Regression and Correlation of Data Apply the least squares method for fitting data with linear regression Methods of regression are used to summarize sets of data in useful form. The values of the data have already been recorded, and as such, are fixed.

Regression and Correlation of Data Regression analysis: a collective name for techniques for the modeling and analysis of numerical data in order to find out the relation between the dependent e.g. y and independent variable(s) e.g. x. Assumption: - There is no error in the independent variable(s). - All errors due to measurement and to approximations in the modeling equations appear in the dependent variable, y.

Regression and Correlation of Data Examples common to engineers: The output of a CSTR changes as the temperature changes within the reactor, and the measurement of output concentration causes additional variation due to measurement errors. The power produced by an electric motor varies with changing input voltage, and any measurement of output voltage includes measuring errors.

Regression and Correlation of Data Application of Regression: - Find the relation between the dependent y and independent variable(s) x. y=a+bx y=axb y=a0+a1x+a2x2+…+anxn or y=a0+a1x1+a2x2+…+anxn or else - Determine the important coefficients. e.g. Determine reaction rate constant k. CAo and CA are the initial reactant concentration and that at time t. Predict the values of the dependent variable using the regressed model and coefficients.

Regression and Correlation of Data Simple Linear Regression: For an example, for the independent variable x (called input or regressor), and the dependent variable y (called response), the following relation exists (y may also be represented as the mean of the probability distribution E(Y)): E(Y)=α+βx α and β are constant parameters, called regression coefficients. From a sample consisting of n pairs of data (xi,yi), we calculate estimates, a for α and b for β. If at x=xi, is the estimated value of E(Y), then the fitted regression line is

Regression and Correlation of Data Simple Linear Regression: How to determine a and b? Method of Least Squares: a and b are determined by minimizing the sum of the squares of errors (SSE), deviation, residuals or difference between the data set and the straight line that approximate it.

Regression and Correlation of Data Simple Linear Regression: Method of Least Squares: Centroidal point:

Regression and Correlation of Data Method of Least Squares: Sum of the squares of errors (SSE), Estimated variance of the points from the line: Estimated standard deviation or standard error of the points from the line: The degrees of freedom=n data points – the number of estimated coefficients

Regression and Correlation of Data Assumptions and graphical checks: For simple linear regression of y on x, represents errors or deviations or residuals. Assumptions: A linear relation between y and x represents the data adequately. The errors ei are entirely in the y-direction and so independent of x. The distribution of errors follows normal distribution. The variance is constant.

Regression and Correlation of Data Assumptions and graphical checks: For simple linear regression of y on x, Graphical checks – satisfactory regression Plot residuals against x or y. There are about as many positive residuals as negatives. Small deviations considerably more frequent than larger ones. There are no outstanding outliers. There is no strong systematic pattern as x or y increases. If it is unsatisfactory, other regression equation should be chosen.