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Data Preprocessing and Neural Network Architecture Selection Algorithms in Cases of Limited Training Sets—On an Example of Diagnosing Alzheimer’s Disease
Data Preprocessing and Neural Network Architecture Selection Algorithms in Cases of Limited...
Alekseev, Aleksandr;Kozhemyakin, Leonid;Nikitin, Vladislav;Bolshakova, Julia
2023-04-25 00:00:00
algorithms Article Data Preprocessing and Neural Network Architecture Selection Algorithms in Cases of Limited Training Sets—On an Example of Diagnosing Alzheimer’s Disease 1 , 2 , 2 , 3 2 , 4 4 , 5 Aleksandr Alekseev * , Leonid Kozhemyakin , Vladislav Nikitin and Julia Bolshakova Administrative Directorate for Organization of Scientific Researches, Perm National Research Polytechnic University, Perm 614990, Russia Perm Decision Making Support Center, Perm 614107, Russia Department of Economics and Finance, Perm National Research Polytechnic University, Perm 614990, Russia Department of Computational Mathematics, Mechanics and Biomechanics, Perm National Research Polytechnic University, Perm 614990, Russia; ybolshakova@pstu.ru Diagnosing Systems, Perm 614101, Russia * Correspondence: aoalekseev@pstu.ru Abstract: This paper aimed to increase accuracy of an Alzheimer ’s disease diagnosing function that was obtained in a previous study devoted to application of decision roots to the diagnosis of Alzheimer ’s disease. The obtained decision root is a discrete switching function of several variables applicated to aggregation of a few indicators to one integrated assessment presents as a superposition of few functions of two variables. Magnetic susceptibility values of the basal veins and veins of the thalamus were used as indicators. Two categories of patients were used as function values. To increase accuracy, the idea of using artificial neural networks was suggested, but a feature of medical data is its limitation. Therefore, neural networks based on limited training datasets may be inefficient. The solution to this problem is proposed to preprocess initial datasets to determine the parameters of the neural networks based on decisions’ roots, because it is known that any can be represented in the Citation: Alekseev, A.; Kozhemyakin, incompletely connected neural network form with a cascade structure. There are no publicly available L.; Nikitin, V.; Bolshakova, J. Data specialized software products allowing the user to set the complex structure of a neural network, Preprocessing and Neural Network which is why the number of synaptic coefficients of an incompletely connected neural network has Architecture Selection Algorithms in been determined. This made it possible to predefine fully connected neural networks, comparable in Cases of Limited Training Sets—On terms of the number of unknown parameters. Acceptable accuracy was obtained in cases of one-layer an Example of Diagnosing and two-layer fully connected neural networks trained on limited training sets on an example of Alzheimer ’s Disease. Algorithms 2023, diagnosing Alzheimer ’s disease. Thus, the scientific hypothesis on preprocessing initial datasets and 16, 219. https://doi.org/10.3390/ neural network architecture selection using special methods and algorithms was confirmed. a16050219 Academic Editors: Sunil Jha, Keywords: integrated rating mechanisms; decisions’ roots; criteria trees; convolution matrices; data Malgorzata Rataj and Xiaorui preprocessing; system-cognitive analysis; neural network structure selection; neural network training Zhang Received: 30 December 2022 Revised: 9 April 2023 Accepted: 18 April 2023 1. Introduction Published: 25 April 2023 Data mining and machine learning methods are used in various areas including engineering, business and management, economics and finance, medicine, etc. Artificial neural networks are actively used in the medical studies [1,2], etc., for example in diagnosing cardiovascular diseases [3–5], intestinal dysbacteriosis [6], oncol- Copyright: © 2023 by the authors. ogy [7–12], in modeling of the prognosis of the outcome of the disease, and the likelihood Licensee MDPI, Basel, Switzerland. of tumor recurrence [13–15]. In [16], it is proposed to use convolutional neural networks This article is an open access article for automatic segmentation of edema in patients with an intracerebral hemorrhage. In [17], distributed under the terms and deep learning models are used to analyze the complication of cerebral edema induced by conditions of the Creative Commons radiation therapy in patients with an intracranial tumor. Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ A feature of medical data is their limitation. This fact is partly due to ethical require- 4.0/). ments, i.e., patients or their legal representatives must consent to the use of their medical Algorithms 2023, 16, 219. https://doi.org/10.3390/a16050219 https://www.mdpi.com/journal/algorithms Algorithms 2023, 16, 219 2 of 25 data for scientific purposes and publication in open press. For example, researchers [18] took 5 years to collect data about 81 subjects, including 59 subjects with clinically diagnosed Alzheimer ’s disease, and 22 elderly subjects did not have problems in cognitive abilities. This limited dataset was chosen for performance verification of neural network training after preprocessing initial datasets and neural network architecture selection using special methods and algorithms on an example of diagnosing Alzheimer ’s disease. The key prob- lem of Alzheimer ’s disease is an accurate diagnosis in the early stages, where cognitive impairments do not yet appear at all or are not noticeable to the patient and others. 1.1. Related Papers The key conclusions of some works of the last two decades [19–22] devoted to the study of the brain are the growth of diseases associated with Alzheimer ’s disease, which does not immediately have distinctive signs for its identification. As shown in [23–25], there is a connection between oxygen saturation of brain tissues and the level of its blood circulation, qualitatively expressed through cerebral metabolism, with the symptoms of Alzheimer ’s disease. In [26], decrease of the dopaminergic activity of the ventral tegmental area (located in the midbrain area) may be of decisive importance for the earliest pathological symptoms of Alzheimer ’s disease. This was verified by data analysis of resting structural and functional magnetic resonance imaging and neuropsychological assessments in 51 healthy adults. Mild cognitive impairments were diagnosed for 30 patients, and Alzheimer ’s disease was diagnosed for 29 patients. A significant functional relationship was also noted between the ventral tegmental area and the hippocampus in pathological symptoms of Alzheimer ’s dis- ease. Changes in the dopaminergic system are often noted among patients with Alzheimer ’s disease and are usually associated with cognitive and noncognitive symptoms [27]. In [28], it is shown that most dopamine-producing neurons are localized in the mid- brain. Their loss is associated with some of the most famous human neurological disorders— Parkinson’s disease and cognitive impairments. In [29,30], the importance of dopamine is noted in participation in the mediated reaction of the body to the environment on different time scales (from fast impulse responses associated with rewards to slower changes with uncertainty, punishment, etc.). In [29], it is shown that the human brain contains about 200,000 dopamine neurons on each side of the brain. In [31–33], it is noted that cerebrovascu- lar diseases largely stimulate the more rapid development of amnestic moderate cognitive impairments and Alzheimer ’s disease and are also characterized by more severe damage to neuronal connections. In [34], the fact of increased cerebral blood flow without changing the oxygen consumption of the brain is described with the use of acetazolamide. This changes the signal observed on MRI images with a gradient echo from areas with increased cerebral blood flow. There is also a relationship between the oxygen content in the blood and the release of dopamine and neurotransmitters in [35,36]. In [37], the dependence of the level of dopamine and the oxygen content into the blood are noted. Since the neuron activity depends on the degree of oxygen uptake, the assessment of the degree of venous blood saturation can become an indicator of the functional state of neurons. Accordingly, we chose the following veins of all involved in the brain blood supply: the basal veins involved in the blood circulation of the striatum and veins of the thalamus (these veins drain the blood from the thalamus, which is closely connected with the midbrain [38]). In [39], there is a fairly pronounced relationship between quantitative magnetic reso- nance imaging (MRI) and spectroscopy (MRS) measurements of energy metabolism (the rate of oxygen metabolism in the brain, CMR (O )), the blood circulation (i.e., cerebral blood flow (CBF) and volume (CBV)) and functional MRI signal over a wide range of neuronal activity. Pharmacological treatments are used to interpret the neurophysiological basis of levels of the blood oxygenation dependent on image contrast at 7 T in glutamatergic neurons in the rat cerebral cortex. Algorithms 2023, 16, 219 3 of 25 1.2. Research Statement The motivation for the present paper was continuation of the previous study [40] on the application of decisions’ roots to diagnosis of Alzheimer ’s disease. “Decision root” is a new name for the integrated rating mechanisms traditionally applicated to aggregation of a few indicators into one integrated assessment [41]. The new term was proposed by Dr. Habil. Korgin, Nikolay A. and his Ph.D. student Sergeev, Vladimir A. [42] in November 2021. Previously, the term “integrated rating mechanisms” was used. Historically, “the integrated rating mechanisms were introduced as multidimen- sional assessment and ranking systems for management and control in organizational and manufacturing systems in the Soviet Union in earlier 80th of the previous century” [43] (p. 610). Nowadays, these systems are used in different areas, including medicine [40]. Decision support system for diagnosis of Alzheimer ’s disease is developed at the LLC “Diagnosing systems”. Mathematically, the integrated rating mechanism is a discrete switching function of several variables presenting as a superposition of a few functions of two variables. Therefore, it can be represented in a hierarchical form, which determines the sequence of operations on variables. Each discrete function of two variables can be represented in matrix form. The tuple of criteria, binary trees and convolution matrices, located in its nodes, gives a graphical representation of such a function. In [44] (p. 1) it was said “integrated rating mechanism belongs to the class of so-called verbal decision analysis approaches”. Verbal decision analysis has been actively used for unstructured decision-making problems [45]. In 2020, two approaches to identification of integrated rating mechanisms based on a training set were suggested. The first one was proposed by Professor, Dr. Habil Burkov, Vladimir N., Dr. Habil Korgin, Nikolay A. and Sergeev, Vladimir A. [44]. This one-hot encoding approach considered the integrated rating mechanism as a sequence of matrix operations with one-hot encoded indicator values. The second approach was proposed by one of the authors of this paper—Alekseev, Aleksandr O. and was based on truth table transformation [46], therefore, it was called the tabular algorithm for the identification of an integrated rating mechanism [47] (p. 599). In 2021, Sergeev, Vladimir A. and Korgin, Nikolay A. [48] proposed to investigate the integrated rating mechanism identification problem on an incomplete dataset using equivalence groups. Approaches [46,47] and [48] are similar, but not the same. In paper [46] (p. 402) it was shown that any integrated rating mechanism can be represented in the form of an artificial neural network. An identified neural network reproduces the training data with discrete values. The obtained function, by using a tabular algorithm for identification [40], establishes the relationship between Alzheimer ’s disease and the magnetic susceptibility value of the basal veins and veins of the thalamus. However, the accuracy of the function ob- tained in a previous study [40] was near 84%, which is not enough for its application in medical practice. To increase accuracy, the idea was suggested to apply an artificial neural network to approximate the relationship between the magnetic susceptibility value of cerebral veins and Alzheimer ’s disease. However, as previously mentioned, a feature of medical data is their limitation. In such cases, neural networks based on limited training datasets may be inefficient. To solve this problem, it is proposed to preprocess initial datasets to determine the parameters of the neural networks based on the integrated rating mechanism (decision’s root). The scientific hypothesis of this paper is that it will be possible to obtain the acceptable accuracy of the neural network by training a neural network on the initial data in a continuous form. Thus, the purpose of this work is performance verification of neural network training in cases of limited training sets on the example of diagnosing Alzheimer ’s disease. The paper is organized as follows: Section 2 provides initial data, methods and algorithms used in this study according to suggested methodology; Section 3 provides Algorithms 2023, 16, 219 4 of 25 results of preliminary data analysis based on interval coding and identifies significant indicators of the study area. The architecture of the neural network based on the found decision’s root is proposed, which is the key result in this study, and the significance of the results is beyond doubt for development and discussion in future works; Section 4 is devoted to a discussion of the obtained results and the insights of this work. Lastly, the main results of this paper and future research directions are briefly discussed in Section 5. 