# Optimization of neural network for nonlinear discrete time system using modified quaternion firefly algorithm: case study of Indian currency exchange rate prediction

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## Abstract

Success of neural networks depends on an important parameter, initialization of weights and bias connections. This paper proposes modified quaternion firefly algorithm (MQFA) for initial optimal weight and bias connection to neural networks. The proposed modified quaternion firefly method is based on updating population, moving fireflies and best solution in quaternion space. The combination of modified quaternion firefly and neural network is developed with the scope of creating an improved balance between premature convergence and stagnation. The performance of the proposed method is tested on two nonlinear discrete time systems, Box–Jenkins time series data and exchange rate prediction of Indian currency. Results of the MQFA with back-propagation neural network (MQFA-BPNN) compared with existing differential evolution-based neural network and opposite differential evolution-based neural network. Results obtain using MQFA-BPNN envisage that this method is effective and provides better identification accuracy. Computational complexity of MQFA-BPNN is deliberated, and validation of proposed method is tested by statistical methods.

## Keywords

Firefly algorithm (FA) MQFA BPNN Nonlinear Box–Jenkins data## Notes

### Acknowledgements

We would like to thank the editors and anonymous reviewers for their valuable suggestions and constructive comments which will really help us improve presentation and quality of paper.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- Adebiyi A, Adewumi A, Ayo CK (2014) Comparison of ARIMA and artificial neural networks model for stock price prediction. J Appl Math 2(1):1–7MathSciNetCrossRefGoogle Scholar
- Adjrad M, Belouchrani A (2007) Estimation of multi component polynomial phase signals impinging on a multisensory array using state-space modeling. IEEE Trans Signal Process 55(1):32–45MathSciNetCrossRefGoogle Scholar
- Beligiannis GN, Skarlas LV, Likothanassis SD, Perdikouri KG (2005) Nonlinear model structure identification of complex biomedical data using a genetic programming-based technique. IEEE Trans Instrum Meas 54(6):2184–2190CrossRefGoogle Scholar
- Bennett C, Stewart RA, Beal CD (2013) ANN-based residential water end used demand forecasting model. Expert Syst Appl 40(4):1014–1023CrossRefGoogle Scholar
- Box GEP, Jenkins GM (1970) Time series analysis. Forecasting and control. Holden Day, San FranciscozbMATHGoogle Scholar
- Chen S, Billings SA (1989) Representation of nonlinear systems: the NARMAX model. Int J Control 49:1013–1032MathSciNetCrossRefzbMATHGoogle Scholar
- Chen L, Narendra K (2004) Identification and control of nonlinear discrete time system based on its linearization: a unified framework. IEEE Trans Neural Netw 15(3):663–673CrossRefGoogle Scholar
- Chen M, Ge SS, How BVE (2010) Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities. IEEE Trans Neural Netw 21(5):796–812CrossRefGoogle Scholar
- Chon KH, Cohen RJ (1997) Linear and nonlinear ARMA model parameter estimation using an artificial neural network. IEEE Trans Boimed Eng 44:168–174CrossRefGoogle Scholar
- Conway JH, Smith DA (2003) On quaternions and octonions: their geometry, arithmetic and symmetry. A.K. Peters, WellesleyzbMATHGoogle Scholar
- Crepinsek M, Mernik M, Liu SH (2011) Analysis of exploration and exploitation in evolutionary algorithm by ancestry trees. Int J Innov Comput Appl 3:11–19CrossRefzbMATHGoogle Scholar
- Eberly D (2002) Quaternion algebra and calculus. Magic Software Inc. http://www.magic-software.com
- Fister I, Yang XS, Brest J, Brest I Jr (2013) Modified firefly algorithm using quaternion representation. Expert Syst Appl 40:7220–7230CrossRefGoogle Scholar
- Girard PR (1984) The quaternion group and modern physics. Eur J Phys 5:25–32CrossRefGoogle Scholar
- Gurusen E, Kayakulu G, Daim T (2011) Using artificial neural network model in stock market index prediction. Expert Syst Appl 38(8):10389–10397CrossRefGoogle Scholar
- Harnandez E, Arkun Y (1993) Control of nonlinear systems using polynomial ARMA models. AICHE J 39(3):446–460CrossRefGoogle Scholar
- Hosseini HG, Luo D, Reynolds KJ (2006) The comparison of different feedforward neural network architecture of ECG signal diagnosis. Med Eng Phys 28(4):372–378CrossRefGoogle Scholar
