Ratio Mathematica Volume 45, 2023 Forecasting of Annual Rainfall using Fuzzy Logic Interval Based Partitioning in Different Intervals Rajan D* Sugunthakunthalambigai R† Abstract Fuzzy time series models have been proposed by many researchers around the world for rainfall forecasting, but the forecasting has not been as accurate as existing methods. Frequency density or ratio-based segmentation methods have been used to represent discourse segmentation. In this paper, to make such predictions, we used interval-based segmentation as the discourse segmentation and the urban mean rainfall in the Trichy district as the discourse universe. Fuzzy models are used for forecasting in many fields such as admissions prediction, stock price analysis, agricultural production, horticultural production, marine production, weather forecasting, and more. Keywords. Mean Square Error; Fuzzy time series; Average Forecast Error Rate. AMS Subject Classification: 05C78‡. *Associate Professor of Mathematics, (TBML College, Affiliated to Bharathidasan University Porayar-609 307, Mayiladuthurai Dist.); dan_rajan@rediffmail.com. †Assistant Professor of Mathematics, HC&RI (W), Trichy (TNAU, Tamilnadu, India); suguntha@tnau.ac.in. ‡ Received on July 10, 2022. Accepted on October 15, 2022. Published on January 30, 2023. doi: 10.23755/rm. v45i0.1006. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY license agreement. 153 Rajan D and Sugunthakunthalambigai R 1. Introduction A forecasting process issued to predict future outcomes. Relevant data and figures are carefully analyzed to make accurate forecasts and make optimal choices regarding the future. There are two main reasons for choosing time series forecasting. First, most of the data that exists in the real world, such as economics, business, and finance, are time series. Second, evaluation of time series data is easy, and many techniques are available for evaluation of time series forecasts. If the future is in doubt, the forecasting process is mandatory. Implement using the fuzzy time series method to forecast precipitation and compare results with other existing techniques. Accurate rainfall information is essential for planning and managing water resources. Moreover, in urban areas, rainfall has a strong impact on transportation, sewerage, N. Q. Hung (nguyenquang.hung@ait.ac.th) systems, and other human activities. Nevertheless, precipitation is best understood and modelled for hydrological cycles because of the complexity of the atmospheric processes that generate precipitation and the tremendous range of variability over a wide range of scales in both space and time. It is one of the complex and difficult factors (French et al., 1992). Accurate rainfall forecasting is thus one of the greatest challenges in operational hydrology, despite many advances in weather forecasting in recent decades (Gwangseob and Ana, 2001). The data for prediction has all the uncertainties. For this purpose, we generate rainfall simulations that reproduce in a distributional sense the set of key rainfall statistics obtained from the observational dataset (Benoit and Mariethoz, 2017). The practical interest of probabilistic rainfall models is, among other things, to