MODELING INVESTOR REASONING USING FUZZY COGNITIVE MAPS Dimitrios E. Koulouriotis, Ioannis E. Diakoulakis* and Dimitrios M. Emiris Department of Production & Management Engineering, Technical University of Crete, Chania, Greece, emails:{jimk,emiris@dpem.tuc.gr} *Athens University of Economics & Business, Athens, Greece ABSTRACT: Modeling in financial markets is an emerging topic of research and development, especially due to the explosive progress and diffusion of artificial intelligent tools, e.g. neural networks, hybrid fuzzy systems etc. From the numerous studies conducted until now, valuable conclusions have been drawn, the most important being the recognition of forecasting limits imposed by the current techniques and the necessity to search for modeling frameworks able to support and exploit common benefit/risk principles. The concept of approximating investors’ reasoning process, which embraces the notions of profit and risk, is a promising direction for decision analysis and forecasting purposes. In search of the appropriate mathematical background to underpin such a scheme, Fuzzy Cognitive Maps constitute an interesting alternative. In the context of this work, a stock price forecasting system, modeling investors’ reasoning, is designed and tested under real-world data, paired with technical modifications of Fuzzy Cognitive Maps in order to enhance the effectiveness and accuracy of the proposed model. KEYWORDS: Investor Reasoning, Stock Price Forecasting, Fuzzy Cognitive Maps, Evolution Strategies 1. INTRODUCTION Artificial Intelligence, and particularly, the Computational Intelligence Techniques, is considered a firm and powerful context for the development of models simulating financial markets. Neural networks and fuzzy systems constitute the two major dimensions of AI already implemented in extent and hybridized for forecasting and trading purposes. Indicative samples are the studies [1],[2],[3],[4] while an extensive review comprises contemporary stock market prediction applications is given in [5]. The core conclusions of these studies are underlined herein: (a) various structures of neural nets, fuzzy systems and hybrid models have been to a high degree exploited and therefore improvements with them are under consideration, (b) the design of simply structured systems, based on profit and risk measures (retaining low dimensionality), is the right beginning for developing an effective forecasting and trading model, and (c) the fusion of diverse techniques and search for innovative paradigms to adopt are encouraged so as to surpass current restrictions and deficiencies. The modeling of investors’ reasoning could be seen as a promising approach for simulating and forecasting stock market, as in general the main force changing the market equilibrium emanates from different investors’ behavior. The significance of such an approach augments as its theoretical background is in accordance to the guidelines extracted by the previous studies. It is important to notice that investors’ reasoning and eventually decision-making involves the notion of cause-effect relationship; such a behavior is exhibited when combining information concerning diverse agents and the perceived value and strength of each one is affected by the other closely related concepts. As Fuzzy Cognitive Maps (FCMs) provide a mechanism, which represents the degrees of causality among concepts/agents/events and construct paths to propagate causality through the use of forward and backward chaining, the simulation of investors’ reasoning with FCMs is considered an interesting approach. This idea formulates the basic topic of this study; theoretical discussions about the nature of decision-making in financial markets, technical modifications on the FCM functional framework and an implementation of the proposed FCM-based forecasting model take place. 2. ANALYSIS OF CMs/FCMs The basis of the contemporary methodological framework underlying Cognitive Maps (CMs) was formulated by Axelrod in 1976 [6], who focused on the analysis of political and social systems. The main idea was about the eunite 2001 386 www.eunite.org representation of environments containing interrelated agents (usually called concepts), where the relations between the constituent entities have a cause-effect form. The causality of such relationships may be positive or negative and moreover several subjects related to the direct, indirect and cumulative effects, as well as the impact of feedbacks, exist. The representation scheme of CMs, established all these years through numerous studies, is a typical network/graph where the nodes/vertices express the system agents and the links/edges the cause-effect relationships between them. The major mathematical support of CMs emanates from the Crisp Binary Relations. In brief, any