2. Materials and Methods 2.1. Initial Data In reference [49], it was shown that synthetic data can be used for neural network training, but in this paper, real medical data were taken for performance verification of neural network training based on preprocessing initial datasets and neural network architecture selection using special methods and algorithms. The initial data were tomographic images of 81 patients examined using the mini- mental state examination (MMSE), Montreal cognitive assessment (MoCA), clock drawing task (CDT), and activity of daily living scale (ADL). Alzheimer ’s disease was clinically diagnosed in 59 subjects (21 males and 38 females). These patients are hereinafter defined as in [18] as the AD group. Twenty-two patients (12 males, 10 females) did not have problems in cognitive abilities and were placed in the control group (defined as CON group). The magnetic susceptibility values (MSV) of the following cerebral veins were quanti- fied based on tomographic images: left and right basal veins (L_BV and R_BV, respectively), left and right internal cerebral veins (L_ICV, R_ICV), left and right veins of the thalamus (L_TV, R_TV), left and right septal veins (L_SV, R_SV), left and right veins of the dentate nucleus (L_DNV, R_DNV). In this study, we exclusively used numerical data given in the appendix to the refer- ence [18]. A fragment of the initial data is presented in Table 1. Table 1. The fragment of the initial data with magnetic susceptibility values of the cerebral veins [18]. Subject L_BV R_BV L_ICV R_ICV L_TV R_TV L_SV R_SV L_DNV R_DNV Group sub001 1 279 288 255 263 140 138 131 131 165 185 sub002 1 274 247 223 243 239 262 190 222 204 102 sub003 1 259 333 236 243 172 159 135 145 152 153 ... ... ... ... ... ... ... ... ... ... ... ... sub079 0 279 288 179 189 141 153 245 214 177 149 sub080 0 249 232 221 232 154 142 165 151 161 143 sub081 0 300 259 295 299 216 197 163 131 162 129 In this Table in column “Group” 1 is the AD group, and 0—the CON group. 2.2. Methodology and Methods Data preprocessing consists in interval coding of initial data and search of integrated rating mechanisms (decisions’ roots). In paper [46] (p. 402), it is shown that any integrated rating mechanism can be represented in the form of an artificial neural network. Thus, it is possible to determine some of the parameters of the required neural network based on the integrated rating mechanism (decision’s root). This mechanism reproduces the encoded data using discrete values. This idea corresponds to the approach presented in reference [50]. The scientific hypothesis lies in the assumption that it will be possible to obtain the acceptable accuracy of the neural network by training a predefined neural network on the initial data in a continuous form. According to [46] (p. 398) the integrated rating mechanism is determined as a tuple (1). < G, M, X, P >, (1) where G is a graph describing a sequence of convolution of particular criteria, M is a set of convolution matrices corresponding to the nodes of a criteria tree, X is a set of the scales for Algorithms 2023, 16, 219 5 of 25 the scoring of the particular criteria, and P is the procedure of aggregation. In this study, discrete procedure is used, but continuous, interval, fuzzy and F–fuzzy procedures are also noted [51]. One of the problems of identifying integrated rating mechanisms (decisions’ roots) is the choice of the structure of the graph G, which is a full binary tree with labeled leaves [52], because their total number is determined according to the equation [43,48]: jGj = (2l 3)!!, (2) where l is the total number of considered variables (leaves in terms of tree graph). The number of G structures for 4 variables is 15, for 5–105, for 6–945 as can be seen from Equation (2), and there are already more than 34 million, for example, for 10 variables. This is a key limitation of the proposed algorithms, as can be seen from the above example. To reduce the number of variables, it is suggested to use methods of the system- cognitive analysis (hereinafter–SCA) [53,54] using the computer program RU 2022615135 “Personal intelligent online development environment “EIDOS-X Professional” (System “EIDOS-Xpro”)” [55], developed by professor Lutsenko, Evgenue V. In this computer program, to solve the classification problem, two integrated similarity criteria are used, known as “Sum of knowledge” (3) and “Semantic resonance of knowledge” (4). I = L I , L = 0; 1 , (3) f g j i i j i i=1 where I —is the amount of information about class j; I—the amount of features (signs, attributes); L —a variable describing the presence (L = 1) or absence (L = 0) of the feature i i i i; I —assessment of informational importance of feature i for class j (amount of information i j about feature i for class j). I = I I L L , L = f0; 1g (4) j å i j j i i s s M j l i=1 where M—number of gradations of descriptive scales (features); s —standard deviation particular criteria for class vector knowledge; s —standard deviation along the vector of the recognized object; I —average informativeness for class vector; L—average for object vector. In the case of diagnosing Alzheimer ’s disease, index j is the ordinary number of patient group {group AD, group CON}, j = 1 correspond to group AD, j = 2 correspond to group CON. Index i is the ordinary number of the intervals of MSV of cerebral veins. We have 10 veins, each of which is divided into 3 equal intervals, so the total amount of MSV intervals is 30 (I = 30). As a result of the implementation of the SC-analysis methods, the following model of determination informational importance (5) were selected as the most reliable. N N i j I = N , (5) i j i j where N —the number of observations of the i-th feature (sign) for objects of the j-th class i j in the training sample; N —the total number of observations of the i-th feature over the entire training sample; N —the total number of features for objects of the j-th class in the training set; N—the volume of the training sample (number of observations). These methods have been used to identify the most significant brain veins and subse- quent reduction of the number of indicators. The algorithm for identification of integrated rating mechanisms [47] has been used for data structuring and representation as a decision root. These algorithms were suggested previously by one of the authors of the present study–Alekseev, Aleksandr O. Algorithms 2023, 16, 219 6 of 25 A specialized computer program called “Neurosimulator 5.0” was used for neural network modeling [56]. It was developed by professor Yasnitsky, Leonid N. and his Ph.D student, Cherepanov, Fedor M. For training the neural network, the computer program “Software package that im- plements the operation of incompletely connected neural networks” [57] was tested, but this program does not implement the conjugate gradient method. The application of this program is shown in the work [58]. Through inputs, the mathematical neuron receives input signals, which it sums up by multiplying each input signal by the appropriate weighting factor w , w , . . . , w : i1 i2 i j Nn s = w x , (6) i å i j i j j=1 where Nn—the number inputs of i-th neuron; x —signals on the input of the i-th neuron; i j w —weights obtained in each iteration when performing neural network training. i j Next, the activation function acts, which is usually an arbitrary monotonically increas- ing function that takes real values in the range from 0 to 1. The activation threshold is a number from the same interval. Nn y = f (s ) = f w x + q = f w x , (7) i i å i j i j i j=1 where y —output signal; q —neural bias b is interpreted as the weight of an additional i i input with a synaptic link strength w , whose signal x = 1; f —activation function. 0 0 The general task of training a neural network is as follows: Let X be a set of objects; R the solution of the algorithm, and then there is an unknown objective function G : X ! R . It is necessary to find such a primary function G : X ! R that restores the assessment G . According to the generally accepted neural network design technology, the entire general population is divided into training, testing and confirming sets in the ratio of 70%: 20%: 10% [59]. 0 0 Based on a set of logical pairs d = (x , r ), where d is n-th precedent, we will distin- n n n n N N Tr 0 Tr Ts 0 Ts guish three relevant subsets of precedents D = d —Train Set, D = d —Test Set, f g f g n n n=1 n=1 Tv 0 Tv Tr Ts Tv D = fd g —Validation Set, provided that data loss is excluded D \ D \ D ? [60] n=1 (p. 426). According to the methodology presented in Figure 1, the training of a non-fully connected neural network will be carried out by the method of conjugate gradients [61], which makes it possible to find the extrema of the function by iterative calculation. The conjugate gradient vector is determined by the following formula: r = rE(w ) + b r , (8) k k k k 1 where the choice of the parameter b responsible for the conjugate direction can be deter- mined according to the Fletcher and Reeves algorithm [62] (10) or algorithm by Polak and Ribiere (11). rE(w ) rE(w ) k k b = , (9) rE(w ) rE(w ) k 1 k 1 (rE(w ) rE(w )) rE(w ) k k k b = , (10) rE(w ) rE(w ) k 1 k 1 Algorithms 2023, 16, 219 7 of 25 Algorithms 2023, 16, x FOR PEER REVIEW 8 of 27 START Sets of convolutional matr ices cor r esponding Initial tr aining to edges of selected data cr iter ia tr ees Methods of system -cognitive analysis per for ming the univer sal Neur al netw or k Algorithm of representation the integrated Preliminary architecture cognitive analytical system rating mechanism in neural netw ork data analysis EIDOS -Xpr o , developed by selection model, developed by associate pr ofessor pr ofessor Lutsenko, Evgenue V . Alekseev, Aleksandr O. Significant indicator s and inter vals for encoding to Specification of neur al discr ete values netw or k: amount of hidden layer s and number of neur ons on each layer Coding initial data to discr ete values Neur al netw or k modeling, per for ming Neur al netw or k Neur osimulator 5.0 , developed by pr ofessor tr aining Yasnitsky , Leonid N. and Cher epanov , Fedor M. Encoded data w ith discr ete values cor r esponding to inter vals number s Algor ithm of identification of the Identification of the integr ated Neur al netw or k m odel integr ated r ating mechanisms , r ating mechanisms (decision s developed by associate pr ofessor r oots) Alekseev, Aleksandr O. END Figure 1. Suggested algorithm of data preprocessing and neural network architecture selection for Figure 1. Suggested algorithm of data preprocessing and neural network architecture selection for neural network training in limited initial dataset cases. neural network training in limited initial dataset cases. 