complement numerical weather models for simulating rainfall heterogeneity at fine scales and to quantify uncertainties associated with rainfall reconstructions. . Indeed, numerical weather models face the challenge of reproducing spatial and temporal rainfall heterogeneity, especially at fine scales (Bauer et al., 2015; Bony et al., 2015). Some of the main applications of probabilistic rainfall models are therefore for local impact studies, e.g. related to hydrology (Paschalis et al., 2014; Caseri et al., 2016). Fuzzy time series forecasting is a smart method in areas where information is explicit, imprecise, and approximate. Fuzzy time series can also tackle situations that do not provide trend investigation and analysis, or visualization of time series patterns. In- depth research work has been done on forecasting problems using this concept. Vikas [1] proposed various techniques for predicting crop yields and used artificial neural networks to predict wheat yields. Did. Adesh [2] conducted a comparative study of Various techniques, including neural networks and fuzzy models. Askar [3] also tried to predict yield using time series models. Sachin [4-5] specifically worked on rice yield prediction using fuzzy time series models. Narendra [6] attempted to predict wheat yields. Pankaj [7] used an adaptive neuro-fuzzy system for predicting wheat yield. W. Qiu, X. Liu, and H. Li proposed a generalized method of forecasting based on fuzzy time series models [30]. The concept and definition of fuzzy time series was devised and published by Song & Chissom. They also delineated concepts and notions of variant and invariant time series [8-9]. First, the time-series data for the University of Alabama were obtained, the 154 Forecasting of Annual Rainfall using Fuzzy Logic Interval Based Partitioning In Different Intervals admission prediction was performed, and after a few years [10] average auto, then Chen [11-12] was the max-min configuration operation previously used by Song &Chissom. We depicted simplified arithmetic operations instead of using, and used higher-order fuzzy time series to organize our forecasting models. Huarng [13-14], Hwang and Chen [15], Lee Wang and Chen [16], Li and Kozma [17] all produced a number of fuzzy prediction methods with slight variations in each. Lee et al. Subsequently, a multivariate heuristic model was designed and implemented to obtain highly complex and complex matrix computations [20]. Research work has been done to ascertain the interval length of fuzzy time series [21]. Wong et al. (2003) used SOM and back propagation neural networks to build a fuzzy rule base and used the rule base to develop a forecast model for Swiss rainfall using spatial interpolation. Bardossy et al. (1995) implemented fuzzy logic for classification of atmospheric circulation patterns. Ozerkan et al. (1996) compared the performance of regression analysis and fuzzy logic in studying the relationship between monthly atmospheric circulation patterns and precipitation. Pesti et al. (1996) implemented fuzzy logic for drought assessment. Baum et al. (1997) developed a cloud classification model using fuzzy logic. Fujibe (1989) classified precipitation patterns in