## 4 Statistical Models of Freeze Length Characteristics  
In this section, we derive models of the progressions of the CCHs with increasing freeze length \(x\). While \(x\) is a positive integer, the progressions shown in Figures 10 and 11 are exponential, which can also be described equivalently by geometric progressions using a straightforward parameter transformation. We are specifically interested in the parameters of the fittings to the measured data, as they reveal properties, similarities, and differences between the CCHs for the various distances and quadrants. First, we focus on the domain of shorter freeze lengths \(x\in\mathcal{X}_{\mathrm{s}}=\{1,\ldots,10\}\), for which a steeper exponential or an equivalent geometric progression is observed. Because the minimum value of the freeze length is given as \(x=1\), we use the following exponential fitting function to serve as statistical model of the freeze length characteristics:  
\[y_{\mathrm{s}}(x)=a_{\mathrm{s}}\,\exp[b_{\mathrm{s}}(x-1)] \tag{4}\]  
where the argument of the operator \(\exp[\cdot]=e^{[\cdot]}\) is written as an exponent of [PERSON]'s number \(e\), parameter \(a_{\mathrm{s}}\) scales the exponential function, and a negative parameter \(b_{\mathrm{s}}\) describes a progression that decays with increasing argument. The _fit_ function of the MATLAB(r) Curve Fitting ToolboxTM was used to create the fit to the freeze length data \(x\) and to determine  
Figure 11: CCH of freeze length observed in the four quadrants. the model parameters \(a_{\rm s}\) and \(b_{\rm s}\). Because the freeze lengths assume discrete values, the statistical model of the exponential progression in (4) can also be described by a geometric sequence using the following relationship:  
\[y_{\rm s}(x)=a_{\rm s}\,q_{\rm s}^{(x-1)} \tag{5}\]  
where \(q_{\rm s}=\exp(b_{\rm s})\) denotes the so-called common ratio. Table 3 shows the coefficients of the fitting functions (4) and (5) and the goodness of fit in terms of sum of squared error (SSE), root mean square error (RSME), coefficient of determination \(\mathcal{R}^{2}\), and adjusted coefficient of determination \(\mathcal{R}^{2}_{a}\). The results for the different distances are shown accumulated over all quadrants and for each quadrant in which a sufficiently large number of freezes were observed to support a meaningful fitting. The adjusted coefficients of determination of the fits are in the range \(0.84<\mathcal{R}^{2}_{a}<0.99\), with the majority of values above 0.95 which indicates a very good fitting performance. The coefficients \(a_{\rm s}\) and \(b_{\rm s}\) of the exponential function (4) for the different scenarios in the domain of short freeze lengths \(\mathcal{X}_{\rm s}\) are obtained in the following ranges:  
\[ \begin{array}{llll}1.06<a_{\rm s}<1.18&\text{and}&-0.33<b_{\rm s}<-0.26&\text {for}&d=10\,\text{m}\\ 1.05<a_{\rm s}<1.13&\text{and}&-0.41<b_{\rm s}<-0.16&\text{for}&d=12.5\,\text{m} \\ 0.98<a_{\rm s}<1.09&\text{and}&-0.29<b_{\rm s}<-0.18&\text{for}&d=15\,\text{m} \end{array} \]  
In all cases, the strongest variation of the decay parameter \(b_{\rm s}\) among the quadrants appears at distance \(d=12.5\,\text{m}\). On the other hand, variations of the decay parameter \(b_{\rm s}\) for the distances \(d=10\,\text{m}\) and \(d=15\,\text{m}\) are much narrower, indicating similar progressions. It may be conjectured that this characteristic is caused by the room topology (see Figure 2) which shows that the transmitter Tx,v at \(d=12.5\,\text{m}\) is positioned at the boundary between the open space multi-purpose lecture hall and the right-hand side narrower extension. Alternatively, considering the geometric sequence models, the common ratio is obtained in the following ranges:  
\[ \begin{array}{llll}0.72<q_{\rm s}<0.77&\text{for}&d=10\,\text{m}\\ 0.66<q_{\rm s}<0.86&\text{for}&d=12.5\,\text{m}\\ 0.75<q_{\rm s}<0.84&\text{for}&d=15\,\text{m}\end{array}