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1 Research Article NON LINEAR MULTIVARIATE REGRESSION ANALYSIS FOR THE PREDICTION OF CHARACTERISTIC DEFLECTION OF FLEXIBLE PAVEMENTS Suneet Kaur Dr V S Ubbobeja Dr Alka Agarwal Address for Correspondence Associate Professor, Department of Civil Engineering, M.A. National Institute of Technology, Bhopal. Director, Mahakal Institute of Technology, Ujjain. Professor, Department of Mechanical Engineering, UIT, RGPV, Bhopal. ABSTRACT Performance of a flexible pavement is closely related to the elastic deflection of the pavement under the wheel loads. Pavement surface deflection is a primary parameter for evaluating the performance of a flexible pavement. Benkelman Beam Deflection (BBD) technique is still widely used for evaluating the structural capacity of an existing flexible pavement, as also for the estimation and design of overlays for strengthening of a weak pavement. But this field method for measuring deflection is expensive and time consuming. An attempt has been made in this paper to develop a model based on Multivariate Regression Analysis to predict the characteristic deflection of a flexible pavement to a reasonable degree of accuracy, based on the thicknesses of different layers of the pavement and the California Bearing Ratio of the sub-grade soil. Field data used for building the model was collected from field tests conducted by various agencies engaged in the development and strengthening of highways in the state of Madhya Pradesh. INTRODUCTION Extensive road development programme has been undertaken to improve the existing highway network in India. Substantial investment is being made towards this end and the amount of investment is likely go up with the continuously rising trends in the volume and magnitude of traffic. Repeated application of the vehicle loads, weather conditions decrease the serviceability of the pavement. In order to prevent further deterioration, a maintenance programme, including strengthening of the weak pavement, needs to be carried out at the right time and right places. Surface deflection data of a flexible pavement, plays an important role in the evaluation of its structural capacity and in the design of pavement rehabilitation programme. Evaluation of structural adequacy and load carrying capacity of an existing pavement layer system is well accomplished by non destructive testing procedures such as Benkelman Beam Deflection method. A.C. Benkelman devised a simple deflection beam in 95 for measurement of pavement surface deflection. Deflection beams are used in India by different organisations to measure surface deflection, the best indicator of the structural condition of a pavement. Benkelman Beam which consists of a slender beam.66m long pivoted at a distance of.m from the tip. By suitably placing the probe between the dual wheels of a loaded truck, it is possible to measure the rebound and residual deflections of the pavement structure. Rebound deflection is related to pavement performance and is used for the overlay design. Overlay design for a given section is based not on the individual deflection values but on the characteristic deflection which is based on the statistical analysis of all the measurement in the section corrected for the temperature and seasonal variations. The field test is relatively expensive and time consuming and it results in substantial cost and time required in the design for strengthening of pavements and preparation of the detailed project reports. Overall economy in time and money, for any project can be achieved, if field tests could be substituted by building a model which could predict the characteristic deflection from the simpler basic observations and tests. Many researchers [6] have attempted to correlate deflection with the soil properties and pavement characteristics. An attempt have been made in this paper to build a non linear multivariate regression model based on the extensive test data collected for road design by different agencies engaged in strengthening and rehabilitation of the state highways in the state of Madhya Pradesh, largely located in black cotton soil areas covering extensive part of the state..0 REGRESSION MODELS Multivariate Data Analysis refers to any statistical technique used to analyze data that arises from more than one variable. This essentially models reality where every situation, product or decision involves more than one variable. The objective is to use one set of variables to predict another for the purpose of optimization, and to find out which variables are important in this relationship. The corresponding analysis is called Multivariate Regression Analysis.. Linear Regression Models Regression analysis is a statistical tool for the investigation of relationships between variables. Regression analysis includes any techniques for modelling and analyzing several variables, when

