# American Institute of Mathematical Sciences

September  2020, 10(3): 355-366. doi: 10.3934/naco.2020007

## An improved ARMA(1, 1) type fuzzy time series applied in predicting disordering

 1 Department of Basic, Shenyang University of Technology, Liaoyang, 111000, China, Department of Mathematics, Northeastern University, Shenyang, 110004, China 2 Department of Mathematics, Northeastern University, Shenyang, 110004, China

* Corresponding author: Tie Zhang

Received  May 2019 Revised  October 2019 Published  February 2020

Fund Project: This work is supported by the State Key Laboratory of Synthetical Automation for Process Industries Fundamental Research Funds, No. 2013ZCX02

Fuzzy time series shows great advantages in dealing with incomplete or unreasonable data. But most of them are based on fuzzy AR time series model, so it is necessary to add MA variables to the fuzzy time series [10] to make it more accurate. An improved ARMA(1, 1) type fuzzy time series based on fuzzy logic group relations including fuzzy MA variables along with fuzzy AR variables was proposed in this paper. To take full account of the errors, the prediction errors were added to the forecast fuzzy sets, and it made the first-order fuzzy logical relationship sets more exact. In order to verify the advantage of the proposed method, it was applied to predict the stock prices of State Bank of India (SBI) and the packet disordering from a common source host in the Northeast University to www. yahoo. com. The experimental results showed that the proposed model was more precise than other models.