3. Results Two directions r and r are defined as conjugate if the following condition is met: k k 1 3.1. Results of the Preliminary Data Analysis r Hr = 0, (11) k k 1 The initial data were processed using SCA methods. The intervals of magnetic sus- ceptibility values of the above-described veins L_BV, R_BV, L_ICV, R_ICV, L_TV, R_TV, In other words, conjugate directions are orthogonal directions in the space of an L_SV, R_SV, L_DNV, R_DNV were used as features. The state of the patient was used as identity Hessian matrix. a class: AD group or CON group, and the domains of observed values were divided into We chose the algorithm of elastic backpropagation (RPROP—Resilent back PROPaga- Three equal intervals for each vein (Table 2). Such a division of the initial dataset into three tion) to train classic single-layer two-layer hidden artificial neural networks with a different intervals was due to expert opinion, since when choosing two intervals, we get the classi- number of neurons on each hidden layer. According to this algorithm, when correcting the cal problem of finding a Boolean function. The choice of four intervals is inappropriate, weight coefficients, only the sign of the gradient matching [63]: since when using 10 variables in the initial data, 80 features will be formed, and the initial dataset, which must be divided into training and testing, contains only 81 examples. Of ¶#(t) Dw (t) = h(t)sign , (12) i j interest is the division of the observation area into different intervals with an almost uni- ¶w i j form number of examples in each of these intervals. In view of the multivariate results of possible partitions, this approach will be considered in a future study. The methodology of this study is presented below (Figure 1). 3. Table Results 2. The intervals for discrete coding. 3.1. Results of the Preliminary Data Analysis Variable Cerebral Veins Domain of MSV 1st Interval 2nd Interval 3rd Interval The initial data were processed using SCA methods. The intervals of magnetic sus- Abbrev. ceptibility values of the above-described veins L_BV, R_BV, L_ICV, R_ICV, L_TV, R_TV, left basal vein L_BV {153.0; 324.0} {153.0; 210.0} {210.0; 267.0} {267.0; 324.0} L_SV, R_SV, L_DNV, R_DNV were used as features. The state of the patient was used right basal vein R_BV {164.0; 357.0} {164.0; 228.3} {228.3; 292.7} {292.7; 357.0} as a class: AD group or CON group, and the domains of observed values were divided left internal cerebral vein L_ICV {179.0; 394.0} {179.0; 250.7} {250.7; 322.3} {322.3; 394.0} into Three equal intervals for each vein (Table 2). Such a division of the initial dataset right internal cerebral vein R_ICV {163.0; 411.0} {163.0; 245.7} {245.7; 328.3} {328.3; 411.0} into three intervals was due to expert opinion, since when choosing two intervals, we left vein of the thalamus L_TV {131.0; 288.0} {131.0; 183.3} {183.3; 235.7} {235.7; 288.0} get the classical problem of finding a Boolean function. The choice of four intervals is Algorithms 2023, 16, 219 8 of 25 inappropriate, since when using 10 variables in the initial data, 80 features will be formed, and the initial dataset, which must be divided into training and testing, contains only 81 examples. Of interest is the division of the observation area into different intervals with an almost uniform number of examples in each of these intervals. In view of the multivariate results of possible partitions, this approach will be considered in a future study. Table 2. The intervals for discrete coding. Cerebral Veins Variable Abbrev. Domain of MSV 1st Interval 2nd Interval 3rd Interval Algorithms 2023, 16, x FOR PEER left REVIEW basal vein L_BV {153.0; 324.0} {153.0; 210.0} {210.0; 267.0}9 of 27 {267.0; 324.0} right basal vein R_BV {164.0; 357.0} {164.0; 228.3} {228.3; 292.7} {292.7; 357.0} left internal cerebral vein L_ICV {179.0; 394.0} {179.0; 250.7} {250.7; 322.3} {322.3; 394.0} right internal cerebral vein R_ICV {163.0; 411.0} {163.0; 245.7} {245.7; 328.3} {328.3; 411.0} left vein of the thalamus L_TV {131.0; 288.0} {131.0; 183.3} {183.3; 235.7} {235.7; 288.0} right vein of the thalamus R_TV {109.0; 286.0} {109.0; 168.0} {168.0; 227.0} {227.0; 286.0} right vein of the thalamus R_TV {109.0; 286.0} {109.0; 168.0} {168.0; 227.0} {227.0; 286.0} left septal vein L_SV {83.0; 310.0} {83.0; 158.7} {158.7; 234.3} {234.3; 310.0} left septal vein L_SV {83.0; 310.0} {83.0; 158.7} {158.7; 234.3} {234.3; 310.0} right septal vein R_SV {73.0; 287.0} {73.0; 144.3} {144.3; 215.7} {215.7; 287.0} right septal vein R_SV {73.0; 287.0} {73.0; 144.3} {144.3; 215.7} {215.7; 287.0} left vein of the dentate nucleus L_DNV {94.0; 244.0} {94.0; 144.0} {144.0; 194.0} {194.0; 244.0} left vein of the dentate nucleus L_DNV {94.0; 244.0} {94.0; 144.0} {144.0; 194.0} {194.0; 244.0} right vein of the dentate nucleus R_DNV {81.0; 269.0} {81.0; 143.7} {143.7; 206.3} {206.3; 269.0} right vein of the dentate nucleus R_DNV {81.0; 269.0} {81.0; 143.7} {143.7; 206.3} {206.3; 269.0} With the help of W ith the the intel help ligent of the analyti intelligent cal system analytical “EIDOS system -Xpro“EIDOS-Xpr ”, all patient o”, s w all ere patients were distributed based distributed on the lev based el of pro on gr the essi level on o of f Al pr zh ogr eimer ession ’s di ofse Alzheimer ase by app ’sly disease ing two by in- applying two integral similarity criteria: “Semantic resonance of knowledge” (4) and “Sum of knowledge” tegral similarity criteria: “Semantic resonance of knowledge” (4) and “Sum of knowledge” (3) (Figure 2). (3) (Figure 2). Figure 2. Recognition results for the AD group of patients in the “Class-objects” mode in the system Figure 2. Recognition results for the AD group of patients in the “Class-objects” mode in the system “EIDOS-Xpro”. “EIDOS-Xpro”. The intervals of MSV were determined as a result of the application of SCA. These The intervals of MSV were determined as a result of the application of SCA. These values had the gre values atest had info the rmation greatest al s informational ignificance for significance classification for of a classification patient in thof e AD a patient in the AD group (Figure 3, see “contributing”), as well as signs that did not characterize this class group (Figure 3, see “contributing”), as well as signs that did not characterize this class to to the greatest extent (Figure 3, see “impending”). Since patients belong to only two the greatest extent (Figure 3, see “impending”). Since patients belong to only two classes classes (AD and CON) in the research dataset, the insignificance for the AD group meant (AD and CON) in the research dataset, the insignificance for the AD group meant signifi- significance for the CON group, respectively. cance for the CON group, respectively. Algorithms 2023, 16, 219 Algorithms 2023, 16, x FOR PEER REVIEW 9 of 25 10 of 27 Contributing values of factors and The impeding values of factors and the strength of their influence-(I) the strength of their influence-(I) I I L_TV 3/3 235.7; 288.0 I = 5.148 L_ICV 1/3 179.0; 250.7 I = 5.123 R_BV 2/3 228.3; 292.7 I = 4.951 R_BV 1/3 164.0; 228.3 I = 4.568 R_TV 3/3 227.0; 286.0 I = 3.235 L_DNV 1/3 94.0; 144.0 I = 4.210 L_ICV 2/3 250.7; 322.3 I = 2.951 L_TV 1/3 131.0; 183.3 I = 3.840 L_DNV 3/3 194.0; 244.0 I = 2.531 L_BV 1/3 153.0; 210.0 I = 3.198 L_BV 3/3 267.0; 324.0 I = 2.247 R_DNV 1/3 81.0; 143.7 I = 2.481 R_DNV 2/3 143.7; 206.3 I = 2.210 R_TV 1/3 109.0; 168.0 I = 2.469 Figure 3. The most significant factors influencing a possible brain pathology in the form of Alz- Figure 3. The most significant factors influencing a possible brain pathology in the form of heimer’s disease (after the name of the variable corresponding to a certain vein, the serial number Alzheimer ’s disease (after the name of the variable corresponding to a certain vein, the serial number of the interval of values out of 3 is indicated, and the interval of MSV values are directly indicated of the interval of values out of 3 is indicated, and the interval of MSV values are directly indicated in in curly brackets) [40] (p. 426). curly brackets) [40] (p. 426). It can be seen from the left part of Figure 3, that the 3rd interval of MSV in the veins It can be seen from the left part of Figure 3, that the 3rd interval of MSV in the veins of the thalamus (L_TV, R_TV) are among the first 3 most contributing factors, and the 1 st of the thalamus (L_TV, R_TV) are among the first 3 most contributing factors, and the 1st interval of MSV in the same veins are impending factors (see right part of Figure 3). Sec- interval of MSV in the same veins are impending factors (see right part of Figure 3). Second ond and third intervals of MSV for both the basal veins (R_BV, L_BV) were included in and third intervals of MSV for both the basal veins (R_BV, L_BV) were included in seven seven contributing factors for classification of a patient in the AD group (see Figure 3), contributing factors for classification of a patient in the AD group (see Figure 3), and the and the first interval of MSV in both left and right basal veins are considered impending first interval of MSV in both left and right basal veins are considered impending factors factors (see Figure 3) [40]. Thus, four indicators (L_BV, R_BV, L_TV, R_TV) were selected (see Figure 3) [40]. Thus, four indicators (L_BV, R_BV, L_TV, R_TV) were selected as the as the most significant from among all available indicators. most significant from among all available indicators. 3.2. Results of Interval Coding Initial Data to Discrete Values 3.2. Results of Interval Coding Initial Data to Discrete Values Veins of the thalamus L_TV, R_TV and in the basal veins L_BV, R_BV have intervals Veins of the thalamus L_TV, R_TV and in the basal veins L_BV, R_BV have intervals performed for coding to discrete values (see, Table 2). performed for coding to discrete values (see, Table 2). The initial data were quantized by replacing the MSV values for a specific patient in The initial data were quantized by replacing the MSV values for a specific patient in a specific vein (L_BV, R_BV, L_TV, R_TV) with the value of the interval obtained using a specific vein (L_BV, R_BV, L_TV, R_TV) with the value of the interval obtained using SCA methods after reducing the number of analyzed parameters. In other words, the orig- SCA methods after reducing the number of analyzed parameters. In other words, the inal initial data set [18] was encoded (Table 3) to discrete values from one to three with original initial data set [18] was encoded (Table 3) to discrete values from one to three with correspondence to the number of MSV intervals (see, Table 2). correspondence to the number of MSV intervals (see, Table 2). Table 3. The fragment of the encoded data. Table 3. The fragment of the encoded data. Encoded Encoded Encoded Encoded Subject Group Subject Group Encoded L_BV Encoded L_TV Encoded R_BV Encoded R_TV L_BV L_TV R_BV R_TV sub001 AD 3 1 2 1 sub001 AD 3 1 2 1 sub002 AD 3 3 2 3 sub002 AD 3 3 2 3 sub003 AD 2 1 3 1 sub003 AD 2 1 3 1 . . . . . . . . . . . . . . . . . . … … … … … … sub079 CON 3 1 2 1 sub080 CON 2 1 2 1 sub081 CON 3 2 2 2 Algorithms 2023, 16, x FOR PEER REVIEW 11 of 27 sub079 CON 3 1 2 1 Algorithms 2023, 16, 219 10 of 25 sub080 CON 2 1 2 1 sub081 CON 3 2 2 2 Subjects with the same vectors of encoded veins were united into 42 subgroups. For Subjects with the same vectors of encoded veins were united into 42 subgroups. For example, subjects 003 has a vector of values of the MSV {259.0; 172.0; 333.0; 159.0}, which example, subjects 003 has a vector of values of the MSV {259.0; 172.0; 333.0; 159.0}, which was encoded to vector {2; 1; 3; 1}, and subjects 057 has a vector of values of the MSV {216.0; was encoded to vector {2; 1; 3; 1}, and subjects 057 has a vector of values of the MSV {216.0; 147.0; 301.0; 140.0}, which was encoded to the same vector {2; 1; 3; 1}. Moreover, there 147.0; 301.0; 140.0}, which was encoded to the same vector {2; 1; 3; 1}. Moreover, there were were 8 contradictory examples in the encoded dataset. For example, from Table 3 we can 8 contradictory examples in the encoded dataset. For example, from Table 3 we can see see that subjects 001 and 079 have the same vector of discrete values {3; 1; 2; 1}, but they that subjects 001 and 079 have the same vector of discrete values {3; 1; 2; 1}, but they belong belong to different groups: AD and CON. To reduce conflict examples, we excluded those to different