2017 Vol: 6 Issue: 2
A Comparative Study of Artificial Neural Network Models for Forecasting USD/EUR-GBP-JPY-NOK Exchange Rates
Cagatay Bal, Serdar Demir,
Faculty of Science, Department of Statistics, Mugla Sitki Kocman University, Turkey.
E-mail: [email protected]
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Abstract
Forecasting exchange rates is one of the most common practical problems in the field of forecasting. Many various approaches for forecasting of exchange rates are discussed by many researchers from various disciplines. In recent years, artificial neural networks have become successful tool to obtain forecasts of the exchange rates. There are many different artificial neural network models which are used and developed for even better and trustable forecasts.
In this study, USD/EUR, USD/GBP, USD/JPY, and USD/NOK exchange rates are forecasted by modelling with different learning algorithms, activations functions, and performance measures. The results show that trial and error method is convenient to use for deciding appropriate parameter combination for artificial neural network modelling of the exchange rate data.
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Key Words: Activation function, Artificial neural network, Exchange rate, Forecasting, Learning algorithm, Performance measure.
1. Introduction
Forecasting future behaviour of exchange rates is one of the most important tasks for economical decision makers. Due to intensive uncertainty of exchange rates, this important task becomes very difficult to perform.
In many studies, it is pointed out that the discouragement in this field has been made after studies of Meese and Rogoff (1983a, 1983b) who stated simple random walk is the best economic model for exchange rate. Their investigation was relied on linear assumption, so that the result of their investigation is occurred naturally. According to uncertain and nonlinear structures of exchange rates, parametric approaches are limited to perform satisfactory results.
Kadilar et al. (2009) stated that researchers and investors hope that the mystery of exchange rate predictions can be solved by themselves by developing artificial neural networks (ANN).
Many study actually proved that the ANN as a type of non-linear model is a strong alternative for predicting exchange rates. ANN models give very favorable solutions, especially in the case of presence of complex, noisy, irrelevant or partial information. (Nagl,2000; Huang et al.,2007;
Kadilar et al.,2009)
Along with ANN, there are many approaches such as heuristic algorithms, soft computing methods, fuzzy inference systems and others for modelling. Conventional nonlinear techniques, such as Markov switching models have been used for modelling. However, the results of many studies generally show that classical nonlinear models do not improve exchange rate forecasts.
(Galeshchuk 2016)
The main purpose of this study is to compare different learning algorithms, activations functions, and performance measures for forecasting the time series data of the USD/GBP, USD/EUR, USD/NOK, and USD/JPY exchange rates. In next section, we introduce the components of ANN and exchange rates. Section 3 gives the properties of the Feed Forward Neural Networks (FNNN). Section 4 reports the information for exchange rates data and the results of the application. Finally, conclusions are expressed in Section 5.
2. Components of Artificial Neural Networks
As an exchange rate forecasting tool, ANN is one of the most popular approach among similar research fields. Learning ability is the most important characteristic of ANN which allows learning from examples, experiences, patterns, functional relation mapping and so on.
Learning algorithms make this process occurs and widely preferred popular algorithm is called Back Propagation (BP) was introduced by Werbos (1974). There are many BP algorithms such as Levenberg-Marquadt, BFGS Quasi-Newton, Resilient Backpropagation, Scaled Conjugate Gradient and many more in literature.
Activation function as main parameter of ANN allows nonlinear mapping within data.
Sigmoidal functions are the most common functions because of their S-shape. Also, there are
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some modified activation functions in the literature to conjugate different functions advantages together such as SigHyper introduced by Abubakar et al. (2014).
Architecture is another major parameter of ANN. Various types of architectures can be found in literature. For forecasting tasks, Feed Forward Neural Networks (FFNN) with one hidden layer is sufficient and widely used by authors in this field. As one of the first important study of ANN, Lapedes and Farber (1987, 1988) designed FFNN in their study for time series forecasting.
Shazly et al. (1997) compared forecasting performances for GBP, Deutsche Mark and JPY.
Zhang et al. (1998) evaluate GBP/USD exchange rate forecasting performance by the effects of different parameters of ANN. Kadilar et al. (2009) employed ANN to TL (Turkish Lira)/USD exchange rate to find best model for forecast accuracy. Majhi et al. (2009) searched efficient ANN models for prediction of exchange rates. Ye (2012) utilized BP neural network for forecasting Chinese currency RMB. Bissoondeeal et al. (2011) compared GBP/USD exchange rate forecasting performance of linear and nonlinear models. Galeshchuk (2016) explored ANN performance of USD/EUR, USD/ JPY, USD/GBP exchange rate series prediction. Lahmiri (2017) showed the effectiveness of ANN over conventional GARCH and EGARCH with the US/Canada and US/Euro exchange rate volatilities.
