CC BY
29UDC 004.81 10.23947/2587-8999-2022-1-2-81-95
METHODS FOR FORECASTING THE DEVELOPMENT OF COMPLEX SYSTEMS USING THE THEORY OF FUZZY COGNITIVE MAP
A.V. Petukhova1, A.V. KovalenkoH2
:Universidade Lusófona de Humanidades e Tecnologías, Lisbon, 1749-024, Portugal 2Kuban State University, Krasnodar territory, Krasnodar, 350000, Russia
The article provides an overview of methods for solving problems of forecasting time series and modeling of complex systems using fuzzy cognitive maps (FCM). The main algorithms used in solving practical problems for complex semi-structured systems are listed, which make it possible to improve the accuracy and reliability of simulation results. For completeness of the review, publications of Russian and foreign researchers working in this area have been studied and described. In addition, the main software tools that implement the existing algorithms were listed and their distinctive features for solving various classes of problems were given. This comparison of software packages allows users to deter-mine the optimal system for further theoretical or practical research.
Keywords: fuzzy cognitive maps, software systems for decision sup-port, decision-making systems, time series forecasting.
Introduction. Fuzzy cognitive maps, proposed by Kosko [1] in 1986, are one of the most convenient means of presenting expert knowledge when modeling dynamic systems containing a variety of concepts from any subject area and their causal relationships.
By its structure, the FCM is an oriented graph describing the behavior of a physical system, representing it in the form of vertices and arcs connecting these vertices. Concepts (i.e. vertices) are understood as fuzzy sets that characterize objects or elements of the system under study.
The marked and weighted arcs of the graph display cause-and-effect relationships between concepts. A specific FCM with its many parameters of factors and the structure of causal interactions of factors and external influences describes not one situation and not one dynamic process, but a variety of processes that differ in their parameters. With the help of the FCM apparatus, multi-purpose models are created, which are successfully used among experts in many fields of science.
Over the past 30 years of research in the field of fuzzy logic and the study of fuzzy cognitive maps, their original architecture has changed significantly. Successes in the study of the applied application of FCM to this day arouse the interest of many researchers around the world. Despite the fact that the main application of the FCM is to model scenarios for the development of complex systems with their help, problems of a different nature are also solved, such as time series forecasting problems and classification problems. Several reviews have been presented in the field of FCM application [2-3] however, they do not cover the full range of research in this area conducted by Russian and international teams.
Most researchers, not having information about the existing software based on the FCM, are forced to develop their own software solutions from scratch. Apart from the fact that this is a fairly
time-consuming task, such solutions are difficult to integrate or even simply compare with each other. This article provides a brief description of the currently existing software tools that can be used for software implementation, testing and validation of models based on FCM. The more general purpose of this article is an attempt to present the theoretical foundations to the reader in a clear form and to provide practical assistance in the implementation of decisions based on the FCM.
1. Fuzzy cognitive maps for predicting time series. The task of forecasting time series.
Fuzzy cognitive maps are widely used to predict time series. Let's consider the formulation of the forecasting problem from the point of view of the FCM apparatus. Let y e R be a variable of real type, set on a discrete time scale during a period t e[1,2,..., N], where N e N means its duration. Thus, a one-dimensional time series is defined as a sequence of observations (y( )} = {y( )'y(2)'. .'y(A°}. Let's denote the series characterizing the period t e[1,2,...,te], where
te < N, as a historical time series. The purpose of forecasting is to predict the following values of the time series, that is y(te+1),y(te+2),...,y(te +H), where u is called the prediction horizon. The prediction task requires the construction of a model F . Let's say the forecast is one-step, that is H = 1, then the model F is used for calculation y(t"+1) = F(y(te)). The problem that needs to be solved is a
construction F that is usually unknown and needs to be inferred using retrospective data.
At each time step (i.e. iteration) of the forecast period t e [te +1,N], separate prediction error
is calculated as E(t) = y(t) - y(t). To determine the total prediction error for the entire period [te +1, N] , can use various metrics to calculate the error. The two most popular are the average absolute Percentage Error (Mean Absolute Percentage Error, MAPE) (see Equation (1)) and the average Squared Error (Root Mean Squared Error, RMSE) (see Equation (2)). The lower the value of these indicators, the more accurate the forecast model.
-- e(t)
1 N
MAPE = - V
n t=1
y(t)
x100% (1)
RMSE =
1 N
- V
N V
E
<t)
y(t)
(2)
The concepts of the FCM are put in accordance with the variables of the time series and then the FM weights are trained; then testing within the prediction horizon follows. Depending on the method of correlation of the concepts of the FCM and the lag variables of the time series, it is possible to use the FCM in forecasting both one-dimensional and multidimensional time series. Also, as will be shown in the following subsections, fuzzy cognitive maps have been adapted to predict approximated time series, interval-valued time series, and time series based on granular calculations.
2. Forecasting of one-dimensional time series. Previously, the use of FCM in time series forecasting tasks was limited only to one-dimensional time series. In such scenarios, FCM concepts were transformed into time series with lag variables; thus, they played a role similar to the role of regressors in autoregressive forecasting models. This approach was proposed in the works of Homenda et al. [4], Lu et al. [5].
In these works, all the values of the training block of the time series are clustered using the
2
fuzzy c-means algorithm, where the number of clusters with is a parameter set by the user. At each time step, the value of the time series is included in each created cluster with some degree of belonging. At the same time, it is assumed that each cluster (considered as a concept of the FCM) plays the role of a fuzzy set. From this we can conclude that the activation value Ai is the degree to which the current value of the time series y(t) belongs to the concept Ct. By performing this operation for all concepts, the value of the output activity of all concepts is obtained, and the FCM weights are trained by using the data generated in the training block.
