1. Excel Part III Curve-Fitting, Regression Section 8 Fall 2013 EGR 105 Foundations of Engineering I

2. Excel Part II Topics Data Analysis Concepts Regression Methods Example Function Discovery Regression Tools in Excel Homework Assignment

3. Analysis of x-y Data Independent versus dependent variables y y = f(x) x independent dependent

4. Common Types of Plots Example: Y=3X 2 log(y) = log(3) + 2*log(x) y = 3x 2 Straight Line on log-log Plot! Cartesian Semi-log : log x log-log : log y-log x Note!

5. What About Other Values? Often have a limited set of data What if you want to know… –Prediction of what occurred before data –Prediction of what will occur after data Many real applications of this… –Discuss this in a little while

6. Finding Other Values Interpolation –Data between known points –Need assume variation between points –May be easier to do for closer points data points

7. Finding Other Values Extrapolation (requires assumptions) –Data beyond the measured range –Forecasting (looking ahead) –Hindcasting (looking behind) Examples (apply equations or models) –Sales –Ocean waves –Stock market –The weather –etc.

8. Stock Market Forecasting – can require complex model(s)

9. Finding Other Values Regression – curve fitting of data –Simple representation of data –Understand workings of system Elements of system behavior are important –How do they affect the overall system? –How important is each one? Can represent these in model(s) –Useful for prediction

10. Excel Part III Topics Data Analysis Concepts Regression Methods Example Function Discovery Regression Tools in Excel Homework Assignment

11. Something Must Be In There…Somewhere….

12. Curve-Fitting - 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

13. What Does “Close” Mean? Want a consistent rule to determine 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

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

15. Coefficient of Determination R 2 = 1.0 –All of the data can be explained by the fit R 2 = 0.0 –None of the data can be explained by the curve fit (Note: R 2 = is sometimes reported as a %)

16. Caution!!! A good fit statistically may not be the correct fit Must always consider the physical phenomenon you are attempting to “model” Does the fit to the data describe reality?

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

18. 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 + …

19. Excel Part III Topics Data Analysis Concepts Regression Methods Example Function Discovery Regression Tools in Excel Homework Assignment

20. Example Function Discovery (How to find the “best” relationship) Look for straight lines on log axes: – linear on semilog x  y = a ln x – 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

21. Excel Part III Topics Data Analysis Concepts Regression Methods Example Function Discovery Regression Tools in Excel Homework Assignment

22. Excel’s Regression Tool Highlight your chart On chart menu, select “add trendline” Choose type: –Linear, log, polynomial, exponential, power Set options: –Forecast = extrapolation –Select y intercept (use zero only if it applies) –Show R 2 value on chart –Show equation of fit on chart

23. Linear & Quartic Curve Fit Example Better fit but does it make sense with expected behavior? Y Y X X

24. Example Applications Look at some curve fitting examples –Examine previous EGR 105 projects Pendulum Elastic bungee cord

25. Previous EGR 105 Project Discover how a pendulum’s timing is impacted by the –length of the string? –mass of the bob? 1.Take experimental data Use string, weights, rulers, and watches 2.Analyze data and “discover” relationships

26. Experimental Setup: Mass Length

27. One Team’s Results Mass appears to have no impact, but length does

28. To determine the effect of length, first plot the data

29. Try a linear fit

30. Force a zero intercept (why?)

31. Try a quadratic polynomial fit

32. Try a logarithmic fit

33. Try a power function fit

34. On log-log axes, nice straight line Power Law Relation: b

35. Question? Which one was the best fit here? Explain why

36. One More Example Another EGR 105 project Elastic bungee cord models –Stretching of an elastic cord Here we have two models to consider –Linear elastic (Hooke’s Law) –Non-linear elastic (Cubic model)

37. Elastic Bungee Cord Models Determined by Curve Fitting the Data Linear Model (Hooke’s Law): Nonlinear Cubic Model: Linear Fit Cubic Fit Better and it Makes Sense with the Physics Force (lb)           Collected Data

38. Homework Assignment #5 See Handout (Excel Part 3) –Analysis of stress-strain data –Plotting of data –Determine equation for best fit to data Regression analysis –Linear elastic model –Cubic polynomial model Discussion of results Remember to email submit using EGR105_5 in Subject Line!