Honshu using the fuzzy C-means method. Garanboshi et al. (1999) Using fuzzy logic, we investigated the effects of ENSO and macro circulation patterns on precipitation over Arizona. Vivekanandan et al. (1999) developed and implemented a fuzzy logic algorithm for water meteor particle identification that is simple and efficient enough to run in real time for operational use. Wuwardi et al. (2006) use a neural fuzzy system to model tropical rainfall during the rainy season. The model results in lower RMSE values, indicating that the forecast model is reliable in representing recent inter annual variability in tropical rainfall during the wet season. 2. Basic Concepts of Fuzzy time Series Modeling Definition 1.1. (Song and Chissom, 1993a, 1994; Liaw, 1997). A fuzzy number on the real line  is a fuzzy subset of  that is normal and convex. Definition 1.2. (Song and Chissom, 1993a, 1994). Suppose that 𝑅1 = ⋃𝑖𝑗 𝑅1𝑖𝑗(𝑡, 𝑡 − 1) and 𝑅2(𝑡, 𝑡 − 1) = ⋃𝑖𝑗 𝑅2𝑖𝑗(𝑡, 𝑡 − 1) are two fuzzy relations between 𝐹(𝑡)and𝐹(𝑡 − 1). If, for any𝑓𝑗 (𝑡) ∈ 𝐹(𝑡), where𝑗 ∈ 𝐽, there exists𝑓𝑖 (𝑡 − 1) ∈ 𝐹(𝑡 − 1), where𝑖 ∈ 𝐼, and fuzzy relations 𝑅1 𝑖𝑗(𝑡, 𝑡 − 1) and 𝑅2 𝑖𝑗(𝑡, 𝑡 − 1) such that 𝑓𝑖 (𝑡) = 𝑓𝑖 (𝑡 − 1) ∘ 𝑅1 𝑖𝑗(𝑡, 𝑡 − 1) and𝑓𝑖 (𝑡) = 𝑓𝑖 (𝑡 − 1) ∘ 𝑅2 𝑖𝑗(𝑡, 𝑡 − 1), then define𝑅1(𝑡, 𝑡 − 1) = 𝑅2(𝑡, 𝑡 − 1). Definition 1.3. (Song and Chissom, 1993a, 1994). Suppose that 𝐹(𝑡) is only caused by 𝐹(𝑡 − 1), 𝐹(𝑡 − 2), ,or𝐹(𝑡 − 𝑚) (𝑚 > 0). This relation can be expressed as the following fuzzy relational equation: 𝐹(𝑡) = 𝐹(𝑡 − 1) ∘ 𝑅0(𝑡, 𝑡 − 𝑚), which is called the first-order model of 𝐹(𝑡). 155 Rajan D and Sugunthakunthalambigai R Definition 1.4 (Song and Chissom, 1993a, 1994). Suppose that 𝐹(𝑡) is simultaneously caused by 𝐹(𝑡 − 1), 𝐹(𝑡 − 2), , and𝐹(𝑡 − 𝑚) (𝑚 > 0). This relation can be expressed as the following fuzzy relational equation:𝐹(𝑡) = (𝐹(𝑡 − 1) × 𝐹(𝑡 − 2) ×  × 𝐹(𝑡 − 𝑚)) ∘ 𝑅𝑎(𝑡, 𝑡 − 𝑚)), which is called the m th-order model of𝐹(𝑡). Definition 1.5 (Chen, 1996). 𝐹(𝑡) is fuzzy time series if 𝐹(𝑡) is a fuzzy set. The transition is denoted as𝐹(𝑡 − 1) ⟶ 𝐹(𝑡). Definition 1.6 (Chou, 2011). Let 𝑑(𝑡) be a set of real numbers𝑑(𝑡) ⊆ 𝑅. A lower interval 𝑑(𝑡) is a number b such that 𝑥 ≥ 𝑏 for all𝑥 ∈ 𝑑(𝑡). The set 𝑑(𝑡) is said to be an interval below if 𝑑(𝑡) has a lower interval. A number, min, is the minimum of 𝑑(𝑡) if min is a lower interval 𝑑(𝑡)andmin ∈ 𝑑(𝑡). Definition 1.7 (Chou, 2011). Let 𝑑 (𝑡) be a set of real numbers 𝑑(𝑡) ⊆ 𝑅. An upper interval 𝑑(𝑡) is a number b such that 𝑥 ≤ 𝑏 for all𝑥 ∈ 𝑑 (𝑡). The set 𝑑 (𝑡) is said to be an interval higher if 𝑑 (𝑡) has an upper interval. A number, max, is the maximum of 𝑑 (𝑡)if max is an upper interval 𝑑 (𝑡)and max ∈ 𝑑(𝑡). 3. Proposed Method In this section, we use real-world rainfall as the universe of discourse and propose a method for forecasting using interval-based segmentation. Relevant concepts and definitions regarding this can be found by referring to a previously published paper [29]. Another method for predicting the values provided in this paper is clearly explained in the following lines. The forecasting process follows these steps: Step 1: First, obviously analysis descriptive statistics. It helps in facilitating data visualization. Next, we describe the discourse