Rainfall Prediction Analysis Using Fuzzy Time Series In Nagapattinam

Abstract

Rainfall is caused by a variety of meteorological conditions and the mathematical model for it is nonlinear. Forecasting is the process of predicting future outcomes, by which decision makers analyse the related data and graphs to decide and take the best decisions for the future. Multiple methods have been proposed to forecast the rainfall distribution but the accurateness is still a concern. Length of intervals greatly affects forecasting results in fuzzy time series (FTS). Hence, an effective length of intervals can significantly improve the forecasting results. The aim of the study is to compare the performance among fuzzy time series methods. In this study 10 years of rainfall data of Nagapattinam region is analysed using the statistical tools Root Mean Squared Error (RMSE) and Average Forecasting Error Rate (AFER).As a result, it is shown that, comparatively, improved Hwang and Chen method is best suited.

Keywords: Fuzzy time series, Forecasting, Difference of rainfall data, Percentage, Inverse Fuzzy number.

1. Introduction

Making decisions on forecasting is a complicatedprocess. Due to the complexity ofmeteorological phenomena, accurate prediction of rainfall is challenging one. To plan our day-to-day activities accurate weather predictions are important. Farmers need information about the weather to plan for planting and harvesting their crops. Weather forecasting helps to keep us out of danger. Fuzzy logic is the most powerful linear model for forecasting of any time series. In 1993, Song &Chissom presented the theory of fuzzy time series. Fuzzy time series model deal with the both linguistic and numerical values.

This paper is organised as follows:

In section 2, the basic concept of fuzzy time series has been discussed. In section 3, review of related works is described in brief. In section 4, areas under study have been discussed. In section 5, fuzzy forecasting models are descried in brief.In section 6, we have compared the forecasting results. The conclusion is given section 7.

2. Basic Concept of Fuzzy Time Series

A time series represents a collection of values of certain events or tasks which are obtained with respect to time. Future prediction of time series events has been attracted people from its beginning. Song and Chissom developed a model, based on uncertainty and imprecise knowledge contained in time series data. They initially used the fuzzy sets concept to represent or manage all these uncertainties and referred this concept as Fuzzy Time Series (FTS). Researchers have developed numerous models based on the FTS concept to deal with the forecasting problems of short term as well as long term events. The following are some definitions related with fuzzy time series.

3. Review of related works

Song and Chissom [5, 6] first introduced the method of fuzzy time series for forecasting humidity and rainfall. In 1994, they proposed a model for forecasting enrollments using fuzzy time series. Later, Chen [2] presented a new method for forecasting university enrollment, which is more efficient than the method proposed by Song and Chissom. Hwang [6] proposed a new method on fuzzification to revise Song and Chissom’s method. The result got a better forecasting error.In 2014, a modified method of forecasting enrollments based on fuzzy time series was developed by HaoFeng and Hongxuwang based on the method proposed bySaxena, Sharma and Easo.

4. Data and area of study

Nagapattinam is a coastal district situated on the eastern side of Tamil Nadu. Nagapattinam district was carved out by bifurcating the erstwhile composite Thanjavur district on October 19, 1991. The town of Nagapattinam is the district headquarters. The district lies on the east coast to the south of Cuddalore district and another part of the Nagapattinam district lies to the south of Karaikkal and Tiruvarur districts. The average maximum temperature of the district as a whole is about 320 C and the average minimum temperature is 24.60C. The Southwest winds sets in during April, it is the strongest in June and continues till September. Northeast monsoon starts during the month of October and blow till January. Cyclonic storm with varying wind velocity affects once in 3 or 4 years during the months of November-December. The storms affect the plantation crop. During Southwest monsoon the air is calm and undisturbed. The Northeast monsoon which starts in October and ends in December contributes about 60% of the total annual rainfall. The southwest monsoon rains occur from June to September. The average normal and actual rainfall is 265.2 and 250.6 mm respectively during south west monsoon while it is 908.8 and 969.2 mm respectively during north east monsoon during 2007-2008.