Another important component of ANN modelling is performance measure. Unlike parametric approaches, ANN has the disadvantage of giving the best model for every case in similar to almost all nonparametric approaches. There is no certain way to obtain best model from ANN. The most common and accepted way is trial and error method by out of sample performance of models. This method is data-aimed by the nature of approach and results hardly can be generalized. Applications must be reutilized again for every different data.
3. Feed Forward Neural Networks
Feed Forward Neural Networks (FFNN) is a type of multi-layer perceptron which has interconnection between all neurons in network and unlike different types of multi-layer perceptron such as recurrent networks, FFNN has no loops or circles within the architecture and signal flows through input layer to output layer in one direction. A FFNN structure can be seen in Figure 1.
Figure 1: A general structure of FFNN
In this study, FFNN has only one neuron in output layer, because our application data contains univariate time series.
As an important stage for ANN modelling, learning process can be described as finding the best weights between the layers. For this study, the training process carried out with 8 different learning algorithms: Levenberg-Marquardt Backpropagation (LM), BFGS Quasi-Newton (BFG), Scaled Conjugate Gradient (SCG), Conjugate Gradient with Powell/Beale Restarts (CGB), Fletcher-Powell Conjugate Gradient (CGF), Polak-Ribiére Conjugate Gradient (CGP), One Step Secant (OSS), Variable Learning Rate Backpropagation (GDX)
Activation functions enable mapping non-linear relationships between input and target values. The performance of neural networks depends on the suitable choice of activation function. In general, the activation function provides a measure of the non-linearity, so it plays important role in ANN modelling. In this study, Tangent-Sigmoid (T) and Logistic-Sigmoid (L) transfer functions was used as activation functions in hidden layers and linear function in output layers for comparing ANN models.
Determining the best architecture of ANN is too important issue in an application. The best architecture minimizes model prediction error. There are many different performance measures (or model selection criteria) which assess prediction errors by different reasoning. (Egrioglu et al. 2008)
The performance measures used for the application in this study are given in Table 1 by classifying based on functional similarities.
Table 1: Model Selection Criteria Based on Absolute, Percentage, Symmetric, Relative, Scaled and Others
Name of Criteria Formula
Mean Absolute Error MAE = 𝑚𝑒𝑎𝑛𝑖=1,𝑛|𝑒𝑖|
Median Absolute Error MdAE = 𝑚𝑒𝑑𝑖𝑎𝑛𝑖=1,𝑛|𝑒𝑖|
Geometric Mean Absolute Error GMAE = 𝑔𝑚𝑒𝑎𝑛𝑖=1,𝑛|𝑒𝑖|
Mean Square Error MSE = 𝑚𝑒𝑎𝑛𝑖=1,𝑛(𝑒𝑖2)
Root Mean Square Error RMSE = √𝑚𝑒𝑎𝑛𝑖=1,𝑛(𝑒𝑖2)
Fourth Root Mean Quadrupled Error R4MS4E = √𝑚𝑒𝑎𝑛4 𝑖=1,𝑛(𝑒𝑖4) Mean Absolute Percentage Error MAPE = 𝑚𝑒𝑎𝑛𝑖=1,𝑛|𝑝𝑖| Median Absolute Percentage Error MdAPE = 𝑚𝑒𝑑𝑖𝑎𝑛𝑖=1,𝑛|𝑝𝑖| Root Mean Square Percentage Error RMSPE = √𝑚𝑒𝑎𝑛𝑖=1,𝑛(𝑝𝑖2) Root Median Square Percentage Error RMdSPE = √𝑚𝑒𝑑𝑖𝑎𝑛𝑖=1,𝑛(𝑝𝑖2) Symmetric Mean Absolute Percentage Error SMAPE = 𝑚𝑒𝑎𝑛𝑖=1,𝑛(𝑠𝑖) Symmetric Median Absolute Percentage Error SMdAPE = 𝑚𝑒𝑑𝑖𝑎𝑛𝑖=1,𝑛(𝑠𝑖) Mean Relative Absolute Error MRAE = 𝑚𝑒𝑎𝑛𝑖=1,𝑛|𝑟𝑖| Median Relative Absolute Error MdRAE = 𝑚𝑒𝑑𝑖𝑎𝑛𝑖=1,𝑛|𝑟𝑖| Geometric Mean Relative Absolute Error GMRAE = 𝑔𝑚𝑒𝑎𝑛𝑖=1,𝑛|𝑟𝑖| Mean Absolute Scaled Error MASE = 𝑚𝑒𝑎𝑛𝑖=1,𝑛|𝑠𝑐𝑖|
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Root Mean Square Scaled Error RMSSE = √𝑚𝑒𝑎𝑛𝑖=1,𝑛(𝑠𝑐𝑖2) Nash Sutcliffe Efficiency NS = 1 −∑𝑛𝑡=1(𝑦𝑡−𝑦̌𝑡)2
∑𝑛𝑡=1(𝑦𝑡−𝑦̅)2
Each model selection criterion in Table 1 has advantages and disadvantages. It can be said that each measure based on mean has the same disadvantage as having high influences for outliers. In addition, it can be said that each measure based on median has the same disadvantages as long time calculation if data set is large. AIC and BIC, which are parameter penalize based in-sample information criterion, are not included into consideration as performance measures because of their disadvantage of ignoring alternative architectures (Bal, Demir and Aladag, 2016). Qi et al. (2001), also stated that these in-sample performance measures are not suitable for out-of-sample forecasting.
