# A Model of Information Visualization Interpretation
## 4 Discussion
### Expected Visual Measurement Error  
Error is typically the difference between an actual and measured value. Knowing how the height of a bar is measured visually is problematic. This may be different person-to-person. Hence, we developed the approach for this course of work. Namely, we make the height of the bar a random variable. Let us work through the concept of this model. First, the viewer looks at the base, leaving a fixation, then moves rapidly to the top of the bar (the next AOI). This is followed by a glance toward the ruler in an effort to obtain a reading on the height. The process is repeated by the path returning to the base of the bar to again obtain a feel for the height of the bar. This creates a family the forms a random triangle. See Figure 5. The circled edges are part of the height estimation process. Figure 3 shows a particular triangle. The edge of the triangle \(\mathbf{P_{1}P_{2}}\) has length \(D_{12}\), which is a random variable with a noncentral chi distribution with noncentrality parameter \(\lambda\), as has been discussed. The random variable is a model of the height of the bar. Admittedly, how the viewer uses the ruler to assign a length to \(D_{12}\) is not clear. From the model's perspective, each triangle would have associated with it a pairing \((r(\mathbf{P_{3}}),D_{12}(\mathbf{P_{1}},\mathbf{P_{2}}))\), where \(r(\mathbf{P_{3}})\) is the reading taken from the ruler based on vertex \(\mathbf{P_{3}}\). In this paper, it is proposed that after viewing the bar graph and creating multiple triangles, the viewer assigns the expected value of \(D_{12}\) as the measured height of the bar. (The exact mechanics of how this is done are not well explained. We envision that for each fixation by the ruler, a number is captured. This becomes the assigned length of the edge. In some sort of limiting process, the mean is recovered from an unbiased statistic. This is being actively investigated and will be an area of future publication.). That is, \(r(\mathbf{P_{3}})=E(D_{12})\). This makes the expected visual measurement error  
\[\text{EVME}=\text{E}(D_{12}-h)=\text{E}(D_{12})-h. \tag{84}\]  
This brings us to the following key theorem. Theorem 6: _For \(\mu\), given in Equation (27),_  
\[\text{EVME}=(\text{E}(D_{12})-h)\sim h(1-\rho)\tau^{2}+\frac{1}{2}h(1-\rho)^{ 2}\tau^{4}+\cdots \tag{85}\]  
_as \(\tau\to 0^{+}\)._  
Proof: We appeal to the paper by [PERSON] [29] for the formula for the asymptotic expansion of the mean as the noncentrality parameter becomes large. In his formula in the paper, we set \(a=1\) for the first moment, \(N=2\) for the number of degrees of freedom, and \(y_{0}=\lambda\) for the noncentrality parameter. This yields, after simplification,  
\[\text{E}\bigg{(}\frac{D_{12}}{\psi};\lambda\bigg{)} \sim\lambda\bigg{[}1+\frac{1}{2}\lambda^{-2}+\frac{1}{8}\lambda^{ -4}+\cdots\bigg{]}\quad\lambda\to\infty\] \[\sim\lambda+\frac{1}{2}\lambda^{-1}+\frac{1}{8}\lambda^{-3}+ \cdots\quad\lambda\to\infty\]  
Figure 5: Conceptual view of the stochastic triangles connecting one fixation in each of the areas of interest. The circled edges have random lengths called \(D_{12}\) that possess an expected value called \(\mu\). so that  
\[\text{E}(D_{12})\sim\psi\lambda+\frac{1}{2}\psi\lambda^{-1}+\frac{1}{8}\psi \lambda^{-3}+\cdots\quad\lambda\to\infty.\]  
Now, we use \(\lambda=h/\psi\):  
\[\text{E}(D_{12}) \sim\psi(h/\psi)+\frac{1}{2}\psi(h/\psi)^{-1}+\frac{1}{8}\psi(h/ \psi)^{-3}+\cdots\quad\tau\to 0^{+}\] \[\sim h+\frac{1}{2}(\psi^{2}/h)+\frac{1}{8}(\psi^{4}/h^{3})+\cdots \quad\tau\to 0^{+}\]  
where \(\tau=1/\lambda\). We substitute \(\psi^{2}=2h^{2}\tau^{2}(1-\rho)\) and rearrange terms to obtain  
\[\text{E}(D_{12})-h=\text{E}(D_{12}-h)\sim h\tau^{2}(1-\rho)+\frac{1}{2}h\tau^{ 4}(1-\rho)^{2}+\cdots\quad\tau\to 0^{+}. \tag{86}\]  
Therefore, we conclude  
\[\text{EVME}=\text{E}(D_{12}-h)=\text{E}(D_{12})-h. \tag{84}\]0  
\(\square\)  
This is a remarkable expression.