universe U and the parcel U in intervals of equal length. Here, according to the data, 379.79 is the minimum value and 1163.59 is the maximum value. We need to specify the discourse universe, the intervals in which all given values of rainfall exist. So in this case the discourse universe would be [300, 1200]. Descriptive statistics and block-by-block rainfall data are shown in Tables I and II. Step 2: Fuzzy partitioning is a methodology for generating fuzzy sets that represent the underlying data. The techniques can be classified into three categories: grid partitioning, tree partitioning, and distributed partitioning. Among the various fuzzy partitioning methods, grid partitioning is the most commonly used in practice, especially in system control applications. Grid partitioning forms partitions by dividing the input space into several fuzzy slices. Next, divide Universe of discourse in 6, 9 and 18 equal intervals these are as following. The discourse universe can be defined by 𝑈 = [300, 1200]. U is then divided into 6 equal length intervals and the midpoint of the 6th interval is calculated as shown below. 156 Forecasting of Annual Rainfall using Fuzzy Logic Interval Based Partitioning In Different Intervals Table 3.1. Annual rainfall in Trichy district (From2004 to 2010) Blocks Rainfall(mm) Andanallur 1085.23 Lalgudi 1136.79 Manachanallur 569.44 Manapparai 837.43 Manikandam 379.79 Marungapuri 942.16 Musiri 794.31 Pullambadi 1163.59 T.pet 710.16 Thiruverumbur 960.9 Thottiam 866.73 Thuraiur 942.11 Uppiliyapuram 473.35 Vaiyampatty 922.27 Table 3.2. Descriptive statistics Minimum = 379.79 Maximum = 1163.59 Range = 783.8 Count = 14 Sum = 11784.26 Mean = 841.733 Median = 894.5 Mode = No mode Standard Deviation = 237.65 Variance = 56479.54 157 Rajan D and Sugunthakunthalambigai R Table 3.3. a) 6 equal intervals Here, U is partitioned into 9 equal length intervals and calculated mid points of 9th intervals given below: Table 3.4. b) 9 equal Intervals with Midpoints Here, U is partitioned into 18 equal length intervals and calculated mid points of 18th intervals given below. Table 3.5. c) 18 equal intervals with Midpoints u1 [300 − 450] 375 u2 [450 − 600] 525 u3 [600 − 750] 675 u4 [750 − 900] 825 u5 [900 − 1050] 975 u6 [1050 − 1200] 375 v1 [300 − 400] 350 v2 [400 −500] 450 v3 [500 − 600] 550 v4 [600 −700] 650 v5 [700 − 800] 750 v6 [800 − 900] 850 v7 [900 −1000] 950 v8 [1000 − 1100] 1050 v9 [1100 −1200] 1150 w1 [300 − 350] 325 w2 [350 − 400] 375 w3 [400 − 450] 425 w4 [450 − 500] 475 w5 [500 − 550] 525 w6 [550 − 600] 575 w7 [600 − 650] 625 w8 [650 − 700] 675 w9 [700 − 750] 725 w10 [750 − 800] 775 w11 [800 − 850] 825 w12 [850 − 900] 875 w13 [900 − 950] 925 w14 [950 − 1000] 975 158 Forecasting of Annual Rainfall using Fuzzy Logic Interval Based Partitioning In Different Intervals Step 3. Define a fuzzy set based on 6, 9, and 18 intervals to fuzz the historical data. a. 6 Equal intervals Let 𝑈 = {𝑢1, 𝑢2, 𝑢3, 𝑢4, 𝑢5, 𝑢6} be the world of discourse. The number of intervals depends on the number of considered linguistic variables (fuzzy sets) 𝐴1, 𝐴2, 𝐴3, 𝐴4, 𝐴5, 𝐴6.Define 6 fuzzy sets 𝐴1, 𝐴2, … , 𝐴6 as linguistic variables in the discourse world U. These fuzzy variables are defined as: Table 3.6. Label Linguistic value of enrolments Fuzzified Linguistic Value A1 very few A2 very very few A3 Moderate A4 High A5 very High A6 very very High b. 9 equal Intervals Let U be the universe of discourse, where𝑈 = {𝑣1, 𝑣2, 𝑣3, 𝑣4, 𝑣5, 𝑣6 … 𝑣9}. The number of intervals will be in accordance with the number of linguistic variables (fuzzy sets) 𝐵1, 𝐵2, 𝐵3, 𝐵4, … 𝐵9, to be considered. Define 9fuzzy sets 𝐵1, 𝐵2, 𝐵3, 𝐵4, … 𝐵9, as linguistic variables on the universe of discourse U. These fuzzy variables are being defined as: Table 3.7. Label Linguistic value of enrolments w15 [1000 − 1050] 1025 w16 [1050 − 1100] 1075 w17 [1100 − 1150] 1125 w18 [1150 − 1200] 1175 Fuzzified Linguistic Value B1 very 3 few B2 very 2 few B3 very 1 few B4 Few B5 Moderate B6 High 159 Rajan D and Sugunthakunthalambigai R c. 18 Equal intervals Let U be the universe of discourse, and let 𝑈 = {𝑤1, 𝑤2, 𝑤3, … 𝑤18}. The number of intervals depends on the number of considered linguistic variables (fuzzy sets) 𝐶1, 𝐶2, 𝐶3, … 𝐶18. Define 18 fuzzy sets 𝐶1, 𝐶2, 𝐶3, … 𝐶18 as linguistic variables in the universe of discourse U. These fuzzy variables are defined as: Table 3.8. Label Linguistic value of enrolments Fuzzified Linguistic Value C1 very 8 few C2 very 7 few C3 very 6 few C4 very 5 few C5 very 4 few C6 very 3 few C7 very 2 few C8 very 1 few C9 Few C10 High C11 Very 1 High C12 very 2 High C13 very 3 High C14 very 4 High C15 very 5 High C16 very 6 High C17 very 7 High C18 very 8 High Step 4: Fuzzy set defined by U (all intervals). A fuzzy set 𝐴𝑖 is represented as 𝐴1 = 1 𝑢1⁄ + 0.5 𝑢2⁄ + 0 𝑢3⁄ + 0 𝑢4⁄ + 0 𝑢5⁄ + 0 𝑢6⁄ 𝐴2 = 0.5 𝑢1⁄ + 1 𝑢2⁄ + 0.5 𝑢3⁄ + 0 𝑢4⁄ + 0 𝑢5⁄ + 0 𝑢6⁄ ……………………………………………………… ………………………………………………………… ………………………………………………………… 𝐴10 = 0 𝑢1⁄ + 0 𝑢2⁄ + 0 𝑢3⁄ + 0 𝑢4⁄ + 0.5 𝑢5⁄ + 1 𝑢6⁄ B7 Very 1 High B8 very 2 High B9 very 3 High 160 Forecasting of Annual Rainfall using Fuzzy Logic Interval Based Partitioning In Different Intervals A fuzzy set 𝐵𝑖 are expressed as follows: 𝐵1 = 1 𝑢1⁄ + 0.5 𝑢2⁄ + 0 𝑢3⁄ + ⋯ + 0 𝑢9⁄ 𝐵2 = 0.5 𝑢1⁄ + 1 𝑢2⁄ + ⋯ + 0 𝑢9⁄ ……………………………………… ………………………………… ………………………………………… 𝐵9 = 0 𝑢1⁄ + 0 𝑢2⁄ + ⋯ + 0.5 𝑢5⁄ + 1 𝑢9⁄ A fuzzy set 𝐶𝑖 are expressed as follows: 𝐶1 = 1 𝑢1⁄ + 0.5 𝑢2⁄ + 0 𝑢3⁄ + 0 𝑢4⁄ + 0 𝑢5⁄ + 0 𝑢6⁄ 𝐶2 = 0.5 𝑢1⁄ + 1 𝑢2⁄ + 0.5 𝑢3⁄ + 0 𝑢4⁄ + 0 𝑢5⁄ + 0 𝑢6⁄ ……………………………………………………… ………………………………………………………… ………………………………………………………… 𝐶18 = 0 𝑢1⁄ + 0 𝑢2⁄ + 0 𝑢3⁄ + ⋯ + 0.5 𝑢17⁄ + 1 𝑢18⁄ Step 5: Fuzzify historical data. Table 3.9. Linguistic values for the enrolments from 2004 to 2010 Block Rainfall (mm) Linguistic value 6th Interval Linguistic value 9th Interval Linguistic value 18th Interval Andanallur 1085.23 C6 D8 E16 Lalgudi 1136.79 C6 D9 E17 Manachanallur 569.44 C2 D3 E6 Manapparai 837.43 C4 D6 E12 Manikandam 379.79 C1 D1 E2 Marungapuri 942.16 C5 D7 E13 Musiri 794.31 C4 D5 E10 Pullambadi 1163.59 C6 D9 E18 T.pet 710.16 C4 D5 E9 Thiruverumbur 960.9 C5 D7 E14 Thottiam 866.73 C4 D6 E12 Thuraiur 942.11 C5 D7 E13 Uppiliyapuram 473.35 C2 D2 E4 Vaiyampatty 922.27 C5 D7 E13 161 Rajan D and Sugunthakunthalambigai R Step 6: Calculate predicted registrations for 6, 9, and 18 intervals given below: Table 3.10. Forecasted value for all intervals Step 7: Calculate MSE and AFER values for 6, 9, and 18 