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Article

Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications in Wheat Production Prediction

by
Rachna Jain
1,
Nikita Jain
1,
Shivani Kapania
1 and
Le Hoang Son
2,3,*
1
Computer Science and Engineering, Bharati Vidyapeeth’s College of Engineering, New Delhi 110012, India
2
Division of Data Science, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
3
Faculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
*
Author to whom correspondence should be addressed.
Symmetry 2018, 10(12), 768; https://doi.org/10.3390/sym10120768
Submission received: 15 November 2018 / Revised: 29 November 2018 / Accepted: 30 November 2018 / Published: 18 December 2018
Browse Figures
Figure 1
<p>Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications DAbFP simulation Workflow.</p> "> Figure 2
<p>(<b>a</b>–<b>f</b>): 9th to 11th Interval for fuzzified degree-based approximation AFER and MSE.</p> "> Figure 2 Cont.
<p>(<b>a</b>–<b>f</b>): 9th to 11th Interval for fuzzified degree-based approximation AFER and MSE.</p> "> Figure 2 Cont.
<p>(<b>a</b>–<b>f</b>): 9th to 11th Interval for fuzzified degree-based approximation AFER and MSE.</p> "> Figure 3
<p>Comparison of MSE among all degrees in 9th interval.</p> "> Figure 4
<p>Comparison of MSE among all degrees in 11th interval.</p> ">
Versions Notes

Abstract

:
Recently, prediction modelling has become important in data analysis. In this paper, we propose a novel algorithm to analyze the past dataset of crop yields and predict future yields using regression-based approximation of time series fuzzy data. A framework-based algorithm, which we named DAbFP (data algorithm for degree approximation-based fuzzy partitioning), is proposed to forecast wheat yield production with fuzzy time series data. Specifically, time series data were fuzzified by the simple maximum-based generalized mean function. Different cases for prediction values were evaluated based on two-set interval-based partitioning to get accurate results. The novelty of the method lies in its ability to approximate a fuzzy relation for forecasting that provides lesser complexity and higher accuracy in linear, cubic, and quadratic order than the existing methods. A lesser complexity as compared to dynamic data approximation makes it easier to find the suitable de-fuzzification process and obtain accurate predicted values. The proposed algorithm is compared with the latest existing frameworks in terms of mean square error (MSE) and average forecasting error rate (AFER).

1. Introduction

Currently, time series having indecision observations are called fuzzy time series, a term originally defined by Song and Chissom [1,2]. The interpretation obtained from time series is then transformed into fuzzy sets. There is a need for data available in numerous forms multiplied over time. Forecasting is suitable for circumstances where vacillation linked to the outcome is tangible. Time series exploration is an essential mechanism for forecasting the unknown on the basis of its past history. The two significant methods that fit this category are time series and regression. Modern approaches to time series forecasting are influenced by the repetition of history itself. Time series include the recorded values of the variable in the past and also include the present value. This method supports the discovery of arrangements and the inference of future events based on the patterns established as the chief focus material of time series analysis. Solutions to various practical problems related to finance, economics, marketing, and business as well as prediction in economic and sales forecasting, information systems forecasting, stock market prediction, the number of outpatient visits, etc., can be determined using time series.
The idea for the exploratory work on this topic came from an extensive study of work previously done in the niche field of extrapolative demonstration using fuzzy logic. In an agrarian country with a primarily tropical climate, a tropical plant like wheat presents itself as a very lucrative and justifiable topic. To put this into perspective, Asia on its own harvests and ingests more than three quarters of the global wheat production. If economists are to be believed, this dominance by Asia in the global wheat market leads to a reduction in poverty in the region. With an improvement in production and quality of yield, wheat becomes more accessible to people from all walks of life at a lower price, which in turn pushes farmers to invest in sophisticated and valued crops. These crops bring additional income and prosperity to the farmers’ families and improve consumer food products. The sustainable computing and management of natural resources has therefore become an imperative field of study [3]. The assessment and forecasting of wheat manufacture certainly require much effort [4].
Various experimental results have been published where prediction has been shown on different datasets based on time series forecasting. Forecasting and predictions help in combating decision-making problems. Askar [5] proposed an autoregressive moving average model to predict wheat crop yield. Sachin [6,7] worked specifically on predicted rice yield for inventory management using a fuzzy time series model. Narendra [8] proposed a model for a terse-period agricultural protraction estimate. Eǧrioǧlu et al. [9,10] and Wangren et al. [11], on the other hand, implemented a generalized equivalent length breaks implanted for improvement. The former used a genetic approach.
In contrast to the above discussed methods, the proposed method in this study focused on diverse and finer levels of partitions with respect to fuzzy series data. Using this degree approximation method based fuzzy portioning, a higher prediction accuracy was observed. The method of fuzzy partitioning involved the creation of newly generated fuzzy sets based on the underlying data. The time series wheat data undertaken consisted of dynamic data whose feature value changes as a function of time. In partitioning, elements that are more similar than others form members of one set, whereas dissimilar elements form different fuzzy sets. Prediction was done under a fuzzy environment that consisted of ambiguity, improbability, and inaccuracy. The fuzzy intervals were divided based on the frequency of number of times series data. Later, historical time series data analysis was performed by computation of higher order logical fuzzy relations based on the universe of discourse. The novelty of this paper is explained below.
The proposed method used the first 9th and 11th interval time series fuzzy partitioning for wheat production prediction. Based on the interval-based fuzzy partition degree, approximation was applied for real-time wheat produce forecasting. De-fuzzified outputs obtained from approximations were estimated for error and compared with four existing methods. The decision to use fuzzy partitioning in comparison to a regression model was due to the fact that relationships become more complex when dealing with time series data. As proposed in our case, the wheat dataset was dynamic as a function of time, and the use of regression would not produce compact sets. Fuzzy partitioning was a better approach that used degrees of memberships rather than a strict rule as in case of regression. Because the relationship in our time series dataset was not sufficient to apply regression, fuzzy partitioning was a better choice. The method of fuzzy partitioning was closer to human observation behavior as compared to a linear regression model. Furthermore, the new method for forecasting wheat production with a fuzzy time series using degree approximation as a fuzzy relation for forecasting provided lesser complexity in the linear order. Such simplicity was extended to cubic and quadratic polynomial approximation which minimized the time needed to generate relational equations based on complex min-max composition operations, as well as the various hits and trials of the defuzzification process that might be required to achieve better accuracy as used in [6,7,8,9,12] as well as by Singh [13]. Two-set partitioning with lower and higher approximation performed over regression analysis finally helped in selecting a best fit line/values that represents the average across all points in graph [14].
The rest of the paper is organized as follows: Section 2 provides the literature overview about the use and progress of time series-based fuzzy partition for prediction problems. Section 3 gives the complete explanation of the proposed framework for the algorithm formulated. Section 3.1 gives a diagram workflow representation of the framework followed by a numerical example explaining the methodology in brief. In Section 3.2, a detailed explanation of the proposed methodology, which we named data algorithm for degree approximation-based fuzzy partitioning (DAbFP), is given with intermediate results. The fuzzy logic relation (FLR) for different intervals is calculated using the wheat yield dataset for different years. Thereafter, average forecasting error rate (AFER) and mean square error (MSE) formulas are also mentioned. Section 4 lists experiments using different degree polynomials and calculating the AFER and MSE for the corresponding polynomials with their respective plots. The proposed algorithm is compared with the existing methods in terms of AFER and MSE with respect to other algorithms. Finally, Section 5 depicts the final conclusions about the method and its implications over the wheat dataset and also emphasizes its future application and scope.

