Improved Approximation and Visualization of the Correlation Matrix

Figures & data

Fig. 1 Displays of the correlation matrix of the Heart attack data obtained in R by using the corrplot, FactoMineR, and factoextra packages. A: Colored tabular display or corrgram. B: Correlation circle or correlation biplot. See Section 2 for the abbreviations of the names of the variables.

Fig. 2 Visualizations of correlation structure, using the Heart attack data. A: PCA biplot of the correlation matrix; B: Trosset’s correlogram; C: MDS map, with negative correlations indicated by dotted lines; D: Biplot obtained by PFA; E: Biplot obtained by WALS; F: Biplot obtained by WALS with scalar adjustment δ. The RMSE of the approximation is given between parentheses in the title of each panel.

Fig. 3 Adjustment of the correlation matrix. All biplots have a calibrated correlation scale for SI. A: PCA biplot; B: PCA biplot using the δ adjustment; C: WALS biplot; D: WALS biplot using the δ adjustment. The interpretation of the biplot origin is shown by its projection (in red) onto the calibrated scale. Black dots on biplot vectors correspond to zero correlation for the corresponding variable. The interpretation of the black dot for SI is shown by its projection (in green) onto the calibrated scale. For positive δ, biplot vectors are extended beyond the biplot origin toward their zero point. For negative δ, tails of biplot vectors are colored in red for the negative part of the correlation scale.

Table 1 Correlation matrix of the Heart attack data.

	CI	SI	VP	Pulse	logPR	DBP	PA
CI	1.000	0.887	–0.282	–0.112	–0.839	–0.361	–0.269
SI	0.887	1.000	–0.201	–0.503	–0.833	–0.483	–0.405
VP	–0.282	–0.201	1.000	–0.085	0.318	0.285	0.244
Pulse	–0.112	–0.503	–0.085	1.000	0.287	0.399	0.370
logPR	–0.839	–0.833	0.318	0.287	1.000	0.761	0.716
DBP	–0.361	–0.483	0.285	0.399	0.761	1.000	0.928
PA	–0.269	–0.405	0.244	0.370	0.716	0.928	1.000

Table 2 Sample correlations ($r_{(SI,i)}$) of SI with all other variables, and estimates of the sample correlations according to four biplots.

		PCA		WALS
	$r_{(SI,i)}$	δ = 0	$\delta =0.14$	δ = 0	$\delta =-0.27$
		$\mathbf{R}$	${\mathbf{R}}_a$	$\mathbf{R}$	${\mathbf{R}}_a$
Pulse	–0.503	–0.264	–0.316	–0.271	–0.340
CI	0.887	0.818	0.905	0.894	0.889
SI	1.000	0.814	1.017	0.842	0.557
DBP	–0.483	–0.609	–0.597	–0.514	–0.497
PA	–0.405	–0.544	–0.546	–0.440	–0.430
VP	–0.201	–0.416	–0.092	–0.252	–0.330
logPR	–0.833	–0.867	–0.717	–0.848	–0.823
RMSE (SI)		0.160	0.116	0.099	0.086

NOTE: The bottom line gives the RMSE of SI for each method. For PCA, the RMSE calculation includes the correlation of SI with itself, for WALS, it does not.

Table 3 RMSE of all variables for four methods.

	PCA		WALS
	δ = 0	$\delta =0.14$	δ = 0	$\delta =-0.27$
	$\mathbf{R}$	${\mathbf{R}}_a$	$\mathbf{R}$	${\mathbf{R}}_a$
Pulse	0.2469	0.1618	0.1345	0.0948
CI	0.0945	0.1078	0.0482	0.0530
SI	0.1598	0.1158	0.0988	0.0857
DBP	0.1212	0.1540	0.0242	0.0239
PA	0.1390	0.1828	0.0196	0.0218
VP	0.3103	0.1336	0.0877	0.0883
logPR	0.0564	0.1275	0.0329	0.0521
All	0.1808	0.1426	0.0755	0.0662

NOTE: The bottom line gives the overall RMSE for each method. For PCA, the RMSE calculation includes the correlations of the variables with themselves whereas for WALS these are excluded.

Fig. 4 Comparison of the double biplots of PCA and WALS. A: PCA biplot, reporting goodness-of-fit of both data and correlation matrix on each axis using eigenvalues. B: WALS biplot (without δ adjustment).

Fig. 5 Biplots of the correlation matrix of the Aircraft data. A: PCA biplot B: WALS biplot. The RMSE of the approximation is given in the title of each panel.

Fig. 6 Biplots of the Swiss banknote data. A: PCA biplot; B: WALS biplot. The RMSE of the approximation is given in the title of each panel.

Table 4 RMSE of all variables for four methods.

	PCA		WALS
	δ = 0	$\delta =0.16$	δ = 0	$\delta =0.07$
	$\mathbf{R}$	${\mathbf{R}}_a$	$\mathbf{R}$	${\mathbf{R}}_a$
Length	0.2725	0.2734	0.0380	0.0170
Left	0.1624	0.1172	0.0616	0.0440
Right	0.1661	0.1855	0.0608	0.0641
Bottom	0.0665	0.0986	0.0071	0.0109
Top	0.2084	0.1080	0.0355	0.0186
Diagonal	0.3352	0.2890	0.0826	0.0790
All	0.2192	0.1949	0.0533	0.0466

NOTE: The bottom line gives the overall RMSE for each method. For PCA, the RMSE calculation includes the correlations of the variables with themselves.

Fig. 7 Biplots of a 10 × 10 equicorrelation matrix. A: PCA biplot; B: WALS biplot. Variable labels for the WALS biplot are not shown, as the 10 vectors coincide.