4. Application
The USD/JPY, USD/GBP, USD/EUR and USD/NOK exchange rate data has been taken from official website of Republic of Turkey Central Bank, http://evds.tcmb.gov.tr/. Time series contains weekly data from January 1999 to October 2016 and consist of 928 observations.
Graph of the data can be seen in Figure 2.
First 878 observations of data used for training and last 50 observations of the data was used for testing. The training set was used for ANN model development and the test set was used to compare model selection criteria. Predicted values obtained from testing compared with real values along with the performance measures to find best ANN model.
Obtained results consists of 8 different learning algorithms with 2 different activation functions and 21 different performance measures for 4 exchange rate time series. Total number of 336 different ANN models for each exchange rate has been achieved.
The results of the application can be seen in Table 2 and Table 3 for each exchange rate time series selected by each performance measure with error, 1-step ahead forecast value, correlation between actual test set and predicted test set values and activation function and learning algorithm parameters.
In this study, best models were chosen according to minimum error value of each performance measure. Overall results show that ANN can give satisfying results in most situation for different data sets. However, performance of ANN models differs according to data shape, activation functions and learning algorithms as parameters of ANN for mapping and learning, performance measures which evaluate data in the aspect of different functional behaviors and 1-step ahead forecast values to show how accurate can model forecast future observations.
For USD/JPY exchange rate, results from Table 2 show that parameter combination T-LM has majority success over all models were chosen by each performance measures. The model
with highest correlation between actual test set and predicted test set values for USD/JPY exchange rate has the parameter combination T-BFG. The best forecast model obtained for USD/JPY exchange rate has only 0.021 difference between actual value and forecasted value.
Related model was chosen by MdAE performance measure and model has the parameter combination T-CGP.
For USD/GBP exchange rate, results from Table 2 show that the parameter combinations of T-CGP and L-CGP has majority success over all models were chosen by each performance measures. The model with highest correlation between actual test set and predicted test set values for USD/GBP exchange rate has L- CGP parameter combination. The best forecast model obtained for USD/GBP exchange rate has only 0.00593 difference between actual value and forecasted value. Related model was chosen by RMSE, MSE, RMSSE, and NS performance measure and model has L-CGP parameter combination.
For USD/EUR exchange rate, results from Table 3 show parameter combinations of T-CGF, T-SCG, and L-GDX have proven nearly same success over all models were chosen by each performance measures. The model with highest correlation between actual test set and predicted test set values for USD/EUR exchange rate has T-CGF parameter combination. The best forecast model obtained for USD/EUR exchange rate has only 0.00557 difference between actual value and forecasted value. Related model was chosen by DA performance measure and model has L-CGF parameter combination.
For USD/NOK exchange rate, results from Table 3 show that T-CGB and L-CGB have majority success over all models were chosen by each performance measures. The model with highest correlation between actual test set and predicted test set values for USD/NOK exchange rate has T-SCG parameter combination. The best forecast model obtained for USD/NOK exchange rate has only 0.00516 difference between actual value and forecasted value. Related model was chosen by MDA performance measure and model has T-SCG parameter combination.
The test set predictions of successful models for each exchange rate can be seen in Figure 3.
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Figure 2: Graphs of USD/JPY, USD/GBP, USD/EUR, and USD/NOK exchange rate weekly
70,0000 80,0000 90,0000 100,0000 110,0000 120,0000 130,0000 140,0000
USD/JPY
1,1800 1,2800 1,3800 1,4800 1,5800 1,6800 1,7800 1,8800 1,9800 2,0800 2,1800
USD/GBP
0,8000 0,9000 1,0000 1,1000 1,2000 1,3000 1,4000 1,5000 1,6000
USD/EUR
4,7 5,2 5,7 6,2 6,7 7,2 7,7 8,2 8,7 9,2 9,7
USD/NOK
Table 2: Best Models Selected for USD/JPY and USD/GBP Exchange Rate Time Series
Error Forecast Correlation Parameter RMSE 0,0254167 103,73992 0,980571207 TLM MSE 0,000646 103,73992 0,980571207 TLM R4MS4E 0,0355067 102,76944 0,97600284 LGDX MASE 0,8847216 104,50902 0,98066374 TBFG RMSSE 1,2118453 103,73992 0,980571207 TLM MAPE 0,0326416 104,50902 0,98066374 TBFG MAE 0,0185557 104,50902 0,98066374 TBFG GMAE 0,0092135 103,82791 0,977901091 LCGF MdAE 0,0104082 103,87555 0,979089098 TCGP MdAPE 0,0188719 103,42136 0,977607814 TLM SMAPE 0,0322447 104,50902 0,98066374 TBFG SMdAPE 0,0188026 103,42136 0,977607814 TLM RMSPE 0,0452779 104,50902 0,98066374 TBFG RMdSPE 0,0188927 103,42136 0,977607814 TLM NS 0,039742 103,73992 0,980571207 TLM MRAE 0,0188834 104,50902 0,98066374 TBFG MdRAE 0,0105734 104,72942 0,977420623 LLM GMRAE 0,0092709 103,82791 0,977901091 LCGF MSMAPE 0,0322447 104,50902 0,98066374 TBFG DA 0,2653061 103,72078 0,980571207 TLM MDA 0,2653061 104,30033 0,980222889 LGDX
JPY
Error Forecast Correlation Parameter RMSE 0,0264605 1,2301125 0,956005133 LCGP MSE 0,0007002 1,2301125 0,956005133 LCGP R4MS4E 0,0429055 1,2339478 0,940146341 LCGB MASE 0,9549396 1,1975367 0,955311831 TCGB RMSSE 1,4852082 1,2301125 0,956005133 LCGP MAPE 77,088844 1,1315014 0,953128448 TCGP MAE 0,0170132 1,1975367 0,955311831 TCGB GMAE 0,0077884 1,2521395 0,946544493 LSCG MdAE 0,0095252 1,2379883 0,933346754 LBFG MdAPE 0,0456974 1,2387098 0,940842522 TCGP SMAPE 0,129028 1,1975367 0,955311831 TCGB SMdAPE 0,0458166 1,2387098 0,940842522 TCGP RMSPE 544,1911 1,1315014 0,953128448 TCGP RMdSPE 0,0457446 1,2387098 0,940842522 TCGP NS 0,0902395 1,2301125 0,956005133 LCGP MRAE 0,0176105 1,1975367 0,955311831 TCGB MdRAE 0,0094554 1,2379883 0,933346754 LBFG GMRAE 0,0078473 1,2521395 0,946544493 LSCG MSMAPE 0,129028 1,1975367 0,955311831 TCGB DA 0,2857143 1,0948088 0,910311469 LLM MDA 0,3265306 1,1975367 0,942631212 TCGB
GBP
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Table 3: Best Models Selected for USD/EURO and USD/NOK Exchange Rate Time Series
Error Forecast Correlation Parameter RMSE 0,0136998 1,1078814 0,860424344 TCGF MSE 0,0001877 1,1078814 0,860424344 TCGF R4MS4E 0,018229 1,1078814 0,860424344 TCGF MASE 0,9420862 1,1096175 0,851592201 TSCG RMSSE 1,2016519 1,1078814 0,860424344 TCGF MAPE 0,0297539 1,1096175 0,851592201 TSCG MAE 0,0107406 1,1096175 0,851592201 TSCG GMAE 0,005495 1,1063798 0,855348286 LGDX MdAE 0,0068068 1,111545 0,818295701 LGDX MdAPE 0,0176373 1,111545 0,818295701 LGDX SMAPE 0,0298659 1,1096175 0,851592201 TSCG SMdAPE 0,0176278 1,111545 0,818295701 LGDX RMSPE 0,0380978 1,1078814 0,860424344 TCGF RMdSPE 0,017648 1,111545 0,818295701 LGDX NS 0,2627242 1,1078814 0,860424344 TCGF MRAE 0,0107067 1,1096175 0,851592201 TSCG MdRAE 0,006773 1,111545 0,818295701 LGDX GMRAE 0,0054963 1,1063798 0,855348286 LGDX MSMAPE 0,0298659 1,1096175 0,851592201 TSCG DA 0,3061224 1,1050893 0,855153746 LCGF MDA 0,3673469 1,1271242 0,854836137 LCGP