groups : AD and CON. To reduce conic fl t examples, we excluded those sub- subgroups or subjects that had high level of similarity to one group but in fact belonged to groups or subjects that had high level of similarity to one group but in fact belonged to another group. The similarity was measured in the System “EIDOS-Xpro” by values of the another group. The similarity was measured in the System “EIDOS-Xpro” by values of integrated similarity criteria using information model INF3. the integrated similarity criteria using information model INF3. The “EIDOS-Xpro” system, according to both integral similarity criteria “Semantic The “EIDOS-Xpro” system, according to both integral similarity criteria “Semantic stability of knowledge” and “Sum of knowledge” with a value of 99.69% similarity, refers stability of knowledge” and “Sum of knowledge” with a value of 99.69% similarity, refers the patient sub007 to the CON groups, i.e., is not ill, while this patient is clinically diagnosed the patient sub007 to the CON groups, i.e., is not ill, while this patient is clinically diag- with Alzheimer ’s disease (Figure 4). nosed with Alzheimer’s disease (Figure 4). Figure 4. Recognition results for CON group of patients in the “Class-objects” mode in the system Figure 4. Recognition results for CON group of patients in the “Class-objects” mode in the system “EIDOS-Xpro”. “EIDOS-Xpro”. Other subjects (Table 4) were excluded in the same way. Other subjects (Table 4) were excluded in the same way. Table 4. Subjects excluded from conflicting examples . Table 4. Subjects excluded from conflicting examples. Encoded Encoded Encoded Encoded Encoded Encoded Encoded Encoded Subjects Group Comments Subjects Group Comments L_BV L_TV R_BV R_TV L_BV L_TV R_BV R_TV The similarity of the subject 007 to CON group is 99.69%, The similarity of the subject 007 to CON group is 99.69%, but sub00 sub0077 AD AD 1 1 1 1 1 1 2 2 in fact is sick but in fact is sick Subject 004 is sick, but the similarity to the CON group is Subject 004 is sick, but the similarity to the CON group is 12.94%, also the subject 080, which has the same vector sub004, sub006 AD 2 1 2 1 sub004, 12.94%, also the subject 080, which has the same vector {2; {2; 1; 2; 1} and is in fact in the CON group, has a similarity to AD 2 1 2 1 the CON group value 38.61% sub006 1; 2; 1} and is in fact in the CON group, has a similarity to The similarity of the subject 059 to the CON group is 12.94%, the CON group value 38.61% also the subjects 070 and 072 which have the same vector sub042, sub059 AD 2 1 2 2 sub042, The si {2; milari 1; 2; 2} ty of t and ar he s e inubject fact in 0 the 59 to t CONhe group, COhave N gro similarity up is to AD 2 1 2 2 the CON group value 62.66% and 38.67%, respectively sub059 12.94%, also the subjects 070 and 072 which have the The similarity of the subject 060 to the AD group is 60.01%, sub060 CON 2 3 2 3 but in fact is not sick The similarity of the subject 063 to the AD group is 19.15%, sub063, sub073 CON 2 2 3 2 but in fact is not sick Algorithms 2023, 16, 219 11 of 25 Table 4. Cont. Encoded Encoded Encoded Encoded Subjects Group Comments L_BV L_TV R_BV R_TV The similarity of the subject 031 to the CON group is 50.23%, sub031 AD 2 2 3 2 but in fact is sick The similarity of the subjects 078 and 068 to the AD group are sub068, sub078 CON 2 3 3 3 63.63% and 39.33%, respectively, but in fact they are not sick The similarity of the subject 079 to the CON group is 10.03%, sub079 CON 3 1 2 1 but the subject 001, which has the same vector {3; 1; 2; 1} and in the AD group, has similarity to the AD group value 28.73% The similarity of both subjects 071 and 081 to the AD group is sub071, sub081 CON 3 2 2 2 33.62%, but in fact they are not sick Subjects 057, 003, 076 and 075 were excluded from the encoded dataset because the vectors {2; 1; 3; 1}, {2; 2; 1; 3} corresponding to them have uncertainty. The training set, after excluding the conflict examples, includes 34 unique, encoded vectors. They corresponded to 63 subjects (patients), which accounted for 78% of the original number of subjects. This training set for identification of the integrated rating mechanism (decision’s root) is presented in Table S1. 3.2.1. The Integrated Rating Mechanism (Decisions’ Roots) Identified Based on the Encoded Training Set In the previous study [40], we used the identification algorithm proposed in [47], which is applicable to any structure of criteria trees without restrictions on the alphabet used inside the convolution matrix. Removing the restriction on the alphabet used to encode the elements of the convolution matrices, it is possible to approximate the initial data with several functions due to the fact that when gluing equivalence groups, the variability of these gluings arises [48]. As was said above, 15 full binary trees with named leaves are possible for 4 factors. From all possible tree structures in [40], it was proposed to convolve to produce indicators characterizing the left veins with each other, as well as the right ones with each other. This is because veins are paired and participate in the process of blood circulation of the two hemispheres separately and because arteries and veins in each of the hemispheres, and in the brain as a whole, have many collaterals that redistribute blood flow both among Algorithms 2023, 16, x FOR PEER REVIEW 13 of 27 themselves at the beginning, within one hemisphere, and then, with a lack of compensatory capabilities within the entire brain (Figure 5). L_BV L_TV R_BV R_TV Figure 5. Criteria tree structure of convolution for the basal and thalamic. Figure 5. Criteria tree structure of convolution for the basal and thalamic. For each criteria tree structure, a few decisions’ roots can be identified ; one example For each criteria tree structure, a few decisions’ roots can be identified; one example was presented in [40] (p. 426). The following decision’s root (Figure 5) was identified based was presented in [40] (p. 426). The following decision’s root (Figure 5) was identified based in the training set presented in Table S2. in the training set presented in Table S2. The founded decision root (see Figure 6) can be used to predefine architecture of the desired neural network. CON AD АD CON CON CON / AD АD АD CON / АD AD CON AD 111 111 2 3 4 1 2 2 2 3 3 4 4 4 L_BV L_TV R_BV R_TV Figure 6. Decision’s root for diagnosing Alzheimer’s disease. The values in the left matrix at the lower level are the average numbers of the rows of the root matrix, the values in the right matrix are the numbers of the columns of the root matrix, respectively. 3.2.2. Neural Network Architecture Selection Using algorithm [47] (p. 402), we can predefine the neurons number of the desired neural network based on the decisions’ root (see, Figure 6). The first hidden layer has 12 neurons, because four input variables are indicated on Set 3 by encoded discrete values {1; 2; 3}. For both convolution matrices from the bottom level of the decisions’ root s, there are nine neurons on the second hidden layer, because matrices have nine elements. Matrix L R M has four values, M has three values, so the third hidden layer has seven neurons. The fourth hidden layer has 12 neurons because root matrix M has 12 elements. The fifth hid- den layer has two neurons because root matrix M has values from set {CON; AD}. Thus, we have obtained a neural network with five hidden layers and 56 neurons on all layers (including one input and one output layer ). The neural network architecture is shown in Figure 7, and the parameters of this network are presented in Table 5. Algorithms 2023, 16, x FOR PEER REVIEW 13 of 27 L R L_BV L_TV R_BV R_TV Figure 5. Criteria tree structure of convolution for the basal and thalamic. Algorithms 2023, 16, 219 12 of 25 For each criteria tree structure, a few decisions’ roots can be identified ; one example was presented in [40] (p. 426). The following decision’s root (Figure 5) was identified based in the training set presented in Table S2. The founded decision root (see Figure 6) can be used to predefine architecture of the The founded decision root (see Figure 6) can be used to predefine architecture of the desi desir red ne ed neural ural netw network. ork. CON AD АD CON CON CON / AD АD АD CON / АD AD CON AD 111 111 1 2 2 2 3 4 2 3 3 4 4 4 L_BV L_TV R_BV R_TV Figure 6. Decision’s root for diagnosing Alzheimer’s disease. The values in the left matrix at the Figure 6. Decision’s root for diagnosing Alzheimer ’s disease. The values in the left matrix at the lower level are the average numbers of the rows of the root matrix, the values in the right matrix lower level are the average numbers of the rows of the root matrix, the values in the right matrix are are the numbers of the columns of the root matrix, respectively. the numbers of the columns of the root matrix, respectively. 3.2.2. Neural Network Architecture Selection 3.2.2. Neural Network Architecture Selection Using algorithm [47] (p. 402), we can predefine the neurons number of the desired Using algorithm [47] (p. 402), we can predefine the neurons number of the desired neural network based on the decisions’ root (see, Figure 6). The first hidden layer has 12 neural network based on the decisions’ root (see, Figure 6). The first hidden layer has neurons, because four input variables are indicated on Set 3 by encoded discrete values 12 neurons, because four input variables are indicated on Set 3 by encoded discrete values {1; 2; 3}. For both convolution matrices from the bottom level of the decisions’ root s, there {1; 2; 3}. For both convolution matrices from the bottom level of the decisions’ roots, there are nine neurons on the second hidden layer, because matrices have nine elements. Matrix L R M are has nine four neur values ons , on M the hassecond three value hidden s, so th layer e th,ird because hidden matrices layer has have sevennine neurelements. ons. The Matrix L R L fourth hidden layer has 12 neurons because root matrix M has 12 elements. The fifth hid- M has four values, M has three values, so the third hidden layer has seven neurons. The den layer has two neurons because root matrix M has values from set {CON; AD}. Thus, fourth hidden layer has 12 neurons because root matrix M has 12 elements. The fifth we have obtained a neural network with five hidden layers and 56 neurons on all layers hidden layer has two neurons because root matrix M has values from set {CON; AD}. (including one input and one output layer ). The neural network architecture is shown in Thus, we have obtained a neural network with five hidden layers and 56 neurons on all Algorithms 2023, 16, x FOR PEER REVIEW 14 of 27 Figure 7, and the parameters of this network are presented in Table 5. layers (including one input and one output layer). The neural network architecture is shown in Figure 7, and the parameters of this network are presented in Table 5. m ∊ M m ∊ M m ∊ M L_BV = 1 m ∊ M 21 m ∊ M L_BV = 2 L m ∊ M m ∊ M st 1 row L_BV = 3 L m ∊ M 23 m ∊ M nd S L_TV = 1 m ∊ M 2 row m ∊ M L_BV S m ∊ M L_TV = 2 m ∊ M rd 22 3 row Group m ∊ M L_TV m ∊ M L_TV = 3 33 23 CON th 4 row S R m ∊ M m ∊ M 31 R_BV = 1 11 R_BV Group st m ∊ M m ∊ M 1 column 32 R_BV = 2 12 AD R_TV S m ∊ M m ∊ M 33 13 nd R_BV = 3 2 column m ∊ M m ∊ M 41 R_TV = 1 rd S 3 column m ∊ M m ∊ M R_TV = 2 S m ∊ M m ∊ M 43 R_TV = 3 R m ∊ M m ∊ M m ∊ M Input Hidden Layers Output Figure Figure 7. 