relation between two crisp sets X=[x1,x2,…,xn] and Y=[y1,y2,…,ym], where X,Y⊆ ℜ , is known as binary relation, usually denoted by: (1) R(X,Y)={((xi,yj), R(xi,yj))|(xi,yj)∈X × Y} where 1≤i≤n, 1≤j≤m, the symbol “ × ” denotes the Cartesian Product and the R(xi,yj) expresses the membership grade (also degree of compatibility or degree of truth [7]]) of the relation between the elements xi and yj. Through the membership grade, the presence or absence of association, interaction or interconnection (interconnectivity) between the elements of the two sets is captured [8]. For crisp binary relations, R(xi,yj)∈{0,1}; if R(xi,yj)=1 then the presence of association between the elements xi and yj is expressed, while in the opposite case, that is if R(xi,yj)=0, the nonexistence of association is denoted. When Y=X, then the relation R(X,X) or R(X2) is called a Directed Graph or Digraph, apparently because of the representation characteristics [7],[8],[9]. In fact, a Cognitive Map resembles a directed graph. This can be easily explained if it is accepted that X=Y=[x1,x2,…,xn] and the elements xi correspond to the CM agents. In this case, the network links express the membership grades of the existing binary (cause-effect) relations. To avoid confusion, a CM is a concise form of a digraph as, in theory, the directed graphs do not have negative membership grades. To clarify this subtle point, Kosko defined each CM concept Ci as: “…the fuzzy union (disjunction) of some fuzzy quantity set Qi and associated dis-quantity set ~Qi, where ~Qi can be thought as the abstract negation, or local fuzzy set complement, of Qi…” [9]. As an extension of the crisp binary relations, the Fuzzy Binary Relations may be defined. This fuzzy version implies that varying degrees of membership are permitted within the relations. In this case, R(xi,yj)∈[0,1], where the higher the membership grade, the stronger the relationship between xi and yj. Despite that the fact that any ordered set of elements could be used for representing the membership function, the interval [0,1] has been widely established as the most useful approach. In case of a fuzzy binary relation R(X,Y) where additionally X=Y, then we refer to Fuzzy Graphs [8]. The FCMs, as the extension of CMs, are built as fuzzy graphs. Obviously, positive (negative) values on the edges of FCMs declare degrees of association of the corresponding positive (negative) cause-effect relationships. Following the logic applied for the links, the nodes take values expressing the degree of belief of their state changes. To become specific, in typical CMs ∀i∈{1,…,n}, Ci∈{0,+1}, while in case of the Kosko’s modification Ci∈{-1,0,+1}. Correspondingly, in FCMs, ∀i∈{1,…,n}, Ci∈[0,+1] or [-1,+1] if the dis-concepts are used. Interpreting the positive (negative) signed nodes, the more positive (negative) the assigned values, the stronger the degree of belief about the increase (decrease) of the corresponding concept states. From a more practical perspective, the operation of FCMs is build on an iterative mechanism, which propagates in the network the initial node stimulations and, through this way, estimates the direct and indirect effects ending in each concept. Analytical presentation of the inference mechanism is given in study [10]. Among the goals of the current work remain the enforcement of functionality and applicability of FCMs, in order to support the investor reasoning model presented in Section 3. Therefore, diverse modifications and extensions on the FCM inference and representation formula are proposed. From a critical perspective of the current approaches in cognitive mapping, it appears that the theoretical (and practical) approaches of FCMs remain an open field of research and development. This remark is easily supported by the observation of many divergent approaches followed by various researchers. To begin with, one significant source of ambiguity underlying FCM theory is the disparity in the meaning of “size of effect” and “degree of association”. As previously noted, the backbone of FCMs is the fuzzy binary relations. This kind of relations refers to degrees of association between the elements of crisp sets; in other words, the edges in FCMs express the degree to which the depicted cause-effect relations take place. Influenced by the aforementioned discussion about the disparity in the meanings of “size of effect” and “degree of association”, the use of real numbers on the network edges, expressing the size of the corresponding effects, is adopted. In parallel, following the prompt of Hagiwara [11], non-linearity is considered the typical extension of the fixed real numbers, as in fact the fixed-valued FCM edges express the constant derivatives of linear functions. To assign nonlinearity on FCM edges, sigmoid functions like the one given in equation (2) could be used. EFFECT = efAB(CA)= + A →B 2 1+ e − α⋅C A −1 (2) where CA: the state of node A, efAB(CA): the cause-effect function and α: a non-negative critical parameter (α≥0). It is important to notice that in case that the “size of effect” approach is adopted then the node values should not be regarded as “degrees of belief” about the corresponding concepts state changes but, to the contrary, they express scaled or fuzzified measures of the real world concept changes. One additional modification proposed in this study concerns the oscillations of the concept states produced during the iterative process of the inference mechanism. The final condition of a CM may be an equilibrium, a limit cycle or a eunite 2001 387 www.eunite.org chaotic behavior. Unfortunately, in complex structured environments, the limit cycles constitute a common phenomenon. The management and effective use of the information extracted by such cycles is not an easy task as, in most cases, the oscillations occurring at each FCM node are very strong, that is the states of the concepts reach extreme values. As a consequence, effectiveness and accuracy are somehow undermined if the limit cycles are accepted as they are. To circumvent this feeble point, a property met at real-world systems may be exploited in order to construct a more appropriate FCM inference mechanism. To become precise, in diverse environments the constituent elements, as time passes, cannot oscillate between extreme values. Their capability to hardly change their states is gradually diminished. Making a metaphor of the aforementioned property, the change of FCM concept states between two subsequent time steps, that is iterations of the inference mechanism, is expected to be gradually decreased and therefore become less than a parameter, which is time-related. This constraint for a potential node Ci is given in equation (3). (3) Δt(Ci)= | Cit − Cit −1 |≤ e − ξ⋅t where t: the symbol of time/iterations and : a non-negative critical parameter which controls how fast must be the convergence of the FCM node values as time passes. Last, a modification of the inference mechanism, already proposed in a previous work [10], is briefly analyzed. This modification is related to the multiple/simultaneous stimulation of diverse concepts. The typical FCM algorithm assumes that the initially stimulated nodes have their states constant during the iterative process. However, this logic enables some weak characteristics on causal reasoning because of the cumulative behavior of the effects. The problematic reasoning becomes even stronger when there are cause-effect relationships between the initially stimulated concepts. The proposed modification acts at two levels: the first level manages the total/final effects accumulated at each FCM node from the discrete activation of the initially stimulated agents, while at a subordinate level, the effects produce at each iteration of the inference mechanism are handled according to the following equations: MAXit = max {efji( C j t ):efji( C j t )>0}- max {| efji( C j t )|:efji( C j t )<0} (4) 1≤ j≤ n 1≤ j≤ n n n j =1 j =1 t t t t t ∑ i = ∑ {efji( C j ):efji( C j )>0}- ∑ {|efji( C j )|:efji( C j )<0} (5) (6) Cit+1 = (1-βi)( MAXit )+ βi( ∑ it ) where n: number of FCM nodes, 1≤ i, j ≤n, t: notation for iterations and βi (0≤βi≤1): the cumulative behavior parameter of node i, which is user-defined and adjusts the degree of influence (significance) of the (positive and negative) maximum effects in relation to the cumulative effects. Values of parameters βi close to 1 reduce the level of significance of the maximum effects (additive behavior is stronger) while small values reinforce the influence of the strongest effects (“winner takes all” or “elitist” behavior). 3. APPROXIMATING INVESTORS’ REASONING The backbone of the current paper concerns the development of a short-term stock price forecasting model exploiting raw market data. The idea behind the proposed scheme is the approximation of investors’ behavior and reasoning, which is considered the most important parameter influencing the market situation. To make decision in short-term horizon, investors are based mainly on the following concepts: (a) trend: the most common concept for describing the current and expected condition of the market, (b) profit: an instant measure of the stock investment expected value, (c) price variability: it is related with the risk and the strength of the market, (d) number of trading orders: it implicitly shows the strength of the market, and (e) supply-demand forces: the critical factors determining the stock price level. One significant characteristic of humans when making decisions is the cumulative, parallel and abstractive evaluation of the available data. For instance, the above listed concepts of stock