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MatLab – Palm Chapter 5 Curve FittingPowerPoint Presentation

MatLab – Palm Chapter 5 Curve Fitting

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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?

ENGR 111A - Fall 2004

Learning Objectives

- Students should be able to:
- Use the Function Discovery (i.e., curve fitting) Techniques
- Use Regression Analysis

ENGR 111A - Fall 2004

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.

ENGR 111A - Fall 2004

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.

ENGR 111A - Fall 2004

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.

ENGR 111A - Fall 2004

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.

ENGR 111A - Fall 2004

Solution

ENGR 111A - Fall 2004

Straight Line Plots

ENGR 111A - Fall 2004

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).

ENGR 111A - Fall 2004

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.)

ENGR 111A - Fall 2004

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.)

ENGR 111A - Fall 2004

In-class Assignment 14.1.1

Given:

x=[1 2 3 4 5 6 7 8 9 10];

y1=[3 5 7 8 10 14 15 17 20 21];

y2=[3 8 16 24 34 44 56 68 81 95];

y3=[8 11 15 20 27 36 49 66 89 121];

- 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.

ENGR 111A - Fall 2004

Solution

ENGR 111A - Fall 2004

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.

ENGR 111A - Fall 2004

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.

- m = pe(1); b = 10^pe(2), best fit line using Briggsian base.

ENGR 111A - Fall 2004

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.

ENGR 111A - Fall 2004

Solution

ENGR 111A - Fall 2004

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.

ENGR 111A - Fall 2004

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.

ENGR 111A - Fall 2004

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?

ENGR 111A - Fall 2004

Solution

ENGR 111A - Fall 2004

Solution

ENGR 111A - Fall 2004

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)

ENGR 111A - Fall 2004

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