2. Related Works

2.1. Literature Review

Fuzzy-based time series forecasting is used to examine information which is neither explicit nor precise. Researchers have developed fuzzy time series perceptions and definitions to deal with imprecise and vague information systems where decisions or predictions could be carried out. This was later proposed by Song and Chissom, who also portrayed a special dynamic forecasting process with linguistic values [1,2,15]. Fuzzy forecasting to predict links in social networks has been described by authors in [16]. Later, the authors in [12] formally defined a fuzzy time series model described in Section 2.2.
Qiu, Liu, and Li [17] proposed a particle swarm optimization technique for similar forecasting. Primarily, data dealing with time series from the University of Alabama [18] were used. An average autocorrelation function was framed to give high forecasting accuracy. In order to analyze time series using computed fuzzy logical relations of higher order, Garg et al. [19], Son [20], Hunrag [21,22], Hwang and Chen [23], Lee Wang and Chen [24], Chu and Kim [25], and Sheta [26] developed extensive fuzzy as well as decision based forecasting methods in order to augment forecasting accuracy, each having minor variations. Lee [27] proposed a fuzzy candlestick [28] pattern to store financial expertise. To obtain highly mosaic matrix computations, a multivariate heuristic model was modelled and implemented in [29]. A determination of the interval over varying length was given by Hiemstra [30]. A number of repetitions of fuzzy relationships were used to determine the weights in fuzzy time series data in [31,32,33]. Regular increasing Monotone (RIM) quantifiers were used by Garg et al. [34,35] to design a priority matrix.
Several distinguished and relative works have been done by Klir et al. [36] and Dostal [37] with some native approaches for prediction. The use of optimization techniques in commercial and communal sector was also demonstrated by Dostal [38]. Li et al. [39] introduced fuzzy logic linking to chaos theory. Peters [40,41] extended it to fractal market analysis in capital markets. Trippi [42] represented fuzzy logic to chaos and non-linear dynamics in financial markets. Altroc [43] applied to business and finance using neuro-fuzzy. Hamam et al. [44] evaluated superiority of understanding of haptic centered uses based on fuzzy logic. Alreshoodi [45] researched an experiential learning established on a fuzzy logic method to measure the QOS/QOE correlation for covered video streaming. Doctor et al. [46] entrenched agent-based method for comprehending ambient intellect. Wang et al. [47] generated fuzzy instructions by learning from instances. In [48] a high order approximation for forecasting tourism demands in turkey using fuzzy time series data and artificial neural network is proposed. Another, new approach using fuzzy type-2 logic and fractal theory was given by Castillo and Melin [49]. The experimental study was done to establish the span of breaks with fuzzy time series [50]. A non-linear optimization with polynomial time series is another work presented by authors [51]. The forecasting models based on Event discretization function were placed forward.
In this paper, the dataset used for forecasting wheat production is taken from a source [52]. Son et al. [53] established a fuzzy clustering method for weather forecasting. Also, a neuro-fuzzy system has been designed and evaluated for insurance forecasting [54]. In [54], the authors have used an ensemble learning technique with limited fuzzy weights. Adaptive neuro-fuzzy [55] framework is another work by [56] in field of wheat production forecasts.
In [57], a different dataset for wheat production forecasts using soil properties has been used. Some of the properties of soil like shear strength has been predicted in [58]. Similarly, an adaptive fuzzy rule-based technique with automatic parameter updating has been used to model financial time series in [59]. A systematic approach has been discussed in [60] for detection of structural breaks in time series, namely the fuzzy transform and other method of fuzzy natural logic. It is based on F- transform to calculate slope of time-series. Another problem of the separable verification of fuzzy binary relations has been addressed in [61] providing necessary conditions and a well-organized algorithm for checking the same.

