Hybrid Modeling Method for the Study of SocioEconomic Systems Abstract. The use of models with a hybrid architecture for data mining in complex socio-economic systems is considered. Approaches to identifying the properties of geo-information space: adaptability, self-learning, self-adjustment and sustainability of development are proposed. Keywords: Complex Systems. Geo-Information Space. Interoperability, Soft Computing, Hybrid Models. 1 Introduction The main advantages of the geo-information spatial approach in the study of distributed economic, social, environmental, and other processes include the possibility of using the advantages of measurement connectivity, a systematic approach, and the effect of self-organization. Currently, there is a serious instrumental base of simulation of the spatial organization of human activity, the main regularities of individual information sectors of the space [1,2], transformation of its individual components in a competitive digital economy. P. A. Minakir noted that: "Spatial economy is a form of existence of the economy as a set of interacting economic agents, distributed in a certain way in geographical space (GIP). At the same time, an economic agent means an individual who participates in at least one of the processes of production, exchange, and consumption" [3]. In the course of system analysis, "methods of lexical and semantic modeling of cognitive knowledge structures were identified that allow us to take into account the features of the geo-information environment"[4]. Methods and approaches of fuzzy lexical and semantic modeling, as shown in the work [3], can be successfully applied in the design of the architecture of the information space, the formation of the structure of technical and software tools for the intellectual analysis of geo data and the integration of thesaurus-type linguistic support in a single GIP. This approach is designed to study methods and approaches of computational linguistics to create a new generation of information technology (IT) GIS-oriented. However, there is a principle of incompatibility L. Zade: “As the complexity of a system increases, our ability to formulate precise, meaningful statements about its behavior decreases to a certain threshold, beyond which accuracy and meaning become mutually exclusive” [5]. At the same time the assessment of the adequacy of the model to the actual object of management in socio-economic systems is related to the set of accepted restrictions on the studied (in fact, fuzzy) dynamic system. System studies of the socio-economic component (SEC) of the UGIS suggest that “as a result, both the conditions for its formation will be optimized, and also the effectiveness of further functioning and development”[6]. Special attention in SEC research "should be paid to the study of the causal relationships of the behavior of the socio-economic system and the identification of its structure and properties that will ensure the effective implementation of the goals of the activities" [7]. 2 Chaos in the socio-economic system Real socio-economic systems have an almost complete set of “NON-factors: inaccuracy, vagueness, incompleteness, undefinability, etc." [8]. However, not all of them can be taken into account using traditional probability theory and mathematical statistics [9.10]. For complex modern systems such as System of Systems (SOS), which, of course, include SEC, which have "strange" attractors in their phase portraits (so-called «butterfly effect»)[11], there is a chaos caused by "deterministic randomness", and, consequently, by the nonlinearities of the general model and a certain set of initial conditions [11,12]. Even the slightest disturbance of such a complex system as the SEC, for example, the weather, not to mention the economic crisis or coronovirus, can lead to a chain of events leading to complete unpredictability. There is no alternative to using soft models in this situation. In the course of modeling, individual properties can be aggregated, for example, by manageability, to ensure clarity and speed in obtaining the result. Among the properties of complex nonlinear SEC, the properties of dynamism, flexibility and adaptability are of particular importance, which, in turn, determine stability and, ultimately, efficiency [11]. The research of UGIS should be based on the principle of consistency in order to consider each layer as an aggregate of components of semantically related types that have heterogeneous properties but co-exist in a certain cognitive space. All types of spaces of UGIS have “a number of common properties: the length in different directions, the mutual location of space objects, nodes (centers), networks, etc. The most important advantage of the spatial approach is the possibility of a multidimensional representation of a spatially localized system” [6]. 3 Lyapunov's Time Lyapunov time is the time during which the system comes to a state of chaos. In other words, this is the time during which you can predict the behavior of the system (the “non-chaos” time). It is possible to calculate if to use the Lyapunov’s exponent for the dynamic SEC, that is, the speed with which the two points in phase space converge or move away from each other: 𝑛 1 𝑑𝑓(𝑥𝑘 ) = lim ∑ log 2 | |. 