Comparative Evaluation of Prediction Model between Inference Fuzzy System and Universal Kriging for Spatial Data

. This paper dealt with one of the spatial interpolation methods in the geostatistics field. The purpose of this research is to get the parameters of unbiased estimators based on regionalized random variables in spatial statistics. In this paper, we used universal kriging with the fuzzy inference system by the Mamdani technique. the objective of this work is to estimate the parameters of covariance functions relying on spatial real for the depth of groundwater in Mosul city, Iraq. The data adopted contains (100) real data with locations representing the depth. From the results we show the best model with the constructs of weights, we illustrate the performance of universal kriging is the best when corresponding with the fuzzy system. In conclusion, the improvement of any method of spatial interpolation or fuzzy system does not depend on more statistical structures but depends on the efficiency of the method which satisfies the conditions of weights and minimum variance errors. All programming is applied by Matlab language.


Introduction
The concept of fuzzy set theory was started by Lotfi A. Zadeh in 1965 when he published his famous research on "fuzzy set". L. A. Zadeh linked the probability theory with the fuzzy sets to lead the mathematical logic. After Zadeh, many studies introduce some concepts of fuzzy set theory dealt with the comparison between the fuzzy inference systems as Mamdani and Sugeno [1], while the other studies were interested in prediction using fuzzy systems [2], [3]. Geostatistics is a branch of statistics sciences, that is interesting to study the regionalized variables which are related the spatial prediction. In (1951) in mining engineering, Krige create the method of prediction process from gold concentrations in the ore of South Africa. Mining engineer D. G. Krige was the first published in spatial statistics, later, a French mathematician G. Matheron developed the method of kriging based on the master thesis of D. G. Krige in geostatistics. Spatial interpolation can be defined as one of the statistical methods for the prediction process for spatial data, where many researchers and statisticians use graphics, curves, and charts that help us give an idea of spatial variation or spatial distribution in the prediction process. There are many different types of interpolation methods, including linear interpolation and multivariate interpolation. Many studies use prediction with variogram functions such as [4], [5], [6]. Matheron was the first scientist to introduce the prediction of spatial statistics using the parameters of covariance functions.

Methods and Materials 2.1 Fuzzy Theory
Lotfi A. Zadeh was the first scientist to define the fuzzy set. Fuzzy set A is defined as: = { , ( ), ∈ }, where ( ) is the membership degree of and X is collection of element numbers x, and : → [0,1]. Fuzzy logic is the theory of fuzzy sets, where the fuzzy logic relies on the idea of degrees of things such as (height, speed, …, etc.) [7].

Mamdani Fuzzy Logic (MFL)
The scientist Ibrahim Mamdani of London university give the first fuzzy system and known as the Mamdani style. The output of the membership function in Mamdani is fuzzy sets, therefore the output variables need processes through the inputs to prediction. There are four steps of Mamdani fuzzy logic defined as the following: [2[, [8].
Step 1: Fuzzification. The purpose of fuzzification is to map the inputs. The crisp input is a numerical value, fuzzy set [Lower, Median, and High].
Step 2: Rule Evaluation. Inputs are applied to a set of (If/Then) control rules The fuzzy operator (AND or OR) is used to get a single number and to obtain the membership function.
Step 3: Aggregation of the rule output Is the process of all outputs of all rules in step 2, for each output variable the result is one fuzzy set.
Step 4: Defuzzification The final step of (MFL) is defuzzification where the output has to be a crisp number. There are several defuzzification methods such as (the centroid technique, center of gravity, …, etc.) The hypothetical fuzzy variogram uses the parameters of variogram function, such as nugget effect, partial variance, and range defined in a fuzzy spherical variogram and also defined as a triangular membership function: Where ≤ ≤ , [3], [9].

Kriging Technique
Ordinary kriging is the most important prediction in spatial statistics. The prediction based on the model:

Universal Kriging
Spatial data can be through resulting from regionalized random variables in the study field on stochastic process: = { ( ): ∈ } Universal kriging assume the model ( ) = ( )+∈ ( ), ∈ ⸦ 2 Where ( ) is some deterministic function and ∈ ( ) is random variation is the errors with the mean is zero. Assume ( ), = 1,2, … , to be regionalized variables at the points ( ), are locations at random field. Let Where ∈ ( ) is the zero mean, −1 is a basic functions be known, and −1 is coefficients to be estimated. And the universal kriging estimator at ( ), defined as: And the variance of universal kriging denoted as 2 and calculate as : we can obtain the minimum universal kriging as the following: [13].

Variogram Function
The random variables are called second-order stationarity with mean μ and covariance C(h) if: When the ( ) is satisfy the second-order stationarity then: [4]. The theoretical variogram function defined by Matheron and Cressie denoted as 2 (ℎ) where: The variogram function assume the stationarity conditions and it's a finite doesn't rely on the location of in domain . Matheron defined the method of moment estimator as: Where (ℎ) is the number of all pairs of points with lag h. To fitting a variogram function, we must getting the three most common parameters defined as the following: • Nugget ( )or discontinuous at origin point, • Sill ( + ) or variance of variogram.
• Range is a distance on x-axis at variogram is stable These parameters used in order to obtain the best fitting covariance model, [14], [15].

Study Data
This research depends on real spatial data on the depth of groundwater in Nineveh governorate, Iraq. These data contain (100) real data with real locations. We will introduce some depth data for groundwater in Nineveh Governorate. Table (1) below shows the data for groundwater where Z is the depth and x, y are the location.    is red curve, And the black curve of (θ=45°,135°). With the shape preserving interpolation, as indicated by three curves of average variogram function.   (4) illustrates the results of variogram function for depth groundwater data after taking the logarithm of depth, the results of G1, G2,G3, and G4) are all thetas and the average of two thetas that have the same lag ( or distance) between the values of depth variables.   (5) show the results and properties of the variogram function after taking the logarithm of groundwater data, these properties represent min, max, median, and range for all basic theta of the compass. These properties gave good parameters for the prediction process. Through the behaviour of curves correspondence between the results of variogram function in all directions, we show that curves nearest to the power model that it's defined as the following:

Results of Kriging Techniques
When the nugget effect ( ) is clear in Table (

Results of Fuzzy System
After we arranged the real data by increasing each value (X, Y, and Z), we classified it into three classes L, M, and H. To find the center of each class for inputs X, Y, and output Z (see Table (6) below). By Matlab language, we applied the real data of levels of groundwater in fuzzy logic, we got the following:    And the rules of second step was taken as the following:

1-if (X is L) and (Y is H) then (Z is H) 2-if (X is L) and (Y is H) then (Z is M) 3-if (X is L) and (Y is M) then (Z is H) 4-if (X is M) and (Y is H) then (Z is H) 5-if (X is M) and (Y is L) then (Z is M) 6-if (X is M) and (Y is M) then (Z is H) 7-if (X is H) and (Y is L) then (Z is M) 8-if (X is H) and (Y is L) then (Z is H) 9-if (X is H) and (Y is M) then (Z is M)
Output Z by using min-max : Figure (7) Figure (7) shows the results of output Z from the value X=515, and Y=829 we get the output of Z=2.26 by using the function min-max.

Conclusions
The data adopted in this paper is from real spatial data of groundwater levels from Mosul city in Iraq. to get the best estimate of the fitting model. Prediction or so-called kriging techniques require programs for distance matrix, mathematical formulas of prediction, and error variance. The fuzzy concepts and tools preparation rely on the probability theory, so the fuzzy model alone is not sufficient except for its integration and linking with the methods of spatial interpolation in order to expand the prediction process. Two methods of fuzzy inference method and spatial interpolation are characterized by uncertain information that appears in spatial variance with there are common criteria indicating the analysis of spatial distribution. The center of mass takes the output distribution found and its center to outcome the crisp, this is computed as COG in the Mamdani inference system. the results obtained show the convergence between the fitting model and the output graphics in a fuzzy system, which has the same properties in the case of spatial prediction.