## 4 Methods
### Variability of Estimators  
The problem of variability is very complex from different points of view. Single segmentation and corresponding fractal-based estimation results are insufficient for the variability analysis. Low variability (small method-induced errors) is required for the user. Semisupervised learning is applied for every image by the user, so the user expects good classification, therefore the proper value of fractal dimension (FD) for classification algorithm should be delivered. Segmented image could be visualized for the user's acceptance or rejection of classifier results. Low variability of system means also that the number of tests and trials is very low for the user. Variability testing is possible by multiple repetitions of training and estimation of fractal parameters (a kind of Monte Carlo test). This process allows estimation of some empirical distribution which is valuable for the analysis of variability. This process could be repeated for the same input image. Neural network learning process is sophisticated, so achieving a global minimum is not usually possible. Multiple runs of the training algorithm will deliver similar but not identical segmentation algorithms. The small data set used in the second phase of a learning of classifier does not guarantee correct convergence but testing of this convergence is needed. The distributions of FD or lacunarity, obtained by multiple tests of the same input image and particular segmentation provided by the user, are the results of Monte Carlo tests. Different images can have similar distributions. Images with the same image content are expected to deliver similar distributions. The question is, how can a pathomorphologist use distributions from a Monte Carlo test without a large database or precise classification? The solution for the abovementioned problems is as follows. The hypothesis about the possible correct structure of neural network and fractal estimator could be checked with a Monte Carlo test, with a decision rule defined as an acceptance by the lack of reasons for the rejection. The rejection of this hypothesis and rejection of a particular system configuration is possible in cases of significant overlapping of all distributions. Two different images with different content should be with two different distributions without overlapping or with very low probability of overlapping. The testing of all distribution pairs provides the quality coefficient. Analysis of bounding boxes defined by means and standard deviations could be proposed as criteria for possible rejection or acceptance of the hypothesis. Three selected mean values define the center of box and standard deviations define nearest distances between center and particular box faces. The lack of significant overlapping for two images, in the standard deviation sense, means that there are box pairs related to two different images without common content even if the learning process is repeated multiple times. Overlapping occurs if there is a common content. Comparison of all pairs/boxes gives an overlapping table \(OT\). This table is diagonally symmetrical, but diagonal results are not used. All possible pairs are tested, and the following formula shows how this table is filled with logical '0' or '1' values:  
\[OT(x,y)=B_{x}\cap B_{y}, \tag{1}\]  
where \(x\) and \(y\) are indexes of boxes \(B_{x}\) and \(B_{y}\). The minimal number of comparisons in \(OT\) is defined by the formula:  
\[T_{n}=\frac{P(  
Relative overlapping \(Q_{\%}\) is defined by formula:  
\[Q_{\%}=\frac{Q}{T_{P}}\cdot 100\%. \tag{4}\]  
and it is a scalar indicator of the system quality that is related to method-induced errors. Minimal value (zero) is the best case (acceptance) and maximal value is 100% (full rejection). It should be noted that classification assignment is not considered in this technique. Significant overlapping of distributions is the source of strong rejection of the system. This is not a proof that such system will function.