intervals given below: Mean Squared Error (MSE) measures the amount of error in a statistical model. Evaluate the mean squared difference between observed and predicted values. If the model has no errors, MSE is equal to zero. MSE formula = (1 𝑛⁄ ) ∗ Σ(𝐴𝑐𝑡𝑢𝑎𝑙 – 𝐹𝑜𝑟𝑒𝑐𝑎𝑠𝑡)2 Mean Absolute Percentage Error (MAPE), also known as Mean Absolute Percentage Deviation (MAPD), is a measure of the predictive accuracy of a forecasting method in statistics. Accuracy is usually expressed as a ratio defined by the following formula: |𝐴𝑐𝑡𝑢𝑎𝑙−𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑| 𝐴𝑐𝑡𝑢𝑎𝑙 × 100% = Average Forecasting Error Rate (AFER) Blocks Rainfall (mm) Forecasted value (6 intervals) Forecasted value (9 intervals) Forecasted value (18 intervals) Andanallur 1085.23 - - - Lalgudi 1136.79 - - - Manachanallur 569.44 - - - Manapparai 837.43 900 900 912.5 Manikandam 379.79 712.5 725 737.5 Marungapuri 942.16 675 675 687.5 Musiri 794.31 750 725 737.5 Pullambadi 1163.59 825 800 812.5 T.pet 710.16 937.5 900 900 Thiruverumbur 960.9 937.5 900 912.5 Thottiam 866.73 937.5 925 937.5 Thuraiur 942.11 900 875 875 Uppiliyapuram 473.35 825 800 812.5 Vaiyampatty 922.27 825 800 800 162 Forecasting of Annual Rainfall using Fuzzy Logic Interval Based Partitioning In Different Intervals 4. Performance Evaluation and Comparative Studies A. Performance rating: Two parameters are used to compare the results of the proposed method with existing methods. These are MSE &AFER. MSE & AFER are the calculated values for intervals 6, 9 and 18 as shown in Tables XI, XII and XIII. Interval-based partitioning is calculated in Table XI. MSE indicates the deviation error from the actual value to the predicted value. The deviation is shown in Figure 4.1 in the form of a graphical representation for better visualization. As we can see the proposed algorithm gives values very close to what is the actual rainfall value. The same is done for the 9th and 18th intervals, as shown in Tables XII & XIII and Fig. 4.2 and 4.3. Table 3.11. MSE and AFER Values (6 Intervals) Block 𝑨𝑖 (Rainfall mm) 𝑭𝑖 MSE (𝑨𝒊 − 𝑭𝒊) 𝟐 AFER |𝑨𝒊 − 𝑭𝒊|/𝑨𝒊 Andanallur 1085.23 - - - Lalgudi 1136.79 - - - Manachanallur 569.44 - - - Manapparai 837.43 900 3915.005 0.074717 Manikandam 379.79 712.5 110695.9 0.876037 Marungapuri 942.16 675 71374.47 0.283561 Musiri 794.31 750 1963.376 0.055784 Pullambadi 1163.59 825 114643.2 0.290987 T.pet 710.16 937.5 51683.48 0.320125 Thiruverumbur 960.9 937.5 547.56 0.024352 Thottiam 866.73 937.5 5008.393 0.081652 Thuraiur 942.11 900 1773.252 0.044698 Uppiliyapuram 473.35 825 123657.7 0.742896 Vaiyampatty 922.27 825 9461.453 0.105468 MSE = 35337.42 AFER = 20.72% 163 Rajan D and Sugunthakunthalambigai R Table 3.12. MSE AND AFER VALUES (9 INTERVALS) Block 𝑨𝑖 (Rainfall mm) 𝑭𝑖 MSE (𝑨𝒊 − 𝑭𝒊) 𝟐 AFER |𝑨𝒊 − 𝑭𝒊|/𝑨𝒊 Andanallur 1085.23 Lalgudi 1136.79 Manachanallur 569.44 Manapparai 837.43 912.5 5635.505 0.089643 Manikandam 379.79 737.5 127956.4 0.941863 Marungapuri 942.16 687.5 64851.72 0.270294 Musiri 794.31 737.5 3227.376 0.071521 Pullambadi 1163.59 812.5 123264.2 0.30173 T.pet 710.16 900 36039.23 0.26732 Thiruverumbur 960.9 912.5 2342.56 0.050369 Thottiam 866.73 937.5 5008.393 0.081652 Thuraiur 942.11 875 4503.752 0.071234 