Twenty patients, who met the DSM-5 (Structured Clinical Interview for DSM-5) criteria for SZ, were involved from the Department of Psychiatry, Psychotherapy and Early Intervention, Medical University of Lublin. Additionally, the other criteria were as follows: age over 18 years; minimum 10 years of regular education; not more than 5 psychiatric hospitalizations associated with exacerbation of psychosis; no markers of structural brain abnormalities visualized on MRI, indicative of surviving craniocerebral trauma or neurovascular episodes; and lack of serious somatic diseases needing intense pharmacotherapy that would impact the EEG recordings. During testing, all patients were on stable doses of atypical antipsychotics. Using anticholinergic agents, benzodiazepines, and mood stabilizers up to 3 months before the assessment was an exclusionary factor for all participants. The patients participated in the study during the last week of psychiatric hospitalization, after obtaining a significant clinical and functional improvement, being fully able to give consent and undergo EEG examination. The control group consisted of HC, demographically matched to the clinical group, who were chosen from the local community. Additionally, HC had no history of psychiatric diagnoses, per the Structured Clinical Interview for DSM-5, brain disease, or neurological injury as well as no family history of psychosis. All patients consented to the study in accordance with the protocol approved by the Bioethical Commission of the Medical University of Lublin. The Commission also validated the methods used in the study. Demographical and clinical data are presented in Table 1. The groups did not differ significantly in terms of age (SZ = 34.41, SD = 8.41; HC = 31.63, SD = 6.42), number of the years of education (SZ = 12.43, SD = 2.94; HC = 14.87, SD = 1.68), and gender (SZ = 50% of men; HC = 50% of men). In the SZ group, the duration of illness lasted for about 12 years.

Using a 21-scalp position, electro-cap electroencephalograph (Electro-Cap International Inc., OH, United States) and Ag/AgCl disk electrodes, in 10 min of resting-state, EEG data were recorded for each participant. Electrodes were distributed according to the 10–20 International system (Fp1, Fp2, F3, F4, C3, C4, P3, P4, O1, O2, A1, A2, F7, F8, T3, T4, T5, T6, Fz, Pz, and, Cz). Subjects were seated with eyes closed and restricted head movement. The electrode impedances were kept below 5 k, and the data were filtered from 0.5 to 70 Hz (with active notch filter set at 50 Hz) when the sampling rate was 512 Hz. The data were exported into ASCII format after recording. Post-processing procedures were carried out in the EEGLAB program, which is a MATLAB toolbox. First, the signal was filtered using the bandpass filter at 0.5–45 Hz (second-order Butterworth filter). Second, the reference was changed offline into the averaged. Next, from the processed signal, 25 epochs lasting for 8 s (4,096 samples) without artifacts were extracted for each patient by a clinical neurophysiologist. Last, EEG signals were divided into six frequency bands using finite impulse response filters: the delta (0.5–4 Hz), theta (4–8 Hz), low alpha (8–10 Hz), high alpha (10–12 Hz), beta (13–30 Hz), and gamma (30–45 Hz).

Several steps were involved in the data processing procedure. Feature extraction was associated with the calculation of functional brain connectivity (FC) measures separately for particular electrodes and each EEG frequency band. The FC was calculated to indicate the statistical dependence between the spatially distributed neurophysiological time series such as EEG signals stemming from separate units of a nervous system (Cheng et al., 2015). There were several metrics used in assessing FC strength, such as classical measures (e.g., Pearson’s correlation coefficient, cross-correlation function, or coherence), phase synchronization indexes (e.g., phase lag index or phase-locking value), and GC measures. We chose the classical linear GC. The idea of GC (Granger, 1969) was based on the assumption that having two simultaneously determined signals (X and Y), the signal X could be better explained using information from the signal Y than using only information from the signal X. In such a situation, signal Y could be specified as “causal” to signal X. The GC measure is widely applied as a statistical tool to detect the influence of particular system components (Nolte et al., 2010). For the GC calculation, we used the Matlab MVGC toolkit (Sackler Centre for Consciousness Science, University of Sussex, Brighton, United Kingdom; Barnett and Seth, 2014), which was based on advanced VAR (vector autoregressive) model theory. To optimize auto-covariance delays, the Akaike information criterion was used to estimate the optimal model order. MVGC algorithms were used to convert EEG signals into auto-covariance data. The observed auto-covariance sequences were then subjected to paired spectral GC. In the case study, the calculated GC measures were used in the next feature extraction procedure step consisting of deriving MST. The MST was built based on Kruskal’s algorithm. First, the weight of all edges was sorted, and then the stronger edges were connected, i.e., those with the highest connectivity values, eliminating those that form loops. These stages were repeated as many times as necessary until the final tree had 19 nodes and 18 edges, which corresponded to the total number of electrodes used in our EEG recording. The MST metrics were generated separately for each frequency band.