At the prediction stage, the sequence of output activity values is reconstructed using a trained model. To obtain numerical values of the predicted time series, each output signal is defuzzified.
In the works of Homenda et al. [6] and Salmeron and Froelich [7], another approach using FCM in time series forecasting was proposed. This approach is based on the direct transformation of lag values of a time series into FCM concepts. Instead of the fuzzification function, the min-max normalization method was used. Training and testing was carried out using data available online by the «sliding window» method. Complementing the previous works, the training method proposed in the work of Salmeron and Froelich [7] considers the optimization of the duration of the training period and the choice of the activation function, as well as its parameters. The authors performed optimization in dynamic mode for each training and testing cycle. The empirical results suggest that the obtained FCM model is quite competitive in forecasting accuracy compared to other models.
3. Prediction of multidimensional time series. In the case of multidimensional time series Y = [Y (1\...,Y (t),..., Y w] the vector includes a sequence of observations for several variables of real type, such that Y(t'' = [y(',...,y(/',...,y^]]. Therefore, one-step forecasting takes the form:
Y(te+1) = F(Y(te)), where F it is used as a predictive model. There are two approaches to the problem of predicting multidimensional time series.
In the first approach, it is assumed that the multidimensional time series Y(t) is synthetic and is generated by the existing cognitive network. It follows that the task of the training block will be to calculate a model simulating the generated time series and determine the method of reconstruction of the prototype of the original FCM. In the work of Stash et al. [8], the authors applied this approach and proposed a new method for predicting time series, implementing it both at the linguistic level and at the level of numerical values. The proposed predictive method combines fuzzy cognitive maps and granular input data calculation models based on fuzzy sets. The results obtained are compared with the results of other predictive models using fuzzy sets. It has been shown that the proposed architecture allows achieving good accuracy.
The second approach focuses on the application of FCM in real time series forecasting tasks. Froelich and Juszczuk [9] consider in detail the predictive possibilities of adaptive and evolutionary FCM. The purpose of their research was to establish which teaching methods can be used to solve a particular predictive problem. The authors demonstrated the predictive capabilities of the FCM by the example of forecasting weather conditions.
Song et al. [10] designed FCM -based system using neural networks and fuzzy sets to predict chaotic time series. This four-layer fuzzy neural network was designed to enhance the learning capabilities of the FCM and is a combination of the logical inference mechanism of traditional FCM and learning using the function of belonging to fuzzy sets. This method used the concept of mutual
subseteness to define and describe cause-and-effect relationships in a cognitive model. Another advantage of this model is the ability to automatically build an FCM based on data and, thus, eliminate the need for expert participation. The simulation results confirmed that this approach in most cases shows better efficiency both in terms of forecasting accuracy and in terms of architectural simplicity, compared with the methods described above.
Vanhoenshoven et al. [11] used an approach based on an autoregressive integrated moving average to solve the problems of convergence of a recurrent network and at the same time preserve its ability to perform multistep forecasting. Optimistic preliminary results were obtained; however, further studies of this approach are needed. In the sequel, Algzawi et al. [12] using a real example of the receipt of funds to the budget of Jordan's social security funds, they demonstrated the negative contribution of convergence to the effectiveness of predictive models based on the FCM.
In the work «Cognitive hybrid decision Support and forecasting systems» [13], a solution approach based on a hybrid system integrating a fuzzy cognitive map and a neuro-fuzzy network was proposed. In the paper, the authors consider the application of this approach based on solving the problem of protecting the interests of the individual. A fuzzy cognitive map is used to build a qualitative forecast of indicators and highlight the factors that most affect the system. This allows you to submit data of the most significant concepts of the cognitive map with the corresponding weights to the inputs of the neural network. In a further study [14], a modification of the hybrid time series forecasting model was proposed, combining a neuro-fuzzy network with regression analysis.
The developed forecasting system is based on a modular architecture, which gives the system additional stability: even if one of the modules fails, the system will continue building the model. The proposed system has three modules: a hybrid neuro-fuzzy network performs a quantitative forecast with verification (assessment of the adequacy of the forecast); a fuzzy cognitive map, working in parallel with the previous module, allows you to identify all the factors influencing a specific predicted indicator and get a forecast with the probability of its implementation; a neural network that aggregates information obtained from previous modules and gives a final forecast.
4. Fuzzy cognitive maps developed by Russian researchers. Russian teams focus on the development of the FCM structure for complex systems. Such systems are used in solving urgent problems of modeling complex systems and are characterized by typical properties of complex systems, namely non-linearity, the presence of flows, positive and negative cycles, as well as adaptability and heterogeneity. The description and structuring of these systems play an important role in the development of a systematic approach and allows for scenario modeling. In this review, we reviewed the maps in the order of their publication and from the point of view of the features of the algorithms used.
The article «Construction and analysis of the cognitive model of the process of choosing a profession by graduates in the system of primary and secondary vocational education», by authors D.G. Lagerev and E.A. Laricheva [17], describes the problem of reducing the number of students. The authors single out R. Axelrod's cognitive modeling methodology [18] as one of the most effective approaches to the study of poorly structured systems and processes.
In the process of applying this technique, experts deduced cause-and-effect dependencies, on the basis of which a cognitive map was built reflecting the cause-and-effect structure of the educational institution selection process, taking into account both microenvironment and
macrofactors. To obtain a forecast of the development of the situation, based on the constructed FCM, the method of impulse processes was used, which allows determining the state of concepts at discrete points in time, analyzing which the expert receives a forecast of changes in the state of the system when implementing the chosen strategy. This made it possible to assess the consequences of the decisions taken and choose the optimal strategy.
Strokova L.A. [19] proposed to apply a cognitive approach to the construction of a computational model of the foundations of engineering structures. The analysis of connectivity was carried out and the process of propagation of disturbances on the graph was studied, which made it possible to determine the reserves for improving the efficiency of the engineering and survey departments of design organizations. The problems that arise when assessing soils as a base were also considered and possible scenarios for improving the quality management system were presented
The article by Meshalkin V.P. and Belozersky A.Yu. [20] analyzes the features of the functioning of industrial enterprises from the point of view of the process and development of the risk situation. The authors give an example of building a risk management system using a system of fuzzy cognitive maps and describe 9 stages of building a model of analysis and decision-making. The stages include analysis of the system under study, assessment of the interrelationships of system factors, determination of the influence of system factors on risk sources, analysis of the impact of identified hazards on risks, assessment and determination of the class of measures based on the results of risk assessment and its possible consequences, selection of measures to reduce risk within the selected class, assessment of the impact of selected measures within the selected class. class, dynamics modeling and analysis of possible risk management scenarios and risk monitoring.
An alternative approach for constructing the model was proposed by Ginis L.A. in the article «Development of cognitive modeling tools for the study of complex systems» [21], the essence of which is to combine various indicators into subsystems both by the object of research and by nature. This made it possible to build both a quantitative and a qualitative forecast of the development of the system. To construct fuzzy oriented graphs, it was proposed to use a minimax basis and conjunctive path strength.
The article «Analysis of the cognitive map of the learning system based on expert assessments» [22] describes the system analysis of the cognitive map for the learning system. The fuzzy map was built on the basis of expert questions and filled with information taken from a specially designed questionnaire.
A systematic analysis of the cognitive map of the learning system was performed on the basis of direct expert surveys and the average complexity of the computational process algorithm as a model of software solution to the problem of evaluating the effectiveness of the learning system was determined. The author also solved the problem of evaluating the effectiveness of the education system in Ukraine. The results of the influence of all concepts are selected according to 4 criteria, which clearly shows which concept has a positive effect on the learning system, which negatively and which strengthens or weakens the learning process of students.
The article also describes the definition of the average complexity of the algorithm of the computational process. The author, using a system of linear algebraic equations, determined the average number of hits of the computational process in the states of the vertices of the graph, which made it possible to calculate the average complexity of the computational process according to the
formula:
d = V Nn (3)
i=1
Another approach to the construction of the Kosko FCM was proposed in the work of Oskin A.F. and Oskin D.A. [23] This algorithm is characterized by a greater formalization of individual steps of the FCM construction process, which allowed to increase the accuracy of modeling, since the model parameters depend less on the subjective opinion of individual experts. When assigning weights of mutual influence of concepts, experts used the point scale of preference intensity proposed by T. Saati [24]. The collective construction of a fuzzy cognitive map by a group of experts made it possible to naturally combine the efforts of experts conducting analysis and modeling of a poorly structured system, simplifies the development of a consolidated view of the processes taking place in the simulated system.
In the work «Cognitive hybrid decision support and forecasting systems» [13], the problem of protecting the interests of the individual is considered and a new solution approach based on a hybrid system integrating a fuzzy cognitive map and ANFIS neuro-fuzzy network is proposed [25]. Fuzzy cognitive map is used to build a qualitative forecast of indicators and identify the most influencing factors, which allows you to submit prepared data of cognitive map concepts to the inputs of the neural network.
In further studies [14], the authors propose a modification of the hybrid time series forecasting model using a neuro-fuzzy network with regression analysis. The developed forecasting system is based on a modular architecture, which gives the system additional stability: even if one of the modules fails, the rest continue to do their work. The system itself has three main modules responsible for the forecasting task. A hybrid neuro-fuzzy network performs a forecast of a time series based on numerical indicators and gives a so-called quantitative forecast, the results of which pass through a verification system (assessment of the adequacy of the forecast), if the forecast corresponds to the required accuracy, then it is transmitted to the next module. In parallel with the neuro-fuzzy network, a module with a fuzzy cognitive map works, which receives input data on the event impact on the time series, a cognitive map is built, which takes into account all the factors influencing a specific predicted indicator. At the output, the cognitive map gives a forecast with the probability of its fulfillment, that is, with the consonance of a factor that says whether the forecast will be fulfilled or not. Then all the data received from these modules are sent to the third module, which operates on the basis of a neural network, which aggregates the information received from the previous modules and gives a final forecast.
5. Software complexes for the construction and training of fuzzy cognitive maps. In the process of compiling the literature review, we drew attention to the lack of diversity of the developed software tools for creating and conducting experimental studies with FCM. In scientific papers devoted to this methodology, there are usually theoretical methods or practical research, rarely supported by a well-described software implementation. In this section we make an overview of the most significant software.
FCM Modeler (Mohr [25]) is the first software tool for designing systems based on FCM. Developed about 20 years ago, it has a simple interface and is designed for collective decision-making based on a high-quality static model.
Some of the characteristics of this software tool are: intuitive user interface, functionality for designing and storing systems based on FCM, logical output of FCM models based on observed concepts and successive outputs.