[image: Description: Image result for nagapattinam district map download]

Figure-1.Map of Nagapattinam.

The rainfall data set provided by the statistical department of Nagapattinam is given in Figure-2. It contains the annual rainfall (in mm) from 2008 to 2017.

Figure-2.Annual rainfall year wise (2008-2017) in Nagapattinam.

5. Fuzzy time series forecasting model

First, we tested the model [2] with rainfall data to forecast the distribution. Then we applied the methods of [6],[3] and [4].

Steps for finding forecasting fuzzy time series (Chen’s Method)

Chen [2] proposed a simple method to forecast the enrollment in fuzzy time series. The six steps involved in the method is given below:

• Step 1 : Define the universe of discourse and partition it into equally lengthy intervals.
• Step 2 : Define fuzzy sets on the universe of discourse.
• Step 3: Fuzzify historical data.
• Step 4 : Identify fuzzy logical relationship (FLR’s).
• Step 5: Establish fuzzy logical relationship groups (FLRG’s).
• Step 6: Defuzzify the forecasted output.

Steps for finding forecasting fuzzy time series (Zhang and Wang’s Method)

They proposed a modified method from Saxena and Easo’s method. In this method the percentage change arrange in increasing order as the universe of discourse and established inverse fuzzy number of consecutive years. The steps followed are given below:

• Step 1: List the historical data.
• Step 2: Calculate the percentage change.
• Step 3 : Construct the discrete domain.
• Step 4 : Establish inverse fuzzy number.
• Step 5 :Defuzzify the forecasted output.

Steps for finding forecasting fuzzy time series (Modified Saxena and Easo’s Method)

In [3], they proposed a modified method from [7]. In this method [3], the percentage change is used as universe of discourse and established inverse fuzzy number of consecutive years. The steps followed are given below:

• Step 1 : List the historical data.
• Step 2 : Calculate the percentage change.
• Step 3 : Establish discrete universe of discourse.
• Step 4 : Establish inverse fuzzy number.
• Step 5 :Defuzzify the forecasted output.

Steps for finding forecasting fuzzy time series (Improved Hwang and Chen and Lee)

In this model, finding a domain of interval and change of percentage for historical data are not necessary. But greater adjustment in fuzzy inverse formula is used.

• Step 1 : List the historical data of annual rainfall.
• Step 2 :Find the yearly difference of actual rainfall data.
• Step 3 :Establish inverse fuzzy number.
• Step 4 :Establish a forecasting formula to forecast.

6. Comparison of differentforecasting models

Forecasting values are obtained using Chen,Zhang and Wang, Modified Saxena and Easo andImproved Hwang &Chenmethods. Moreover, their RMSE and AFER are compared. The results are shown in Table-1.

Table-1 Comparison of different forecasting models

Year

Actual rainfall

Chen

Zhang and Wang method

Modified method of Saxena

Improved method of Hwang &Chen

2008

1885.69

2009

1861.09

1800

1739.43

1861.09

1861.09

2010

1758.24

1800

1769.50

1758.29

1758.25

2011

1145.96

900

1186.09

1146.84

1146.40

2012

1011.40

1200

1014.16

1011.26

1010.81

2013

1014.64

1200

1015.01

1014.64

1014.64

2014

1348.32

1200

1375.62

1341.67

1344.96

2015

1737.38

1700

1425.46

1737.51

1737.43

2016

691.94

900

726.92

690.96

691.35

2017

1714.12

1100

1568.42

1714.83

1714.22

RMSE

240.7750

117.0600

2.1580

1.1045

AFER

14.3080%

4.4510%

0.0776%

0.0430%

From Table 1, it can be seen the small value of RMSE and AFER, this confirms the goodness of forecasting model.

Figure-3.forecasting observation.