7. Ne Neur urons ons of of artificial artificial neural neural network network (wher (wher e: vec e: tor vector of neuof ron neur s on ons the Inpu on the t Lay Input er ∊ R Layer , 2 R , 12 18 12 18 vector of neurons on the Hidden Layer (1) ∊ R , vector of neurons on the Hidden Layer (2) ∊ R , vector of neurons on the Hidden Layer (1) 2 R , vector of neurons on the Hidden Layer (2) 2 R , 7 12 vector of neurons on the Hidden Layer (3) ∊ R , vector of neurons on the Hidden Layer (4) ∊ R , 2 1 vector of neurons on the Hidden Layer (5) ∊ R , vector of neurons on the Output Layer ∊ R ). Table 5. Numbers of neurons in hidden layers of the desired artificial neural network . Hidden Number Group Comments Layer of Neurons of Neurons 1 12 6 and 6 Neurons corresponds to intervals on input signals Input Neurons corresponds to elements of matrices M and 4 signals 2 18 9 and 9 L R 3 7 4 and 3 Neurons corresponds to values of matrices M and M Output 4 12 12 Neurons corresponds to elements of matrix M 1 signal 5 2 2 Neurons corresponds to values of matrix M The synaptic connections (Figure 8) are determined in full accordance with the ele- ments of convolutional matrices in the decisions’ roots (see, Figure 6). Algorithms 2023, 16, 219 13 of 25 7 12 vector of neurons on the Hidden Layer (3) 2 R , vector of neurons on the Hidden Layer (4) 2 R , 2 1 vector of neurons on the Hidden Layer (5) 2 R , vector of neurons on the Output Layer 2 R ). Table 5. Numbers of neurons in hidden layers of the desired artificial neural network. Number of Group of Hidden Layer Comments Neurons Neurons Input 1 12 6 and 6 Neurons corresponds to intervals on input signals L R 4 signals 2 18 9 and 9 Neurons corresponds to elements of matrices M and M L R 3 7 4 and 3 Neurons corresponds to values of matrices M and M Output 4 12 12 Neurons corresponds to elements of matrix M 1 signal 5 2 2 Neurons corresponds to values of matrix M Algorithms 2023, 16, x FOR PEER REVIEW 15 of 27 The synaptic connections (Figure 8) are determined in full accordance with the ele- ments of convolutional matrices in the decisions’ roots (see, Figure 6). X x , x , x , x Y 1 2 3 4 Input Output Hidden Layers Figure 8. Incompletely connected artificial neural network identified based on the decision root. Figure 8. Incompletely connected artificial neural network identified based on the decision root. Numbers from 1 to 5 designations of serial numbers of hidden layers of the neural network. Numbers from 1 to 5 designations of serial numbers of hidden layers of the neural network. The total number of unknown parameters (synaptic coefficients) of the incompletely The total number of unknown parameters (synaptic coefficients ) of the incompletely connected neural network predefined, based on the decision’s root is 158. connected neural network predefined , based on the decision’s root is 158. The structure of the predefined neural network model had 12 connections between The structure of the predefined neural network model had 12 connections between the input signals and the first hidden layer (each connection generates two parameters of the input signals and the first hidden layer (each connection generates two parameters of the mathematical model, which means that we have 24 parameters in total). the mathematical model, which means that we have 24 parameters in total). The next layer, in which the neurons correspond to the elements of the matrix, has The next layer, in which the neurons correspond to the elements of the matrix, has three parameters each (two inputs from the previous layer and one unknown constant). three parameters each (two inputs from the previous layer and one unknown constant). Since we have two matrices of nine elements each at the bottom level of the criteria tree, this Since we have two matrices of nine elements each at the bottom level of the criteria tree, generates 18 neurons on the second layer, which means a total of 54 unknown parameters. this generates 18 neurons on the second layer, which means a total of 54 unknown param- The next layer consists of the neurons corresponding to the matrix estimates, and eters. there are exactly as many connections as there were neurons on the previous layer, i.e., in The next layer consists of the neurons corresponding to the matrix estimates, and our case, 18, but we need to add seven parameters to them (determined by the number there are exactly as many connections as there were neurons on the previous layer, i.e., in of neurons on the 3rd layer), which also means we have eighteen and seven parameters our case, 18, but we need to add seven parameters to them (determined by the number of (25 parameters on the third layer). neurons on the 3rd layer ), which also means we have eighteen and seven parameters (25 Then, there is a matrix containing 12 elements, where each element corresponds to parameters on the third layer ). three parameters (two inputs and one unknown constant). It means 36 parameters on the Then, there is a matrix containing 12 elements, where each element corresponds to 4th layer. three parameters (two inputs and one unknown constant). It means 36 parameters on the 4th layer. There are two neurons on the last hidden layer, where 14 input signals (10 neurons on the previous layer have one signal and two neurons have two signals; the last ones correspond to elements in the root matrix with value CON/AD, see Figure 6) plus two unknown constants (the number of neurons on the current layer ) are taken. There is only one signal at the output. This signal is associated with the last two neu- rons. It means there are three parameters (two inputs plus one unknown constant). Thus, the total number of unknown parameters is determined by the sum (24 + 54 + 25 + 36 + 16 + 3 = 158). Nowadays, it is worth noting that there are no publicly available specialized software products allowing the user to set the neural network structure, for example, to load the adjacency matrix or the incidence matrix of the graph corresponding to the neural net- work architecture. We tried to use the computer program “Software package that imple- ments the operation of incompletely connected neural networks” [57]. It implemented in- completely connected neural networks only for one hidden layer, but as you can see from Algorithms 2023, 16, 219 14 of 25 There are two neurons on the last hidden layer, where 14 input signals (10 neurons on the previous layer have one signal and two neurons have two signals; the last ones correspond to elements in the root matrix with value CON/AD, see Figure 6) plus two unknown constants (the number of neurons on the current layer) are taken. There is only one signal at the output. This signal is associated with the last two neurons. It means there are three parameters (two inputs plus one unknown constant). Thus, the total number of unknown parameters is determined by the sum (24 + 54 + 25 + 36 + 16 + 3 = 158). Nowadays, it is worth noting that there are no publicly available specialized software products allowing the user to set the neural network structure, for example, to load the Algorithms 2023, 16, x FOR PEER REVIEW 16 of 27 adjacency matrix or the incidence matrix of the graph corresponding to the neural network architecture. We tried to use the computer program “Software package that implements the operation of incompletely connected neural networks” [57]. It implemented incompletely connected neural networks only for one hidden layer, but as you can see from Figure 8, the Figure 8, the required neural network has incompletely connected neurons on each hid- required neural network has incompletely connected neurons on each hidden layer. Thus, den layer. Thus, this program is not applicable for this case. this program is not applicable for this case. Though there are no publicly available specialized software products allowing the Though there are no publicly available specialized software products allowing the user to set the structure of the neural network, we can find fully connected neural net- user to set the structure of the neural network, we can find fully connected neural networks works comparable in terms of the number of unknown parameters and compare the ef- comparable in terms of the number of unknown parameters and compare the effectiveness fectiveness of diagnosing Alzheimer’s disease knowing the number of unknown parame- of diagnosing Alzheimer ’s disease knowing the number of unknown parameters that will ters that will be determined in the process of training the neural network. be determined in the process of training the neural network. Therefore, a fully connected single-layer neural network (Figure 9) with four input Therefore, a fully connected single-layer neural network (Figure 9) with four input signals and one output signal has the following rule—each neuron on the hidden layer signals and one output signal has the following rule—each neuron on the hidden layer corresponds to four input-unknown variables corresponding to the input signals and one corresponds to four input-unknown variables corresponding to the input signals and one unknown constant (i.e., 5x unknown parameters); the output signal receives signals unknown constant (i.e., 5x unknown parameters); the output signal receives signals from x from x neurons at the input and has one unknown constant. neurons at the input and has one unknown constant. Hidden layer 1 X = x , x , x , x 1 2 3 4 4xx + + x+ 1 Figure 9. Schematic representation of a single-layer neural network structure with x neurons and Figure 9. Schematic representation of a single-layer neural network structure with x neurons and numbers of unknown parameters. numbers of unknown parameters. Thus, the number of unknown parameters in a fully connected network with one Thus, the number of unknown parameters in a fully connected network with one hidden layer can be expressed using the Equation (14) hidden layer can be expressed using the Equation (14) 4x x 1 158, x N (13) 4x + x + 1 158, x 2 N (13) According to (3), the minimum number of neurons on the hidden layer for a single- According to (3), the minimum number of neurons on the hidden layer for a single- layer network is 27, where this number is comparable to the number of unknown param- layer network is 27, where this number is comparable to the number of unknown parame- eters with the found incompletely connected neural network (see Figure 8). ters with the found incompletely connected neural network (see Figure 8). The following equation will be obtained for a two-layer neural network structure The following equation will be obtained for a two-layer neural network structure (Figure 10): (Figure 10): 5x xy 2y 1, x, y N 5x + xy + 2y + 1, x, y 2 N (14) (14) Hidden layer 1 Hidden layer 2 X = x , x , x , x 1 2 3 4 + xy+ y + y+ 1 4xx + Algorithms 2023, 16, x FOR PEER REVIEW 16 of 27 Figure 8, the required neural network has incompletely connected neurons on each hid- den layer. Thus, this program is not applicable for this case. Though there are no publicly available specialized software products allowing the user to set the structure of the neural network, we can find fully connected neural net- works comparable in terms of the number of unknown parameters and compare the ef- fectiveness of diagnosing Alzheimer’s disease knowing the number of unknown parame- ters that will be determined in the process of training the neural network. Therefore, a fully connected single-layer neural network (Figure 9) with four input signals and one output signal has the following rule—each neuron on the hidden layer corresponds to four input-unknown variables corresponding to the input signals and one unknown constant (i.e., 5x unknown parameters); the output signal receives signals from x neurons at the input and has one unknown constant. Hidden layer 1 X = x , x , x , x Y 1 2 3 4 4xx + + x+ 1 Figure 9. Schematic representation of a single-layer neural network structure with x neurons and numbers of unknown parameters. Thus, the number of unknown parameters in a fully connected network with one hidden layer can be expressed using the Equation (14) 4x x 1 158, x N (13) According to (3), the minimum number of neurons on the hidden layer for a single- layer network is 27, where this number is comparable to the number of unknown param- eters with the found incompletely connected neural network (see Figure 8). The following equation will be obtained for a two-layer neural network structure (Figure 10): Algorithms 2023, 16, 219 15 of 25 5x xy 2y 1, x, y N (14) Hidden layer 1 Hidden layer 2 X = x , x , x , x 1 2 3 4 Algorithms 2023, 16, x FOR PEER REVIEW 17 of 27 4xx + + xy+ y + y+ 1 Figure 10. Schematic representation of the structure of a fully connected two-layer neural network and the number of unknown parameters. Figure 10. Schematic representation of the structure of a fully connected two-layer neural network and the number of unknown parameters. These minimal integer values x were found by changing the number of neurons on the second hidden layer y, starting from one, subject to the fulfillment of inequality (4). These minimal integer values x were found by changing the number of neurons on These values determine the number of neurons on the fir st hidden layer (Figure 11). Thus, the second hidden layer y, starting from one, subject to the fulfillment of inequality (4). various two-layer neural networks were found, where they are comparable in terms of the These values determine the number of neurons on the first hidden layer (Figure 11). Thus, number of unknown neural network parameters (this network is built by the decision’s various two-layer neural networks were found, where they are comparable in terms of the root number ) (see of Fig unknown ure 6). neural network parameters (this network is built by the decision’s root) (see Figure 6). 