Uppiliyapuram 473.35 812.5 115022.7 0.716489 Vaiyampatty 922.27 800 14949.95 0.132575 MSE=35914.42 AFER=21.39% 164 Forecasting of Annual Rainfall using Fuzzy Logic Interval Based Partitioning In Different Intervals Table 3.13. MSE AND AFER values (18 intervals) The following figures (Fig.4.1, Fig 4.2 and Fig 4.3) are compared in Forecasted and Actual rainfall for the corresponding intervals respectively 6,9&18.And also Fig. 4.4 compares the MSE for all intervals. Fig. 4.1. Forecasted vs. Rainfall (6 intervals) Block 𝑨𝑖 (Rainfall mm) 𝑭𝑖 MSE (𝑨𝒊 − 𝑭𝒊) 𝟐 AFER |𝑨𝒊 − 𝑭𝒊|/𝑨𝒊 Andanallur 1085.23 - - - Lalgudi 1136.79 - - - Manachanallur 569.44 - - - Manapparai 837.43 900 3915.005 0.074717 Manikandam 379.79 725 119169.9 0.90895 Marungapuri 942.16 675 71374.47 0.283561 Musiri 794.31 725 4803.876 0.087258 Pullambadi 1163.59 800 132197.7 0.312473 T.pet 710.16 900 36039.23 0.26732 Thiruverumbur 960.9 900 3708.81 0.063378 Thottiam 866.73 925 3395.393 0.06723 Thuraiur 942.11 875 4503.752 0.071234 Uppiliyapuram 473.35 800 106700.2 0.690081 Vaiyampatty 922.27 800 14949.95 0.132575 MSE=35768.45 AFER=21.13% 165 Rajan D and Sugunthakunthalambigai R Fig. 4.2. Forecasted vs. Rainfall (9 intervals) Fig. 4.3. Forecasted vs. Rainfall (18 intervals) 166 Forecasting of Annual Rainfall using Fuzzy Logic Interval Based Partitioning In Different Intervals Fig. 4.4. Graph of MSE 6, 9 and 18 intervals B. Results and Discussion. The MSE and AFER calculated in Tables 3.11, 3.12 and 3.13 above were analyzed. The paper shows working with different intervals such as 6 , 9 and 18 . The majority of recently published papers work in one of these intervals. The focus of this paper was to propose a new algorithm and check its predicted variability over all these intervals. The results show that prediction works best on the 6th interval among all other intervals. All results are presented in an easy-to-understand bar chart format to reduce the complexity of this study and present it in a more understandable manner. To prove that this algorithm is efficient, a comparison is made with other existing methods proposed by Heuristic and FTS first order in Table XIV. As can be seen from Fig. 4.5, the proposed algorithm was able to achieve significantly lower MSE compared to other methods. This model not only provides a lower MSE, but also explains why researchers making fuzzy logic predictions choose the 6th interval for their work. All other intervals do not give better results than the 6th interval partitioning. A possible reason for increasing the number of intervals is that the data is overcrowded. As such, the relevant data between intervals is not included in the forecasting algorithm and affects forecasting results. Keeping the interval lower than the 6th interval will spread the data too much. Therefore, the 6th interval partitioning seems to be the best overall for fuzzy logic-based predictive models. 167 Rajan D and Sugunthakunthalambigai R Table 4.1. Comparison to prove efficiency 5. Conclusion First, divide the data set into 6, 9, and 18 intervals and compute the values for each block. Use these midpoint values to compute rainfall forecasts for all blocks. 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