EURO
Error Forecast Correlation Parameter RMSE 0,0194397 8,1644675 0,92175714 LCGP MSE 0,0003779 8,1644675 0,92175714 LCGP R4MS4E 0,0240017 8,1644675 0,92175714 LCGP MASE 1,0021052 8,1488547 0,918508465 TCGB RMSSE 1,2299027 8,1644675 0,92175714 LCGP MAPE 0,0210118 8,1488547 0,918508465 TCGB MAE 0,0158391 8,1488547 0,918508465 TCGB GMAE 0,0081155 8,1488547 0,918508465 TCGB MdAE 0,011925 8,1150584 0,922639936 LGDX MdAPE 0,015153 8,2296341 0,908700799 LCGB SMAPE 0,0210707 8,1488547 0,918508465 TCGB SMdAPE 0,01504 8,2296341 0,908700799 LCGB RMSPE 0,0260979 8,1644675 0,92175714 LCGP RMdSPE 0,0151592 8,2296341 0,908700799 LCGB NS 0,1526599 8,1644675 0,92175714 LCGP MRAE 0,0158065 8,1488547 0,918508465 TCGB MdRAE 0,0120771 8,1150584 0,922639936 LGDX GMRAE 0,0081249 8,1488547 0,918508465 TCGB MSMAPE 0,0210707 8,1488547 0,918508465 TCGB DA 0,3469388 8,1859107 0,920696865 TCGP MDA 0,3877551 8,1699386 0,923048838 TSCG
NOK
Figure 3: The Test Set Predictions of Successful Models for Each Exchange Rate
100 105 110 115 120 125
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51
USD/JPY
USD/JPY TLM TBFG TCGF
1,23 1,28 1,33 1,38 1,43 1,48 1,53
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51
USD/GBP
USD/GBP LCGP
1,058 1,068 1,078 1,088 1,098 1,108 1,118 1,128 1,138 1,148
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51
USD/EUR
USD/EUR TCGF LCGF
8 8,1 8,2 8,3 8,4 8,5 8,6 8,7 8,8 8,9
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51
USD/NOK
USD/NOK TCGB TSCG
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5. Conclusions
This paper includes the performance investigation of the parameters for forecasting USD/JPY, USD/GBP, USD/EUR, and USD/NOK exchange rate time series. Generally, all models give satisfying forecasts when comparing the actual values of 103.898, 1.22418, 1.09952, and 8.1751 for USD/JPY, USD/GBP, USD/EUR, and USD/NOK exchange rates, respectively.
The procedure of measuring performance is a key issue about selecting the best model for ANN. To obtain the more feasible solutions for each exchange rate, the results in the way of desired findings such as error, forecast, and correlation can be pruned by using alternative performance measures.
In this study, we consider forecasts for determination of the parameter sets. For USD/JPY exchange rate, ANN selected by MdAE performance measure with TCGP parameter set forecasted 103.875. For USD/GBP exchange rate, ANN selected by RMSE, MSE, RMSSE, NS performance measures with LCGP parameter set forecasted 1.23011. For USD/EUR exchange rate, ANN selected by GMAE, GMRAE performance measures with LGDX parameter set forecasted 1.10637. For USD/NOK exchange rate, ANN selected by MDA performance measure with TSCG parameter set forecasted 8.1699.
The results show that ANN parameters are highly associated with data characteristics. In every kind of exchange rate data has its own ideal parameters of ANN to give the appropriate results. Therefore, unfortunately there is not a distinctive parameter set for ANN to give best results in every situation. However, the results of this study can be a guide for the users for whom going to work with the same exchange rates. Thus, trial and error is still the most reliable method for modelling with ANN.
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