trend, variability etc. are all taken into account and combined, according to undefined rules and abstract structures, in order to produce the ultimate trade decision. Trying to simulate this procedure, the adoption of linear schemes is the most naive approach, which obviously constituted the first manifest attempt in the scientific community. As an extension, the formulation of non-linear functions and moreover the application of intelligent techniques (e.g. neural networks, fuzzy and neurofuzzy methods) improved the accuracy but achievements are still far away from the real human reasoning mechanism [5]. Influenced by the previous applications of FCMs in stock market, such as those of Koulouriotis et al. [12] and Lee and Kim [13], an innovative connection of FCM framework with the problem of modeling investors’ reasoning is attempted. The basic notion, which is able to support such a link, springs from the observation, and generic conception, that humans formulate their opinions about various concepts in an environment with many factors directly or indirectly affecting these opinions. In other words, humans adapt their objective perception about diverse things when related factors are activated. Such a behavior is also valid in stock market. As an example, if an investor, who pursues his profit maximization, analyzes a stock with an extremely favorable profit during the prior period, then his opinion about the stock profitability and of course his final decision will be undoubtedly influenced when the size of risk attached to the eunite 2001 388 www.eunite.org specific stock is learned. The higher the risk, the less attractive and profitable will the stock be considered and, eventually, the less willing will the investor be to buy the stock. Generally, decision-making encloses the notion of implicit cause-effect relationships that underlie the ability to perceive and evaluate situations and things. Following the aforementioned rationale, an abstract model of investors’ reasoning can be built on an FCM including the concepts of: (a) stock trend, (b) stock profit, (c) market profit, (d) supply, (e) demand, and (f) stock price distortions. Indeed, each of the constituent elements when evaluated is strongly affected by the remaining ones and, ultimately, all of them comprise a mechanism determining the trading decision, which may be seen as an additional FCM node (output node) and used as a prediction signal for the forthcoming stock price movements. In parallel, from a strictly mathematical perspective, the constructed FCM is considered as a non-linear system with many variables and degrees of freedom. In order to avoid the drawbacks imposed by the use of qualitative terms for the representation of the FCM concepts, the alternative of quantitative measures was imperative; the trend, supply etc. were estimated or better approximated with indices produced only by daily stock market data. Analytically, the concepts correspond to the following indices: TrendT= m1 , where Pricei=m0+m1·i, t≤i≤t+T α1 ⋅ Pr ice t +1 Stock_ProfitT= ((Pr ice t + T − Pr ice t +1 ) / Pr ice t +1 ) /(α2·Τ) Market_ProfitT= ((Market t + T − Market t +1 ) / Market t +1 ) /(α3·Τ) (7) (8) (9) T DemandT= ( ∑ {(High t + i − Pr icet t + i −1 ) / Pr ice t + i −1}) /(α4·Τ) (10) SupplyT= ( ∑ {( Low t + i − Pr icet t + i −1 ) / Pr ice t + i −1}) /(α5·Τ) (11) i =1 T i =1 (12) Price_DistortionsT=ρ (OrdersT, VariabilityT) where α1,…,α5≥0, T: a period of days, OrdersT: the vector of daily buy and sell orders during the period T, VariabilityT: the vector of daily percentage price changes during the period T, and ρ(.): the correlation coefficient. The parameters α1,…,α5 are user-defined and their role mainly concerns the adjustment of the indexes level so as to be appropriate for the use in the FCM reasoning mechanism. The parameters of the Investor FCM (IFCM), such as the type of the non-linear edge functions or the membership grades assigned to the existing relationships, may not be constant but variable under different stock market situations or for specific categories of stocks, such as banks, insurance companies and others. This simply means that a learning algorithm is needed in order to determine the optimum, case-sensitive, configurations. Evolution Strategies (ES) constitute the most appropriate methodology for the specific application due to their favorable representation capabilities and the powerful evolution and self-adaptation mechanism. ES imitate the principles of natural/biological evolution, with ultimate goal, in most cases, the solution of optimization problems. In detail, ES assume the existence of a population of individuals, each of which represents a search point in the space of potential solutions. The individuals take their initial values in a random manner and afterwards evolve successively to better regions of the search space (according to their fitness value, which correspond to the objective function of the optimization problem). The core evolution processes are recombination, mutation and selection. During recombination, pairs of individuals combine their characteristics through a random exchange of genetic information while during mutation, the individuals undergo random changes in a portion of their characteristics. Finally, selection aims to discern the best individuals and form the parental population of the next generation. The fine-tuning of the IFCM is explicitly assigned to this combinatorial mechanism. More explanatory details may be found in [14]. 