0 5 10 15 20 25 30 35 40 45 50 55 Figure Figure11. 11. The The de dependence pendence of ofthe thenu number mber of ofneu neur ron ons s betw between een the thetwo two la layers yers (x (— x—first first hid hidden den laylayer er, , y—second hidden layer ). y—second hidden layer). 3.2.3. Results of Neural Network Training 3.2.3. Results of Neural Network Training According to the generally accepted neural network design technology, the entire According to the generally accepted neural network design technology, the entire general population is divided into training, testing and confirming sets in the ratio of general population is divided into training, testing and confirming sets in the ratio of 70:20:10% [52]. In our problem, we neglect a strict formal approach to determining the 70:20:10% [52]. In our problem, we neglect a strict formal approach to determining the percentage ratio between training, testing and confirming sets due to a fairly small number percentage ratio between training, testing and conr fi ming sets due to a fairly small num- of examples, which is acceptable [63], thereby dividing the entire initial set only into ber of examples, which is acceptable [63], thereby dividing the entire initial set only into training and testing in the ratio of 90:10%. training and testing in the ratio of 90:10%. We performed multiple training and testing of neural networks for diagnosing Alzheimer’s We performed multiple training and testing of neural networks for diagnosing Alz- disease on limited raw data using one-layer and two-layer neural networks. Several results heimer’s disease on limited raw data using one-layer and two-layer neural networks. Sev- of one-layer neural networks training and testing are shown below (Figure 12, Table 6). eral results of one-layer neural networks training and testing are shown below (Figure 12, Table 6). 30 30 10 10 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 -10 -10 Train Error Test Error Train Error Test Error Iterations Iterations (a) (d) 50 50 30 30 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 -10 -10 Train Error Test Error Iterations Train Error Test Error Iterations uadratic error uadratic error uadratic error uadratic error Algorithms 2023, 16, x FOR PEER REVIEW 17 of 27 Figure 10. Schematic representation of the structure of a fully connected two-layer neural network and the number of unknown parameters. These minimal integer values x were found by changing the number of neurons on the second hidden layer y, starting from one, subject to the fulfillment of inequality (4). These values determine the number of neurons on the first hidden layer (Figure 11). Thus, various two-layer neural networks were found, where they are comparable in terms of the number of unknown neural network parameters (this network is built by the decision’s root) (see Figure 6). 0 5 10 15 20 25 30 35 40 45 50 55 Figure 11. The dependence of the number of neurons between the two layers (x—first hidden layer, y—second hidden layer ). 3.2.3. Results of Neural Network Training According to the generally accepted neural network design technology, the entire general population is divided into training, testing and confirming sets in the ratio of 70:20:10% [52]. In our problem, we neglect a strict formal approach to determining the percentage ratio between training, testing and confirming sets due to a fairly small num- ber of examples, which is acceptable [63], thereby dividing the entire initial set only into Algorithms 2023, 16, 219 training and testing in the ratio of 90:10%. 16 of 25 We performed multiple training and testing of neural networks for diagnosing Alz (a) (d) Algorithms 2023, 16, x FOR PEER REVIEW 18 of 27 (b) (e) (c) (f) Figure 12. Changing of the square errors at training and testing depending on iterations of calcula- Figure 12. Changing of the square errors at training and testing depending on iterations of calculation tion synaptic coefficients at fully connected one-layer neural networks with different numbers of synaptic coefficients at fully connected one-layer neural networks with different numbers of neurons neurons on the hidden layer (hereinafter the following designation will be used HL1 stands for on the hidden layer (hereinafter the following designation will be used HL1 stands for neural network neural network with 1 hidden layer): HL1-5 has 5 neurons (a); HL1-10 has 10 neurons (b); HL1-15 with 1 hidden layer): HL1-5 has 5 neurons (a); HL1-10 has 10 neurons (b); HL1-15 has 15 neurons (c); has 15 neurons (c); HL1-20 has 20 neurons (d), HL1-25 has 25 neurons (e), and fully connected one- HL1-20 has 20 neurons (d), HL1-25 has 25 neurons (e), and fully connected one-layer neural network layer neural network with predefined number of neurons HL1-27 has 27 neurons (f). with predefined number of neurons HL1-27 has 27 neurons (f). Figure 12a–e shows five neural networks, distinguished by five neurons. Figure 12f Table 6. Results of a trained different fully connected one-layer neural networks. shows the training and testing errors for the neural network HL1-27 with the 27 neurons satisfying inequality (14). Figure 12 shows that the quadratic error becomes less than 10% Neural Network Q Q Q Q Q Q Q Q 1 2 3 4 5 6 7 8 in all cases. Therefore, we should compare the results of the training with the other crite- HL1-5 1 layer with 5 neurons 17.068 12.582 24.517 21.047 2.913 6.011 0.849 0.744 ria: Q1–average quadratic relative (%); Q2–average relative error; Q3–average quadratic HL1-10 1 layer with 10 neurons 11.557 8.718 17.329 13.899 1.334 3.003 0.930 0.872 relative (Test, %); Q4–average relative error (Test); Q5–quadratic train error; Q6–quadratic HL1-15 1 layer with 15 neurons 9.268 7.000 21.897 18.348 0.859 4.795 0.955 0.795 2 2 test error; Q7–R train set; Q8–R test set (Table 6). HL1-20 1 layer with 20 neurons 5.829 4.422 21.262 18.988 0.034 4.521 0.982 0.807 HL1-25 1 layer with 25 neurons Comparing 2.047 the perfo 1.440 rman 21.827 ce by all the c 16.692 riteria 0.042 in Table 6, 4.764 it can be 0.997 seen that the 0.797 pro- HL1-27 1 layer with 27 neurons 1.987 1.392 21.138 14.462 0.039 4.468 0.997 0.809 posed neural network HL1-27 is bett er on almost all the criteria. At the same time, the best neural network training results from the multiple experiments we showed on both Figure 12a–e a Figur nd e the 12a–e first sev shows en ro five ws of T neural able networks, 6. distinguished by five neurons. Figure 12f In our experiments, we noticed that the quality and speed of neural network learning shows the training and testing errors for the neural network HL1-27 with the 27 neurons satisfying could be improv inequality ed by incr (14). Figur easing the number e 12 shows that of neur the quadratic ons. The neur erroral networ becomes k with prede- less than 10% in fined number all cases. Ther of neurons efore, we should HL1-27 compar (see Figur e the e 12f results ) waof s trained the training rather quic with the kly, other and the re criteria: - Q –average quadratic relative (%); Q –average relative error; Q –average quadratic relative sults were more robust compared to variants HL1-5, HL1-10, HL1-15, HL1-20, which often 1 2 3 (Test, %); Q –average relative error (Test); Q quadratic train error; Q –quadratic test error; over-trained the neural network. 4 5– 6 2 2 Q –R train set; Q –R test set (Table 6). 7 8 Table 6. Comparing Results of a traine the performance d different fully by all connected one-la the criteria y in er neural net Table 6, it wcan orks. be seen that the proposed neural network HL1-27 is better on almost all the criteria. At the same time, the Neural Network Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 best neural network training results from the multiple experiments we showed on both HL1-5 1 layer with 5 neurons 17.068 12.582 24.517 21.047 2.913 6.011 0.849 0.744 Figure 12a–e and the first seven rows of Table 6. HL1-10 1 layer with 10 neurons 11.557 8.718 17.329 13.899 1.334 3.003 0.930 0.872 HL1-15 1 layer with 15 neurons 9.268 7.000 21.897 18.348 0.859 4.795 0.955 0.795 HL1-20 1 layer with 20 neurons 5.829 4.422 21.262 18.988 0.034 4.521 0.982 0.807 HL1-25 1 layer with 25 neurons 2.047 1.440 21.827 16.692 0.042 4.764 0.997 0.797 HL1-27 1 layer with 27 neurons 1.987 1.392 21.138 14.462 0.039 4.468 0.997 0.809 We propose to consider nine variations of two-layer neural networks with different numbers of neurons on first and second hidden layers. The network structures are chosen so that the number of neurons on the first and the second hidden layer form a cross of gridlines with steps of five neurons—selected combinations of neuron number on the first and second layers are highlighted by black points (Figure 13, points 1–9). We also propose to consider several variations of neural networks whose structures are comparable to dependence (see, Figure 11). Selected eight combinations of neuron number on first and second are highlighted by red points (Figure 13, points a–h). This is necessary to assess the quality and ability to recognize and predict our neural network, the structure of which is determined according to the decision’s root and convolution ma- trices. Algorithms 2023, 16, 219 17 of 25 Algorithms 2023, 16, x FOR PEER REVIEW 19 of 27 In our experiments, we noticed that the quality and speed of neural network learning could be improved by increasing the number of neurons. The neural network with pre- defined number of neurons HL1-27 (see Figure 12f) was trained rather quickly, and the the structure of which is determined according to the decision’s root and convolution ma- results were more robust compared to variants HL1-5, HL1-10, HL1-15, HL1-20, which trices. often over-trained the neural network. The dotted line in Figure 13 divides all neural networks into two types: in the upper We propose to consider nine variations of two-layer neural networks with different left corner, there are neural networks with fewer neurons on the rst fi layer than on the numbers of neurons on first and second hidden layers. The network structures are chosen second layer. In the lower right area, on the contrary, there are neural networks with more so that the number of neurons on the first and the second hidden layer form a cross of neurons on the first layer than on the second layer. gridlines with steps of five neurons—selected combinations of neuron number on the first and second layers are highlighted by black points (Figure 13, points 1–9). (1) - HL2-5-5 (a) - HL2-22-2 (a) 19 (2) - HL2-5-10 (b) - HL2-17-4 (3) - HL2-5-15 (c) - HL2-14-6 (4) - HL2-10-5 (d) - HL2-10-10 (b) 15 (5) - HL2-10-10 (e) - HL2-6-17 (7) (9) (8) (6) - HL2-10-15 (f) - HL2-3-29 (c) (7) - HL2-15-5 (g) - HL2-2-41 (8) - HL2-15-15 8 (h) - HL2-1-51 (6) (4) (9) - HL2-15-15 (d) (5) (e) (1) (2) (3) (f) (g) (h) 0 5 10 15 20 25 30 35 40 45 50 55 Figure 13. Selected two-layers neural networks with different numbers of neurons on first and sec- Figure 13. Selected two-layers neural networks with different numbers of neurons on first and second ond hidden layers (hereinafter the following designation will be used–HL2 stands for two hidden hidden layers (hereinafter the following designation will be used–HL2 stands for two hidden layers, layers, the next number after the hyphen stands for the number of neurons on the first hidden l ayer, the next number after the hyphen stands for the number of neurons on the first hidden layer, and the and the last number stands for the number of neurons on the second hidden layer, respectively). last number stands for the number of neurons on the second hidden layer, respectively). The network HL2-10-10 refers to both a set of nine neural networks (see Figure 13, We also propose to consider several variations of neural networks whose structures point 5) chosen by varying the number of variables on the first and second layer of five are comparable to dependence (see, Figure 11). Selected eight combinations of neuron neurons (see Figure 13, black points 1–9), and a set of eight predefined neural networks number on