# Haar Wavelet-Based Classification Method for Visual Information Processing Systems
## 3 Case Study  
The effect of the interval width changing for the coefficient of the separating surface on the relative value of the average risk R was investigated. The sample was synthesized artificially to study the capabilities of the method for the simplest case: linearly separable classes. The authors illustrate the classification method using the example of separating objects in the feature space into two classes for clarity. The article considers a rather simple case: the division by a straight-line segment, when only two coefficients are calculated. According to this, the research was conducted with variance in two classes of 70 objects-values of parameters (features) per each, separated into the two-dimensional feature space by the segment \(\hat{f}(x,c)=c_{0}+c_{1}x\). Such a situation can occur when the dispersion of classes in the operating mode increases due to changes in operating conditions. A similar result can be obtained if the classifier is trained based on a small sample of data. The article investigates a method that allows us to determine a set of nested ranges for coefficient values. In this case, the classes of patterns in the feature space can be divided not by a segment, but by a \"range\". Next, the situation is investigated when classes of patterns become less compact in the working mode of classification due to an increase of dispersion. If the pattern comes into the range, it is difficult to determine which class it belongs to. The classification error is also registered when the pattern belongs to the wrong class. Thus, the influence of the \"range\" width and variance on the classification result is evaluated. Figure 3 shows the result of division in the two-dimensional space of the parameter features, X1, X2 by three intervals: 1, 2, 3. The result was obtained by studying the influence of changing \(q\)--relative mean-square deviation (RMSD)--in the working mode of classification. Here, \(q=\frac{q_{p}}{q_{0}D}\), where \(q_{p}\) and \(q_{0}\) are the RMSDs in classes in working mode and \"teaching\" mode, respectively; \(D\) is the distance between the centers of the classes for the training sample; \(\mathrm{H}=\frac{\delta}{D}\), where \(\delta\) represents a width of coefficient interval \(c_{1}\); \(\mathrm{R}=\frac{P_{1}+P_{2}}{P_{1}}\), where \(P_{1}\) is a probability of wrong classification and \(P_{2}\) is a probability of hitting the interval correspondingly. Figure 3: The result of separating the two classes by three intervals: having a first interval \(\Delta_{1}c_{1}\) (1); the second interval \(\Delta_{3}c_{1}\) (2); and the third one \(\Delta_{7}c_{1}\) (3). Based on the research results, it can be concluded that with the relative interval width \(\mathrm{H}=0.007\) and \(q_{p}=4q_{0}\) (Table 1, column 2), the amount of the relative value of the average risk R is practically equal to one. That is, for further research, and the choice of system parameters for a given increase of noise in the data, an interval of this width can be selected without loss of reliability. To estimate the time consumption depending on classification error and speed, the authors conducted a second series of experiments. Here, the time for determining the ranges for coefficients was validated experimentally using the example of two classes for 15 values of parameter features per each class. When classifying, the search for the minimum of the functional was performed. The functional \(Q(x,c)\) in this case is the probability of the incorrect classification over the vector of coefficients \(c=(c_{0},c_{1})\). The probability of incorrect classification was estimated as the ratio of the number of incorrectly recognized objects to the total number of objects in the sample. The classes were divided in a two-dimensional feature space by the segment \(\hat{f}(x,c)=c_{0}+c_{1}x\). When calculating for given values of error \(\delta_{1}=0.1\) and the training step \(\gamma=0.7\), one range for \(c_{0}\) and two ranges for \(c_{1}\) were obtained.