ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 26 Regression Analysis-Chapter 17

Curve Fitting Often we are faced with the problem… what value of y corresponds to x=0.935?

Curve Fitting Question 2 : Is it possible to find a simple and convenient formula that represents data approximately ? e.g. Best Fit ? Approximation

Experimental Measurements Strain Stress

Experimental Measurements Strain Stress

BEST FIT CRITERIA Strain y Stress Error at each Point

Best Fit => Minimize Error Best Strategy

Best Fit => Minimize Error Objective: What are the values of a o and a 1 that minimize ?

Least Square Approximation In our case Since x i and y i are known from given data

Least Square Approximation 2 Eqtns 2 Unknowns

Least Square Approximation

Example

Quantification of Error Average

Quantification of Error Average

Quantification of Error Average

Quantification of Error Standard Deviation Shows Spread Around mean Value

Quantification of Error

“Standard Deviation” for Linear Regression

Quantification of Error Better Representation Less Spread

Quantification of Error Coefficient of Determination Correlation Coefficient

Linearized Regression The Exponential Equation

Linearized Regression The Power Equation

Linearized Regression The Saturation-Growth-Rate Equation

Polynomial Regression A Parabola is Preferable

Polynomial Regression Minimize

Polynomial Regression

3 Eqtns 3 Unknowns

Polynomial Regression Use any of the Methods we Learned

Polynomial Regression With a 0, a 1, a 2 known the Total Error Standard Error Coefficient of Determination

Polynomial Regression For Polynomial of Order m Standard Error Coefficient of Determination