Class Prep

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Class Prep

• Bring to Class
• In-Class Exercises
• Paper for Classroom Printer
• Copy Files to MATLAB Directory
• Run MATLAB

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Copy Files

• While we are waiting for class to start, copy
  beam1.m, beam2.m, beam3.m, interp2_Demo.m,
  PolynomialRoots.m from week 10 in the course
  folder to your username subfolder in My Documents
• Run MATLAB, set correct Current Directory

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Week 10-a(15.1-15.2)

• Interpolation
• Curve Fitting
• Roots of Polynomials

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The Interpolation Problem
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Interpolated Data
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Interpolation

• Interpolation is a process for estimating values
  that lie between known data points. It has
  important applications in areas such as image
  processing (smoother jpg's).
• Also recall shading interp of surf plots
• MATLAB provides a number of interpolation
  techniques that let you balance the smoothness of
  the data fit vs. speed of execution and memory
  usage.

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Cantilever Beam Deflection Model
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Hands-On DEMO Linear Interpolation

• Interpolate missing values in Cantilever Beam
  Deflection
• See beam1.m then beam2.m

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Hands-On DEMO interp1 Function(that's a one not
an L)

• Add the following to beam2 (uncomment)
• interp1(x,y,400)
• (NOTE x, y variables are force and deflection)
• interp1 performs one-dimensional interpolation,
  an important operation for data analysis and
  curve fitting.

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Cubic Spline Interpolation
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Polynomial Interpolation

• interp1 uses polynomial techniques, fitting the
  supplied data with polynomial functions between
  data points and evaluating the appropriate
  function at the desired interpolation points.
• Its most general form is
•  yi interp1(x, y, xi, method)

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Polynomial Interpolation Methods

• method is an optional string specifying an
  interpolation method
• Nearest neighbor interpolation (method
  'nearest'). This method sets the value of an
  interpolated point to the value of the nearest
  existing data point.
• Linear interpolation (method 'linear'). This
  method fits a different linear function between
  each pair of existing data points, and returns
  the value of the relevant function at the points
  specified by xi. This is the default method for
  the interp1 function.
• Cubic-Spline interpolation (method 'spline').
  This method fits a different cubic function
  between each pair of existing data points, and
  uses the spline function to perform cubic spline
  interpolation at the data points.

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Hands-On DEMO Comparing Methods

• This example compares two-dimensional
  interpolation methods on a matrix of data.
• As opposed to vectors in previous examples.
• Generate the peaks function at low resolution.
•     gtgt x,y meshgrid(-313)
•     gtgt z peaks(x,y)
•     gtgt surf(x,y,z)

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Example
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DEMO interp2 for 2-Dimensional
Interpolation(see interp2_Demo.m)

• Generate a finer mesh for interpolation.
• xi,yi meshgrid(-30.253)  
• Interpolate using nearest neighbor interpolation.
  
• zi1 interp2(x,y,z,xi,yi,'nearest')  
• Interpolate using bi-linear interpolation zi2
  interp2(x,y,z,xi,yi,'bilinear')
• Interpolate using bi-cubic interpolation.zi3
  interp2(x,y,z,xi,yi,'bicubic')  
• Compare the surface plots for the different
  interpolation methods.

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Hands-On DEMO Surface Plots
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Hands-On DEMO Contour Plots
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Data Approximation with a Function

• Data points (xi, yi) can be approximated by a
  function y f(x) such that the function passes
  close to the data points but does not
  necessarily pass through them.

• Data fitting is necessary to model data with
  fluctuations such as experimental measurements.

• The most common form of data fitting is the
  least squares method.

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Fitting Polynomials to Data

• Fitting analytical curves to experimental data is
  a common task in engineering.
• The simplest is linear interpolation (straight
  lines between data points). Polynomials are
  often used to define curves, and there are
  several options
• Fit curve through all data points
  (polynomialspline interpolation),
• Fit curve through some data points,
• Fit curve as close to points as possible
  (polynomial regression, also called least square
  fitting)

polynomial interpolation
polynomial regression
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Multi-Valued Data Points

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Fitting Data with a Linear Function
y

x
Using the least squares algorithm, ensure that
all the data points fall close to the straight
line/function.
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Linear Curve Estimate
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Linear Curve Fit
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Higher-Order Polynomials
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Linear Least Squares Algorithm

• Data points

• Interpolating function (linear)

• Errors between function and data points

• Sum of the squares of the errors

• Compact notation

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Why Least Squares?

• Why not just add distances that approximation is
  from actual measured data points?
• Eliminates the need for absolute value function
• Squaring differences makes one point that is3
  off worth more than three points that are 1 off.
• Consider the case of computer dating
• 3 questions such as music preferences are off by
  only one point from a potential date less
  important than responses to one question that is
  far off

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Plot of Linear Fit
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Choosing the Right Polynomial

• The degree of the correct approximating function
  depends on the type of data being analyzed.
• When a certain behavior is expected, such as the
  linear response of beam deflection under
  increasing loads, we know what type of function
  to use, and simply have to solve for its
  coefficients.
• When we dont know what sort of response to
  expect, ensure your data sample size is large
  enough to clearly distinguish which degree is the
  best fit.

