# A Model of Information Visualization Interpretation
## 4 Discussion
### Remarks on the Correlation Parameter  
One of the main contributions of this paper is the analysis of the influence of the correlation. Let us look at it more closely. Recall that the general model originally proposed had seven correlations in it. In order to proceed, we had to consider cases that were tractable with the available mathematical machinery. Hence, we made decisions that uncorrelated \(\eta\) and \(\zeta\) and give them identical variances. In particular, we set  
\[\rho=\rho_{1}=\rho_{2}=\rho_{12x}=\rho_{12y}=\rho_{1y2x}=\rho_{1x2y}\]  
with  
\[\sigma=\sigma_{1x}=\sigma_{2x}=\sigma_{1y}=\sigma_{2y}\]  
making \(Cov(\eta,\zeta)=0\) (see Equation (17) and \(\psi=2\sigma^{2}(1-\rho)\), which has allowed interesting analysis, but notice what happened in particular. Making \(Cov(\eta,\zeta)=0\) essentially removed four correlations from the model: namely, \(\rho_{1}\), \(\rho_{2}\), \(\rho_{1y2x}\), and \(\rho_{1x2y}\). The two correlations that remained in the model under the name of \(\rho\) were \(\rho_{12x}\) and \(\rho_{12y}\). Furthermore, we required them to have the same sign and magnitude.  
Figure 6 visualizes the consequences of the assumption we made regarding the correlation. Consider situations for which  
\[\text{corr}(d_{2x},d_{1x})=\text{corr}(d_{2y},d_{1y})>0.\]  
This corresponds to fixation clouds that are either shifted to the right and up or to the left and down. Since we have assumed  
\[\sigma=\sigma_{1x}=\sigma_{2x}=\sigma_{1y}=\sigma_{2y}, \tag{88}\]the fixation clouds have roughly similar shapes at each end of the bar. This results in what is essentially a rigid body shift of the bar without a significant change of length. This is not the case with negative correlation. As shown in Figure 6, the fixation cloud shifts rotate the bar and either compress or stretch it.  
Does the model support these suspicions? Recall Corollary 1 that shows the mean is decreasing for \(\rho\) over the interval \((-1,1)\). The mean is the largest for fixation clouds with correlations near \(-1\). This means the expected visual measurement error will be worse for negative correlations. This seems to be consistent with the foregoing discussion. Similarly, translations with little to no dilation would change the mean little. These correspond to strongly positive correlations. Furthermore, we know from Corollary 2 that \(\mu\sim h\) as \(\rho\to 1^{-}\), so the EVME will become small as the correlation becomes strongly positive. This would seem to be consistent with the previous observations shown in Figure 6.  
What remains to question is whether there is a significant difference caused by including the correlation. We have calculated the derivative of the mean with respect to the correlation in Equation (62). Furthermore we know that as \(\rho\to 1^{-}\),  
\[\frac{d\mu}{d\rho}\sim-h\tau^{2}. \tag{89}\]  
As we normally consider small values of \(\tau\), this says the derivative in these cases is small, particularly since \(\tau\) is squared. Therefore, for positive correlation, there might not be much gained by including this parameter. However, for negatively correlated fixations, the relative size of the derivative is unknown.