1. EGR 105 Foundations of Engineering I Fall 2007 – week 7 Excel part 3 - regression

2. Analysis of x-y Data Independent versus dependent variables y y = f(x) x independentdependent

3. Finding Other Values Interpolation –Data between known points Regression – curve fitting –Simple representation of data –Understand workings of system –Useful for prediction Extrapolation –Data beyond the measured range data points

4. Regression Useful for noisy or uncertain data – n pairs of data (x i, y i ) Choose a functional form y = f(x) polynomial exponential etc. and evaluate parameters for a “close” fit

5. What Does “close” Mean? Want a consistent rule Common is the least squares fit (SSE): (x 1,y 1 ) (x 2,y 2 ) (x 3,y 3 ) (x 4,y 4 ) x y e3e3 e i = y i – f(x i ), i =1,2,…,n sum squared errors

6. Quality of the Fit: Notes: is the average y value 0  R 2  1 closer to 1 is a “better” fit x y

7. Linear Regression Functional choice y = m x + b slope intercept Squared errors sum to Set m and b derivatives to zero

8. Further Regression Possibilities: Could force intercept: y = m x + c Other two parameter ( a and b ) fits: – Logarithmic: y = a ln x + b – Exponential: y = a e bx –Power function:y = a x b Other polynomials with more parameters: – Parabola: y = a x 2 + bx + c – Higher order:y = a x k + bx k-1 + …

9. Function Discovery or How to find the best relationship Look for straight lines on log axes:  linear on semilog x  y = a ln x + b  linear on semilog y  y = a e bx  linear on log log  y = a x b No rule for 2 nd or higher order polynomial fits (not very useful toward real problems)