell }_{k+1})}(k=1,\dots ,p-1)$(8) )) applied to Swiss Army real data datasets [37].

	1st PC		2nd PC
Dataset	Mean	95%BCI	Mean	95%BCI
Male soldiers	0.19	[0.15, 0.29]	0.28	[0.18,0.48]
Female soldiers	1.06	[0.33, 3.46]	0.72	[0.33,1.94]

Flury B. Flury: data sets from flury, 1997; 2012. R package version 0.1-3. Available from: https://CRAN.R-project.org/package=Flury.

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