FCM Modeler was intended to be used as a modeling tool for a wide range of users, but the project did not develop in this direction. Despite the fact that there are no other references to this software package in the literature, its creation laid the foundation for other developments in this area.
A similar software package FCM Designer (Aguilar and Contreras [26]), develops FCM Modeler, and is a more successful software implementation, but has a rather complex interface to work with. The main functionality of FCM Designer includes: interactive graph visualization, user graphical interface for designing models based on FCM, the ability to simulate new scenarios using available information about cause-and-effect relationships.
Also, FCM Designer makes it possible to determine the rule for updating cause-and-effect weights by selecting the activation function and the stop criterion. The main disadvantage of this software is the lack of training algorithms for calculating the parameters that define the system.
Another developed tool, Mental Modeler (Gray et al. [27]), is a web application with support for collective decision-making, designed for collective presentation and testing of expert judgments about the system. The main users of Mental Modeler are probably non-specialists in the field of IT. As a rule, these are subject experts or other interested persons who have a need to develop a simple cognitive map (with qualified and balanced connections) and simulate its behavior in some scenarios.
The disadvantages of this tool are the lack of training algorithms and a limited set of experimental tools. Nevertheless, the focus on web technologies was highly appreciated by users.
JFCM is a system that implements the construction of fuzzy cognitive maps in the Java programming language. It is a small and simple open source library (De Franciscis [28]) and can be used to create a variety of models based on the FCM. JFCM has the ability to load data from XML files, which increases the convenience of the user interface. The main idea of the project is to create modules that can be loaded while solving tasks based on the FCM. This means that if the set of standard components is insufficient, the library allows the source code to be expanded. This advantage turns into a disadvantage for users who do not have programming skills, since it requires a deep understanding of the source code.
ISEMK is a software designed for modeling decision—making systems based on FCM and artificial neural networks (Poczketa et al. [29]; Papageorgiou et al. [30]). The ISEMK software tool consists of 4 basic blocks, namely: knowledge processing, FCM analysis, neural network toolkit and graphical user interface.
The first block contains the FCM module and training algorithms based on the gradient descent method using retrospective data and a population-based approach to training (namely, a genetic algorithm with real coding and a genetic algorithm for optimizing the structure) as optimizers. The neural network module allows to obtain multilayer neural networks used in time series forecasting tasks, as well as to use two training algorithms: the Levenberg-Marquardt algorithm (Hagan and Menhaj [31]) and the error back propagation algorithm (Haykin [32]). In addition, the interface of the ISEMK software product supports visualization of simulation results.
FCM Tool was first introduced in the work of León et al. [33] as a tool for modeling decision-making tasks in the field of public transport in Belgium. This software allows to develop complex
models based on the FCM, thanks to interactive graphical visualization, customize the update rule by selecting the desired type of activation function and the required stop criterion, analyze scenarios and their impact on the system.
The FCM Tool complex includes a population-based learning algorithm that allows you to automatically generate cause-and-effect weights based on retrospective data. Another popular function is the inclusion of aggregation operators that allow combining several systems based on the FCM into a single «smart» model. Subsequently, the FCM Tool was transformed into the FCM Expert system (Napoles et al. [34]), designed for non-specialists and is a more complete software platform for modeling systems based on the FCM. As already mentioned, the purpose of creating the FCM Tool was to solve a specific decision-making problem, so the implemented learning algorithms could not be used to solve more general problems of pattern classification. FCM Expert has the strongest functions of the FCM Tool, and several unsupervised (teaching without a teacher) and supervised (teaching with a teacher) training algorithms have been added to adjust the weight matrix. This software tool contains methods for optimizing the network topology (Napoles et al. [15]) and improving the convergence of the system (Napoles et al. [35-36]). Moreover, there is the possibility of custom configuration of model parameters (for example, activation functions, inference rules and stopping criteria).
Russian research teams actively engaged in the use of FCM for decision support and situation analysis have created their own software products to automate various stages of the decision-making process. The V.A. Trapeznikov Institute of Management Problems of the Russian Academy of Sciences has made a great contribution to the creation of software products for FCM modeling. The following complexes were developed:
«Situation» and «Situation-2» [37] used to analyze the development of the situation and trends in its development, taking into account the combined influence of external influences. The second version of the program expanded the original functionality by including the possibility of choosing strategies that contribute to the development of a complex system in a given direction, as well as adding a report generator to systematize modeling results. Both programs allow you to work with deterministic FCM and use the method of obtaining a forecast of the development of the situation with summation of increments.
«Compass» and «Compass-2» [38], which are used for cognitive modeling of a wide class of situations with a focus on systematization of experts' reasoning about a problem situation. To do this, the software package uses a «fuzzy linguistic model of the situation». An additional distinctive feature is the ability to identify «points of effective management of the development of the situation. «Compass» and «Compass-2» allow you to simulate deterministic FCM using the method of maximizing increments. The forecast of the situation is presented in the form of a chain of triggered rules.
The integrated system «Course» [39], which combined the subsystems «Situation», «Compass-2» and «KIT» [39] for the task of modeling the dynamics of weakly structured situations in solving strategic management problems.