2.2. Mathematical Preliminary

This section presents the preliminaries needed to understand any problem of time series forecasting
Definition 1 [62].
Given F(t) as the group of all possible values of fuzzy time series at time t, F(t − 1) is group of all possible values at t − 1 having Z as a fuzzy relation between F(t) and F(t − 1) where Z is a union of all fuzzy relations defined as:
Z = Z ( t , t − 1 ) = f i 1 ( t − 1 ) X f j o ( t ) ∪ f i 2 ( t − 2 ) X f j 1 ( t − 1 ) ∪ … … ∪ f i n ( t − n ) X f j n − 1 ( t − n + 1 )  
Then a first order time invariant series model is expressed as
F ( t ) = F ( t − 1 ) ° Z ( t , t − 1 )
Definition 2 [62].
Let be U the universe of discourse, U = {u1, u2, u3…} and U be a finite set A fuzzy set F of U can be expressed as follows:
∑ i = 1 n μ A ( u i ) / u i
= μ A ( u 1 ) u 1 + μ A ( u 2 ) u 2 + μ A ( u 3 ) u 3 + μ A ( u 4 ) u 4 … … + μ A ( u n ) u n .
where “+” is operator ∪ and ”/” is separator.
Definition 3 [63].
Assume that F(t) is a fuzzy time series, and Z (t, t − 1) is a first order model of time series F(t). If
Z ( t , t − 1 ) ∈ Z ( t − 1 , t − 2 ) ∀   time   ( t ) then   F ( t ) → Time   invariant   fuzzy   time   series
Z ( t , t − 1 ) ∈ t   &   ≠ Z ( t − 1 , t − 2 )   ∀   time   ( t ) then   F ( t ) → Time   variant   fuzzy   time   series
Definition 4 [64].
Given F(t) as the time series data D, with Ft(I) as fuzzy set, then defuzzified value Fd is defined as the z-value with the highest membership degree.
μ A ( z o ) ≥ μ A ( z )   ∀   z ∈ F
z 0 = max ( d e g ( μ A ) ) i n   F
Definition 5 [64].
Given F(t) as the time series data D, with Ft(I) as fuzzy set, a quasi-arithmetic mean for fuzzified output is:
F ( t ) [ I = 1 … n ] = [ 1 / n ∑ i = 1 n x i ] 1 / α
α = 1 for arithmetic means
Forecasting models are categorized as follows:
AR (Autoregressive) Models,
MA (Moving Average) Models,
ARMA (Autoregressive Moving Average) Models,
ARIMA (Autoregressive Integrated Moving Average) Models.
Definition 6 [65].
AR model of a given order r is defined as:
A t = ρ 1 W t − 1 + … … + ρ r W t − r + n o i s e ( ϵ )
where W t − 1 … … . are independent variables and ρ 1 … ρ r are model parameters.
AR(r) model =
(a) A 2 = ρ 1 W 1 + e 2
(b) A 3 = ρ 1 W 2 + ρ 2 W 1 + e 3
(c) A n = ρ 1 W n − 1 + ρ n − 1 W 1 + e n
Y ( ( n − 1 ) w 1 ) = W ( ( n − 1 ) w ( n − 1 ) ) ρ ( ( n − 1 ) w ) + e ( ( n − 1 ) w 1 )
Substituting values for ρ parameter aids in prediction.

3. The Proposed Framework

3.1. The Need of This Framework

Recent studies on wheat production forecasts have been conducted in [56,57,58]. Here, the later of an artificial neural network with fuzzy systems have been used for predicting forecast for a 5-degree polynomial in only two periods. In another work, ensemble learning with limited fuzzy weights was used. While the former uses another artificial neural network to forecast production based on energy inputs, another decision making analysis has been done in [66]. Several prediction procedures on case basis has been done by authors in [67,68,69,70]. The above stated method provided prediction using support vector machines based on soil properties. A similar prediction was performed in [7,8,9] where data are not partitioned and fuzzified as per time series.
The proposed method in this paper will take the yield data in reference to time series fuzzified in diverse partitions and give precise prediction. The precision comes from the 9 or 11-level linguistic partition carried out over large time series scale. Our method outperforms the existing 4 methods in terms of RMSE and AFER. Hence, a consolidated framework to perform predictions over multiple and diverse linguistic partitions is needed.

3.2. The Workflow Diagram

In this section, an overview of the proposed framework with simulation steps is given in Figure 1. Table 1 gives the linguistic fuzzy set partitioning while Table 2 gives the frequency distribution over 9 interval partitioning.

3.3. DAbFP Algorithm

The proposed algorithm is performed on the source dataset taken under following steps:
Step 1: Let D denotes the source dataset variable.
U = [ D m i n − x , D m a x + y ]
Using Definition 2, Universe of Discourse (U) is defined as
D m i n , D m a x ∈ M a x , M i n { d a t a s e t ( D ) }
x , y ∈ R + ,   given   R   as   real   numbers .
Step 2: Partition the dataset D into suitable four frequencies to perform subsequent forecasting steps to each group:
D → D f i , { i ∈ 1 , 2 … 4 }
Step 3: Using above partitioned data as Dnew, we define fuzzy sets as F1, F2…F7 linguistically mapped over the universe of discourse U defined as follows:
F 1 = 1 q 1 + 0.5 q 2 + 0 q 3 + 0 q 4 + 0 q 5 + 0 q 6 + 0 q 7 + 0 q 8 + 0 q 9
F 2 = 0.5 q 1 + 1 q 2 + 0.5 q 3 + 0 q 4 + 0 q 5 + 0 q 6 + 0 q 7 + 0 q 8 + 0 q 9
F 3 = 0 q 1 + 0.5 q 2 + 1 q 3 + 0.5 q 4 + 0 q 5 + 0 q 6 + 0 q 7 + 0 q 8 + 0 q 9
F 4 = 0 q 1 + 0 q 2 + 0.5 q 3 + 1 q 4 + 0.5 q 5 + 0 q 6 + 0 q 7 + 0 q 8 + 0 q 9
F 5 = 0 q 1 + 0 q 2 + 0 q 3 + 0.5 q 4 + 1 q 5 + 0.5 q 6 + 0 q 7 + 0 q 8 + 0 q 9
F 6 = 0 q 1 + 0 q 2 + 0 q 3 + 0 q 4 + 0.5 q 5 + 1 q 6 + 0.5 q 7 + 0 q 8 + 0 q 9
F 7 = 0 q 1 + 0 q 2 + 0 q 3 + 0 q 4 + 0.5 q 5 + 1 q 6 + 0.5 q 7 + 0 q 8 + 0 q 9
F 8 = 0 q 1 + 0 q 2 + 0 q 3 + 0 q 4 + 0 q 5 + 0.5 q 6 + 1 q 7 + 0.5 q 8 + 0 q 9
F 9 = 0 q 1 + 0 q 2 + 0 q 3 + 0 q 4 + 0 q 5 + 0 q 6 + 0.5 q 7 + 1 q 8 + 0.5 q 9
Every partition obtained in the partitioning based on frequency is represented by F(I), where (I) indicates the intervals inside its value exist. The value of the outcome increases on increasing the value of “I”. The same taxonomy helps to provide an evocative vision to the researchers. For instance, every interval can be signified by fuzzy partitions if we are operating on 9 partitions, as presented beneath:
Hence, growth in the suffix (I) is evidently related through greater harvest in the production of wheat and having the same taxonomy. Subsequently, Fuzzy Logic Relationships (FLR) is recognized for the specified group of values. It can be elucidated over the particular instance. Here, q1, q2…q7 ∈ fixed length intervals.
Step 4: From above partitioned data as Dnew, we define 11 fuzzy sets as F1, F2…F11 over U. Similar equations (as 5 to 13) are observed for 11 intervals. Here, q1, q2…q11 ∈ fixed length intervals.
Step 5: Mean of middle values of fuzzy partitions on the Right-Hand Side of Fuzzy logic relation (FLR) is calculated. This calculation is performed for degree approximation. For instance, in the 2nd order FLR, F4 <- F2, F7. If P and Q are the centers of Interval F2 and F7 respectively then
R = P + Q/2
where for fuzzy partition F4, R is the center. Likewise, for 3rd order FLR:
If F <- F2, F7, F3 where P, Q, R are the centers for Interval F2, F7 and F3 respectively then
S = (P + Q + R)/3
Here, S is the mean fuzzy value for a particular forecast year. It is used in Linear Regression Model as a variable, for thorough de-fuzzification. From this, the results can be used to calculate the forecast value:
Mean   Fuzzy   Value ( MFV ) = ∑ i = 1 i = n v a l u e ( F i ( d ) ) /
Here, n is total number of values while v a l u e ( F i ( d ) ) is the fuzzy value at degree.
As per the steps followed in the proposed algorithm, Table 3, Table 4 and Table 5 give the intermediate results. In this section, the concluding part of the devised algorithm is explained with the results presented in Table 6, Table 7, Table 8 and Table 9.
Step 6: After degree approximation based on fuzzy logic relation, defuzzification is performed using regression analysis. On plotting the points, we select a Best Fit line that represents average across all points in graph. Thereafter, the equation of line is estimated which can be linear or polynomial of higher degrees 2, 3, 4, 5 or 6. In the consequent section, we use two important constraints to associate the outcome as stated below:
Average Forecasting Error Rate (AFER) =
∑ i = 1 i = n ( ( m o d ( X I − Y i ) / X i ) / n ) ∗ 100
Mean Square Error (MSE).
= ∑ i = 1 n ( ( X − Y ) ^ 2 ) / n
Here, Xi is the actual production cost whereas Yi is the predicted value.

3.4. Numerical Example

In year 1981, Produce = 3552 (fits to F3).
In year 1982, Produce = 4177 (fits to F7).
In year 1983, Produce = 3372 (fits to F2).
In year 1984, Produce = ? (Assume this request to be forecast, let F be the partition where value is contained).
Hence, the above Logical Relationships can assist forecast for a specific year by means of the values obtained for earlier years and then creating a relationship amongst values. Fuzzy Logical Relationship of Order 3 is: F = F2, F7, F3, here F is the forecast partition for produce in year 1984.
Now, an appropriate defuzzification procedure can be functional on these values to forecast value of the harvest in year 1984 (conferred in step 5), agreed that appropriate calculations are done for fuzzy sets which resemble to the fuzzy partitions in the previous years.
By means of formulation stated above, the results are shown in Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 is calculated for 9 and 11 partitions in together of order two and order three FLR.

4. Results and Discussion

4.1. Linear Polynomial

A linear polynomial relation is defined as:
Y = m X + C   w h e r e   C   ∈ c o n s t a n t
Here, variable Y provides the value that is predicted. Output for year and the input variable X fed to equation using form (14, 15, 16, which relates to years 1981, 1982, 1983, 1984…). By means of Figure 2a,b, one can calculate the yearly predicted results and then estimating the AFER and MSE as given in Table 10.

4.2. Quadratic Polynomial

A linear polynomial relation is defined as:
Y = A x 2 + B x + C
Here, variable Y will give the value that is predicted Output for year and the input variable X fed to equation using form (1, 2, 3, 4…., which relates to years 1981, 1982, 1983, 1984…). By means of Figure 2c,d, one can calculate the yearly predicted results and then estimating the AFER and MSE was calculated matching to figure as given in Table 11.

4.3. Cubic Polynomial

The cubic polynomial relation is given as:
Y = A x 3 + B x 2 + C x + D
Here, variable Y will give the value that is predicted output for each year and input variable X fed to equation using form (1, 2, 3, 4…., which relates to years 1981, 1982, 1983, 1984…). By means of Figure 2e,f, one can calculate the yearly predicted results and then estimating the AFER and MSE matching to figure as given in Table 12.

4.4. Results

From the above analysis, we have computed the mean MSE and AFER values for final predicted values where the degree approximation is computed accordingly. The proposed algorithm is initially compared with baseline method such as Chissom [1,2] on benchmark data for forecasting the enrollments of University of Alabama. The superiority in values in terms of MSE and AFER marks it as a probable candidate for predicting wheat production in future as shown in Table 13. For further performance analysis, the proposed method is hereby compared with existing methods as shown in Table 14 and Table 15 for both 9 and 11 intervals. The proposed algorithm outperforms the existing ones in terms of MSE and AFER; thereby proving to be a best fit for wheat produce prediction. The MSE and AFER of the proposed algorithm comes out to be 362,119.88 and 5,107,713.738 for 3rd degree and 2nd degree polynomial in 9th interval as compared to the MSE of 36,559.88 and AFER AS 11.92547975 for 3rd degree polynomial of Yalaz et al. [64]. Similarly, the values of MSE and AFER are compared in Table 13 and Table 14 for 9th interval 2nd degree polynomial. Also, the evaluation statistics of our proposed algorithm outperforms in 11th interval.
In Figure 3, the FLR 3rd degree MSE is generally higher than FLR 2nd degree MSE except in case of polynomial degree 3. We can infer that Linear FLR 2nd degree polynomial has the lowest MSE among all the cases for 9th interval. It is convenient to estimate a particular case is the best among all others. As it can be inferred from the graph, total 10 cases for 9th interval has been monitored. We have also worked on 8 cases in 11th interval partitioning as shown in Figure 4.