7. Conclusion

Song and Chissom proposed the first forecasting model of fuzzy time series in 1993. In 1994, they proposed a model for forecasting enrollments using fuzzy time series. Later, Chen[1996] presented a new method for forecasting university enrollment, which is more efficient than the method proposed by Song and Chissom, as the proposed method uses simplified arithmetic operation rather than the complicated Max-Min composition operation.Up to now, many literatures and researchers research and develop this theory. In this paper, we calculated and compared the forecasted values of the rainfall data of kanyakumari district using the forecasting models, namely Chen, modified Saxena and Easo and improved Hwang and Chen and Lee. From Table 1, we see that AFER and RMSE values are less in improved Hwang and Chen and Lee.

References:

1. Arumugam , P, Karthik S.M, “Prediction of Seasonal Rainfall Data in India using Fuzzy Stochastic modelling”, Journal of pure and Applied mathematics ,Vol. 13, No.9 (2017), PP 6167 – 6174.
2. Chen, S.M., “Forecasting Enrollments Based on Fuzzy Time Series”. Fuzzy sets and systems, Vol. 81, PP.311 -319, 1996.
3. HaoFeng, Hongxu Wang, “A Modified Method of Forecasting Enrollments Based on Fuzzy Time Series”. SCICT 2014 .
4. Hongxu Wang, Hui Wang, “A Fuzzy Time Series Forecasting Model Based on Yearly Difference of the Students Enrollment Number” , SCICT 2014.
5. Hwang, J.R. ,Chen,S.M., and Lee ,C .H .,”Handling Forecasting Problems using Fuzzy Time Series ”,Fuzzy Sets and Systems,Vol.100 ,PP.217-228,1998.
6. Kun Zhang, Zhung Li,Hai-fengWang,Hong-xu Wang, “Fuzzy Time Series Prediction Model and Applications based on Fuzzy Inverse” , International Journal of Signal Processing, Vol.8, No. 10(2015), PP. 121-128.
7. PreetikaSaxena, SanthoshEaso, “A New Method for Forecasting Enrollments based on Fuzzy Time Series with Higher Forecasting Accuracy Rate” Vol. 3(6), 2033-2037.
8. Saxena, P., Sharma, K. and Easo, S., “Forecasting Enrollment Based on Fuzzy Time Series with Higher Forecast Accuracy Rate”, International journal of computer Technology and Applications, Vol. 3, No. 3, (2012).
9. SonalDani and Sanjay Sharam , “Forecasting Rainfall of a Region by using Fuzzy time series”. Vol. 2013 , Article ID ama0065, 10 pages.
10. Song, Q. and Chissom ,B.S .,”Forecasting Enrollments with Fuzzy Time Series – Part I”, Fuzzy Sets and Systems , Vol. 54, pp.1-9,1994.
11. Song, Q. and Chissom ,B.S .,”Forecasting Enrollments with Fuzzy Time Series – Part II”, Fuzzy Sets and Systems , Vol. 62, pp.1-8,1994.
12. Zadeh , L.A., “ The Concept of a Linguistic Variable and its Application to Approximate Reasoning-part I”. Information sciences, Vol. 8, PP.199 -249, 1975.

Actual rainfall 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 1885.6899999999998 1861.09 1758.24 1145.96 1011.4 1014.64 1348.32 1737.3799999999999 691.93999999999983 1714.12 Chen 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 0 1800 1800 900 1200 1200 1200 1700 900 1100 Zhang and Wang method 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 0 1739.43 1769.5 1186.0899999999999 1014.16 1015.01 1375.62 1425.46 726.92 1568.42 Modified method of Saxena 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 0 1861.09 1758.29 1146.8399999999999 1011.26 1014.64 1341.6699999999998 1737.51 690.95999999999981 1714.83 Improved method of Hwang & Chen 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 0 1861.09 1758.25 1146.4000000000001 1010.81 1014.64 1344.96 1737.43 691.3499999999998 1714.22