The «Canvas» system [40]. The focus of the developed system is determined by the possibility of searching and testing hypotheses about the mechanisms of development and management of the situation. The Canvas system can be used for conceptual analysis and modeling of complex and
poorly defined political, economic or social situations, development of management strategies and mechanisms for their implementation, development of program documents for the strategic development of a country, region, enterprise, firm, etc., as well as tools for continuous monitoring of the state of the situation, generation and testing hypotheses of development mechanisms and mechanisms of situation management. The system works with deterministic maps, in the construction of which absolute linguistic scales are used. The results of the forecast are verbalized and also presented in the form of a chain of triggered rules.
Let's also consider the research of other teams in this area: Kosmos [41], a system developed by Data C under the leadership of V.B. Silov, which is the first developed FCM modeling system and largely determined the path of development of the FCM theory in Russia. It considered non-deterministic maps as fuzzy sets and built a forecast of the development of the situation in the form of a chain of triggered rules.
The software system of cognitive modeling of sociotechnical systems, created in the scientific team of the Southern Federal University under the leadership of G.V. Gorelova [42], which has the ability to determine the format of incoming information, the function of interaction with the database, the function of pulse modeling, the function of solving the inverse problem, stability, structural analysis.
The «Strategist» system [43], Volgograd State Technical University, developers M.A. Zabolotsky and others. The program establishes mutual influences between factors based on available statistical data based on correlation analysis. Also, the establishment of the initial trends of changes in factors, the determination of indirect influences and their levels, Pareto analysis of the indirect influences of a factor (and on a factor) in order to identify a group of factors that have the greatest or least influence on this factor (or are experiencing from this factor). This mechanism is very much in demand when solving problems related to quality problems. Due to this, the software tool developed by the authors can be used in building quality management systems and solving related tasks.
The «IGLA» system [44], Bryansk State Technical University, developers D. A. Korostelev and others. It refers to systems focused on modeling nondeterministic cognitive maps using fuzzy set theory. The system provides support for the group construction and coordination of a fuzzy cognitive map, the calculation and analysis of its system indicators, as well as dynamic modeling of scenarios for the development of the situation. A feature of the system is the verification mechanisms of fuzzy cognitive maps, which allow you to check the adequacy of the model, if there is data on changes in the most significant factors for a certain period.
Alf-ZDr [45] was presented in the work of Pylkin and etc. is an intelligent inventory management system based on fuzzy cognitive analysis. It consists of five main blocks: a block of static modeling, a block of collecting, correcting and issuing information, a block of converting information for input and output, a block of dynamic modeling, a block of knowledge base management. The proposed system makes it possible to use various types of material analysis at the enterprise, with the possibility of expanding the functionality to other subject areas.
In the work of M.M. Putyato [46], an approach to the construction of the Silov FCM is proposed, based on the application of methods for specifying membership functions of discrete fuzzy sets and providing the possibility of taking into account and coordinating the opinions of a group of
experts, which, according to the author, allows to increase the adequacy and validity of the cognitive model. A software package of the PC «X» was developed, the distinctive feature of which is the implementation of an algorithm for solving the inverse problem of cognitive analysis based on the method of sorting out the effects on controlled concepts. In addition, the developed software package provides support for fuzzy cognitive models for the development, research and monitoring of management strategies for weakly structured systems. The software package was focused on the task of developing and substantiating decisions on the management of public authorities and was used in the development of the Situation Center of the President of the Russian Federation and the Situation Center of the Governor of the Krasnodar Territory.
Table 1 shows a comparison of the considered software tools. The comparison is carried out according to several frequently demanded indicators, such as the availability of tools for conducting a computer experiment, the availability of machine learning algorithms and a user graphical interface.
Table 1. Comparison of existing software tools for FCM modeling.
Title Year Simulation capabilities Training algorithms User graphical interface
FCM Modeler 1997 no only one unsatisfactory
FCM Designer 2005 limited no good
FCM Tool 2011 few only one improved
JFCM 2013 for developers no no
Mental Modeler 2013 limited no good
ISEMK 2015 few few good
FCM Expert 2017 few few improved
Situation + Situation-2 1999 few only one there is no way to determine
Compass + Compass-2 1998 few only one there is no way to determine
Course 2001 few few good
Canvas 2002 few only one good
Space 1995 few only one unsatisfactory
PSK MSS 2004 few only one there is no way to define
Strategist 2007 few only one good
IGLA 2008 few only one good
Alf-ZDr 2012 few only one there is no way to define
X 2010 few only one good
The evaluation results allow us to conclude that FCM Designer, Mental Modeler, FCM Tool, Course, Canvas, Strategist, IGLA and X have a user-friendly graphical interface that is convenient for experts when analyzing scenarios and conducting experiments with new situations, whereas JFCM is suitable for developing FCM modules with the possibility of their reuse in more complex scenarios. However, these software implementations still lack the means of conducting experiments and/or they do not allow solving machine learning problems, which significantly hinders their applicability in real situations. FCM Expert and ISSEMK are the most convenient software tools for developing systems based on FCM. The first of the above is intended for simulation and pattern classification tasks, while the main purpose of the second is time series forecasting.
The Russian systems presented in the review are closed source systems, which does not allow the possibility of reusing the code. For this reason, new researchers in this field need to develop a separate system, which explains the large number of software products with overlapping
functionality. A potential direction of development in this area will be the transition to modular systems with open-source code, which will allow combining algorithms developed by Russian teams and accelerate the involvement of new researchers in the field. This approach will allow us to focus on the development of new areas of application, methods of modernization of the structure and algorithms of FCM training, as well as introduce a standard interface with the possibility of adapting it to specific user requests.