first and second are highlighted by red points (Figure 13, points a–h). This is (Figure 13, point d) that contain a number of synaptic coefficients comparable to a de ci- necessary to assess the quality and ability to recognize and predict our neural network, the sion-root-based neural network (see Figure 8). structure of which is determined according to the decision’s root and convolution matrices. Therefore, out of nine neural networks, Figure 14 and Table 7 show the training and The dotted line in Figure 13 divides all neural networks into two types: in the upper testing results of eight networks: HL2-5-5-5 (Figure 14a); HL2-5-10 (Figure 14b); HL2-5-15 left corner, there are neural networks with fewer neurons on the first layer than on the (Figure 14c); HL2-10-5 (Figure 14d); HL2-10-15 (Figure 14e); HL2-15-5 (Figure 14f); HL2- second layer. In the lower right area, on the contrary, there are neural networks with more 15-10 (Figure 14g); and HL2-15-15 (Figure 14h). neurons on the first layer than on the second layer. The network HL2-10-10 refers to both a set of nine neural networks (see Figure 13, point 5) chosen by varying the number of variables on the first and second layer of five neurons (see Figure 13, black points 1–9), and a set of eight predefined neural networks 50 50 (Figure 13, point d) that contain a number of synaptic coefficients comparable to a decision- 30 30 root-based neural network (see Figure 8). Therefore, out of nine neural networks, Figure 14 and Table 7 show the training and 10 10 testing results of eight networks: HL2-5-5-5 (Figure 14a); HL2-5-10 (Figure 14b); HL2- 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 -10 -10 5-15 (Figure 14c); HL2-10-5 (Figure 14d); HL2-10-15 (Figure 14e); HL2-15-5 (Figure 14f); Train Error Test Error Iterations Train Error Test Error Iterations HL2-15-10 (Figure 14g); and HL2-15-15 (Figure 14h). (a) (e) 70 70 30 30 10 10 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 -10 -10 Train Error Test Error Train Error Test Error Iterations Iterations uadratic error uadratic error uadratic error uadratic error Algorithms 2023, 16, x FOR PEER REVIEW 19 of 27 The dott ed line in Figure 13 divides all neural networks into two types: in the upper left corner, there are neural networks with fewer neurons on the first layer than on the second layer. In the lower right area, on the contrary, there are neural networks with more neurons on the first layer than on the second layer. Figure 13. Selected two-layers neural networks with different numbers of neurons on first and sec- ond hidden layers (hereinafter the following designation will be used–HL2 stands for two hidden layers, the next number after the hyphen stands for the number of neurons on the first hidden layer, and the last number stands for the number of neurons on the second hidden layer, respectively). The network HL2-10-10 refers to both a set of nine neural networks (see Figure 13, point 5) chosen by varying the number of variables on the first and second layer of five neurons (see Figure 13, black points 1–9), and a set of eight predefined neural networks (Figure 13, point d) that contain a number of synaptic coefficients comparable to a deci- sion-root-based neural network (see Figure 8). Therefore, out of nine neural networks, Figure 14 and Table 7 show the training and testing results of eight networks: HL2-5-5-5 (Figure 14a); HL2-5-10 (Figure 14b); HL2-5-15 Algorithms 2023, 16, 219 18 of 25 (Figure 14c); HL2-10-5 (Figure 14d); HL2-10-15 (Figure 14e); HL2-15-5 (Figure 14f); HL2- 15-10 (Figure 14g); and HL2-15-15 (Figure 14h). (a) (e) Algorithms 2023, 16, x FOR PEER REVIEW 20 of 27 (b) (f) (c) (g) (d) (h) Figure 14. Changing of the square errors at training and testing depending on iterations of calcula- Figure 14. Changing of the square errors at training and testing depending on iterations of calcula- tion synaptic coefficients at two-layers neural networks with different numbers of neurons on the tion synaptic coefficients at two-layers neural networks with different numbers of neurons on the hidden layers: HL2-5-5 (a); HL2-5-10 (b); HL2-5-15 (c); HL2-10-5 (d); HL2-10-15 (e); HL2-15-5 (f); hidden layers: HL2-5-5 (a); HL2-5-10 (b); HL2-5-15 (c); HL2-10-5 (d); HL2-10-15 (e); HL2-15-5 (f); HL2-15-10 (g); HL2-15-15 (h). HL2-15-10 (g); HL2-15-15 (h). Figure 14 shows the networks HL2-10-5 (Figure 14d); HL2-15-5 (Figure 14f ) and HL2- Table 7. Results of a trained different fully connected two-layer neural networks. 15-10 (Figure 14g) required fewer iterations to training. Neural Network Q Q Q Q Q Q Q Q 1 2 3 4 5 6 7 8 Table 7. Results of a trained different fully connected two-layer neural networks. 2 layers with 5 neurons on first layer and HL2-5-5 4.132 2.755 16.518 12.429 0.145 2.210 0.996 0.884 5 neurons on second layer Neural Network Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 2 layers with 5 neurons on first layer and HL2-5-10 4.003 2.922 17.234 11.922 0.160 2.970 0.992 0.873 2 layers 10 wit neur hons 5 ne onurons second on layer first layer HL2-5-5 4.132 2.755 16.518 12.429 0.145 2.210 0.996 0.884 2 layers with 5 neurons on first layer and and 5 neurons on second layer HL2-5-15 4.581 3.163 17.720 15.341 0.210 3.140 0.989 0.866 15 neurons on second layer 2 layers with 5 neurons on first layer 2 layers with 10 neurons on first layer and HL2-10-5 0.715 0.480 13.094 8.362 0.005 1.714 0.999 0.934 HL2-5-10 4.003 2.922 17.234 11.922 0.160 2.970 0.992 0.873 5 neurons on second layer and 10 neurons on second layer 2 layers with 10 neurons on first layer and HL2-10-15 0.775 0.458 16.823 9.647 0.006 2.830 0.999 0.888 2 layers with 5 neurons on first layer 15 neurons on second layer HL2-5-15 4.581 3.163 17.720 15.341 0.210 3.140 0.989 0.866 2 layers and 15 ne with 15 uro neur ns on second ons on first layer layer and HL2-15-5 0.665 0.479 9.918 7.110 0.004 0.797 0.999 0.958 5 neurons on second layer 2 layers with 10 neurons on first layer 2 layers with 15 neurons on first layer and HL2-10-5 0.715 0.480 13.094 8.362 0.005 1.714 0.999 0.934 HL2-15-10 0.904 0.548 14.145 10.509 0.007 1.621 0.999 0.915 and 5 neurons on second layer 10 neurons on second layer 2 layers with 15 neurons on first layer and 2 layers with 10 neurons on first layer HL2-15-15 1.010 0.654 12.276 9.647 0.008 1.221 0.999 0.936 15 neurons on second layer HL2-10-15 0.775 0.458 16.823 9.647 0.006 2.830 0.999 0.888 and 15 neurons on second layer 2 layers with 15 neurons on first layer HL2-15-5 0.665 0.479 9.918 7.110 0.004 0.797 0.999 0.958 and 5 neurons on second layer 2 layers with 15 neurons on first layer HL2-15-10 0.904 0.548 14.145 10.509 0.007 1.621 0.999 0.915 and 10 neurons on second layer 2 layers with 15 neurons on first layer HL2-15-15 1.010 0.654 12.276 9.647 0.008 1.221 0.999 0.936 and 15 neurons on second layer Table 7 shows that the HL2-15-5 network is bett er than the others by all criteria except Q2–average relative error. The HL2-15-5 network is insignificantly inferior to the HL2-10- 15 network in this criterion. At the same time, the HL2-10-15 network is inferior to other networks in many criteria. It is of interest to compare the training results of randomly selected neural networks and predefined neural networks. Table 8 and Figure 15 show the results of training and testing of eight selected neural networks with predefined number of synaptic coefficients: HL2-22-2 (Figure15a); HL2-17-4 (Figure15b); HL2-14-6 (Figure15c); HL2-10-10 Algorithms 2023, 16, 219 19 of 25 Figure 14 shows the networks HL2-10-5 (Figure 14d); HL2-15-5 (Figure 14f) and HL2-15-10 (Figure 14g) required fewer iterations to training. Table 7 shows that the HL2-15-5 network is better than the others by all criteria except Q –average relative error. The HL2-15-5 network is insignificantly inferior to the HL2-10-15 network in this criterion. At the same time, the HL2-10-15 network is inferior to other networks in many criteria. It is of interest to compare the training results of randomly selected neural networks and predefined neural networks. Table 8 and Figure 15 show the results of training and testing of eight selected neural networks with predefined number of synaptic coefficients: HL2-22-2 (Figure 15a); HL2-17-4 (Figure 15b); HL2-14-6 (Figure 15c); HL2-10-10 (Figure 15d); HL2-6-17 (Figure 15e); HL2-3-29 (Figure 15f); HL2-2-41 (Figure 15g); HL2-1-51 (Figure 15h). Table 8. Results of a trained predefined fully connected two-layer neural networks. Neural Network Q Q Q Q Q Q Q Q 1 2 3 4 5 6 7 8 2 layers with 22 neurons on first layer and HL2-22-2 1.043 0.556 20.344 15.716 0.011 4.139 0.998 0.823 2 neurons on second layer 2 layers with 17 neurons on first layer and HL2-17-4 0.774 0.477 16.971 11.897 0.006 2.888 0.999 0.877 4 neurons on second layer 2 layers with 14 neurons on first layer and HL2-14-6 0.543 0.348 14.759 9.564 0.003 2.178 1.000 0.907 6 neurons on second layer 2 layers with 10 neurons on first layer and HL2-10-10 0.809 0.507 14.293 12.834 0.005 1.655 0.999 0.913 10 neurons on second layer 2 layers with 6 neurons on first layer and HL2-6-17 1.121 0.818 21.026 16.886 0.010 3.581 0.998 0.811 17 neurons on second layer 2 layers with 3 neurons on first layer and HL2-3-29 6.006 3.763 14.313 12.314 0.292 1.659 0.981 0.913 29 neurons on second layer 2 layers with 2 neurons on first layer and HL2-2-41 15.952 6.950 32.393 17.504 2.061 8.499 0.868 0.552 41 neurons on second layer 2 layers with 1 neuron on first layer and HL2-1-51 27.104 16.408 34.519 23.725 5.951 9.652 0.618 0.492 51 neurons on second layer The network HL2-15-5 is close to the predefined networks HL2-17-4 and HL2-14-6 by number of neurons, precisely these two networks with the HL2-10-10 network that have Q values less than one (see Table 8). The HL2-14-6 network has the lowest Q , Q and 1 2 4 Q values (see Table 8). HL2-10-10 has the lowest Q , Q values (see Table 8). In terms 5 3 6 of the coefficient of determination, the network HL2-14-6 has one, which means 100% of the variance of the resultant quantity in the training set is explained by the influence of selected variables. Figure 15 shows that neural networks with a predefined number of neurons required significantly fewer iterations to train as compared to the brute-force selected networks. It is worth recalling again that Figures 14 and 15 show the best results from the number of multiple computational experiments. A smaller number of iterations was required to train the predefined neural networks. Having learnt the results of training two-layer networks, we noticed the following— the quality of neural network models is better if the number of neurons on the first layer is greater than on the second (Figures 14d–g and 15a–d). The results of training and testing two-layer neural networks are not entirely consistent with the observation in the case of single-layer neural networks. In the case of single layer neural networks, we observed the following—as the number of neurons on the hidden layer increases, the quality of neural network models becomes better. With two-layer networks, this is not true, e.g., HL2-22-2 with 24 neurons or HL2-17-4 with 21 neurons are in many ways (see Table 8) worse than the 20-neuron HL2-14-6 and HL2-10-10. Algorithms 2023, 16, x FOR PEER REVIEW 21 of 27 HL2-22-2 (Figure15a); HL2-17-4 (Figure15b); HL2-14-6 (Figure15c); HL2-10-10 (Fig- ure15d); HL2-6-17 (Figure15e); HL2-3-29 (Figure15f); HL2-2-41 (Figure15g); HL2-1-51 (Figure15h) Table 8. Results of a trained predefined fully connected two-layer neural networks. Neural Network Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 2 layers with 22 neurons on first layer HL2-22-2 1.043 0.556 20.344 15.716 0.011 4.139 0.998 0.823 and 2 neurons on second layer 2 layers with 17 neurons on first layer HL2-17-4 0.774 0.477 16.971 11.897 0.006 2.888 0.999 0.877 and 4 neurons on second layer 2 layers with 14 neurons on first layer HL2-14-6 0.543 0.348 14.759 9.564 0.003 2.178 1.000 0.907 and 6 neurons on second layer 2 layers with 10 neurons on first layer HL2-10-10 0.809 0.507 14.293 12.834 0.005 1.655 0.999 0.913 and 10 neurons on second layer 2 layers with 6 neurons on first layer HL2-6-17 1.121 0.818 21.026 16.886 0.010 3.581 0.998 0.811 and 17 neurons on second layer 2 layers with 3 neurons on first layer HL2-3-29 6.006 3.763 14.313 12.314 0.292 1.659 0.981 0.913 and 29 neurons on second layer 2 layers with 2 neurons on first layer HL2-2-41 15.952 6.950 32.393 17.504 2.061 8.499 0.868 0.552 and 41 neurons on second layer 2 layers with 1 neuron on first layer HL2-1-51 27.104 16.408 34.519 23.725 5.951 9.652 0.618 0.492 and 51 neurons on second layer The network HL2-15-5 is close to the predefined networks HL2 -17-4 and HL2-14-6 by number of neurons, precisely these two networks with the HL2-10-10 network that have 1 values less than one (see Table 8). The HL2-14-6 network has the lowest 2, 4 and 5 values (see Table 8). HL2-10-10 has the lowest 3, 6 values (see Table 8). In terms of the coefficient of determination, the network HL2 -14-6 has one, which means 100% of Algorithms 2023, 16, 219 20 of 25 the variance of the resultant quantity in the training set is explained by the influence of selected variables. 