1. EGR 105 Foundations of Engineering I
Fall 2008 – Session 4
Excel – Plotting, Curve-Fitting, Regression
TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.: AA

2. EGR105 – Session 4 Topics Review of Basic Plotting
Data Analysis Concepts
Regression Methods
Example Function Discovery
Regression Tools in Excel
Homework Assignment

3. Analysis of x-y Data Independent versus dependent variables dependent

4. Simple Plotting
Generate X and Y data to Plot

5. Common Types of Plots: Y=3X2
Normal
log-log: log y-log x
Semi-log: log x
logy = log3 + 2logx
y = 3x2
Straight Line on log-log Plot!

6. Finding Other Values Interpolation Regression – curve fitting
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

7. Curve-Fitting - Regression
Useful for noisy or uncertain data
n pairs of data (xi , yi)
Choose a functional form y = f(x)
polynomial
exponential
etc.
and evaluate parameters for a “close” fit

8. What Does “Close” Mean? Want a consistent rule
squared
errors
Want a consistent rule
Common is the least squares fit (SSE):
sum
(x1,y1)
(x2,y2)
(x3,y3)
(x4,y4) x y e3
ei = yi – f(xi), i =1,2,…,n

9. Quality of the Fit: Notes: is the average y value 0  R2  1
closer to 1 is a “better” fit x y

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

11. 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 x2 + bx + c
Higher order: y = a xk + bxk-1 + …

12. 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
Show R2 value on chart
Show equation on chart

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

14. Example Function Discovery How to find the best relationship
Look for straight lines on log axes:
à   linear on semilog x  y = a ln x + b
à   linear on semilog y  y = a e bx
à   linear on log log  y = a x b
No rule for 2nd or higher order polynomial fits

15. Previous EGR105 Project
Discover how a pendulum’s timing is impacted by the:
length of the string?
mass of the bob?
Take experimental data
string, weights, rulers, and watches
Analyze data and “discover” relationships

16. Experimental Setup:

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

18. To determine the effect of length, first plot the data:

19. Try a linear fit:

20. Force a zero intercept:

21. Try a quadratic polynomial:

22. Try logarithmic:

23. Try power function:

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

25. 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

26. Homework Assignment
See passed out sheet or course web site