Citation: Zhi Liu, Tie Zhang. An improved ARMA(1, 1) type fuzzy time series applied in predicting disordering. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 355-366. doi: 10.3934/naco.2020007
##### References:
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##### References:
 [1] I. Abdullah, D. Daw and K. Seman, Traffic forecasting and planning of wimax under multiple priority applying fuzzy time series analysis, J. Appl. Math. Phys., 3 (2015), 68-74. [2] G. Box and G. Jenkins, Time series analysis: forecasting and control, Journal of the Operational Research Society, 37 (1976), 238-242. [3] M. Chen and B. Chen, A hybrid fuzzy time series model based on granular computing for stock price forecasting, Information Sciences, 294 (2015), 227-241.  doi: 10.1016/j.ins.2014.09.038. [4] S. Chen, Forecasting enrollments based on fuzzy time series, Fuzzy Sets and Systems, 81 (1996), 311-319.  doi: 10.1016/S0165-0114(98)00266-8. [5] S. Chen, Forecasting enrollments based on high-order fuzzy time series, Cybernetics and Systems: An International Journal, 33 (2002), 1-16. [6] C. Cheng, T. Chen and C. Chiang, Trend-weighted fuzzy time-series model for taiex forecasting, in Int. Conf. Neural. Hong Kong: Inf. Process, 2006. [7] H. Y. Guo, P. Witold and X. D. Liu, Fuzzy time series forecasting based on axiomatic fuzzy set theory, Neural Comput. Appl., 26 (2018), 2807-2817. [8] J. R. Hwang, S. M. Chen and C. H. Lee, Handling forecasting problems using fuzzy time series, Fuzzy Sets and Systems, 100 (1998), 217-228.  doi: 10.1016/S0165-0114(98)00266-8. [9] T. Jilani, S. Burney and C. Ardil, Fuzzy metric approach for fuzzy time series forecasting based on frequency density based partitioning, Int. J. Comput. Intell., 4 (2008), 112-117. [10] C. Kocak, Arma(p, q) type high order fuzzy time series forecast method based on fuzzy logic relations, Applied Soft Computing, 58 (2017), 92-103. [11] W. Lee and J. Hong, A hybrid dynamic and fuzzy time series model for mid-term power load forecasting, Int. J. Electr. Power Energy Syst., 64 (2015), 1057-1062. [12] B. S. Lian, Q. L. Zhang and J. N. Li, Sliding mode control for non-linear networked control systems subject to packet disordering via prediction method, Int. Control Theory and Applications, 11 (2017), 3079-3088.  doi: 10.1049/iet-cta.2016.1591. [13] W. Lu, X. Y. Chen, W. Pedrycz, X. D. Liu and J. H. Yang, Using interval information granules to improve forecasting in fuzzy time series, Int. J. Approx. Reason, 57. (2015), 1–18. [14] N. Rahman, M. Lee and M. Latif, Artificial neural networks and fuzzy time series forecasting: an application to air quality, Quality and Quantity, 49 (2015), 2633-2647. [15] P. Saxena, K. Sharma and S. Easo, Forecasting enrollments based on fuzzy time series with higher forecast accuracy rate, Int. J. Comput. Technol., 3 (2012), 95-961.  doi: 10.1002/for.1185. [16] P. Singh, High-order fuzzy-neuro-entropy integration-based expert system for time series, Neural Computing and Applications, 28 (2016), 3851-3868. [17] Q. Song and B. Chissom, Forecasting enrollments with fuzzy time series - part Ⅰ, Fuzzy Sets and Systems, 54 (1993), 1-9.  doi: 10.1016/0165-0114(93)90372-O. [18] T. H. K. Yu, Weighted fuzzy time series models for TAIEX forecasting, Physica A: Statistical Mechanics and Its Applications, 349 (2005), 609-624. [19] G. Yule, On a method of investigating periodicities in disturbed series with special reference to wolfer's sunspot numbers, Philosophical Transactions of the Royal Society of London, 226 (1927), 26-298. [20] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
A comparison between the proposed model and other models
A comparison between the proposed model and other models
A comparison between the delays of the proposed model and the real delays
Example of weights.
 Time FLR Weight $\; t=1$ $A_i \rightarrow A_j$ 1 $\; t=2$ $A_i \rightarrow A_k$ 1 $\; t=3$ $A_i \rightarrow A_j$ 1 $\; t=4$ $A_i \rightarrow A_l$ 1 $\; t=5$ $A_i \rightarrow A_j$ 1
 Time FLR Weight $\; t=1$ $A_i \rightarrow A_j$ 1 $\; t=2$ $A_i \rightarrow A_k$ 1 $\; t=3$ $A_i \rightarrow A_j$ 1 $\; t=4$ $A_i \rightarrow A_l$ 1 $\; t=5$ $A_i \rightarrow A_j$ 1
The predication values.
 Year Actual price The rate of change PRC Predicted $t$ (in rupee) $r(t)\%$ values 6/5/2012 2080.25 6/6/2012 2159.45 3.81 3.81 6/7/2012 2167.85 0.39 0.38 2167.66 6/8/2012 2180.05 0.56 0.61 2181.07 6/11/2012 2164.55 -0.71 -0.65 2165.88 6/12/2012 2206.90 1.96 1.91 2205.89 6/13/2012 2222.25 0.70 0.68 2221.91 6/14/2012 2154.25 -3.06 -3.12 2152.92 6/15/2012 2182.80 1.33 0.99 2175.58 6/18/2012 2087.65 -4.36 -4.36 2087.63 ⋮ ⋮ ⋮ ⋮ ⋮ 7/17/2012 2198.85 0.20 0.16 2197.96 7/18/2012 2185.95 -0.57 -0.65 2184.56 7/19/2012 2157.75 -1.29 -1.29 2157.75 7/20/2012 2134.55 -1.08 -1.03 2135.53 7/23/2012 2092.55 -1.97 -1.96 2092.71 7/24/2012 2094.80 0.11 0.09 2094.43 7/25/2012 2070.65 -1.15 -1.03 2073.22 7/26/2012 2017.15 -2.58 -2.40 2020.95 7/27/2012 1941.20 -3.77 -3.74 1941.71 7/31/2012 2005.20 3.30 3.36 2006.42
 Year Actual price The rate of change PRC Predicted $t$ (in rupee) $r(t)\%$ values 6/5/2012 2080.25 6/6/2012 2159.45 3.81 3.81 6/7/2012 2167.85 0.39 0.38 2167.66 6/8/2012 2180.05 0.56 0.61 2181.07 6/11/2012 2164.55 -0.71 -0.65 2165.88 6/12/2012 2206.90 1.96 1.91 2205.89 6/13/2012 2222.25 0.70 0.68 2221.91 6/14/2012 2154.25 -3.06 -3.12 2152.92 6/15/2012 2182.80 1.33 0.99 2175.58 6/18/2012 2087.65 -4.36 -4.36 2087.63 ⋮ ⋮ ⋮ ⋮ ⋮ 7/17/2012 2198.85 0.20 0.16 2197.96 7/18/2012 2185.95 -0.57 -0.65 2184.56 7/19/2012 2157.75 -1.29 -1.29 2157.75 7/20/2012 2134.55 -1.08 -1.03 2135.53 7/23/2012 2092.55 -1.97 -1.96 2092.71 7/24/2012 2094.80 0.11 0.09 2094.43 7/25/2012 2070.65 -1.15 -1.03 2073.22 7/26/2012 2017.15 -2.58 -2.40 2020.95 7/27/2012 1941.20 -3.77 -3.74 1941.71 7/31/2012 2005.20 3.30 3.36 2006.42
MSE of different methods.
 Errors [4] [18] [16] Proposed model RMSE 54.1958 82.3197 32.2586 1.6498
 Errors [4] [18] [16] Proposed model RMSE 54.1958 82.3197 32.2586 1.6498
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