A neuro-fuzzy system for portfolio evaluation

Transcription

1 A neuro-fuzzy system for portfolio evaluation Christer Carlsson IAMSR, Åbo Akademi University, DataCity A 320, SF Åbo, Finland ccarlsso@ra.abo.fi Robert Fullér Comp. Sci. Dept., Eötvös Loránd University, Muzeum krt. 6-8, H-088 Budapest, Hungary rfuller@ra.abo.fi Abstract Suppose that the value of our portfolio depends on the currency fluctations on the global finance market. Our knowledge is given in the form of fuzzy IF-THEN rules, where all of the linguistic values for the exchange rates and the portfolio values are represented by sigmoidal fuzzy numbers. It is relatively easy to create fuzzy IF-THEN rules for portfolio evaluation, however it is time-consuming and difficult to fine-tune them. In this paper we compute the crisp portfolio values by Tsukomoto s inference mechanism and introducing some reasonable interdependences among the linguistic terms we show a simple method for tuning the membership functions in the rules. Introduction Suppose that our portfolio value depends on the currency fluctations on the global finance market. For modeling the partially known causal link between the exchange rates and the portfolio value we employ fuzzy IF-THEN rules of the following type R i :ifx is A i and... and x n is A in then PV is C i where PV is the linguistic variable for the portfolio value, x,...,x n are the linguistic variables for exchange rates having effects on the portfolio value. Each x i has two linguistic terms low and high, denoted by L i and H i, which satisfy the equality L i (t)+h i (t) = for each t. The portfolio value can have four terms: very big (VB), big (B), small (S) and very small (VS). It is clear that the value of the portfolio can not be negative (in the worst case we lose everything). The membership functions for the portfolio are supposed to satisfy the properties B(t)+ S(t) =, VS(t) =S(t + c) and VB(t) =B(t c) for some constant c and for each t. We believe that two linguistic terms {low, high} are sufficient for exchange rates, because the term exchange rate is medium can be derived from the terms exchange rate is low and exchange rate is high. The final version of this paper appeared in: R.Trappl ed., Cybernetics and Systems 96, Proceedings of the Thirteenth European Meeting on Cybernetics and Systems Research, Vienna, April 9-2, 996, Austrian Society for Cybernetic Studies, Vienna, Presentlyvisiting professor at Institute for Advanced Management Systems Research.

2 In a similar manner we consider our portfolio value as small if its value is smaller or exceeds a little bit the value of our investment, and big if its value is definitively bigger than our investment. The term portfolio value is medium is rapidly changing and can be derived from other terms. Under these assumptions it seems to be reasonable to derive the daily portfolio values from the actual exchange rates and from the rule-base R by using Tsukomoto s reasoning mechanism, which requires monoton membership functions for all linguistic terms. Consider a simple case with the following three fuzzy IF-THEN rules in our knowledge-base: R : R 2 : R 3 : if x is L and x 2 is L 2 and x 3 is L 3 then PV is VB if x is H and x 2 is H 2 and x 3 is L 3 then PV is B if x is H and x 2 is H 2 and x 3 is H 3 then PV is S where x, x 2 and x 3 denote the exchange rates between USD and DEM, USD and SEK, and USD and FIM, respectively. The rules are interpreted as: R : If the US dollar is weak against the German mark Swedish and the Finnish mark then our portfolio value is very big. R 2 : If the US dollar is strong against the German mark and the Swedish crown and the US dollar is weak against the Finnish mark then our portfolio value is big. R 3 : If the US dollar is strong against the German mark the Swedish crown and the Finnish mark then our portfolio value is small. The fuzzy sets L = USD/DEM is low and H = USD/DEM is high are given by the following membership functions L (t) = + exp(b (t c )), H (t) = + exp( b (t c )) It is easy to check that the equality L (t)+h (t) = holds for all t. Figure Initial membership functions of x is low and x is high, b = 6 and c =.5. The fuzzy sets L 2 = USD/SEK is low and H 2 = USD/SEK is high are given by the following membership functions H 2 (t) = + exp(b 2 (t c 2 )), L 2(t) = + exp( b 2 (t c 2 )) It is easy to check that the equality L 2 (t)+h 2 (t) = holds for all t. 2

3 Figure 2 Initial membership functions for x 2 is low and x 2 is high, b 2 = 6 and c 2 =7.5. The fuzzy sets L 3 = USD/FIM is low and H 3 = USD/FIM is high are given by the following membership function L 3 (t) = + exp(b 3 (t c 3 )), H 3(t) = + exp( b 3 (t c 3 )) It is easy to check that the equality L 3 (t)+h 3 (t) = holds for all t. Figure 3 Membership functions for x 3 is low and x 3 is high, b = 6 and c =4.5. The fuzzy sets VB = portfolio value is very big and VB = portfolio value is very small are given by the following membership functions VS(t) = + exp(b 4 (t c 4 c 5 )), VB(t) = + exp( b 4 (t c 4 + c 5 )), Figure 4 Initial membership functions for PV is very big and PV is very small with b 4 =0.5, c 4 = 5 and c 5 =5. 3