110 190 -10 0 100 200 300 400 500 600 700 800 900 1000 -10 0 100 200 300 400 500 600 700 800 900 1000 Train Error Test Error Train Error Test Error Iterations Iterations (a) (e) Algorithms 2023, 16, x FOR PEER REVIEW 22 of 27 0 100 200 300 400 500 600 700 800 900 1000 -10 -10 0 100 200 300 400 500 600 700 800 900 1000 Train Error Test Error Iterations Train Error Test Error Iterations (b) (f) 70 70 50 50 30 30 10 10 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 -10 -10 Train Error Test Error Train Error Test Error Iterations Iterations (c) (g) -10 0 100 200 300 400 500 600 700 800 900 1000 -10 0 100 200 300 400 500 600 700 800 900 1000 Train Error Test Error Train Error Test Error Iterations Iterations (d) (h) Figure 15. Changing of the square errors at training and testing depending on iterations of calcula- Figure 15. Changing of the square errors at training and testing depending on iterations of calcula- tion synaptic coefficients at two -layers neural networks with different numbers of neurons on the tion synaptic coefficients at two-layers neural networks with different numbers of neurons on the hidden layers: HL2-22-2 (a); HL2-17-14 (b); HL2-14-6 (c); HL2-10-10 (d); HL2-6-17 (e); HL2-3-29 (f); hidden layers: HL2-22-2 (a); HL2-17-14 (b); HL2-14-6 (c); HL2-10-10 (d); HL2-6-17 (e); HL2-3-29 (f); HL2-2-41 (g); HL2-1-51 (h). HL2-2-41 (g); HL2-1-51 (h). Figure 15 shows that neural networks with a predefined number of neurons required 4. Discussion significantly fewer iterations to train as compared to the brute -force selected networks. It Having compared all obtained results of training and testing of neural networks for is worth recalling again that Figures 14 and 15 show the best results from the number of diagnosing Alzheimer ’s disease on limited data, it is possible to make a general conclusion— multiple computational experiments. A smaller number of iterations was required to train the constructed neural networks whose structure is chosen according to a predefined the predefined neural networks. number of synaptic coefficients according to inequalities (14) and (15) have acceptable Having learnt the results of training two-layer networks, we noticed the following— accuracy in comparison with neural networks whose structure is chosen by brute force. the quality of neural network models is better if the number of neurons on the first layer In our experiments, we noticed that neural networks with a predefined number of is greater than on the second (Figures 14d–g and 15a–d). neurons needed less training to obtain acceptable accuracy, and the results were more The results of training and testing two-layer neural networks are not entirely con- robust compared to neural networks with a random number of neurons. In the latter case, sistent with the observation in the case of single-layer neural networks. In the case of sin- if the number of neurons was very different from the predefined number of neurons, we gle layer neural networks, we observed the following—as the number of neurons on the quite often observed retraining of neural networks. hidden layer increases, the quality of neural network models becomes better. W ith two- The key value of the approach proposed in this paper for analyzing data and deter- layer networks, this is not true, e.g., HL2-22-2 with 24 neurons or HL2-17-4 with 21 neu- mining the structure of a neural network is to reduce the time spent on finding the optimal rons are in many ways (see Table 8) worse than the 20-neuron HL2-14-6 and HL2-10-10. 4. Discussion Having compared all obtained results of training and testing of neural networks for diagnosing Alzheimer’s disease on limited data, it is possible to make a general conclu- sion—the constructed neural networks whose structure is chosen according to a prede- fined number of synaptic coefficients according to ineq ualities (14) and (15) have accepta- ble accuracy in comparison with neural networks whose structure is chosen by brute force. In our experiments, we noticed that neural networks with a predefined number of neurons needed less training to obtain acceptable accuracy, and the results were more robust compared to neural networks with a random number of neurons. In the latter case, if the number of neurons was very different from the predefined number of neurons, we quite often observed retraining of neural networks. The key value of the approach proposed in this paper for analyzing data and deter- mining the structure of a neural network is to reduce the time spent on finding the optimal neural network, since based on the results of data preprocessing, we obtain an encoded data set, which is already a graph-matrix representation of the possible structure of the uadratic error uadratic error uadratic error uadratic error uadratic error uadratic error uadratic error uadratic error Algorithms 2023, 16, 219 21 of 25 neural network, since based on the results of data preprocessing, we obtain an encoded data set, which is already a graph-matrix representation of the possible structure of the neural network. Moreover, the prospects for such an approach to data processing and analysis are reduced to improve the quality and accuracy of models on a limited data set, which we encountered in the study of brain disease in the form of Alzheimer ’s disease. Thus, it is possible to determine some of the parameters of the incomplete neural network with cascade structure based on the integrated rating mechanism (decision root). In future studies, we will use the term decisions-root-based neural network (DRB NN). We would also like to point out that the best distribution for two-layer neural networks appears to be in the upper left corner of the space, separated by a dotted line (see Figure 13), which means that there should be fewer neurons on the first layer than on the second in the case of limited sets of training examples. Based on this, we can assume that in the case of multilayer neural networks, a similar ratio of neurons on upstream and downstream layers will also affect the quality of neural networks. This is fundamentally important for the proposed approach of building incomplete neural networks. Figure 8 clearly shows that the decisions-root-based neural network on layers three and five has a greater number of neurons than layers two and four, respectively. This could potentially limit the learning effectiveness of decision-root-based neural networks. However, the pattern described above regarding the correlation of neurons on previous and subsequent layers is true for fully connected neural networks. We will study incompletely connected neural networks in the future. This is of interest, and we plan to explore such a pattern in more detail in the following studies. There are no publicly available specialized software products allowing the user to get the incompletely connected neural network structure. For example, the computer program “Software package that implements the operation of incompletely connected neural networks” [57] can create incompletely connected neural networks, but with only one hidden layer with incompletely connected neurons. It is not enough. Therefore, the authors intend to create a special computer program on software as a service available online for all researchers. This work is performed at the LLC “Perm Decision Making Support Center”. 5. Conclusions It should be noted that the purpose of this study was achieved, and the hypothesis was confirmed. The neural network, preidentified, based on preprocessing initial datasets using suggested methods and algorithms, has acceptable accuracy in a case with a very limited training set. The suggested methodology consisting of interval coding of initial data and search of integrated rating mechanisms, following representation in the form of an artificial neural network and training using initial data, is effective. Based on the results of this study, it should be noted that with the help of special meth- ods based on the mechanism of complex assessment, it was possible to obtain the optimal structure of the neural network model capable of describing the study area with sufficient accuracy, using the example of Alzheimer ’s disease. This, once again, confirms that neural network technologies are quite successful in medicine, which is especially valuable with a limited set of initial data. Similar conclusions about the successful application of neural network modeling in medicine and, in particular, the study of the brain, can be found in references [16,17] etc., where it was proposed to use convolutional neural networks and deep learning models in medicine. Supplementary Materials: The following supporting information can be downloaded at: https://www. mdpi.com/article/10.3390/a16050219/s1, Table S1: The training set for identification of integrated rating mechanism (decisions’ root). Author Contributions: Conceptualization, A.A. and L.K.; methodology, A.A.; software, L.K.; valida- tion, A.A., L.K., V.N. and J.B.; formal analysis, A.A., L.K., V.N. and J.B.; investigation, A.A. and L.K.; resources, L.K.; data curation, A.A. and L.K.; writing—original draft preparation, A.A., L.K. and V.N.; Algorithms 2023, 16, 219 22 of 25 writing—review and editing, A.A., L.K., V.N. and J.B.; visualization, L.K. and J.B.; supervision, A.A.; project administration, L.K.; funding acquisition, A.A. and L.K. All authors have read and agreed to the published version of the manuscript. Funding: This research was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation in the framework of the program of activities of the Perm Scientific and Educational Center “Rational Subsoil Use”. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: In this study, we used exclusively numerical data given in the appendix to the reference [18]. The computer programs we used by license CC BY-SA 4.0. The computer pro- gram RU 2022615135 “Personal intelligent online development environment “EIDOS-X Professional” (System “EIDOS-Xpro”)” [55] available at: http://lc.kubagro.ru/aidos/_Aidos-X.htm (accessed on 30 December 2022). The computer program RU 2014618208 “Neurosimulator 5.0” [56] available at: https://www.lbai.ru/#;show;install (accessed on 30 December 2022). Acknowledgments: The authors are grateful to Valerii Yu. Stolbov from the Perm National Research Polytechnic University (Perm, Russia) for scientific supervising of the project ”New materials and technologies for medicine” at the Perm Scientific and Educational Center “Rational Subsoil Use”; Evgenue V. Lutsenko from the Kuban State Agrarian University (Krasnodar, Russia) for free access to the computer program “EIDOS-Xprofessional” [55]; Leonid N. Yasnitsky from the Perm State University (Perm, Russia) and Fedor M. Cherepanov from the Perm State Humanitarian Pedagogical University (Perm, Russia) for free access to the computer program “Neurosimulator 5.0” [56], and also to Nikolay A. Korgin from V.A. Trapeznikov Control Sciences Institute (Moscow, Russia), proposed term “decisions’ roots” [42], for statement of integrated rating mechanisms identification problem. We also sincerely thank all reviewers for their thoughtful comments and recommendations, especially, the first reviewer for suggesting the term “Decisions-Root-based Neural Network”, We will use such term in our future studies and papers. Conflicts of Interest: The authors declare no conflict of interest. References 1. Yasnitsky, L.N.; Dumler, A.A.; Cherepanov, F.M.; Yasnitsky, V.L.; Uteva, N.A. Capabilities of neural network technologies for extracting new medical knowledge and enhancing precise decision making for patients. Expert Rev. Precis. Med. 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