1. 11/20/2015ENGR 111A - Fall 20041 MatLab – Palm Chapter 5 Curve Fitting Class 14.1 Palm Chapter: 5.5-5.7

2. 11/20/2015ENGR 111A - Fall 20042 Learning Objectives Students should be able to: Use the Function Discovery (i.e., curve fitting) Techniques Use Regression Analysis

3. 11/20/2015ENGR 111A - Fall 20043 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) = bx m Exponential:y(x) = be mx (Naperian) y(x) = b(10) mx (Briggsian) The corresponding plot types are explained at the top of p. 299.

4. 11/20/2015ENGR 111A - Fall 20044 Steps for Function Discovery 1. Examine data and theory near the origin; look for zeros and ones for a hint as to type. 2. Plot using rectilinear scales; if it is a straight line, it’s linear. Otherwise: a) y(0) = 0 try power function b) Otherwise, try exponential function 3. If power function, log-log is a straight line. 4. If exponential, semi-log is a straight line.

5. 11/20/2015ENGR 111A - Fall 20045 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.

6. 11/20/2015ENGR 111A - Fall 20046 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.

7. 11/20/2015ENGR 111A - Fall 20047 Solution

8. 11/20/2015ENGR 111A - Fall 20048 Straight Line Plots Forms of EquationStraight Line Systems MatLab Syntax Linear Equation y = mx + b Rectilinear System plot(x,y) Power Equation y=bx m Loglog System loglog(x,y) Exponential Equation y = be mx or y=b10 mx Semilog System semilogy(x,y)

9. 11/20/2015ENGR 111A - Fall 20049 Why do these plot as lines? Exponential function: y = be mx Take the Naperian logarithm of both sides: ln(y) = ln(be mx ) 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).

10. 11/20/2015ENGR 111A - Fall 200410 Why do these plot as lines? Exponential function: y = b10 mx Take the Briggsian logarithm of both sides: log(y) = log(b10 mx ) 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.)

11. 11/20/2015ENGR 111A - Fall 200411 Why do these plot as lines? Power function: y = bx m Take the Briggsian logarithm of both sides: log(y) = log(bx m ) log(y) = log(b) + log(x m ) 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. 11/20/2015ENGR 111A - Fall 200412 In-class Assignment 14.1.1 Given: x=[1 2 3 4 5 6 7 8 9 10]; y 1 =[3 5 7 8 10 14 15 17 20 21]; y 2 =[3 8 16 24 34 44 56 68 81 95]; y 3 =[8 11 15 20 27 36 49 66 89 121]; 1. Use MATLAB to plot x vs each of the y data sets. 2. Chose the best coordinate system for the data. 3. Be ready to explain why the system you chose is the best one.

13. 11/20/2015ENGR 111A - Fall 200413 Solution

14. 11/20/2015ENGR 111A - Fall 200414 Be Careful 1. What value does the first tick mark after 10 0 represent? What about the tick mark after 10 1 or 10 2 ? 2. Where is zero on a log scale? Or -25? 3. See pages 282 and 284 of Palm for more special characteristics of logarithmic plots.

15. 11/20/2015ENGR 111A - Fall 200415 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. ORpe = polyfit(x,log(y),1) m = pe(1); b = exp(pe(2)), best fit line using Naperian base.

16. 11/20/2015ENGR 111A - Fall 200416 In-class Assignment 14.1.2 Determine the equation of the best-fit line for each of the data sets in In-class Assignment 14.1.1 Hint: use the result from ICA 14.1.1 and the polyfit( ) function in MatLab. Plot the fitted lines in the figure.

17. 11/20/2015ENGR 111A - Fall 200417 Solution

18. 11/20/2015ENGR 111A - Fall 200418 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.

19. 11/20/2015ENGR 111A - Fall 200419 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.

20. 11/20/2015ENGR 111A - Fall 200420 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?

21. 11/20/2015ENGR 111A - Fall 200421 Solution

22. 11/20/2015ENGR 111A - Fall 200422 Solution