1. MatLab – Palm Chapter 5 Curve Fitting
Class Palm Chapter:
12/1/2018
ENGR 111A - Fall 2004

2. RAT 14.1
As in INDIVIDUAL you have 1 minute to answer the following question and another 30 seconds to turn it in. Ready?
When (day and time) and where is Exam #3?
The answer is: Thursday at 6:30 pm,
Bright 124
Do we have any schedule problems?
12/1/2018
ENGR 111A - Fall 2004

3. Learning Objectives Students should be able to:
Use the Function Discovery (i.e., curve fitting) Techniques
Use Regression Analysis
12/1/2018
ENGR 111A - Fall 2004

4. 5.5 Function Discovery
Engineers use a few standard functions to represent physical conditions for design purposes. They are:
Linear: y(x) = mx + b
Power: y(x) = bxm
Exponential: y(x) = bemx (Naperian)
y(x) = b(10)mx (Briggsian)
The corresponding plot types are explained at the top of p. 299.
12/1/2018
ENGR 111A - Fall 2004

5. Steps for Function Discovery
Examine data and theory near the origin; look for zeros and ones for a hint as to type.
Plot using rectilinear scales; if it is a straight line, it’s linear. Otherwise:
y(0) = 0 try power function
Otherwise, try exponential function
If power function, log-log is a straight line.
If exponential, semi-log is a straight line.
12/1/2018
ENGR 111A - Fall 2004

6. Example Function Calls
polyfit( ) will provide the slope and y-intercept of the BEST fit line if a line function is specified.
Linear: polyfit(x, y, 1)
Power: polyfit(log10(x),log10(y),1)
Exponential: polyfit(x,log10(y),1); Briggsian
polyfit(x,log(y),1); Naperian
Note: the use of log10( ) or log( ) to transform the data to a linear dataset.
12/1/2018
ENGR 111A - Fall 2004

7. Example 5.5-1: Cantilever Beam Deflection
First, input the data table on page 304.
Next, plot deflection versus force (use data symbols or a line?)
Then, add axes and labels.
Use polyfit() to fit a line.
Hold the plot and add the fitted line to your graph.
12/1/2018
ENGR 111A - Fall 2004

8. Solution
12/1/2018
ENGR 111A - Fall 2004

9. Straight Line Plots Forms of Equation Straight Line Systems
MatLab Syntax
Linear Equation
y = mx + b
Rectilinear System
plot(x,y)
Power Equation
y=bxm
Loglog System
loglog(x,y)
Exponential Equation
y = bemx or y=b10mx
Semilog System
semilogy(x,y)
12/1/2018
ENGR 111A - Fall 2004

10. Why do these plot as lines?
Exponential function: y = bemx
Take the Naperian logarithm of both sides:
ln(y) = ln(bemx)
ln(y) = ln(b) + mx(ln(e))
ln(y) = ln(b) + mx
Thus, if the x value is plotted on a linear scale and the y value on a log scale, it is a straight line with a slope of m and y-intercept of ln(b).
12/1/2018
ENGR 111A - Fall 2004

11. Why do these plot as lines?
Exponential function: y = b10mx
Take the Briggsian logarithm of both sides:
log(y) = log(b10mx)
log(y) = log(b) + mx(log(10))
log(y) = log(b) + mx
Thus, if the x value is plotted on a linear scale and the y value on a log scale, it is a straight line. (Same as Naperian.)
12/1/2018
ENGR 111A - Fall 2004

12. Why do these plot as lines?
Power function: y = bxm
Take the Briggsian logarithm of both sides:
log(y) = log(bxm)
log(y) = log(b) + log(xm)
log(y) = log(b) + mlog(x)
Thus, if the x and y values are plotted on a on a log scale, it is a straight line. (Same can be done with Naperian log.)
12/1/2018
ENGR 111A - Fall 2004

13. In-class Assignment 14.1.1 Given: x=[1 2 3 4 5 6 7 8 9 10];
y1=[ ];
y2=[ ];
y3=[ ];
Use MATLAB to plot x vs each of the y data sets.
Chose the best coordinate system for the data.
Be ready to explain why the system you chose is the best one.
12/1/2018
ENGR 111A - Fall 2004

14. Solution
12/1/2018
ENGR 111A - Fall 2004

15. Be Careful
What value does the first tick mark after 100 represent? What about the tick mark after 101 or 102?
Where is zero on a log scale? Or -25?
See pages 282 and 284 of Palm for more special characteristics of logarithmic plots.
12/1/2018
ENGR 111A - Fall 2004

16. How to use polyfit command.
Linear: pl = polyfit(x, y, 1)
m = pl(1); b = pl(2) of BEST FIT line.
Power: pp = polyfit(log10(x),log10(y),1)
m = pp(1); b = 10^pp(2) of BEST FIT line.
Exponential: pe = polyfit(x,log10(y),1)
m = pe(1); b = 10^pe(2), best fit line using Briggsian base.
OR pe = polyfit(x,log(y),1)
m = pe(1); b = exp(pe(2)), best fit line using Naperian base.
12/1/2018
ENGR 111A - Fall 2004

17. In-class Assignment
Determine the equation of the best-fit line for each of the data sets in In-class Assignment
Hint: use the result from ICA and the polyfit( ) function in MatLab.
Plot the fitted lines in the figure.
12/1/2018
ENGR 111A - Fall 2004

18. Solution
12/1/2018
ENGR 111A - Fall 2004

19. 5.6 Regression Analysis
Involves a dependent variable (y) as a function of an independent variable (x), generally: y = mx + b
We use a “best fit” line through the data as an approximation to establish the values of: m = slope and b = y-axis intercept.
We either “eye ball” a line with a straight-edge or use the method of least squares to find these values.
12/1/2018
ENGR 111A - Fall 2004

20. Curve Fits by Least Squares
Use Linear Regression unless you know that the data follows a different pattern: like n-degree polynomials, multiple linear, log-log, etc.
We will explore 1st (linear), … 4th order fits.
Cubic splines (piecewise, cubic) are a recently developed mathematical technique that closely follows the “ship’s” curves and analogue spline curves used in design offices for centuries for airplane and ship building.
Curve fitting is a common practice used my engineers.
12/1/2018
ENGR 111A - Fall 2004

21. T5.6-1 Solve problem T5.6-1 on page 318.
Notice that the fit looks better the higher the order – you can make it go through the points.
Use your fitted curves to estimate y at x = 10. Which order polynomial do you trust more out at x = 10? Why?
12/1/2018
ENGR 111A - Fall 2004

22. Solution
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ENGR 111A - Fall 2004

23. Solution
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ENGR 111A - Fall 2004

24. Assignment 14.1 Prepare for Exam #3.
Group Projects are due at Exam #3 (parts 1 through 3 required; parts 4 and 5 as extra credit)
12/1/2018
ENGR 111A - Fall 2004