Global MST parameters were used as the features for the classification procedure, which included the following:

After the feature extraction procedure, the classification was done using classical classifiers.

The classification procedure was performed to assign observations into one of two classes: schizophrenic patient and HC. Several classical classification methods were used: cubic SVM, linear SVM, decision tree, logistic regression, multilayer perceptron (MLP), random forest, and k-nearest neighbors (kNN). Data were split into training and testing datasets based on the 80:20 ratio.

The results of this classification process, in the form of the probabilities of belonging to considered classes, were taken as the inputs to the aggregation functions. The next step of the analysis procedure was to generate fuzzy measure density values to apply aggregation operators. The aggregation operators allow combining the predictions of multiple classifiers to further improve the results. The fuzzy measures can be interpreted as the degree of trust (weights or level of importance) to predictions of the individual classifier.

In general, there are several methods of fuzzy measure generation, such as expert assumption, optimization, and the heuristic one. In our study, the cross validation-based heuristic one was applied. N-fold cross-validation was run on the training set to obtain a density measure for a classifier.

It is a well-known fact that the results of different classifications can be aggregated. This situation can be easily illustrated by the example of various sports competitions held in the form of a Grand Prix cycle, where the points are added together to determine the final winner. EEG signal-based features can be similarly aggregated. In general, the results obtained using different kinds of classifiers can be added, averaged, or, generally speaking, transformed with the help of different kinds of aggregation operators. Typical operators are median, minimum, and maximum functions, etc. The general scheme of classification using aggregation methods is presented in Figure 1. It is worth noting that the values that are input to the aggregation operator can change the distances between the training and testing representation of an EEG signal in the case of k-nearest neighbor-based classifiers, likelihoods of belonging to a class in the case of neural network-like methods, etc. Here, it is worth stressing that the weights presented in this diagram can be obtained from experts but also based on the quality of classification of individual classifiers (e.g., their accuracy measures).

One of the best known and most efficient classifiers is the Choquet integral. Hence, let us recall the main properties of the fuzzy measure, Choquet integral, and its generalizations. Let us denote a set as X. Then P(X) = 2^X is a family of all its subsets. 2^X is a σ-algebra; i.e., the empty set belongs to it, the complement of a set belonging to σ-algebra belongs to it, and the sum of countable many sets from the σ-algebra also belongs to it. Generally speaking, in the context of classification tasks, the elements of the set X are the individual classifiers (methods, parts of the images under consideration, etc.). In the context of this specific study, they are particular classifiers, see the experimental section for details of the methods. These classifiers are denoted as x₁,x_n,n1. Now, one can define (Sugeno, 1974) a fuzzy measure as a set function g:P(X)→ℝ satisfying the following conditions:

Here, {U_n}, n = 1, 2, … means increasing set sequence. Recall that Sugeno λ-fuzzy measure realizes the above conditions and

with λ >− 1. Here, U and W are not overlapping. In addition, we have

for U_i={x₁,…,x_i}, U_i+1={x₁,…,x_i+1}. The following notation is used commonly: g_i = g({x_i}), i = 1, …, n. Now, let us introduce a function h(x) and let the series h(x_i), i = 1, …, n, be ordered non-increasingly and let h(x_n+1) = 0. In the context of this study, the function h(⋅) represents a value of classifier describing the probability of belonging to a specific class. Next, the U_i set is, in fact, only an abstract object. The real importance has the value of g(U_i) appearing in (5) which can be easily found recursively starting from the values of g_i. The value g_i represents a significance (or importance) of a particular classifier x_i. Its value can be commonly defined twofold: (1) based on the opinions of experts and (2) based on initial tests. In this study, we applied the second method. Finally, n is a number of classifiers. The last parameter to be found is λ, which can be obtained from the following equation:

see Sugeno (1974).

For such assumptions, the Choquet integral is defined as

From this function, many generalizations and extensions can be delivered as follows:

and for any t-norm M (⋅,⋅), see Lucca et al. (2014),

(Lucca et al., 2014, 2015),

see (Lucca et al., 2017), C_Min (Lucca et al., 2015), where the role of the function M is played by the minimum, or C_O [see (Lucca et al., 2016)] with a so-called overlap function under the integral sign. Newer functions were proposed in Karczmarek (2018) and Karczmarek et al. (2019a). They are

and the integrals inspired by some numerical analysis formulae such as

and

It is worth noting that M(⋅,⋅) can be any triangular norm which, as an intersection or conjunction operator in many application areas, is a counterpart to a classic product operator appearing in the original Choquet integral.