2 the focus is on the relationship between a dependent variable and one or more independent variables. Multivariate regression takes into account several predictive variables simultaneously, thus modeling the property of interest with more accuracy. A linear regression model fits a linear function to a set of data points. The form of the function is: Y = β 0 +β.x + β.x + β n X n Where Y is the target variable, X, X, X n are the predictor variables, and,β β, β n, are coefficients that multiply the predictor variables. β 0, is a constant. The goal of regression analysis is to determine the values of the β parameters that minimize the sum of the squared residual values for the set of observations. This is known as a least squares regression fit. Since linear regression is restricted to fitting linear (straight line/plane) functions to data, it rarely works as well on realworld data as more general techniques such as neural networks which can model non-linear functions. However, linear regression has a number of strengths: Linear regression is the most widely used method and it is well understood. Training a linear regression model is usually much faster than methods such as neural networks. Linear regression models are simple and require minimum memory to implement, so they work well on embedded controllers that have limited memory space. By examining the magnitude and sign of the regression coefficients (β) we can infer how predictor variables affect the target outcome.. Non Linear Regression Models In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data are fitted by a method of successive approximations. Some of the examples of nonlinear functions [0] include exponential functions, logarithmic functions, trigonometric functions and power functions. Some functions, such as the exponential or logarithmic functions, can be transformed so that they are linear. When so transformed, standard linear regression can be performed but must be applied with caution. In transformed state these models are fit using linear regression in order to evaluate the constant coefficients. They could then be transformed back to their original state and used for prediction of parameter defined by the model. A nonlinear multivariate regression model is developed in the present study. The form of the function is : Y = β 0 X β X β X β β X where, Y = Characteristic deflection. X = Thickness of bituminous macadam. X = Thickness of water bound macadam / water mixed macadam. X = Thickness of granular subbase. X = California bearing ratio of subgrade soil. β 0, β, β, β and β are the coefficients of each predictor variable. The above equation is made linear by taking Log of both sides. Log (Y) = log(β 0 ) + β log(x ) + β log(x ) + β log(x ) + β log(x ) The predictions and some of the estimated coefficients are transformed to anti-logs to get back to the parameters in the original non-linear equation. Performance of the model is indicated by the coefficient of determination (R ), root mean square error (RMSE) and mean absolute relative error (MARE). R-Square, also known as the coefficient of determination is a commonly used statistical parameter to evaluate model fit. R-square is minus the ratio of residual variability. When the variability of the residual values around the regression line relative to the overall variability is small, the predictions from the regression equation are good. The R-square value is an indicator of how good a predictor might be constructed from the modeled values. R value is calculated as the square of the correlation coefficient between the original and the modeled data values. The R-square value in the present model is calculated for the measured deflection and the predicted deflection for the model data set as well as for the validation data values..0 MODEL PARAMETERS A thorough understanding of the factors affecting the characteristic deflection is needed in order to obtain an accurate deflection prediction. Database used in the present study is based on the data collected from the state government agencies in Madhya Pradesh, pertaining to Morena - Porsa Road, Shivpuri- Sheopur Road, Khalghat-Kasrawat Road, Barwah-Dhamnod Road, Gaitarganj - Silwani Road, Bareli - Piparia Road, Deshgaon - Khargon Road and Khasravad - Khargone Road in Madhya Pradesh. The following four variables are selected as predictor variables:. Thickness of Bituminous macadam layer. Thickness of Water bound macadam layer. Thickness of Granular subbase layer. California Bearing Ratio (CBR) Characteristic deflection is the target variable.

3 . Data Division Available data is divided into two sets, one employed to develop the nonlinear multivariate model and the other independently to validate the performance of the developed model. This would ensure a universally applicable model for the result estimation. In this study, 70% of data, consisting of 88 samples is used to build the model and 0%, consisting of 8 samples is used in the validation of the model..0 ANALYSIS AND RESULTS. Comparison of Measured and Model Predicted Deflection.5 MEASURED AND PREDICTED DEFLECTION (REGRESSION MODEL) Figure- exhibits the measured deflection and the predicted deflection by the regression model. Figure- similarly exhibits the measured and the predicted deflection for the validation data set. Figure- exhibit the Error graph for each sample between measured and predicted deflection. Figure- exhibit the Error graph for each sample between measured and predicted deflection for validation data set. Figure-5 exhibits the relative measured deflection versus the predicted deflection by the regression model. Figure-6 exhibit the relative measured deflection versus the predicted deflection of the validation set. Def (measured) Def (predicted) Measured / Predicted deflection (mm) Fig. : Measured and Predicted Deflection.5 Measured and Predicted deflection (mm) Validation Set Def Measured Def Predicted Measured / Predicted Deflection (mm) FIG : Measured and Predicted Deflection (Validation Set )

