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MatLab – Palm Chapter 5Curve Fitting

Class 14.1 Palm Chapter: 5.5-5.7

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Learning Objectives

Students should be able to: Use the Function Discovery (i.e.,

curve fitting) Techniques Use Regression Analysis

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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.

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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 functionb) Otherwise, try exponential function

3. If power function, log-log is a straight line.4. If exponential, semi-log is a straight line.

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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.

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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.

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Solution

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Straight Line Plots

Forms of Equation Straight Line SystemsMatLab Syntax

Linear Equationy = mx + b

Rectilinear Systemplot(x,y)

Power Equationy=bxm

Loglog Systemloglog(x,y)

Exponential Equationy = bemx or y=b10mx

Semilog Systemsemilogy(x,y)

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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).

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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.)

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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.)

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In-class Assignment 14.1.1Given:

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];

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.

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Solution

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Be Careful

1. What value does the first tick mark after 100 represent? What about the tick mark after 101 or 102?

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.

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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.

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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.

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Solution

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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.

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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.

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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?

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Solution

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Solution