5. Conclusions

Various researchers in the past have tried to explore this prediction modeling field using fuzzy logic. Further research is needed by researchers around the world. In this paper, we proposed a novel algorithm using fuzzy linear regression to forecast wheat production. The results demonstrated the efficiency of the suggested method. Further studies will focus on accelerating computational time of this method by GPU and examining other wheat problems or exploring advanced methods [70,71,72,73,74,75,76,77,78,79,80,81,82].

Author Contributions

The article conceptualization was done by R.J. and N.J. The methodology was proposed by L.H.S. Software feasibility was validated and tested by S.K. Further the formal analysis with Investigation was performed by R.J. and N.J. The data curation for the experiments was done by S.K. Writing-Original Draft Preparation was done by R.J. and N.J. along with Writing-Review & Editing performed by L.H.S. The project was visualized by N.J. and S.K. under the supervision of R.J. and L.H.S.

Funding

This research received no external funding.

Conflicts of Interest

The authors declare no conflict of interest.

Compliance with Ethical Standards

The authors declare that they do not have any conflict of interests. This research does not involve any human or animal participation. All authors have checked and agreed the submission.

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Symmetry 10 00768 g001 550
Figure 1. Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications DAbFP simulation Workflow.
Figure 1. Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications DAbFP simulation Workflow.
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Symmetry 10 00768 g002a 550Symmetry 10 00768 g002b 550Symmetry 10 00768 g002c 550
Figure 2. (a–f): 9th to 11th Interval for fuzzified degree-based approximation AFER and MSE.
Figure 2. (a–f): 9th to 11th Interval for fuzzified degree-based approximation AFER and MSE.
Symmetry 10 00768 g002aSymmetry 10 00768 g002bSymmetry 10 00768 g002c
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Figure 3. Comparison of MSE among all degrees in 9th interval.
Figure 3. Comparison of MSE among all degrees in 9th interval.
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Figure 4. Comparison of MSE among all degrees in 11th interval.
Figure 4. Comparison of MSE among all degrees in 11th interval.
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Table
Table 1. Fuzzy linguistic partitions.
Table 1. Fuzzy linguistic partitions.
F1very meagre produce
F2meagre produce
F3better than poor produce
F4not so quality produce
F5average production
F6superior produce
F7very superior produce
F8Very very superior produce
F9tremendous produce
Table
Table 2. Frequency Distribution (9 Interval).
Table 2. Frequency Distribution (9 Interval).
Fuzzy SetsUpperLowerFrequency
F1233,200343,3553
F2233,355343,5112
F3233,511343,6662
F4233,666343,8223
F5233,822344,1334
F7234,133344,2883
F8234,288344,4442
F9234,444344,6001
Table
Table 3. Frequency Centered Partitioning (9 Interval).
Table 3. Frequency Centered Partitioning (9 Interval).
Fuzzy SetsUpperLowerNew Fuzzy Sets
AF1A932,0073252.76Z1
3253.763303.8Z2
3303.83356.66Z3
AF2A3356.663432.435Z4
3432.4353512.2Z5
AF3A3512.23589.985Z6
3589.9853677.75Z7
AF4A3677.753729.5Z8
3729.53771.45Z9
3771.453823.3Z10
AF5A3823.33862.1985Z11
3862.19853900.175Z12
3900.1753949.9725Z13
3949.97253988.865Z14
AF7A4234.34285.25Z15
4285.254238Z16
42384289.95Z17
AF8A4289.954367.735Z18
4367.7354445.5Z19
AF9A4445.54600Z20
Table
Table 4. Frequency Distribution (11 Interval).
Table 4. Frequency Distribution (11 Interval).
Fuzzy SetsUpperLOWERFrequency Uency
A1320033273
A2332734541
A3345435812
A4358137093
A5370938361
A6383640914
A7393741203
A8409142182
A9421843452
A10434544721
A11447246001
Table
Table 5. Frequency Centered Partitioning (11 Interval).
Table 5. Frequency Centered Partitioning (11 Interval).
Fuzzy SetsUpperLowerNew Fuzzy Sets
A13200.0003253.423NF1
3253.4233295.847NF2
3295.8473330.270NF3
A23330.2703460.540NF4
A33460.5403521.175NF5
3521.1753579.810NF6
A43579.8103630.233NF7
3630.2333670.657NF8
3670.6573711.080NF9
A53711.0803841.350NF10
A63841.3503870.168NF11
3870.1683900.985NF12
A73900.9853929.803NF13
3929.8033970.720NF14
A84091.9904149.525NF15
4149.5254220.160NF16
A94220.1604290.795NF17
4290.7954351.430NF18
A104351.4304469.700NF19
A114469.7004600.000NF20
Table
Table 6. (9 INTERVALS, Fuzzy Logic Relation 2nd DEGREE).
Table 6. (9 INTERVALS, Fuzzy Logic Relation 2nd DEGREE).
YearProductFuzzy SetsFLR RelationsAvg.Mid Fuzzy Value
19813552Z6--3549.9875
19824177Z15--4159.225
19833372Z4Z4<-Z15,Z63854.606253394.4375
19843455Z5Z5<-Z4,Z153776.831253472.2125
19853702Z8Z8<-Z5,Z43433.3253692.575
19863670Z8Z8<-Z8,Z53582.393753692.575
19873865Z12Z12<-Z8,Z83692.5753880.5315
19883592Z7Z7<-Z12,Z83786.553253627.7625
19893222Z1Z1<-Z7,Z123754.1473225.925
19903750Z9Z9<-Z1,Z73426.843753744.425
19913851Z11Z11<-Z9,Z13485.1753841.644
19923231Z1Z1<-Z11,Z93793.03453225.925
19934170Z15Z15<-Z1,Z113533.78454159.225
19944554Z20Z20<-Z15,Z13692.5754522.2
19953872Z12Z12<-Z20,Z154340.71253880.5315
19964439Z19Z19<-Z12,Z204201.365754405.5125
19974266Z17Z17<-Z19,Z124143.0224262.925
19983219Z1Z1<-Z17,Z194334.218753225.925
19994305Z18Z18<-Z1,Z173744.4254327.7375
20003928Z13Z13<-Z18,Z13776.831253919.419
Table
Table 7. (11 INTERVALS, Fuzzy Logic Relation 2nd DEGREE).
Table 7. (11 INTERVALS, Fuzzy Logic Relation 2nd DEGREE).
YearProductFuzzy SetsFLR relationAvg.Fuzzy
19813552F6--3549.9925
19824177F16--4186.3425
19833372F4F4<-F16,F63868.16753390.905
19843455F5F5<-F4,F163788.623753486.3575
19853702F9F9<-F5,F43438.631253687.868325
19863670F9F9<-F9,F53587.1129133687.868325
19873865F11F11<-F9,F93687.8683253852.25875
19883592F7F7<-F11,F93770.0635383603.021665
19893222F1F1<-F7,F113727.6402083221.211665
19903750F10F10<-F1,F73412.1166653772.714995
19913851F11F11<-F10,F13496.963333852.25875
19923231F2F2<-F11,F103812.4868733263.634995
19934170F16F16<-F2,F113557.9468734186.3425
19944554F20F20<-F16,F23724.9887484536.35
19953872F12F12<-F20,F164361.346253884.07625
19964439F19F19<-F12,F204210.2131254409.065
19974266F17F17<-F19,F124146.5706254249.9775
19983219F1F1<-F17,F194329.521253221.211665
19994305F18F18<-F1,F173735.5945834313.6125
20003928F13F13<-F18,F13767.4120833915.89375
Table
Table 8. 9 INTERVALS, Fuzzy Logic Relation 3rd DEGREE.
Table 8. 9 INTERVALS, Fuzzy Logic Relation 3rd DEGREE.
YearProductFuzzy SetsFLR RelationsAvgMid Fuzzy Value
19813552Z6--3549.9875
19824177Z15--4159.225
19833372Z4--3394.4375
19843455Z5Z5<-Z4,Z15,Z63701.2166673472.2125
19853702Z8Z8<-Z5,Z4,Z153675.2916673692.575
19863670Z8Z8<-Z8,Z5,Z43519.7416673692.575
19873865Z12Z12<-Z8,Z8,Z53619.1208333880.5315
19883592Z7Z7<-Z12,Z8,Z83755.2271673627.7625
19893222Z1Z1<-Z7,Z12,Z83733.6233225.925
19903750Z9Z9<-Z1,Z7,Z123578.0733744.425
19913851Z11Z11<-Z9,Z1,Z73532.7041673841.644
19923231Z1Z1<-Z11,Z9,Z13603.9983225.925
19934170Z15Z15<-Z1,Z11,Z93603.9984159.225
19944554Z20Z20<-Z15,Z1,Z113742.2646674522.2
19953872Z12Z12<-Z20,Z15,Z13969.11866673880.5315
19964439Z19Z19<-Z12,Z20,Z154187.3188334405.5125
19974266Z17Z17<-Z19,Z12,Z204269.4146674262.925
19983219Z1Z1<-Z17,Z19,Z124182.9896673225.925
19994305Z18Z18<-Z1,Z17,Z193964.78754327.7375
20003928Z13Z13<-Z18,Z1,Z173938.86253919.419
Table
Table 9. 11 INTERVALS, Fuzzy Logic Relation 3rd DEGREE.
Table 9. 11 INTERVALS, Fuzzy Logic Relation 3rd DEGREE.
YearProductFuzzy SetsFLR RelationAvgFuzzy
19813552F6--3549.9925
19824177F16--4186.3425
19833372F4--3390.905
19843455F5F5<-F4,F16,F63709.083486.3575
19853702F9F9<-F5,F4,F163687.8683333687.868325
19863670F9F9<-F9,F5,F43521.7102753687.868325
19873865F11F11<-F9,F9,F53620.698053852.25875
19883592F7F7<-F11,F9,F93742.6651333603.021665
19893222F1F1<-F7,F11,F93714.3829133221.211665
19903750F10F10<-F1,F7,F113558.8306933772.714995
19913851F11F11<-F10,F1,F73532.3161083852.25875
19923231F2F2<-F11,F10,F13615.3951373263.634995
19934170F16F16<-F2,F11,F103629.5362474186.3425
19944554F20F20<-F16,F2,F113767.4120824536.35
19953872F12F12<-F20,F16,F23995.4424983884.07625
19964439F19F19<-F12,F20,F164202.256254409.065
19974266F17F17<-F19,F12,F204276.4970834249.9775
19983219F1F1<-F17,F19,F124181.0395833221.211665
19994305F18F18<-F1,F17,F193960.0847224313.6125
20003928F13F13<-F18,F1,F173928.2672223915.89375
Table
Table 10. MSE and AFER values for all intervals.
Table 10. MSE and AFER values for all intervals.
9th Interval11th Interval
FLR 2nd DegreeFLR 3rd DegreeFLR 2nd DegreeFLR 3rd Degree
----
----
42,986.55822-44,818.16021-
22,492.800584800.82694423,809.724425074.567696
5095.104420,567.862234501.73902519,945.9129
188.6776965947.18592490.1360365579.492416
33,522.6804656,558.8280432,002.7054555,301.16624
13,352.26474826.91457614,330.005265237.706384
261,321.3504224,460.8555265,543.3655227,439.3328
78.1456397.2049166.6681274.2336
4424.3782567505.2766893904.8751216893.316676
335,389.2404322,242.4169340,019.6045326,621.3941
111,708.356113,595.2875109,089.5024110,861.0298
479,672.5971471,614.5746474,288.4066465,702.5158
226.8036873.498025357.7772251163.4921
276,987.4796253,157.9099272,989.5303248,358.7027
107,355.833187,527.8142104,900.197784,577.43568
555,013.0801616,925.4189560,578.6435625,225.4669
99,454.452570,892.7900597,145.0457667,992.64852
7617.798421,036.60168266.446422,734.6084
MSE = 130,938.2001MSE = 134,290.0745MSE = 130,933.4741MSE = 134,057.8249
AFER = 7.352165941AFER = 7.50564575AFER = 7.360701563AFER = 7.497227115
Table
Table 11. MSE and AFER values for all intervals.
Table 11. MSE and AFER values for all intervals.
9th Interval11th Interval
FLR 2nd DegreeFLR 3rd DegreeFLR 2nd DegreeFLR 3rd Degree
----
----
86,872.72867-86,973.79553-
42,656.7888431,205.3638243,329.2533930,070.88937
1748.4942255821.3085061515.54495985.730056
53.418557442014.17849611.203747841939.204525
38,394.9132955,270.082236,517.2225954,101.41093
7616.541622309.1484738616.889062698.84406
222,169.1256187,981.991227,928.7106192,375.311
1499.23845409.60251044.58244573.8169
13,907.9094521,993.8094712,407.5538820,095.30221
278,480.7993253,320.5535285,381.6062260,326.8975
145,760.7025158,971.1792141,014.3692153,420.3511
536,451.9471548,144.1831527,917.3339538,011.1009
174.636225113.582306363.520917.53515625
290,677.1154275,185.8551285,894.6794269,126.8781
103,181.13285,931.00097100,960.48783,089.84966
599,950.6294668,580.3041603,535.0764674,659.1905
66,975.266139,664.3471166,480.021738,764.05762
30,520.0963,695.664430,317.774463,988.7616
MSE = 137,060.6376MSE = 141,506.5973MSE = 136,661.6458MSE = 140,779.1254
AFER = 7.687795338AFER = 7.758800407AFER = 7.653515775AFER = 7.720197268
Table
Table 12. MSE and AFER values for all intervals.
Table 12. MSE and AFER values for all intervals.
9th Interval11th Interval
FLR 2nd DegreeFLR 3rd DegreeFLR 2nd DegreeFLR 3rd Degree
----
----
290,632.1524-313,062.5185-
77,523.93313103,695.334781,762.1411114,607.7066
7830.03765616008025.9201561299.6025
15,835.504266608.64940117,384.106067268.858358
120,277.245598,004.56003125,760.2382103,028.674
4043.2302652068.7214824999.2818712928.454871
119,470.9499119,186.2386116,370.5638114,544.0704
14,713.6919,909.2114,859.6120,793.64
21,494.4041334,105.8159420,329.9974433,467.62901
309,726.0861253,318.1376319,561.3766260,429.7685
89,440.41254131,289.695981,819.51089122,710.9307
367,945.1196452,673.8343348,552.3228431,268.5495
18,985.395166037.2924,176.362669254.44
150,648.9335185,762.3792137,798.9429170,127.8712
41,022.0870346,271.7958435,822.3379739,841.27777
674,680.875729,631.6726684,819.8035744,583.9843
109,253.584555,372.86685112,612.392756,584.23018
4830.2511,491.848172.167779.24
MSE = 135,464.105MSE = 132,766.3554MSE = 136,438.3104MSE = 131,795.231
AFER = 7.752071496AFER = 8.228273107AFER = 7.744400101AFER = 8.305847824
Table
Table 13. Comparison of MSE and AFER values for one set of intervals with Chissom [1,2] on enrollment data.
Table 13. Comparison of MSE and AFER values for one set of intervals with Chissom [1,2] on enrollment data.
YearEnrollement DataChissom [1,2]Proposed Method (DAbFP)
2nd Degree3rd Degree
197113,055-13,56113,261
197213,56314,00013,75613,786
197313,86714,00013,75613,776
197414,69614,00014,45114,431
197515,46015,50015,36115,271
197615,31116,00015,36115,661
197715,60316,00015,72115,321
197815,86116,00015,90015,887
197916,80716,00017,08517,067
198016,91916,81317,08517,067
198116,38816,81316,48716,480
198215,43316,78915,38515,371
198315,49716,00015,38515,371
198415,14516,00015,02915,012
198515,16316,00015,02915,012
198615,98416,00015,88515,780
198716,85916,00017,06917,054
198818,15016,81317,98117,934
198918,97019,00018,80218,780
199019,32819,00018,90418,800
199119,33719,00018,90418,800
199218,876-18,81618,800
MSE 775,687415,382323,421
AFER 37.487616.6114.43
Table
Table 14. Comparison of MSE and AFER values for 9 intervals with existing frameworks.
Table 14. Comparison of MSE and AFER values for 9 intervals with existing frameworks.
YearJilani and Burney [67]Qiu et al. [11]Yalaz et al. [64]Khoshnevisan et al. [57]Proposed Method DAbFP
2nd Degree3rd Degree2nd Degree3rd Degree2nd Degree3rd Degree2nd Degree3rd Degree2nd Degree3rd Degree
1981----------
1982----------
198344,312.75772-45,322.7237-35,332.72372-35,212.72372-35,312.72372-
198414,926.988,729.995612,827.562588,721.885611,826.562591,721.885616,726.562581,721.885611,826.562581,721.8856
19851893.19890239,129.69061862.118936,122.64061765.11890227,122.64061772.11890226,122.640631762.11890226,122.64063
19862250.7027296459.780752200.592735955.775752090.5927295045.775753090.5927295135.7757542090.5927295035.775754
198730,182.0507950,014.098129,982.050835,014.098128,892.0507932,014.098128,982.0507931,014.0981128,882.0507931,014.09811
1988900.253993419,558.9699792.25377315,558.0697772.253773415,560.0697782.253773416,559.0697782.253773415,559.0697
1989108,560295,969.786109,856225,968.38699,859205,968.38699,857215,968.385699,856205,968.3856
199066,850.4021931,73663,839.400120,73663,849.4000920,72683,829.4000920,73663,839.4000920,736
1991109,770.807993,938.686104,965.53893,532.67611,565.537983,531.676103,565.537983,532.67601103,565.537983,532.67601
1992178,169.5971229,062.574169,167.487229,062.574165,176.4871130,062.574165,166.4871129,062.5743165,166.4871129,062.5743
1993190,709.579297,812.898154,309.577297,812.898150,309.5771217,812.898160,309.5771217,812.8982140,309.5771207,812.8982
1994380,483.1606592,830.047369,362.141592,830.047364,363.1406393,810.047364,363.1406392,810.0471364,363.1406392,810.0471
199527,937.24568104,571.39129,438.2497104,571.39138,438.23968107,571.39136,438.23968114,571.390626,438.23968104,571.3906
1996256,945.9024767.2593226,733.902767.2593206,734.9024761.2593226,733.9024760.2592998206,733.9024760.2592998
1997281,348.3152169,575.758271,340.315169,575.758290,339.3151179,676.758271,339.3151189,575.7582250,339.3151179,575.7582
199835,892.380042,632,778.8435,689.37882,632,778.8455,682.378842,732,778.8457,682.378842,632,778.83735,682.378842,632,778.837
19991,650,121441,151.6631,590,721441,151.6631,891,121441,151.6631,600,121441,151.66291,590,121431,151.6629
2000100,011,7761,607,82488,811,7891,607,82488,911,7761,707,82488,811,7761,607,82488,811,7761,607,824
MSE = 5,744,057.738MSE = 394,230.0844MSE = 5,112,788.738MSE = 388,116.21MSE = 5,129,438.738MSE = 376,067.88MSE = 5,114,874.349MSE = 3,651,259.88MSE = 5,107,713.738MSE = 362,119.88
AFER = 23.95793579AFER = 13.90547975AFER = 22.95793579AFER = 13.8052AFER = 21.95793579AFER = 11.92547975AFER = 21.865793579AFER = 12.10547975AFER = 20.95793579AFER = 11.80547975
Table
Table 15. Comparison of MSE and AFER values for 11 intervals with existing frameworks.
Table 15. Comparison of MSE and AFER values for 11 intervals with existing frameworks.
YearJilani and Burney [67]Qiu et al. [11]Yalaz et al. [64]Khoshnevisan et al. [57]Proposed Method DAbFP
2nd Degree3rd Degree2nd Degree3rd Degree2nd Degree3rd Degree2nd Degree3rd Degree2nd Degree3rd Degree
1981----------
1982----------
198337,375.72372-35,412.72472-35,312.72372-32,417.72572-32,312.72371-
198411,830.5880,731.885611,827.562581,821.885611,826.562581,721.885611,728.512781,729.887611,726.512580,721.8856
19851781.11910226,328.640641762.11891126,122.64061762.11890226,122.640631757.11760326,125.648631756.11750225,122.64063
19862200.5927295200.781762093.5937295037.775752090.5927295035.7757542085.5942245038.7857562081.5922235030.775754
198728,982.058833,017.0991128,694.0517931,014.098128,882.0507931,014.0981128,372.0496231,016.0971128,375.0496531,012.09811
1988789.270773415,561.0698783.253773415,559.0697782.253773415,559.0697776.254763415,859.0698775.253763215,520.0665
198999,896205,969.395699,857205,968.38699,856205,968.385697,854205,988.395797,853205,940.3346
199063,850.4000920,73763,850.410092074663,839.4000920,73661,828.420,74061,820.399920,732
1991103,570.539983,539.67701103,566.537984,532.676103,565.537983,532.67601104,563.525983,633.67604103,561.523983,512.66201
1992165,170.4971135,250.575165,167.4871129,062.574165,166.4871129,062.5743165,242.4931130,063.5843165,040.4831128,061.5443
1993140,319.5781207,825.9152140,410.5871207,812.898150,309.5771207,812.8982140,299.5671207,914.8992140,289.5661206,812.7182
1994364,373.1506303,016.048364,364.1506372,820.047364,363.1406392,810.0471364,333.1256392,811.0472364,323.1206391,810.0465
1995264,390.2407104,585.400726,441.24068104,571.39127,438.23968104,571.390626,437.23769104,566.380626,433.23568104,565.3206
1996206,740.9024760.2693206,736.9034772.2594206,733.9024761.2592998206,725.92764.2602998206,723.901745.2452998
1997250,350.3151199,577.7583250,441.3151179,576.768260,339.3151179,575.7582250,325.315179,576.7782250,320.312179,545.3682
199835,689.478892,692,780.83735,682.378842,932,788.8635,682.378842,632,778.83734,687.378372,642,798.84534,681.378342,632,765.817
19991,590,630481,157.66291,590,123431,151.6631,690,121431,151.66291,590,108431,156.6631,590,100431,051.6569
200088,811,7801,607,87088,811,7761,707,82488,811,7761,607,82488,811,7401,607,83088,811,7321,607,310
MSE = 5,121,095.58MSE = 364,935.8833MSE = 5,107,721.684MSE = 384,540.1758MSE = 5,114,435.96MSE = 362,119.88MSE = 5,107,293.457MSE = 362,800.8246MSE = 5,107,217.009MSE = 361,780.0106
AFER = 22.85793579AFER = 13.00547975AFER = 21.95793579AFER = 12.8052AFER = 20.95793579AFER = 11.80547975AFER = 20.865793579AFER = 11.7807960AFER = 19.75793272AFER = 11.75647975

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Jain, R.; Jain, N.; Kapania, S.; Son, L.H. Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications in Wheat Production Prediction. Symmetry 2018, 10, 768. https://doi.org/10.3390/sym10120768

AMA Style

Jain R, Jain N, Kapania S, Son LH. Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications in Wheat Production Prediction. Symmetry. 2018; 10(12):768. https://doi.org/10.3390/sym10120768

Chicago/Turabian Style

Jain, Rachna, Nikita Jain, Shivani Kapania, and Le Hoang Son. 2018. "Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications in Wheat Production Prediction" Symmetry 10, no. 12: 768. https://doi.org/10.3390/sym10120768

APA Style

Jain, R., Jain, N., Kapania, S., & Son, L. H. (2018). Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications in Wheat Production Prediction. Symmetry, 10(12), 768. https://doi.org/10.3390/sym10120768

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