𝑛→∞ 𝑛 𝑑𝑥𝑘 𝑘=0 The meaning of this indicator: when the distance 𝑑𝑓(𝑥𝑘 ) changes at the k-step in comparison with the corresponding parameter 𝑑𝑥𝑘 in a larger direction, the value of the Lyapunov’s exponent > 0, it means that there is chaos or instability of the SEC system. When 𝑑𝑓(𝑥𝑘 ) is at the k-step, in comparison with the corresponding parameter 𝑑𝑥𝑘 in the smaller direction, the logarithm of a number less than one is negative, 0 - SEC - is stable. Modeling of socio-economic systems The solution of this type of multi-criteria modeling problems in the UGIS involves the use of mathematical systems that describe the main processes of functioning of this SoS. As a hybrid, they usually use [4,13]: - neuro computing+fuzzy logic (NF); - fuzzy logic+chaos theory (FCh); - neural networks+chaos theory (NCh), etc. 4 Modeling Let's consider an example of chaos formation in an absolutely deterministic Lorentz model in the Cauchy form [2], given by a system of three nonlinear firstorder differential equations. «The next slide shows a phase portrait of the system's behavior, where the presence of a strange attractor is obvious even visually. Such qualitative research is convenient for rapid analysis of the state of society for the operational forecast of chaos. For a more accurate forecast, it is desirable to obtain information about the stability margin of the current trend. The algorithm for quantifying the state of the system model assumes the following sequence of actions»[15]: 1) forming the Cauchy model, for example: 𝑥̇ = −𝑘(𝑥 − 𝑦) {𝑦̇ = −𝑥𝑧 − 𝑦 + 𝑝𝑥 ; 𝑧̇ = −𝑞𝑧 + 𝑥𝑦 2) obtaining a Jacobi matrix composed of partial derivatives of the right-hand sides of the corresponding differential equations −10 10 0 −𝑘 𝑘 0 −𝑧 + 28 −1 −𝑥 −𝑧 + 𝑝 −1 −𝑥 [ ]=[ ]; 𝑦 𝑥 −11 𝑦 𝑥 −𝑞 3) substitution of initial conditions −10 [−𝑧0 + 28 𝑦0 10 −1 𝑥0 0 −𝑥0 ]; −11 4) finding the eigenvalues of the resulting numerical matrix - for n=3 there will be 3 of them: 𝞵1, 𝞵2, 𝞵3 ; 5) finding first-order Lyapunov exponents as the real part of the eigenvalue 𝜆𝑗1 =Re 𝜇1𝑗 ; 6) finding a one-dimensional Lyapunov exponent 𝜆1 = max 𝜆𝑗1 ; 𝑗 7) finding 𝜆21 = 𝜆11 + 𝜆12 ; 𝜆22 = 𝜆11 + 𝜆13 ; 𝜆23 = 𝜆13 + 𝜆12 ; 8) finding 𝜆2 = max 𝜆𝑗2 ; 𝑗 9) finding 𝜆31 = 𝜆11 + 𝜆12 + 𝜆13 In this case, the information space of the socio-economic system is formed, within which the zones of its predictive behavior are found, using Lyapunov indicators [4]. 5 A model for evaluating UGIS as a complex information SoS The complexity of modeling SEC as a subsystem of the UGIS is due to [3.14]: - the complexity of the research object, the non-linearity and undefinability of processes and initial conditions, the presence of threshold effects, bifurcations and time lags (differential models of SEC with a delay); - the effect of interaction of SEC model variables, which mostly implement NON-factors; - the complexity of measuring fuzzy variables of the model; - fuzzy and unstable relationships in the model; - significant influence of the human factor on all socio-economic processes. As a result of “hybridization of methods of intellectual data processing, combining several artificial intelligence technologies, the term soft computing appeared” [8], which was introduced by L. Zadeh in 1994. "Soft computing is a set of computational methodologies that provide a framework for understanding, designing, and developing intelligent systems. Soft Computing combines areas such as probabilistic reasoning and evolutionary algorithms, artificial neural networks (NN), and fuzzy logic (FL). These areas complement each other and are used in various combinations to create hybrid intelligent systems" [9]. Hybrid models that combine the main advantages of soft computing can be used to reliably assess the stability of geo-information SEC. “Among them, we will highlight such methods that can implement the properties of adaptability and the ability to learn, self-tune. These are primarily neural networks and fuzzy logic. Both technologies are modeling tools and work after the learning or knowledge extraction stage. Neural networks are used in cases when dependent and independent variables are connected by complex nonlinear relations”[13], therefore, they have the ability to generate chaos. The general “structure of a system using fuzzy logic and neural networks contains the main blocks, the synergetic effect of their joint interaction determines the intelligence of the system: the knowledge base, the decision block, the blocks of fuzzification (𝑖А (𝑥𝑗 )), aggregation and defuzzification (yk (x))” [6]. (Fig. 1) Fig.1 - Classic neuro-fuzzy (NF) process research model [2,3] 6 Reservoir calculations Classical neural networks consist of an input vector, several layers of neurons connected to each other in a certain way, and an output vector. Each neuron is a function of a linear combination of inputs (Fig. 2a). The network learning process - is a reduction in the network output error relative to the expected output of the training sample [2б]. The problem of reducing the learning error or optimization is solved by adjusting the coefficients of a linear combination on each of the neurons, using a training sample, using one of the modifications of the gradient descent method. The more layers, the longer the setup takes. Reservoir calculations are based primarily on the use of output, final layers in a multi-layer neural network. In other words, you don't need to configure the internal layers. There was a so-called “dynamic reservoir” of nonlinear neurons connected to each other, in general, randomly. The tank has an entrance and exit. The output is a simple layer of linear neurons. And the reservoir, in fact, is an extensive set of different nonlinear functions, from which you can “collect” any function that is needed at the moment. This approach has many interesting properties. 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