4 The fuzzy sets B = portfolio value is big and S = portfolio value is small are given by the following membership function B(t) = + exp( b 4 (t c 4 )), S(t) = + exp(b 4 (t c 4 )) It is easy to check that the equality B(t)+S(t) = holds for all t. Figure 5 Initial membership functions for S and B, b 4 =0.5 and c 4 = 5. We evaluate the daily portfolio value by Tsukamoto s reasoning mechanism, i.e. L L2 L3 H H2 L3 α z H H2 H3 z 2 α2 a a2 a3 Figure 6 min α3 z 3 Tsukamoto s reasoning mechanism with three inference rules. The firing levels of the rules are computed by α = L (a ) L 2 (a 2 ) L 3 (a 3 ), α 2 = H (a ) H 2 (a 2 ) L 3 (a 3 ), α 3 = H (a ) H 2 (a 2 ) H 3 (a 3 ), The individual rule outputs are derived from the relationships z = VB (α )=c 4 + c 5 + b 4 ln α α, z 2 = B (α 2 )=c 4 + b 4 ln α 2 α 2 () z 3 = S (α 3 )=c 4 b 4 ln α 3 α 3 (2) 4

5 The overall system output is expressed as z 0 = α z + α 2 z 2 + α 3 z 3 α + α 2 + α 3 where a, a 2 and a 3 are the inputs to the system. 2 Tuningthe membership functions by gradient descent method We describe a simple method for learning of membership functions of the antecedent and consequent parts of fuzzy IF-THEN rules. A hybrid neural net [] computationally identical to our fuzzy system is shown in the Figure 7. a Layer Layer 2 Layer 3 Layer 4 Layer 5 L H L(a) H(a) α T N β βz a2 L2 T N β2 β2z2 z 0 H2 a3 L3 L3(a3) T α3 N β3 β3z3 H3 H3(a3) Figure 7 A hybrid neural net (ANFIS architecture [3]) which is computationally equivalent to Tsukomato s reasoning method. Layer The output of the node is the degree to which the given input satisfies the linguistic label associated to this node. Layer 2 Each node computes the firing strength of the associated rule. The output of top neuron is α = L (a ) L 2 (a 2 ) L 3 (a 3 ), the output of the middle neuron is α 2 = H (a ) H 2 (a 2 ) L 3 (a 3 ), and the output of the bottom neuron is α 3 = H (a ) H 2 (a 2 ) H 3 (a 3 ). 5

6 All nodes in this layer is labeled by T, because we can choose other t-norms for modeling the logical and operator. The nodes of this layer are called rule nodes. Layer 3 Every node in this layer is labeled by N to indicate the normalization of the firing levels. The output of the top, middle and bottom neuron is the normalized firing level of the corresponding rule β = α α + α 2 + α 3, β 2 = α 2 α + α 2 + α 3, β 3 = α 3 α + α 2 + α 3, Layer 4 The output of the top, middle and bottom neuron is the product of the normalized firing level and the individual rule output of the corresponding rule β z = β VB (α ), β 2 z 2 = β 2 B (α 2 ), β 3 z 3 = β 3 S (α 3 ), Layer 5 The single node in this layer computes the overall system output as the sum of all incoming signals, i.e. z 0 = β z + β 2 z 2 + β 3 z 3. Suppose we have the following crisp training set {(x,y ),...,(x K,y K )} where x k is the vector of the actual exchange rates and y k is the real value of our portfolio at time k. We define the measure of error for the k-th training pattern as usually E k = 2 (y k o k ) 2 where o k is the computed output from the fuzzy system R corresponding to the input pattern x k, and y k is the real output, k =,...,K. The steepest descent method is used to learn the parameters of the conditional and the consequence parts of the fuzzy rules. We show now how to tune the shape parameters b 4, c 4 and c 5 of the portfolio value. From () and (2) we get the following learning rule for the slope, b 4,of the portfolio values b 4 (t +)=b 4 (t) η E k = b 4 (t) η α + α 2 α 3 b 4 b 2 δ k, 4 α + α 2 + α 3 In a similar manner we can derive the learning rules for the center c 4 and for the shifting value c 5 c 4 (t +)=c 4 (t) η E k c 4 = c 4 (t)+ηδ k α + α 2 + α 3 α + α 2 + α 3 = c 4 (t)+ηδ k, c 5 (t +)=c 5 (t) η E k c 5 = c 5 (t)+ηδ k α α + α 2 + α 3 where δ k =(y k o k ) denotes the error, η>0 is the learning rate and t indexes the number of the adjustments. The learning rules for the shape parameters of the antecedent part of the rules can be derived in a similar way. 6

7 Figure 8 Final membership functions for S and B. Table shows some mean exchange rates, the computed portfolio values (using the initial membership functions) and real portfolio values from 995. Date USD/DEM USD/SEK USD/FIM Computed PV Real PV January, May 9, August, August 28, Table performance of the fuzzy system before the training. Table shows some mean exchange rates, the computed portfolio values (using the final membership functions) and real portfolio values from 995. Date USD/DEM USD/SEK USD/FIM Computed PV Real PV January, May 9, August, August 28, Table Performance of the fuzzy system after the training. References [] J.J.Buckley and Y. Hayashi, Neural nets for fuzzy systems, Fuzzy Sets and Systems, 7(995)

8 [2] S. Horikowa, T. Furuhashi and Y. Uchikawa, On identification of structures in premises of a fuzzy model using a fuzzy neural network, in: Proc. IEEE International Conference on Fuzzy Systems, San Francisco, [3] J.-S. Roger Jang, ANFIS: Adaptive-network-based fuzzy inference system, IEEE Trans. Syst., Man, and Cybernetics, 23(993)