4. EXPERIMENTAL RESULTS Before presenting the results produced by our tests, a brief description of the IFCM settings is needed. Firstly, the data selected for the current application were two stock time series and a composite index, all from the Athens Stock Exchange. One of the stocks and the composite index formulated the IFCM training set, while the other stock was selected as the testing set. In addition, the IFCM analysis concerned only 5-days stock price forecasting and the node values were indexes produced from 10-days periods (T=10). From equations (7)-(11), α1=α2=α3=0.04 and α4=α5=0.08. The mechanism presented in equation (3) was applied for all IFCM nodes with =0.03, and the multi-stimulus mechanism, that is equations (4)-(6), applied for all IFCM nodes. Focusing on the training procedure, the following settings must be clarified: - Two IFCM structures were analyzed. The first one considers that all causal functions were linear functions while the second IFCM structure considers that the causal functions are sigmoids such that of equation (2). - Viewing the training procedure as an optimization problem, the object variables to be found are in case of linear causal functions the derivatives of these functions while in case of sigmoids, the parameter α of equation (2). eunite 2001 389 www.eunite.org - The set of object variables for each IFCM structure contains also the cumulative effect parameters βi (1≤i≤7, 6 input and 1 output node). - Six different example sets were used for conducting training. The first two sets include random points extracted from the whole training stock time series, while the other four include points extracted from subsequent time series constituting the whole training stock time series. Considering the evolution strategies: - The default configurations are ( + ) strategy, generations=20, population=5 individuals, offspring/parents=6. - Two extreme cases were also tested (only for the example set 1), one with generations=50 and one with population=10 individuals. As far as the in-sample and the out-of-sample IFCM performance evaluation is concerned, the following measures were used: - In-Sample: the mean absolute error over the index= ((Pr ice t +5 − Pr ice t +1) / Pr ice t +1) /(0.04·5). - Out-Of-Sample: the percentage of accurate predictions of the sign of price change in a weekly basis. The results produced by our tests are given in Tables I-VIII. Due to the stochastic nature of evolution strategies, for each set of configurations, the training was conducted twice so as to draw more valid conclusions. Table I: Example Set 1 Linear Functions Table V: Example Set 5 Sigmoid Functions Linear Functions Sigmoid Functions In-Sample 0,2370 0,2104 0,1882 0,1827 In-Sample 0,2555 0,2555 0,1582 0,2264 Out-of-Sample 58,75% 71,25% 71,25% 72,50% Out-of-Sample 46,25% 45,00% 61,25% 48,75% Table II: Example Set 2 Linear Functions Table VI: Example Set 6 Sigmoid Functions Linear Functions Sigmoid Functions In-Sample 0,1830 0,1847 0,1978 0,2044 In-Sample 0,2136 0,1986 0,1684 0,2070 Out-of-Sample 52,50% 56,25% 43,75% 36,25% Out-of-Sample 30,00% 36,25% 30,00% 50,00% Table III: Example Set 3 Linear Functions Table VII: Set: 1 & Double Population Sigmoid Functions Linear Functions Sigmoid Functions In-Sample 0,1355 0,1616 0,1596 0,1208 In-Sample 0,2343 0,1938 0,2245 0,2019 Out-of-Sample 40,00% 36,25% 40,00% 30,00% Out-of-Sample 56,25% 70,00% 66,25% 67,50% Table IV: Example Set 4 Linear Functions Table VIII: Set: 1 & Double Generations Sigmoid Functions Linear Functions Sigmoid Functions In-Sample 0,2172 0,2365 0,1869 0,1713 In-Sample 0,2345 0,2331 0,1621 0,1583 Out-of-Sample 47,50% 37,50% 56,25% 60,00% Out-of-Sample 63,75% 63,75% 72,50% 68,75% Beginning the analysis of experimental results with issues related to the algorithmic dimension, we may underline that the computational enhancement imposed by the non-linear causal functions and the mechanism managing FCM node oscillations proved valuable. Indeed, when the advanced approach was adopted, the in-sample performance improvement, that is (Errornon-linear-Errorlinear)/Errorlinear, was on average 12.5% while the