The singled out singularities e_MS dataset are finally classified by using the Kohonen LVQ algorithm. The new datasets (e_MS dataset) comprised of the singled out singularities, with the matrix size (64 × 64), are finally classified by using the Kohonen LVQ algorithm.
The steps of Figure 11.44 are explained as follows:
Step (i) Application to MS dataset multifractal Brownian motion synthesis 2D. Multifractal Brownian motion synthesis 2D can be described as follows: primarily you have to calculate the fractal dimension D(D = [x1, x2, ..., x304]) of the original MS dataset. Fractal dimension is computed with eq. (11.15). As the next step, you have the new dataset (e_MS Dataset) formed by selecting attributes that bring about the minimum change of current dataset’s fractal dimension until the number of remaining features is the upper bound of fractal dimension D. The newly produced multifractal Brownian motion synthesis 2D dataset is
Step (ii) Application to data multifractal Brownian MOTION SYNTHESIS 2D (see [50] for more information).
We can carry out the classification of the e_MS dataset. For this, let us have a closer glance at the steps provided in Figure 11.45:
Step (iii) The new dataset made up of the singled out singularities, e_MS dataset, is finally classified using the Kohonen LVQ algorithm in Figure 11.45.
Figure 11.45 explains the steps of LVQ algorithm as follows:
Step (1) Initialize reference vectors (several strategies are discussed shortly). Initialize learning rate, γ is default (0).
Step (2) While stopping condition is false, follow Steps (2–6).
Step (3) foreach training input vector x new follow Steps (3–4).
Step (4) Find j so that 
is a minimum.
Step (5) Update wij.
Steps (6–8) if (Y = yj)
else
Step (9) The condition is likely to postulate a fixed number of iterations (i.e., executions of Step (2)) or learning rate reaches an acceptably small value.
Thus, the data in the new MS dataset with 33.33% portion are allocated for the test procedure, being classified as Y = [PPMS, SPMS, RRMS, healthy] with an accuracy rate of 83.5% grounded on the LVQ algorithm.
11.4.2.3Exponential Hölder function with LVQ algorithm for the analysis of mental functions
As presented in Table 2.19, the WAIS-R dataset has data with 200 belonging to patients and 200 samples to healthy control group. The attributes of the control group are data regarding school education, gender, …, D.M. Data are made up of a total of 21 attributes. By using these attributes of 400 individuals, we know whether the data belong to patient or healthy group. How can we make the classification as to which individual belongs to which patient or healthy individuals and those diagnosed with WAIS-R test (based on the school education, gender, the DM, vocabulary, QIV, VIV, …, D.M)? D matrix has a dimension of 400 × 21. This means D matrix includes the WAIS-R dataset of 400 individuals along with their 21 attributes (see Table 2.19 for the WAIS-R dataset). For the classification of D matrix through LVQ the first step training procedure is to be employed.
Our purpose is to ensure the classification of the WAIS-R dataset by the multifractal Brownian motion synthesis 2D multifractal analysis of the data. Our method follows the steps shown in Figure 11.46:
| (i) | Multifractal Brownian motion synthesis 2D is applied to the WAIS-R dataset (400 × 21). |
| (ii) | Brownian motion Hölder regularity (exponential) for analysis is applied to the data for the purpose of identifying on the data and new dataset made up of the singled out singularities regarding e_WAIS-R dataset: |
| (iii) | The new dataset made up of singled out singularities e_WAIS-R dataset has the matrix size (16 × 16) yielding the best classification result and they are classified using the Kohonen LVQ algorithm. |
The steps in Figure 11.46 are explained as follows:
Step (i) Application to WAIS-R dataset multifractal Brownian motion synthesis 2D. Multifractal Brownian motion synthesis 2D can be described as follows: primarily you have to calculate the fractal dimension D(D = [x1, x2, . . . , x400]) of the original WAIS-R dataset. Fractal dimension is computed with eq. (11.15). As the next step, you have the new dataset (e_WAIS-R dataset) formed by selecting attributes that bring about the minimum change of current dataset’s fractal dimension until the number of remaining features is the upper bound of fractal dimension D. The newly produced multifractal Brownian motion synthesis 2D dataset is
Step (ii) Application to data multifractal Brownian motion synthesis 2D (see [50] for more information).
We can do the classification of the e_WAIS-R dataset. For this, let us have a closer look at the steps provided in Figure 11.47:
Step (iii) The singled out singularities e_WAIS-R dataset are finally classified by using the Kohonen LVQ algorithm in Figure 11.47.
Figure 11.47 explains the steps of LVQ algorithm as follows:
Step (1) Initialize reference vectors (several strategies are discussed shortly).
Initialize learning rate, γ is default (0).
Step (2) While stopping condition is false, follow Steps (2–6).
Step (3) foreach training input vector xnew follow Steps (3–4).
Step (4) Find j so that is a minimum.
Step (5) Update wij.
Steps (6–8) if (Y = yj)
else
Step (9) The condition is likely to postulate a fixed number of iterations (i.e., execution of Step (2)) or learning rate reaches an acceptably small value.
Thus, the data in the new e_WAIS-R dataset with 33.33% portion are allocated for the test procedure, being classified as Y = [Patient, Healthy] with an accuracy rate of 82.5% based on the LVQ algorithm.
New datasets have been obtained by the application of multifractal method Hölder regularity functions (polynomial and exponential function) on MS dataset, economy (U.N.I.S.) dataset and WAIS-R datasets.
–New datasets, comprised of the singled out singularities, p_MS Dataset, p_Economy Dataset and p_WAIS-R Dataset, have been obtained from the application of polynomial Hölder functions.
–New datasets, comprised of the singled out singularities, e_MS Dataset, e_Economy Dataset and e_WAIS-R Dataset, have been obtained by applying exponential Hölder functions.
LVQ algorithm has been applied in order to get the classification accuracy rate results for our new datasets as obtained from the singled out singularities (see Table 11.1).
Table 11.1: The LVQ classification accuracy rates of multifractal Brownian motion Hölder functions.
| Datasets | Hölder functions | |
| Polynomial (%) | Exponential (%) | |
| Economy (U.N.I.S.) | 84.02 | 80.30 |
| MS | 80 | 83.5 |
| WAIS-R | 81.15 | 82.5 |
As shown in Table 11.1, the classification of datasets such as the p_Economy dataset that can be separable linearly through multifractal Brownian motion synthesis 2D Hölder functions can be said to be more accurate compared to the classification of other datasets (p_MS Dataset, e_MS Dataset; p_Economy Dataset, e_Economy Dataset; and e_WAIS-R Dataset).
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