A_A0008: FUZZY MODELLING APPROACH FOR PREDICTING GOLD PRICE BASED ON RATE OF RETURN

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1 Section A - Mathematics / Statistics / Computer Science 13 A_A0008: FUZZY MODELLING APPROACH FOR PREDICTING GOLD PRICE BASED ON RATE OF RETURN Piyathida Towwun,* Watcharin Klongdee Risk and Insurance Research Group, Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand * piyathida_t@kkumail.com Abstract: This paper we wish to forecast the gold price based on the data from the London gold market, using recent methodologies such as fuzzy modelling. It is very important to raise an accurate model that is suitable to value of gold price as it develops. In this paper, we develop a new procedure based on fuzzy theory to forecast the gold price from January 2008 to May 2013 by applying the gold price in London January 1990 to December 2007.We compare the accuracy with Markov chain, moving average, and fuzzy sets. We found that the fuzzy set forecasting is more accurate than Markov chain and moving average. Introduction: Gold is one of the precious metals which is the most popular item for investment. Gold has been widely used throughout the world as jewelry since the past of humanity that existence over 6000 years ago. Furthermore, it has been also used as currency in the economic processes through both local and international trade until nowadays. Generally, most investors buy gold in order to prevent any economic, political, or currency crisis, as well as inflation, war, and social unrest. Nowaday, as we look at the situation of world economic, we see that the price of gold has slightly increased by demand and supply of consumption. Our research is base on the gold price of London market from January 1990 to December From this data, we see that gold price has significantly fluctuated monthly. Especially in 2010, the gold price rapidly increased but fell down in In the point of view of most investors, they wish to predict and analyze the trend of gold price and also its rate of return. In our research, we wish to forecast the gold price based on the data from London market as mentioned above using recent methodologies, such as neuro fuzzy. It is very important to raise the accuracy model that is suitable to value of gold price as it develops. In this thesis, we develop a new procedure based on fuzzy theory to forecast the gold price from January 2008 to May 2013 by applying the gold price in London January 1990 to December Fuzzy sets are also used in modeling the gold price. Fuzzy sets were invented by Zadeh in 1965 in his doctoral thesis, which discussed the uncertainty and vagueness of the definition of natural phenomenon. Zedeh therefore defined the nature of those in the form of fuzzy sets, for example, the set of people categorized as childhood, adolescence, and old age is, people in each age range. We cannot determine whether one is in early middle age and old age, and how the separation is. By fuzzy sets, it is based on the membership function. This will break up the membership level to a value between 0 and 1. The membership function is a function which is defined as the level of the variable to use. Assigning values to members that are not clear, uncertain and ambiguous is the most important operation of fuzzy. The shape of the membership functions are critical thinking and problem solving. The membership function can be either asymmetric or symmetric in all respects. Since the pioneering work of Zadeh, fuzzy set theory has been applied to a myriad of areas. Song and Chissom 3 introduced the concept of fuzzy time series and applied some methods to the enrollments of the University of Alabama. In recent years, a number of techniques based on fuzzy set theory have been proposed for forecasting. These methods have either used enrollment numbers or differences of enrollments as the universe of discourse. The propose method results is better forecasting accuracy than existing models.

2 14 Section A - Mathematics / Statistics / Computer Science We compare the accuracy of the prediction using the testing data from January 2008 to May 2013 with two criterions, namely, mean square error (MSE) and mean absolute error (MAE), which are given by: (1) and (2) In this paper, we consider the gold price from January 1990 to May 2013, then study and separate the data into two parts, namely, training data (from January 2005 to December 2007) and testing data (from January 2008 to May 2013). We compare the accuracy of four membership functions, namely, trapezoid function, triangular function, gable function and lotus function, the same average rule find the best function in terms of using in order to MSE and MAE. We divide the data into two parts using the 70:30 ratios to obtain training and testing data, respectively. The rate of return from January 1990 to May 2013 can be explained by the graph in Figure 1. Figure 1. Rate of return of gold price from January 1990 December 2013 Methodology: Modelling We separate the data series of gold price from January 1990 to May 2013 into two parts; the first part, data series from January 1990 to December 2007, is considered for training, and the second part, data series from January 2008 to May 2013, is considered for testing the accuracy of the Markov chain, moving average, and fuzzy sets. We assume that the gold price is described by the random process, where is the gold price per unit weight at time. Therefore, rate of return of gold price at time is defined by the random variable Obviously, the gold price at time can be written by

3 Section A - Mathematics / Statistics / Computer Science 15 Next, let be the sequence of rate of return, be a number of states, and If ( ], then we say that is in the state where = L + ( j-1 ) for j=1,2,,s. Moreover, if =, then we say that is in the state. Thus, we obtain the middle point of the states as Fuzzy prediction Fuzzification. step The gold price from January 1990 to December 2007 is used to determine the rate of return. Then the rate of return is used to build the relative transition matrix. The membership function can be obtained by triangular function, and trapezoid function. We set the number of states s = 9. Triangular function. Let be the membership function of the state defined by where j=1, 2, 3,,9. Trapezoid function. Let be the membership function of the state defined by where j=1, 2, 3,, 9. In this paper, we construct two new membership functions, namely, gable membership function and lotus membership function. Gable function. Let be the membership function of the state defined by where j=1,2,3,,9.