4 0 5 Percentage Error Graph Between Measured and Predicted Deflection PerError 0 Percentage Error FIG : Percentage Error Graph for Measured and Predicted deflection Percentage Error Graph Between Measured and Predicted Deflection (Validation Set) PerError 0 Percentage Error FIG :Percentage Error Graph for Measured and Predicted deflection (validation set)

5 Measured Deflection v/s Predicted deflection of regression model Predicted Deflec tion (m m ) Regression M odel.5.5 Measured Deflection (mm) FIG 5 : Measured versus Predicted Deflection Measured Deflection v/s Predicted Deflection for Validation Set Predicted Deflection (mm) Validation Set.5.5 Measured Deflection (mm) FIG 6 : Measured versus Predicted Deflection (Validation Set)

6 . Statistical Analysis Performances of the model is indicated by the coefficient of determination (R ), Root mean square error (RMSE) and mean absolute relative error (MARE).The R-square value is calculated for the measured deflection and the predicted deflection for the model creating data values as well as for the validation data values. These values for the regression model data set and for the validation data set are compiled in the Table. Table No.: Result Analysis Parameter Model Creation Data set Validation Set R RMSE MARE CONCLUSIONS In regression, the size of the coefficient for each independent variable give us the size of the effect that variable is having on the dependent variable, and the sign on the coefficient (positive or negative) gives the direction of the effect. The coefficients generated by the model clearly indicate that the thickness of Bituminous Macadam has the greatest influence on the characteristic deflection. This agrees with the FEM analysis carried out by other Researchers [8], wherein the deflection of flexible pavement was found to be predominantly governed by the thickness of BM layer. Nonlinear multivariate regression model can be a consistent and quick tool for reasonably precise estimation of characteristic deflection from the basic soil and pavement structure data. Its use for the prediction of characteristic deflection can result in substantial savings in cost and time. A large number of test data is available from the field and laboratory testing carried out for many highway projects in the country. This available data can be used to further improve and generalize the nonlinear multivariate regression model. 6. ACKNOWLEDGEMENT The assistance of the officials of the M.P. Road Development Corporation and the State Public Works Department in the collection of field data is gratefully acknowledged. REFERENCES. Chapman, Stephen J., MATLAB Programming for Engineers, rd Edition, CENGAGE Learning.. Croney D., The design and performance of road pavements, London, HMSO, Dr. P Kumar, V Patel, Effect of various parameters on performance of PMGSY Roads, IE (I) Journal, Volume 90, August IRC , Guidelines for design of flexible pavements, Indian Road Congress, New Delhi. 5. IRC , Guidelines for strengthening of flexible road pavements using Benkelman beam deflection technique, Indian Road Congress, New Delhi. 6. Mehmet Saltan, Serdal Terzi, Comparative analysis of using artificial neural network(ann) and gene expression programming (GEP) in back calculation of pavement layer thickness, Indian Journal of Engineering & Material Sciences,Vol.,February, Musharraf Zaman, Pranshoo Solanki, Ali Ebrahimi, Luther White, Neural Network Modeling of Resilient Modulus Using Routine Subgrade Soil Properties, International journal of geomechanics ASCE / January / February00 / 8. Ranadive M.S., Katkar A.G., Finite Element Analysis of flexible pavements, Indian Highways- Indian Road Congress, VOL 8, No.6, June 00, pp Shahin, M. A. et al., Predicting settlement of shallow foundations using neural networks, Journal of Geotechnical and Geoenvironmental Engineering, Vol. 8, No. 9, September, 00, pp Steven C.Chapra, Raymond P.Canale, Numerical Methods for engineers with personal computer applications,mcgraw-hill.