1. Introduction

The top layer of earth is dirt, but as we get deeper, the dirt changes into layers of solid rock. Each of these layers has many small spaces and cracks filled with water. Ground water is the water that is found beneath the earth's surface in the patterns of rock formations. Groundwater has an important role in the environment: it replenishes streams, rivers, and wetlands and helps to support wildlife habitat; it is used as primary source of drinking water and also in agricultural and industrial activities. Regression analysis can be used to model the relationship between one or more independent or predictor variables and a dependent or response variable (which is continuous-valued). In other words, a regression model is a mathematical model that predicts a continuous response variable. For example, a regression model could be developed to predict actual sales volume or the temperature resulting from an experiment. Bacterial foraging optimization is first proposed by Passino in 2002. This algorithm is inspired by the foraging and chemotactic behavior of bacteria Escherichia Coli (E. coli). The process of searching nutrients is known as foraging. The original Bacterial Foraging Optimization system consists of three principal mechanisms, namely, chemotaxis, reproduction, and elimination-dispersal. Chemotactic consists of Swim and Tumble.

2. Review of Literature

1. Chandni et. al. proposed linear regression and multiple-Linear regression are efficient prediction technique to predict ground water possibility and author has developed a system which can predict ground water level at any place without drilling holes by knowing its geological features only ^[4]. The other possible applications may be processing medical data where integration of Bacteria Foraging and CBR system can be used to diagnose the disease of patients.

2. Ranu Rani Sethi et. al presented a procedure which determines the factors that influence and control the water table fluctuation in a specific geomorphologic situation. Author hasdeveloped a forecasting model and examines its potential in predicting water table depth using limited data and also developed different models for prediction of water table depth based on artificial neural networks (ANNs) ^[5].

3. Sara Maatta presents two models which predict groundwater levels in an unconfined shallow aquifer in the Sears Ville basin, part of the Jasper Ridge Biological Preserve. She took input data (i.e. features) for the models as local weather, lake stage, and stream flow data, and moving averages of the weather, stage and stream flow data taken over time-frames of one week, one month, three months and six months. The author conclude that linear regression does a good job of predicting groundwater levels in the summer, when water levels are low, while the neural network does a good job of predicting groundwater levels in the winter, when water levels are high ^[6].

4. MAO Xiaomin et. al. gave the idea that the prediction of groundwater level is important for the use and management of ground water resources. Author uses the artificial neural networks (ANN) to predict groundwater level in the Dawu Aquifer of Zibo in Eastern China ^[7].The author suggests that the first step was an auto-correlation analysis of the groundwater level which showed that the monthly groundwater level was time dependent.

5. Jianhua Tao et. al. developed a predicting method of ground water level based on Back Propagation neural network. His experimental results that show the predicting model has high precision. This paper is able to describe complicated system of ground water, and reveals nonlinear characteristics of ground water movement ^[8].

6. Ritanjali Majhi et. al. introduces the Bacterial Foraging Optimization(BFO) technique to develop an efficient forecasting model for prediction of various stock indices. Here she wants to show the difference between BFO and multilayer perceptron base model in stock market prediction. Finally she conclude that BFO model is computationally more efficient, prediction wise more accurate and takes less training time compared to the standard MLP based model ^[9].

7. Dharminder Kumar et. al. proposed an optimization technique, Bacterial Foraging Optimization (BFO) to tackle complex search problems of the real world. It is also used to find the optimal solution of the problem. His research work suggests that images quantized with Bacteria Foraging Optimization technique gives good results ^[10]. He explains the Bacteria Foraging Optimization for Colour Image Quantization and Presents the evaluation for various file formats.

8. Dr Shyam.S.Pattnaik et. al. developed a novel technique by hybridizing Bacterial Foraging Optimization technique in conjunction with Particle Swarm Optimization and is applied to the expressions developed by M.Kara to calculate accurately the desired resonant frequency of rectangular micro strip antenna of any dimension and of any substrate thickness ^[11]. After his experiment, the result shows promising improvement in the accuracy, while achieving drastic reduction in computational time.

3. Preliminary Concepts

Prediction is a form of data analysis that can be used to extract models describing important data classes or to predict future data trends. A predictive model will calculate an estimate for one or more variables (responses), based on other variables (descriptors).Prediction is used where an estimate or forecast is required, for example, to project sales or forecast the weather. Prediction is used to understand the relationship between the input descriptor variables and the output response variables ^[12]. There are different types of prediction techniques that are used for prediction:

•  Regression Techniques

•  Numeric Prediction

Regression analysis can be used to model the relationship between one or more independent or predictor variables and a dependent or response variable (which is continuous-valued). In other words, a regression model is a mathematical model that predicts a continuous response variable. Regression is of three types:

a. Linear Regression

b. Non-Linear Regression

c. Multiple-Linear Regression

•  Linear Regression analysis involves a response variable y, and a single predictor variable, x.

•  It is the simplest form of regression, and models y as a linear function of x.

That is,

(1)

Where,

~Variance of is assumed to be constant.

~ and are regression coefficients.

•  The regression coefficients, w and b, can also be thought of as weights, so that we can equivalently write,

(2)

•  Let be a training set consisting of values of predictor variable, x, for some population and their associated values for response variable, y.

•  The regression coefficients can be estimated using this method with the following equations:

(3)

(4)

Where,

~ = Training Set.

~ = Total Number of predicted variables.

~ = Mean value of

~ = Total number of actual variables.

~ = Mean value of ^[13]

•  Multiple-Linear regression is an extension of straight-line regression so as to involve more than one predictor variable.

•  It allows response variable y to be modelled as a linear function of, say, n predictor variables or attributes, A1, A2…, An, describing a tuple, X. (That is, X = .)

•  Our training data set, D, contains data of the form , …, , where the Xi are the n-dimensional training tuples with associated class labels, y_i.

•  An example of a multiple linear regression model based on two predictor attributes or variables, A1 and A₂, is

(5)

Where x1 and x2 are the values of attributes A1 and A2, respectively, in X. The method of least squares shown above can be extended to solve for .

Bacterial foraging optimization is first proposed by Passino in 2002. This algorithm is inspired by the foraging and chemotactic behavior of bacteria Escherichia coli (E. coli). This optimization is used in many areas like: Ground-water prediction, Stock-market prediction, Weather prediction, Disease prediction etc ^[15].

The original Bacterial Foraging Optimization system consists of three principal mechanisms, namely, chemotaxis, reproduction, and elimination-dispersal. Chemotactic consists of Swim and Tumble. Each of these processes is described as follows.

In the original BFO, a unit walk with random direction represents a “tumble” and a unit walk with the same direction in the last step indicates a “run”. Suppose (j,k,l) represents the bacterium at j^th chemotactic, kth reproductive, and ith elimination-dispersal step. C(i) is the chemotactic step size during each run or tumble (i.e., run-length unit). Then in each computational chemotactic step, the movement of the ith bacterium can be represented as:

Where is the direction vector of the jth chemotactic step. when the bacterial movement is run, is the same with the last chemotactic step; otherwise, is a random vector whose elements lie in [-1,1]. With the activity of run or tumble taken at each step of the chemotaxis process, a step fitness, denoted as J(i,j,k,l), will be evaluated.

The health status of each bacterium is calculated as the sum of the step fitness during its life, that is, , where Nc is the maximum step in a chemotaxis process. All bacteria are sorted in reverse order according to health status. In the reproduction step, only the first half of population survives and a surviving bacterium splits into two identical ones, which are then placed in the same locations. Thus, the population of bacteria keeps constant.

The chemotaxis provides a basis for local search, and the reproduction process speeds up the convergence which has been simulated by the classical BFO. While to a large extent, only chemotaxis and reproduction are not enough for global optima searching. Since bacteria may get stuck around the initial position or local optima, it is possible for the diversity of BFO to change either gradually or suddenly to eliminate the accidents of being trapped into the local optima. In BFO, the dispersion even happens after a certain number of production processes. Then some bacteria are chosen, according to a preset probability Ped, to be killed and moved to another position within the environment.

The original BFO algorithm is briefly outlined step by step as follows.

4. Experimental Evaluation and Result Analysis

All the details about the area, where we are going to predict the ground water posibility is collected. Details are followed by the attributes like Geology, Landform, Soil, Land use, Slope and Lineament. From those details a data set is prepared. The six attributes have their own value. The six attributes and its values are representing in the Table 1 given below:

Table 1. Six attributes of ground water possibility detection

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Tables index

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The predictions are done by using Linear Regression and Multiple Linear Regression technique. Then all the errors are calculated. After analysis it is found that Multiple Linear Regression is best prediction technique. Then an optimization technique is proposed to optimize the error values. After optimization it is found that BFO is best. The figure given below shows the errors in all the techniques. The errors are as follows, For place 6 error of BFO is 2.5, for Multiple Linear Regression the error is 3.0 and for Linear Regression it is 15.0 in Figure 2. Therefore it is analyzed that BFO has less error and hence BFO is best technique. Groundwater possibility is high with 81.5%.

Figure 1. Error plot in different prediction techniques

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5. Conclusion

In this paper, we have presented linear regression and Multiple Linear regression as an efficient prediction technique to predict ground water possibility in a given area. The results obtained in this work are comparable to the work already done in this field. A system has been developed which can predict ground water level at any place without drilling holes by knowing its geological features only. The system developed is of great significance as the presence or absence of ground water in a given area has a direct effect on its real estate values. So the system can play a crucial role in economic sector of a nation. Moreover, it is very useful in military applications during the time of combats to find groundwater possibility in inaccessible areas like areas across the border of a nation. Movement of troops depends largely on such information as water is a basic need for survival. Linear regression does a good job of predicting groundwater levels in the summer, when water levels are low, while the multiple-linear regression does a good job of predicting groundwater levels in the summer as well as in winter, when water levels are high. This result supports the combination of linear regression and multiple-linear regression for predicting hydrologic response up to one year in advance. We found some errors in previous prediction techniques i.e. our actual values are much different from predicted values. Therefore we proposed a optimization technique Bacterial Foraging Optimization to optimize the previous error values. This algorithm is inspired by the foraging and chemotactic behavior of bacteria Escherichia coli (E. coli).The E. coil can move to the nutrient area and escape from poison area by running and tumbling in the environment. This optimization is used in many areas like: Ground-water prediction, Stock-market prediction, Weather prediction, Disease prediction etc.

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