4 16 Section A - Mathematics / Statistics / Computer Science Lotus function. Let be the membership function of the state defined by where j=1,2,3,,9. Next we construct the fuzzy controls (transition matrices) of each membership functions as mentioned previously by the following conditions: Let be the membership function of state j=1,2,,t. Case 1 If, then we say that the state move to the state at time step n, denoted by j, respectively. Case 2 If, then we say that the state move to the state and the state at time step n, denoted by j and j respectively. Case 3 If, then we say that the state move to the state and the state move to the at time step n, denoted by i and j. Case 4 If and, then we say that the state move to the state and the state move to the state move to the state and the state move to the state at time step n, denoted by i, i j j Now, we can define the relative transition matrix, T= by

5 Section A - Mathematics / Statistics / Computer Science 17 Table 1. Relative transition matrix (rate of return of gold price) A1 A2 A3 A4 A5 A6 A7 A8 A9 A A A A A A A A A Step 2 Inference. We use the data from Table 1 to make the rules defined by the relative transition matrix: if x is A, then y is B, where A and B are linguistic values of the fuzzy set of x and y, respectively. Rule A If is the rate of return at time, then the forecasting value is where is the membership degree of state and is the middle point of state. Step 3. Accuracy Compare the error of rate of return for each function with rate of return of testing data. Step 4. Defuzzification Choose the most accurate function and predict the rate of return of June Then, convert it back to the silver price of June Moving average prediction A moving average is commonly used with time series data to smooth out shortterm fluctuations and highlight longer-term trends or cycles. The threshold between shortterm and long-term depends on the application, and the parameters of the moving average will be set accordingly. For example, it is often used in technical analysis of financial data, like stock prices, returns or trading volumes. It is also used in economics to examine gross domestic product, employment or other macroeconomic time series. Mathematically, a moving average is a type of convolution and so it can be viewed as an example of a low-pass filter used in signal processing. When used with non-time series data, a moving average filters higher frequency components without any specific connection to time, although typically some kind of ordering is implied. Viewed simplistically it can be regarded as smoothing the data. In the next rule, we will use the moving average for 3-month defined as follows. Rule 3MA. The forecasting value at time n+1 is given by where is the rate of return at time k = n-2, n-1, n.

6 18 Section A - Mathematics / Statistics / Computer Science Markov chain forecasting We classify the rate of return of gold price into 9 states to construct the transition matrix and the rule for forecasting, next we test the rule by using the gold price from January 2008 to May 2013 and using the historical gold price from January 1990 to December 2007 (204 months) to create transition matrix. Based on the rate of return, we obtain the transition matrix as The middle point matrix is obtained by Next, we present the rule of forecasting. Rule MC. If is in the state, then the forecasting value is where is the middle point of the state, j=1, 2,,9. Results and Discussion: Converting the rate of return to forecast the gold price Next, we show the gold price prediction which are obtained from the rate of return of each model. Comparison of the accuracy of fuzzy set, Markov Chain, and moving average Next, we shall compare the accuracy of Markov chain, moving average and fuzzy set, by using testing data from 2008 to We use two criterions, namely, MSE, and MAE as Equation (1) and (2), respectively, and we obtain Table 3.

7 Section A - Mathematics / Statistics / Computer Science 19 Table 2. Gold price prediction Date (USD) a (USD) b (USD) c (USD) d Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun a Original data b Markov Chain (Rule2) c moving average d Fuzzy (trapezoid) Table 3. Comparison of the accuracy of fuzzy set, Markov Chain, and moving average Sign Accuracy Criterions MSE AME Fuzzy (trapezoid) 96.20% Markov Chain 84.62% Moving Average(3MA) 77.77%

8 20 Section A - Mathematics / Statistics / Computer Science Figure 2. Left is fuzzy (trapezoid) and right is Markov Chain. Figure 3. Moving average. In the three figures above, the blue cross represents the original data and the red circles represent the prediction. Figure 4. Prediction gold price by Trapezoid blue line represent the original data and the red line represents the prediction.

9 Section A - Mathematics / Statistics / Computer Science 21 Figure 2 and 3 show the comparison of the predictions rate of return using fuzzy set, Markov Chain, and moving average. The data on the left is the training and the data on the right is the testing data, respectively. The success this fuzzy (trapezoid) forecasting is more accurate than Markov-chain and moving average. Conclusions: The purpose of this research is to develop a model to predict the price of gold. We use the real price of gold in London from January 1990 to May 2013 and use data from January 1990 to December 2007 for training in the data. We check the error by using MSE, and AME and compared to other methods. We conclude that using trapezoid forecasting is more accurate than triangular, gable and lotus function by Rule A. Moreover, this fuzzy (trapezoid) forecasting is more accurate than Markov-chain and moving average. References: 1. Chen S. IEEE Systems. 2000;30: Stevenson M, Porter J. World Acad Sci, Eng Technol. 2009; 55: Song Q, Chissom S. Fuzzy Set Syst. 1993;54: