The Comparison of Grey System and the Verhulst Model for Rainfall and Water in Dam Prediction

A time series of data of rainfall in+ailand between the years 2005 and 2015 was employed to predict possible future rainfall based on Julong Deng’s grey systems theory and the grey Verhulst model to see which model can predict more accurately with uncertain and limited data. Firstly, the rainfall data were arranged to display the overall patterns of rainfall volume along with its frequency as well as the temperature during +ailand’s rainy seasons. +is makes it possible to see the cycle of rainfall, which is too long for people to intuitively understand the nature of precipitation. One puzzling phenomenon that has made rainfall forecast elusive is the unpredictability of the haphazard nature of rainfall in +ailand. A more precise prediction would certainly result in a better control of water volume in rivers and dams for fruitful agricultural business and adequate human consumption. +is can also prevent the flooding that can devastate the economy and transportation of the whole country and also tremendously improve the future water management policy in many ways. +is effective prediction could also be employed elsewhere around the globe for similar benefits. Hence, the grey systems theory and the grey Verhulst model are juxtaposed to determine a better prediction possible.


Introduction
Water is extremely important to ailand's agriculture as well as people largely because the country's physical terrain is rather flat and consequently appropriate for agriculture while the densely populated cities require ample supply of water for daily use and consumption.Such needs for water would not have been a serious problem if there had been adequate reservoirs and means to conserve rain water for later uses.What has exacerbated the problems is the haphazard nature of precipitation patterns during the rainy seasons.Such uncertainty results in prolonged droughts and severe floods.Fortunately, to relieve his people's suffering from having too much or too little water, the late King Bhumibol Adulyadej initiated various Royal projects of water resource development and management.Some examples are water resources development at Huay Hong Krai National Park, Doi Saket, and Huay Jo Reservoir, Chiang Mai; drainage projects from lowland and swamp areas of Bacho, Bacho, Narathiwat; building water reservoirs in strategic regions of ailand, for instance Pa Sak River Dam; and flood relief projects in Lopburi and Saraburi.e king had also invented a method to resolve water shortage problems in other ways such as the cloud seeding procedure for artificial rains [1].
With technological and theoretical advances in science and mathematics during this decade, however, the outlook of rainfall prediction has been improved.Beginning in 1982, Professor Julong Deng's grey systems theory [2] has attracted worldwide attention of researchers and has been utilized in diversified fields of study such as natural science, engineering science, and many others [3].Grey systems theory focuses mainly on systems that have partially known and unknown information.Other solutions similar to grey systems were initiated through technological progression.Solutions to uncertain systems have become a challenge for further development in any associated fields.However, one possible means can be provided by different variations of grey systems theory [4].GM(1,1) model is an important model in grey models family.e procedure to construct the GM(1,1) model, however, is neither a differential equation nor a difference equation.It is an approximate model which has dual characteristics of both differential and difference equations.However, it is inevitable for the approximate model to yield some errors in practical applications.To increase the prediction accuracy of GM(1,1) model, researchers concentrate on the improvements of GM (1,1) model [5] in three main aspects, one of which is the adjustments of grey derivative.Although types of improvements of grey models may increase the precision in some practical way, there are still some existing gaps to improve the precision of any prediction with the use of GM(1,1) model.A novel approach to optimize the initial condition in time response function for general grey prediction model involving the GM(1,1) is proposed, and the grey Verhulst model is opted as its special cases.
e optimization approach comprised the detailed item in the time response function from the original GM(1,1) model and the nth item of X (1) [6].e corresponding weighted coefficients of the two parts, forming the initial condition in new optimization method, are derived from minimizing errors in the summation of square in terms of time response function [7].e combination of newly initial conditions of fuzzy set and rough set can now express the principle of new information priority which is fully emphasized on in grey systems theory.Hence, the full utilization of new pieces of information can be as follows.

The Process Data Sequence of Grey Model
Actual data (1) From the matrix above, the optimized data for computing from group data are selected initially.e group of rainfall and temperature data points that are close (or similar) to each other are clustered to identify such groupings (or clusters), using Mathlab [8].
Mathlab graphic [8] representation below indicates that the ranges of temperature that rainfall occurs lie between approximately 25 and 35 degrees Celsius.It is interesting to note that most of the rainfall clusters between 30 and 35 degrees Celsius, while there is very little rainfall (outliers) between the temperatures of 25 and 30 degrees Celsius.As a result, it is possible to forecast the quantity of rainfall using Mathlab to graphically represent when and in what condition the rainfall occurs [7,9,10].(Figure 1).In other words, the density of the clusters is a very good indicator of overflowing, adequate, or insufficient water from the rainfall, the information of which can be very helpful to water management, especially in the prevention of droughts or floods in ailand.

Methodology
Starting, such a number instead of its range whose exact value is unknown is referred to as a grey number.In applications, a grey number in fact stands for an indeterminate number that takes its possible value within an interval or a general set of numbers.
is grey number is generally written using the symbol "⊗." ere are several types of grey numbers [6]: ( Another fundamental rainfall and water in dam of uncertain systems was the inaccuracy naturally existing in the available data.After choosing the quantity to reflect the GM(1,1) of the system concerned, one needs to determine the factors that influence the behavior of the system.If a quantitative analysis is considered, one needs to process the chosen actual data and the effective factors using sequence operators so that the available data are converted to their relevant nondimensional values of roughly equal magnitudes.
e calculation of GM(1,1) is performed as follows: Assume Y n and Matrix B Y n � x (0) (1) and then find a, b, or a ⌢ : We take to substitute in the following equation: 2

Verhulst Model
e GM(1,1) model is suitable for sequences that show an obvious exponential pattern and can be used to describe monotonic changes [6].As for nonmonotonic wavelike development sequences or saturated sigmoid sequences, one can consider establishing GM(2,1) and Verhulst models, and then, the comparison between two di erent grey prediction models of rainfall and water in dam is made.
Assume that x (0) is a sequence of actual data, x (1) the sequence of accumulation generation of x (0) , and z (1) the adjacent neighbor mean generation of x (1) , then, where And α ≠ 0, λ ∈ [0, 1] are parameters.e parameter λ is called a mean value parameter.e model ( 4) is also called the GM(1,1) power model [9].Often, λ 0.5 is selected.e whitened equation of the above equation is as follows: e solution of the whitenization (9) power model is as follows: where c is a constant and a and b are the parameters estimated from the method of least square.If α 0, then we have general form of GM(1,1) model: If α 2, then we have general form of Verhulst GM(1,1) model: From the general solution of ( 7) if we let t 1 and t n, respectively, then we can obtain the following equations: From ( 13) and ( 14), we substitute the values in the following equation: And then where β ∈ [0, 1] is called the initial condition factor.From the above, we obtain the constant determined by the initial condition: For α � 0, that is, GM(1,1) model, we substitute the values in the following equation: which is the same as c of [2].When β � 0.5, c defined by ( 16) is the same as c of [10].For α � 2, that is, grey Verhulst model, we substitute the values in the equation as follows: Substitution is made in the following function: Calculation of the effect of the initial data comparison and prediction of the result is made in the approach of f 1 (c).e improved value of the decimal point is continuously increased.
e second calculation of the effect of the same data comparison and prediction of the result is changed in the approach of f 2 (c).e improved values of the smooth curve and square function are continuously increased.
We can obtain the estimate of c so that it minimizes the f 1 (c) or f 2 (c).
For α � 0, that is, GM(1,1) model, if and only if  n k�1 (ce −ak + (b/a) − x (1) (k))e −ak � 0 from which we obtain the least square estimate: From ( 20) and ( 22), we obtain the initial condition parameter β as follows: which is the same as one of [4].en, we can obtain the following optimized time response function for GM (1, 1) model from ( 12), (18), and (22): e restored values for GM(1,1) model can be derived by first-order inverse accumulating generation operator on the time response function (23): e optimized grey systems model, the optimization of initial condition for general form of grey prediction model, is considered, and some optimal forecasting of rainfall is derived.is section takes a look at the optimization of mean values and model parameters for the general grey prediction model defined.
e time response function for grey Verhulst model is obtained by ( 20) and (21) as follows: where the parameter β can be derived by substituting c that minimizes f 1 (c) or f 2 (c) from (26).If β � 1, then we have x (1) (t) � ax (1) (1) bx (1) (1) + a − bx (1) (1) e −a(t−1) . ( which is just the time response function of the original grey Verhulst model [8].e restored values for grey Verhulst model can be derived by first-order inverse accumulating generation operator from the time response function (27).
Here, a represents the development coefficient and u denotes the grey action quantity. 4 Advances in Meteorology e solution of the parameter vector a ⌢ � a b   can be obtained by using the least square method. Here And find  a: )e a(k−2) bx (0) (1) + a − bx (0) (1) e a(k−1) bx (0) (1) + a − bx (0) (1) e a(k−2)  .
(30) e model ( 26) is called the modified grey Verhulst model.e restored values of grey Verhulst model from (27) can be obtained by substituting (28) instead of x (1) (k).

Mean absolute percentage error
e basic idea for adjusting and optimizing regional structure of agriculturalists is to select and optimize the development of the dominant cultivators so that they can effectively perform the task of farming using the principle of "Sufficient Economy" according to the late King Rama 9th of ailand [13] (Table 1).

Results and Discussion
e different results of the water and rainfall forecasting are value x (0) (i) - x (0) (i) transformed to absolute value.As a result, the error of each prediction can be measured.e error is in the form of a percentage for a more accurate measurement, while the levels of tolerance do not exceed 5%.

Optimization of Mean Values of Water and Rainfall of Prediction Models.
e optimized grey systems model and the adjustment of precision grey model for the general form of grey prediction model are employed, and some optimal forecasting of rainfall is derived.is section takes a look at the optimization of mean values and model parameters (MAPE) for the general grey prediction [14] model as defined by ( 27), (30), and (31).
In Table 2, the discrepancy of relative errors between GM (1,1) and GVM is obvious.And the relative errors of GVM indicate that it is a more effective predictor than GM(1,1) (Figure 2).
From the actual data above, the grey Verhulst model's forecasting power of rainfall data of each year is better than the GM(1,1) model because most of the relative errors are as low as or lower than those forecasted by the GM(1,1).However, such comparison cannot give the whole picture of the water remaining in the dams for use by agriculturalists.
e accuracy of the calculation is approximately less than 5% (MAPE � 0.04249 in Table 3).And here, the two models, GM (1,1) and Verhulst model, are employed to compare the accuracy of prediction once again.
is section takes a look at the ordering of the forecasting power, from A to D, while A is the best and D is the worst according to MAPE, respectively.A < A * , which means that although A is highly accurate in terms of predictability, A * is even a more accurate indicator of predictability.
In Table 3, the point of using grey forecasting will be to predict the next year's water volume in ailand as well as rainfall.During the forecasting process, as compared to Table 2, the grey forecasting model should be operated in accordance with the principle of keeping the same dimension rainfall in dam data series (Table 6) [15].And it appears that the relative errors of GVM indicate that this model is a more effective predictor.So, the importance about the 2012 result difference predicted from data in Tables 2 and 3 is that it has a high error in GVM bar in a point on the graphs.e major causes of annual precipitation and the number of dams are strongly correlated by the observation of the floods in late 2011 to early 2012 (Figure 3) [15].
Finally, by using MAPE to pinpoint the minute differences in the predictive power of both models, it can be clearly seen that Table 3 shows a better picture of the superior power of the Verhulst model to predict the water remaining in the dam for actual use because the MAPE rankings predicted by grey Verhulst model are much better in most cases.

Conclusion
Agriculture depends on rainfalls and the quantity of water available for farmers.Good planning will result in proper and adequate use of water for agriculturalists.However, the unpredictability of the quantity of rainfalls each year is affected by many factors unknown to agriculturalists, and by using MAPE and the comparison between grey GM(1,1) and grey Verhulst model, it is possible to predict the total rainfalls in ailand.By applying MAPE again, the calculation of the water remaining in the dam for use turned 6 Advances in Meteorology out to be even more accurate, judging from the MAPE value of each data range.is can provide a way to deal with unpredictability in the best manner.erefore, the grey model systems have been proposed to deal with uncertainty of data in the most systematic way.
is paper demonstrates how the grey systems theory has been employed to deal with the prediction problems with incomplete or unknown information with large samples.It is also an attempt to bridge the gap between the immeasurably great works of the late King Rama 9th of ailand [13], who had devoted a large part of his entire life to improve the welfare of his citizens, especially agriculturalists through various means.In conclusion, the performance of the grey Verhulst model is more superior to the GM(1,1) model because it has the bene ts of simplicity for application and a higher forecasting precision [16].Advances in Meteorology erefore, we suggest the use of the grey Verhulst model to predict the volume of water in dams of ailand and other countries for water management planning for agriculture and other similar purposes.

(A.1)
Step 3. Determine whether or not it complies with the law of quasiexponentiality.
From Table 5, both the quasismoothness and quasiexponentiality are satisfied (Table 6).
Using Relation Information.A search of the Chinese Data Base of Scholarly Periodicals (CNKI) shows that, from 1990 to 2008, the number of scholarly publications with one of the keywords "fuzzy mathematics," "grey systems," and "rough set" also shows an uptrending development.See Tables 7-9 for more details [6].
e research of uncertain (fuzzy, grey, and rough) systems can be categorized into the following three aspects: (1) e mathematical foundation of the uncertain systems theories; (2) e modeling of uncertain systems and computational schemes, including various uncertain system modeling, modeling combined with other relevant methods, and related computational methods; and (3) e wide-range applications of uncertain systems theories in natural and social sciences.

Figure 1
Figure 1: e volume of rainfall and the ranges of temperatures.Source: ailand Meteorological Department [11].

Table 2 :
e simulation value of rainfall and relative error from actual data of the two models.

Table 3 :
e comparison of water in dam and relative errors of the two models from actual data gathering.

Table 4 :
Calculation of c-value of GM and GVM.