Conclusion. The article discussed in detail the various classes of tasks for which the use of FCM is possible. New methods of solution were presented and the latest achievements in the field of modeling of fuzzy weakly structured systems were described. Algorithms used to predict one-dimensional, multidimensional and approximated time series based on the FCM are described. Examples of FCM and algorithms adapted for the construction of complex systems are given, as well as a list of available software complexes and practical recommendations for their use in the development of systems based on FCM.
Currently, the results of the use of FCM in time series forecasting methods are of the greatest interest to researchers. Unfortunately, low-level predictive models based on FCM tend to converge to a point attractor when implementing the logical inference process. And although this property is very attractive in the conditions of simulation and classification of patterns, the manifestation of this effect in scenarios using time series is less desirable. If the network converges to a point attractor, then this predictive model will be unsuitable for a time series. This observation suggests that further research in this direction is needed.
When considering the FCM developed by Russian researchers, it is worth noting the variety of tasks and areas of application of this modeling method. Special attention is also paid to balancing expert opinion and statistical analysis of components. This approach is probably due to the need to solve real problems and the practical use of the FCM at various levels of government in Russia. Many of the developed systems are implemented in the management process of enterprises and are used to predict potential scenarios and ways of development of companies. It is worth noting the lack of a common knowledge base about existing maps, both developed by Russian and foreign teams. The creation of such a base would simplify the implementation of FCM and expand the scope of their application.
In this study, we conclude that FCM Expert and ISEMK contain the most complete set of tools for designing, training and simulating systems based on FCM. FCM Expert implements algorithms for calculating the weight matrix, simplifying the topology of high-density FCM-based systems and improving convergence without losing important information, while ISEMK contains several training methods for time series forecasting tasks. Despite their functionality, both software products are still far from eliminating the lag in the development of fully functional practical applications of the FCM from the level of theoretical knowledge in this area.
References
1. B. Kosko, Fuzzy cognitive maps, Int. J. Man-Mach. Stud. (1986), 24 (1), pp. 65-75.
2. Felix Benjamin, Gerardo & Napoles, Gonzalo & Falcon, Rafael & Froelich, Wojciech & Vanhoof, Koen & Bello,
Rafael. (2019). A Review on Methods and Software for Fuzzy Cognitive Maps. Artificial Intelligence Review. 52.
3. Kulinich Alexander Alekseevich. «Computer systems for modeling cognitive maps: approaches and methods» Problems of Management, 2010, № 3, pp. 2-16.
4. Homenda, Wladyslaw & Jastrzebska, Agnieszka & Pedrycz, Witold. (2014). Joining Concept's Based Fuzzy Cognitive Map Model with Moving Win-dow Technique for Time Series Modeling. 397-408. 10.1007/978-3-662-45237-0_37.
5. Wei Lu, Jianhua Yang, Xiaodong Liu, Witold Pedrycz, The modeling and prediction of time series based on synergy
of high-order fuzzy cognitive map and fuzzy c-means clustering, Knowledge-Based Systems, 2014, V. 70, pp. 242255, 10.1016/j.knosys.2014.07.004.
6. Homenda, Wladyslaw & Jastrzebska, Agnieszka & Pedrycz, Witold. Time Series Modeling with Fuzzy Cognitive
Maps: Simplification Strategies, 2014, pp. 409-420. 10.1007/978-3-662-45237-0_38.
7. Jose L. Salmeron and Wojciech Froelich. Dynamic optimization of fuzzy cognitive maps for time series forecasting.
Know. -Based Syst. 105, 2016, pp. 29-37. 10.1016/j.knosys.2016.04.023.
8. W. Stach, L.A. Kurgan, W. Pedrycz, Numerical and linguistic predic-tion of time series with the use of fuzzy cognitive maps, IEEE Trans. Fuzzy Syst., 2008, 16 (1), pp. 61-72. 10.1109/TFUZZ.2007.902020.
9. Wojciech, Froelich & Juszczuk, Przemyslaw., Predictive Capabilities of Adaptive and Evolutionary Fuzzy Cognitive Maps - A Comparative Study., 2009, 10.1007/978-3-642-04170-9_7.
10. Song, Hengjie & Miao, Chunyan & Roel, Wuyts & Shen, Zhiqi & Catthoor, Francky., Implementation of Fuzzy Cognitive Maps Using Fuzzy Neu-ral Network and Application in Prediction of Time Series. IEEE T. Fuzzy Systems. 2010, 18, pp. 233-250. 10.1109/TFUZZ.2009.2038371.
11. Vanhoenshoven, Frank & Napoles, Gonzalo & Bielen, Samantha & Vanhoof, Koen. Fuzzy Cognitive Maps Employing ARIMA Components for Time Series Forecasting. 2018, pp. 255-264. 10.1007/978-3-319-59421-7_24.
12. Alghzawi, Ahmad & Napoles, Gonzalo & Sammour, George & Vanhoof, Koen., Forecasting Social Security Revenues in Jordan Using Fuzzy Cognitive Maps., 2018, pp. 246-254 10.1007/978-3-319-59421-7_23.
13. Averkin A.N., Yarushev S.A., Pavlov V.Yu., Cognitive hybrid decision support and forecasting systems. 2017, 36, pp. 632-642. 10.15827/0236-235X.120.632-642.
14. Yarushev S.A., Averkin A.N., Efremova N.A., Hybrid fuzzy cognitive maps in decision support and forecasting tasks. International Journal of Software Products and Systems., 2017, 19. 10.15827/2311-6749.25.291.
15. Napoles, Gonzalo & Grau, Isel & Bello, Rafael & Grau, Ricardo, Two-steps Learning of Fuzzy Cognitive Maps for Prediction and Knowledge Dis-covery on the HIV-1 Drug Resistance. Expert System with Applications, 2014, 41. 821-830. 10.1016/j.eswa.2013.08.012.
16. Wojciech Froelich, Towards improving the efficiency of the fuzzy cognitive map classifier, Neurocomputing, 2017, vol. 232, 83-93, 10.1016/j.neucom.2016.11.059.
17. Laricheva E.A., Lagerev D.G., Construction and analysis of a cognitive model of the process of choosing a profession by graduates in the primary education systemsecondary vocational education, Economic psychology of innovation management: proceedings of the mezhregion. scientific-practical. Internet Conference - Bryansk: BSTU, 2008. pp. 47-51.
18. R. Axelrod, The analysis of cognitive maps, in: Structure of Decision: The Cognitive Maps of Political Elites, 1976, 55-73. https://press.princeton.edu/books/hardcover/9780691644165/structure-of-decision.
19. Strokova L.A., The use of fuzzy cognitive maps in the development of computational models of foundations, Proceedings of Tomsk Polytechnic University, Georesources Engineering, vol. 314, No. 5, 2009. https://cyberleninka.ru/article/n/ispolzovanie-nechetkih-kognitivnyh-kart-pri-razrabotke-raschetnyh-modeley-osnovaniy.
20. Meshalkin V.P., Belozersky A.Yu., Methodological foundations of an integrated risk management system of an industrial enterprise, Transport Business of Russia, No. 2, 2011, pp. 189-191. https://cyberleninka.ru/article/n/metodologicheskie-osnovy-kompleksnoy-sistemy-upravleniya-riskami-promyshlennogo-predpriyatiya.
21. Ginis L.A., Development of cognitive modeling tools for the study of complex systems, Engineering Bulletin of the Don, 2013, vol. 26, No. 3 (26), p. 66. https://cyberleninka.ru/article/n7razvitie-instrumentariya-kognitivnogo-modelirovaniya-dlya-issledovaniya-slozhnyh-sistem
22. Marigodov V. K., Analysis of the cognitive map of the learning system based on expert assessments, Visnik SevNTU. Ser.: Pedagogika. 2013. V. 144. pp. 77-80. http://www.irbis-nbuv.gov.ua/cgi-bin/irbis_nbuv/cgiirbis_64 exeC21 COM=2&I21 DBN=UJRN&P21 DBN=UJRN&IMAGE_FILE_DOWNLOA D=1&Image_file_name=PDF/Vsntup_2013_144_15.pdf
23. Oskin A.F., Oskin D.A., Application of fuzzy cognitive maps for modeling poorly structured systems, Bulletin of Polotsk State University. Series C, Fundamental Sciences. 2017. № 4. pp. 15-20. http://elib.psu.by:8080/handle/123456789/20350
24. Thomas L. Saaty "Axiomatic foundation of the analytic hierarchy process ", Management science, 1986, 32 (7), pp. 841-855.
25. Mohr S, Software design for a fuzzy cognitive map modeling tool. Tensselaer Polytechnic Institute, Troy, 1997 https://www.researchgate.net/publication/267701424_Software_Design_For_a_Fuzzy_Cognitive_Map_Modelin g_Tool
26. Aguilar, Jose & Contreras, José., The FCM designer tool, 2010 10.1007/978-3-642-03220-2_4
27. S. A. Gray, S. Gray, L. J. Cox and S. Henly-Shepard, Mental Model-er: A Fuzzy-Logic Cognitive Mapping Modeling Tool for Adaptive Environmen-tal Management, 46th Hawaii International Conference on System Sciences, 2013, 965-973. 10.1109/HICSS.2013.399
28. Franciscis, Dimitri. JFCM: A Java library for fuzzy cognitive maps. Intelligent Systems Reference Library, 2014, 54. 199-220. 10.1007/978-3-642-39739-4_12
29. Poczeta, Katarzyna & Yastrebov, Alexander & Papageorgiou, Elpin-iki. (2015). Learning Fuzzy Cognitive Maps using Structure Optimization Genetic Algorithm, 2015, 10.15439/2015F296
30. Papageorgiou, Elpiniki & Poczeta, Katarzyna & Laspidou, Chrysi. (2016). Hybrid Model for Water Demand Prediction based on Fuzzy Cognitive Maps and Artificial Neural Networks. 2016 IEEE International Conference on Fuzzy Systems (FUZZ). 10.1109/FUZZ-IEEE.2016.7737871
31. M. T. Hagan and M. B. Menhaj, "Training feedforward networks with the Marquardt algorithm," in IEEE Transactions on Neural Networks, 1994, vol. 5, no. 6, 989-993, 10.1109/72.329697
32. Simon Haykin, Neural Networks: A Comprehensive Foundation (2nd. ed.). Prentice Hall PTR, USA., 1998, 10.5555/521706
33. León, M., Nápoles, G., Rodriguez, C., Lorenzo, M., Bello, R., Vanhoof, K., A Fuzzy Cognitive Maps Modeling, Learning and Simulation Framework for Studying Complex System. IWINAC, 2011. https://www.semanticscholar.org/paper/A-Fuzzy-Cognitive-Maps-Modeling%2C-Learning-and-for-Le%C3%B3n-N%C3%A1poles/7e7f4de3fefi656c4a9170f4424262b839889c3#citing-papers
34. Nápoles, Gonzalo & Leon Espinosa, Maikel & Grau, Isel, Vanhoof, Koen, Fuzzy Cognitive Maps Tool for Scenario Analysis and Pattern Classifica-tion, 2017 10.1109/ICTAI.2017.00103
35. Nápoles, Gonzalo, Bello, Rafael, Vanhoof, Koen, How to Improve the Convergence on Sigmoid Fuzzy Cognitive Maps?. Intelligent Data Analysis, 2014, 18. 77-88. https://www.researchgate .net/publication/271832015_How_to_Improve_the_Convergence_on_Sigmoid_Fuzzy_ Cognitive_Maps
36. Nápoles, Gonzalo, Papageorgiou, Elpiniki, Bello, Rafael, Vanhoof, Koen, On the Convergence of sigmoid Fuzzy Cognitive Maps. Information Sci-ences, 2016, 349. 10.1016/j.ins.2016.02.040
37. Maksimov V.I., Grigoryan A.K., Kornoushenko E.K., Software package "Situation" for modeling and solving weakly formalized problems, International Conf. On management issues. Moscow, IPU RAS, June 29 — July 2, 1999 Moscow, 1999. Vol. 2. pp. 58-65.