In this study, we described particular classifiers that were considered in the series of numerical experiments and determined their accuracy. We applied the following classical machine learning models: SVM with linear and cubic kernels, logistic regression, kNN, decision tree, random forest, and MLP. The classical machine learning models were used due to a low number of observations available for training and testing. In order to obtain the fuzzy density that is necessary for aggregation, the following approach can be adapted. According to the holdout validation procedure, the data were split into training and validation subsets, where 20% of the dataset was used for validation. The fuzzy density was calculated as a mean accuracy measure obtained in the process of a fivefold cross-validation run using the training data. The resulting classification quality of separate models was tested on the validation subset after fitting the models on the complete training set. The classification accuracy values obtained with separate classifiers are presented in Figure 2.

The experimental results of the aggregation scheme used for the classifiers discussed in the previous section, namely decision tree, k-nearest neighbor, quadratic SVM, cubic SVM, linear SVM, logistic regression, random forest, and MLP, are discussed. The accuracies of the individual classifiers obtained in the initial series of experiments are the input to establish the fuzzy measure densities g_i. The values of the function h are the results of the classification of testing elements being the probabilities of belonging to the two classes, namely, healthy and SZ patients. The validation procedure described in the previous section was repeated 200 times. After each run, a value of fuzzy density and 8 probability vectors (20% out of 40 observations) were obtained per classification model. The details of aggregation algorithm implementation required a single estimation of classification accuracy per model and aggregation method. Hence, all 1,600 (200 × 8) classification results were analyzed. It is worth noting that the models were fitted and the fuzzy densities were obtained independently in every separate run of the experiment. In the series of experiments, we have evaluated 25 classes (families) of popular and commonly considered in the literature triangular norms (Alsina et al., 2006, page 72). The monograph can be treated as a compendium of the t-norms to be applied in more advanced aggregation operators. In this particular approach, the t-norms serve as the integer functions M with parameters −10, −9.9, …, 0, …, 9.9, 10, but only if the parameter is in the range allowed for the t-norm. Such a choice of the parameter range seems to be optimal and emphasizes the most important properties of each of the t-norm classes. The maximal accuracy was obtained for the classifier C_D2 for triangular norm from the family no. 8, namely

In this case, the best option to choose is α = 0.1. The plot illustrating the recognition rate for the combination of C_D2 and the function (18) is given in Figure 3. It is obvious that satisfying accuracy can be obtained only for relatively small values of the parameter α.

It is important to understand the process of calculation of the value of the aggregation function. Let us consider, for example, the process of C_D2 finding. Here, the individual classifiers x_i,i = 1,…,n = 5 (five classifiers are discussed in this experiment) should be analyzed, and for their significance measures (simply, weights) g_i = g({x_i}), see Figure 2. Next, the parameter λ appearing in (6) is calculated. Based on the value of λ, the values of g(U_i) appearing in Eq. (5) are found recursively. Next, using the values of h(x_i−1), which are the likelihoods of belongings of a given probe to a specific class, the final sum (16) can be obtained, taking into account that M(⋅,⋅) is any t-norm, in particular a function given by (19).

Very good results were obtained also for the aggregation functions C_M and very similar C_FM. Maximal yielded values were 98.81% for the function number 12 serving as integer function and α = 0.2. The formula of the t-norm is as follows:

Table 2 supplements the above discussion by showing for which triangular norms and their parameters the classification rate exceeding 98.81% was reached.

It is worth stressing that the best average result among the operators C_M, C_FM, C_CM, C_MC, C_MMin, C_MMin2, C_D1, C_D2, and C_D3 was also obtained for the triangular norm no. 8 given by the formula (18) and its parameter α = 0.1. The plot presenting the values of the combination of all the aggregation functions with this t-norm and parameter is presented in Figure 4. It is obvious that the function (18) works well with almost all of the aggregation operators except C_CM and C_D3.

As a supplement to the results, it is worth noting that Table 3 presents the information for which the best results of the t-norms were obtained by aggregation operators. It can help match t-norms and generalizations of the Choquet integral by experts conducting similar research. For instance, function no. 4, see Alsina et al. (2006), works well when it is combined with a few Choquet integral-based operators.

Finally, what should be emphasized, we have conducted the same tests for over 1,000 functions that are not Choquet-like integrals. The functions that were used in this competition were selected based on the studies by Alsina et al. (2006), Beliakov et al. (2007); Grabisch et al. (2009), and Calvo et al. (2012). The best results were obtained for the ordinary weighted averaging operator at the level of 98.81%.

where y_j is the j-th largest of the x_i, and the weights are ω₁ = 1, ω₂ = 1 – 1n, …, ω_n = 1nwith or without normalization to their sum. Despite it being a very good result, it is obvious that it is hard to find the function giving the results more satisfying than Choquet integral-based operators. Moreover, to find the proper form of OWA, similar to the Choquet integral case, a proper heuristic can be used.