- Irani R, Nasimi R (2011) Evolving neural network using real coded genetic algorithm for permeability estimation of reservoir. Expert Syst Appl 38(8):9862–9866CrossRefGoogle Scholar
- Kwon YK, Moon BR (2007) A hybrid neurogenetic approach for stock forecasting. IEEE Trans Neural Netw 18(3):851–864CrossRefGoogle Scholar
- Lee TT, Jeng JT (1998) The Chebyshev polynomial based unified model neural network for functional approximation. IEEE Trans Syst Man Cybern B 28:925–935CrossRefGoogle Scholar
- Li YM, Tong SC, Liu Y, Li T (2014) Adaptive fuzzy robust output feedback control of nonlinear systems with unknown dead zones based on a small-gain approach. IEEE Trans Fuzzy Syst 22(1):164–176CrossRefGoogle Scholar
- Narendra KS, Parthaasarathy K (1990) Identification and control of dynamical systems using neural networks. IEEE Trans Neural Netw 1:4–27CrossRefGoogle Scholar
- Sadegh N (1993) A perceptron based neural network for identification and control of nonlinear systems. IEEE Trans Neural Netw 4:982–988CrossRefGoogle Scholar
- Sakhre V, Singh UP, Jain S (2016) FCPN approach for uncertain nonlinear dynamical system with unknown disturbance. Int J Fuzzy Syst 18:1–18. doi: 10.1007/s40815-016-0145-5 MathSciNetGoogle Scholar
- Schetzmen M (1980) The voltera and winner theories on nonlinear systems. Wiley, New YorkGoogle Scholar
- Simon Haykin S (2009) Neural networks and learning machines, vol 3. Pearson Education, Upper Saddle River. http://www.mif.vu.lt/~valdas/DNT/Literatura/Haykin09/Haykin09.pdf
- Subudhi B, Jena D (2011) A differential evolution based neural network approach to nonlinear system identification. Appl Soft Comput 11:861–871CrossRefGoogle Scholar
- Theofilatos K, Beligiannis G, Likothanassis S (2009) Combining evolutionary and stochastic gradient techniques for system identification. J. Comput. Appl. Math. 227(1):147–160MathSciNetCrossRefzbMATHGoogle Scholar
- Ticknor JL (2013) A Bayesian regularized artificial network for stock market forecasting. Expert Syst Appl 40(14):5501–5506Google Scholar
- Tong SC, Li YM, Zhang HG (2011) Adaptive neural network decentralized backstepping output-feedback control for nonlinear large scale systems with time delays. IEEE Trans Neural Netw 22(7):1073–1086Google Scholar
- Wang J, Zeng YR, Zhang JL, Huang W, Bao YK (2006) The criticality of spare parts evaluating model using an artificial neural network approach. Lect Notes Comput Sci 3991:728–735CrossRefGoogle Scholar
- Wang L, Zeng Y, Chen T (2015) Backpropagation neural network with adaptive differential evolution algorithm for time series forecasting. Expert Syst Appl 42:855–863CrossRefGoogle Scholar
- Watanbe K, Matsuura I, Abe M, Kebota M, Himelblau DM (1989) Incipient fault diagnosis of chemical processes via artificial neural networks. AICHE J 35(11):1803–1812CrossRefGoogle Scholar
- Xie Y, Guo B, Xu L, Li J, Stoica P (2006) Multistatic adaptive microwave imaging for early breast cancer detection. IEEE Trans Boimed Eng 53(8):1647–1657CrossRefGoogle Scholar
- Yam JY, Chow TWS (2000) A weight initialized method for improving training speed in feedforward neural network. Neurocomputing 30(1):219–232CrossRefGoogle Scholar
- Yang XS (2010) Engineering optimization: an introduction with metaheuristic applications. Wiley, HobokenCrossRefGoogle Scholar
- Yang XS (2014) Nature-inspired optimization algorithms, Elsevier Insights Series, 1st edn. pp 111–123Google Scholar
- Zhang L, Subbarayan G (2002) An evaluation of back-propagation neural network for the optimal design of structural systems: part I training procedures. Comput Methods Appl Mech Eng 191(25):2873–2886CrossRefzbMATHGoogle Scholar
- Zhang YQ, Wan X (2007) Statistical fuzzy interval neural network for currency exchange rate time series prediction. Appl Soft Comput 7(4):1149–1156CrossRefGoogle Scholar
- Zhang GP, Patuwo BE, Hu MY (2001) A simulation study of artificial neural network for nonlinear time series forecasting. Comput Oper Res 28(4):381–396CrossRefzbMATHGoogle Scholar
- Zhang JR, Zhang J, Lok TM, Lyu MR (2007) A hybrid particle swarm optimization–back-propagation algorithm for feedforward neural network training. Appl Math Comput 185(2):1026–1037zbMATHGoogle Scholar
- Zhang Y, Chai T, Wang H (2011) A nonlinear control method based on ANFIS and multiple models for a class of SISO nonlinear systems and its application. IEEE Trans Neural Netw 22(11):1783–1795CrossRefGoogle Scholar