MatLab

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MatLab Palm Chapter 5Curve Fitting

• Class 14.1 Palm Chapter 5.5-5.7

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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 630 pm,
• Bright 124
• Do we have any schedule problems?

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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

• 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, its 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.

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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 5.5-1Cantilever 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 Systems MatLab Syntax
Linear Equation y mx b Rectilinear System plot(x,y)
Power Equation ybxm Loglog System loglog(x,y)
Exponential Equation y bemx or yb10mx Semilog System semilogy(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 14.1.1

• Given
• x1 2 3 4 5 6 7 8 9 10
• y13 5 7 8 10 14 15 17 20 21
• y23 8 16 24 34 44 56 68 81 95
• y38 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.

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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 10pp(2) of BEST FIT line.
• Exponential pe polyfit(x,log10(y),1)
• m pe(1) b 10pe(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 ships 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
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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)