38. Kulinich A.A., Maksimov V.I., The system of conceptual modeling of socio-political situations "Compass", Collection of dokl. "Modern management technologies", Scientific and practical. seminar "Modern management technologies for the administration of cities and regions". M., 1998. pp. 115-123
39. Avdeeva Z.K., Maksimov V.I., Rabinovich V.M., Integrated system "COURSE" for cognitive management of the development of situations, Tr. IPU RAS. M., 2001. Vol. XIV. pp. 89-114
40. Kulinich A.A., Cognitive decision support system "Canvas", Software products and systems. 2002. No. 3. pp. 2528.
41. Silov V.B. Strategic decision-making in a fuzzy environment. Moscow: INPRO-RES, 1995. 228 p.
42. Gorelova G.V., Radchenko S.A., Software system of cognitive modeling of sociotechnical systems, Izv. TRTU. Tem.. Actual problems of economics, management and law. Taganrog, 2004. № 4 (39). pp.218-227
43. Zabolotsky M.A., Polyakova I.A., Tikhonin A.V., Application of cognitive modeling in quality management of specialist training, Management of large systems. 2007. № 16. pp. 91-98
44. Korostelev D.A., Lagerev D.G., Suspensovsky A.G., Decision support system based on fuzzy cognitive models "IGLA", The Eleventh National Conference on Artificial Intelligence with international participation CII-2008, Dubna, September 28 - October 3, 2008. M., 2008. Vol. 3. pp. 327-329
45. Pylkin A.N., Kroshilin A.V., Kroshilina S.V., Methodology of cognitive analysis in the automation of material flow management, Computer science and Control systems. 2012.
46. Putyato M.M., Development of methods and algorithms for intellectual decision-making support based on fuzzy cognitive maps, dissertation of Candidate of Technical Sciences: 05.13.01 / Putyato Mikhail Mikhaylovich; [Place of defense: Kuban State Technical University. un-t]. Krasnodar, 2010. 152 p.: ill. RGB OD, 61 11-5/1597
Authors:
Alina V. Petukhova, Senior engineer, JohnSnowLabs, Stavropolskaya st., 149, Krasnodar, Russian Federation, petukhova. [email protected]
Anna V. Kovalenko, Dr.Sci. (Eng.), associate professor, Head of the Department of Data Analysis and Artificial Intelligence, Kuban State University (KubSU), Stavropolskaya st., 149, Krasnodar, Russian [email protected]
УДК 004.81 10.23947/2587-8999-2022-1-2-81-95
МЕТОДЫ ПРОГНОЗИРОВАНИЯ РАЗВИТИЯ СЛОЖНЫХ СИСТЕМ С ПРИМЕНЕНИЕМ ТЕОРИИ НЕЧЕТКИХ КОГНИТИВНЫХ КАРТ
А.В. Петухова1, А. В. Коваленкон2
Университет гуманитарных наук и технологий Люсофона, Лиссабон, 1749-024, Португалия 2ФГБОУ ВО «Кубанский государственный университет», Краснодарский край, Краснодар, 350040, Россия
В статье представлен обзор методов решения задач прогнозирования временных рядов и моделирования сложных систем с использованием нечетких когнитивных карт (НКК). Перечислены основные алгоритмы, используемые при решении практических задач для сложных слабоструктурированных систем, позволяющие улучшить точность и надежность результатов моделирования. Для полноты обзора изучены и описаны публикации российских и зарубежных исследователей, работающих в данной области. Помимо вышеупомянутого, были перечислены основные программные средства, реализующие существующие алгоритмы и приведены их отличительные особенности для решения различных классов задач. Такое сравнение программных комплексов позволит исследователям определить оптимальную систему для проведения дальнейших теоретических или практических работ.
Ключевые слова: нечеткие когнитивные карты, программные комплексы для поддержки принятия решений, системы принятия решений, прогнозирование временных рядов.
Авторы:
Петухова Алина Владимировна, Старший инженер, JohnSnowLabs, РФ, г. Краснодар, ул. Ставропольская,149, [email protected]
Коваленко Анна Владимировна, Доктор технических наук, доцент, Заведующий кафедрой анализа данных и искусственного интеллекта, Кубанский государственный университет (КубГУ), РФ, г. Краснодар, ул. Ставропольская,149, [email protected]