out-of-sample forecasting accuracy was increased about 4% (that is Accuracynon-linear-Accuracylinear=0.04). All applied evolution strategies could not efficiently optimize the IFCM parameters; even the increase of population size and generations horizon could not offer significantly better results. Additionally, over-training problems exhibited. Even though a large number of generations during the evolution process hardly improve the in-sample performance, the out-of-sample behavior does not follow the same trend. Intense training on specific examples affects the generalization capabilities of the IFCM; however, it is important to notice that over-training proved quite favorable in several anomalous periods where typical linear or non-linear techniques would definitely fail. Focusing on the specific application properties, the IFCM demonstrated a quite interesting forecasting ability comparatively to other well-known methodologies, such neural networks and neural fuzzy systems [5]. Intense nonlinearities and complex patterns were revealed. This observation motivates the development of an analytical comparison with the currently advanced techniques, in order to extract conclusions about the most appropriate research directions so as to enhance stock price predictability; furthermore, exploiting information from many time periods with different stock market behaviors and conducting training on it is the optimum choice in order to make profitable predictions. To become precise, if we select a training set based on some expected generic market characteristics, such as the long-term trend, this is not enough to achieve a satisfactory performance even in case that the predicted behavior is fulfilled. This is the outcome of the inherent market attributes; a stable long-term trend may include intense price reversals and abrupt oscillations in short-term horizons, which finally strongly deteriorate the overall forecasting ability. As solution, the eunite 2001 390 www.eunite.org systematic selection/construction of training sets /patterns, capturing many different stock market behaviors is proposed. Last to mention, a performance exceeding 70% accuracy in a weekly basis is considered satisfactory when compared with other analogous studies. This observation paired with an optimism for achieving even better results are easily supported when enumerating the diverse degrees of freedom available (and able to be fine-tuned with a powerful optimization technique such as evolution strategies) and recognizing the great amount of information able to be extracted from different kinds of stock market indices and rules. 5. DISCUSSION Recognizing the necessity for innovative approaches in investor reasoning modeling, a Fuzzy Cognitive Map-based model is developed and tested under real world conditions. The basic idea behind the proposed simulation procedure is the attempt to approximate investors’ reasoning manner and as a consequence estimate a significant source of information about the future market trend. Among the basic assumptions behind the proposed model is the existence of cause-effect relationships in the generic decision making process, which can be managed appropriately by FCMs. In parallel, to support the developed stock market model, several theoretical and practical issues of FCMs are investigated and modified. The results of the real world application of the Investor FCM prove the significance of the new approach, shed light on important aspects in the domain of stock price forecasting and financial market modeling in general, like the role of examples used for training and the kind of information used for conducting simulation, and motivate the systematic research towards various directions, such as the search for more and better indexes indicating diverse market characteristics, the automation and optimization of the initially designed training-test procedure and the development of a database including patterns for many different situations able to provide the necessary information for a generic stock price forecasting model. REFERENCES [1] Koulouriotis D.E., Emiris D.M., Diakoulakis I.E. and Zopounides C. (2001), “Comparative Analysis and Evaluation of Intelligent Methodologies for Short-Term Stock Price Forecasting”, Fuzzy Economic Review, submitted. 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(1997), “A Fuzzy Cognitive Map-Based Bi-Directional Inference Mechanism: An Application to Stock Investment Analysis”, Intelligent Systems in Accounting, Finance & Management, vol. 6, pp. 41-57. [14] Koulouriotis D.E., Diakoulakis I.E. and Emiris D.M. (2001), “Learning Fuzzy Cognitive Maps using Evolution Strategies: a novel schema for modeling and simulating high-level behavior”, IEEE Congress on Evolutionary Computation, pp. 364-371. eunite 2001 391 www.eunite.org