## 2 State-of-the-Arts
\begin{split} Y(x_{1},\dots,x_{n})&=\theta_{i}+ \sum\limits_{i=1}^{n}\theta_{i}x_{i}+\sum\limits_{i=1}^{n}\frac{n}{j_{i}} \theta_{ij}x_{i}x_{j}+\sum\limits_{i=1}^{n}\sum\limits_{j=l}^{n}\theta_{i,j,l }x_{i}x_{j}x_{l}+\dots\\ &\dots+\sum\limits_{i=1}^{n}\sum\limits_{i=l}^{n}\sum\limits_{i= j}^{n}\dots\sum\limits_{z=k-1}^{n}\theta_{i,j,l,\dots z}x_{i}x_{j}x_{l}\dots x_{z}\end{split} , \tag{1}\]  
where \(n\) is a number of features \(x_{1},\dots,x_{n}\), and \(k\) is the Ito decomposition degree, \(\theta\) is the Ito decomposition coefficient. Figure 2: Flowchart of the PNN-Logit system that used PNN-based dimensionality reduction. According to experimental studies, a modified approach using Ito decomposition increased accuracy up to 2% (from 90% for the scheme of Figure 2 to 92% for the scheme of Figure 3). Despite the relatively high accuracy of the two approaches described above, an error of 8% is significant, especially in the case of solving the tasks of diagnosing the properties of the alloy from which biomedical implants will be made. In addition, in the case of a few labeled classes, particularly for a two-class problem, they will not be as effective because the actual classification is based on PNN-derived probabilities (only two attributes). The authors of [38] investigate the problem of machine learning methods processing short sets of data in the Materials Science domain. The authors proposed an intuitive scheme to expand the input data space of the problem to improve the prediction accuracy. It consists of adding to the initial dataset additional attributes that are essentially a \"rough estimate\" of the material's properties. This approach is also based on [PERSON]'s theorem and should increase the accuracy of classifiers or regressors. The authors have shown a significant increase in the accuracy of the proposed method compared to existing ones. However, the main problem here is the need for a \"rough assessment\" of the material's properties as a scheme to expand the space of input features of a given dataset. An expert should do it. Therefore, in such a model appears a human factor. Secondly, the expert's opinion may be subjective, which will significantly affect the data in the form of outliers or anomalies--and consequently, the accuracy of the machine learning method. Third, for medium-sized datasets, such an approach will require significant time and human resources in the form of expert work on each vector of the specified dataset. To avoid such shortcomings, Ref. [39] developed a new smart system. It is also based on PNN summation layer outputs, but the fundamental difference here is that they are included as additional features to the initial dataset. The dataset expanded in this way is fed to the selected classifier. The flowchart of the proposed approach is shown in Figure 4 [39]. In [39], SVM was used by the authors as a classifier. This machine learning algorithm demonstrates the highest accuracy of existing machine learning methods to solve the stated task [13]; allows the process of short datasets efficiently, and provides a high-speed training process. In the case of creating a hybrid system, where the element is a neural network without training, the latter factor plays an essential role in developing an algorithm for learning the entire hybrid scheme. Figure 3: Flowchart of the PNN-Ito-Logit system. Experimental studies were based on SVM with _rbf_-kernel. This approach for nonlinear input expansion significantly increases the classification accuracy. The results showed that the proposed scheme provides a 15% higher accuracy (91%) than a single SVM with _rbf_-kernel (74%). However, this scheme did not increase accuracy compared to the scheme in Figure 2.  
This paper proposed an advanced PNN-SVM smart system that will provide the best classification accuracy. In addition, the paper proposed a user-friendly smart web service that implements all of the above methods.