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polyfit Function

• Polynomial curve fitting
• coef polyfit(x,y,n)
• A least-square fit with polynomial of degree n.
• x and y are data vectors, and coef is vector of
  coefficients of the polynomial,
• coef an, an-1, , a1, a0

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Hands-On DEMO Polynomial Curve Fitting

• x and y are vectors containing the x and y data
  to be fitted, and n is the order of the
  polynomial to return. For example, consider the
  x-y test data.
•     x 1 2 3 4 5
• y 5.5 43.1 128 290.7 498.4
• A third order (n3) polynomial that approximately
  fits the data is
• p polyfit(x,y,3)
• p -0.1917 31.5821 -60.3262 35.3400
• In other words, the polynomial function would be
• Y -0.1917X3 31.5821X2 -60.3262X 35.34

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Polynomial Curve Fitting, Graph
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Hands-On DEMO Polynomial Evaluation

• Going the other way, if p(x) x3 - 2x - 5
• To enter this polynomial into MATLAB, use
•            gtgt p 1 0 -2 -5
• The polyval function evaluates a polynomial at a
  specified value or vector.To evaluate p at x
  5, use
• gtgt polyval(p,5)
•     ans 110

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Hands-On DEMO Polynomial Roots(see
PolynomialRoots.m)

• Draw graph of p 1 0 -9
• x -100.0110, ypolyval(p,x), plot(x,y), grid
  on
• Note that the Roots are /- 3. Try r roots(p)
  
• The roots function calculates the roots of a
  polynomial.
• gtgt p 1 0 -2 -5
• gtgt r roots(p)                  
• r 2.0946
• -1.0473 1.1359i
•      -1.0473 - 1.1359i  
• By convention, MATLAB stores roots in column
  vectors. The function poly returns the polynomial
  coefficients given the roots.
• gtgt p2 poly(r)
• p2   1  8.8818e-16   -2   -5
• poly and roots are inverse functions, up to
  ordering, scaling, and roundoff error.

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Hands-On DEMO Cantilever Beam Deflection(see
beam3.m)

• Uses polyfit() to fit a straight line.
• Hold the plot and add the fitted line to your
  graph.
• Note that with we can now extrapolate as well as
  interpolate
• The program extracts the coefficients for the
  various powers of x, and calculates the function
• The program also uses polyval to evaluate the
  polynomial directly

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beam3 Results
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Hands-On DEMO Basic Curve-Fitting Tools

• Given x05, y0,1,60,40,41,47.
• Plot the data, then from within Figure, select
  Tools ? Basic Fitting
• Try fitting polynomials of various degrees...

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Solution
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Curve Fitting First thru Fourth Degree

• Notice that the fit looks better the higher the
  order you can make it go through more points.
• Use the fitted curves to estimate y at x 10.
  Which order polynomial do you trust more out at x
  10? Why?

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Interpolation vs. Approximation

• INTERPOLATION
• When you wish to find an unknown intermediate
  point within a dataset with well-defined
  behavior.
• When a relatively small data set exists.
• APPROXIMATION
• When evaluating fluctuating data.
• To create a mathematical model of a process.
• Deciding between interpolation and approximation
  depends on the type of data, not on the data
  values themselves.
• Recall that, by definition, multi-valued data
  points cannot be interpolated.

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Homework 10a Spacecraft Accelerometer(Problem
8, page 417)

• NEVER COLLECTED
• The guidance and control system for a spacecraft
  often uses a sensor called an accelerometer,
  which is an electromechanical device that
  produces an output voltage proportional to the
  applied acceleration. Assume that the experiment
  has yielded the data on the exercise sheet.
• Determine the linear equation that best fits this
  set of data. Plot the data points and the linear
  equation.
• Determine the sum of squares of the distances of
  these points from the line of best fit determined
  above.
• Compare the error sum for linear fit (above) with
  the same error sum computed from the best
  quadratic fit. What do these sums tell you about
  the two models for the data?

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In-Class Exercise 10a Interpolation (interp1)

• Generate f(x) x2 for x -3 -1 0 2 5 6
• Compute and plot the linear and cubic-spline
  interpolation of the data points over the range
  -30.056
• Compute the value of f(4) using linear
  interpolation and cubic-spline interpolation.
  What are the respective errors when the answer is
  compared to the actual value or f(4) 42?
• Plot the original points as black circles,
  'nearest' values with dotted line in red,
  'linear' with dashed line in blue, and 'spline'
  with a solid line in green

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END
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EXTRA MATERIAL
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Bad Data

• What about
• this point?
• Is it really a
• good data point?
• What do you know about the data?
• Is it monotonic?

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A linear fit ignoring one data point
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Second degree polynomial fit, ignoring a
different data point

• Ignoring a different data point allows us to
  approximate the data pretty well with a second
  degree polynomial.

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Exponential fit

• Not all experimental data can be approximated
  with polynomial functions.
• Exponential data can be fit using the least
  squares method by first converting the data to a
  linear form.

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Relationship between exponential and linear form

• This graph shows a population that doubles every
  year. Notice that for the first fifteen years,
  growth is almost imperceptible at this scale. It
  would be difficult to approximate this raw data.

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Relationship between exponential and linear form

• What if we plot the same data with the y-axis
  against a logarithmic scale?
• Notice that the change in population is not
  simply perceptible, but is clearly linear.
• If we can transform the numeric exponential data,
  it would be simple to approximate with the
• least-squares method.

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Relationship between exponential and linear form

• This graph shows what happens if we take the
  natural logarithm of each y value.
• Notice that the shape of the graph did not change
  from the last example, only the scaling of the
  y-axis.

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Relationship between Exponential and Linear

• An exponential function,
• y aebx
• can be rewritten as a linear polynomial by
  taking the natural logarithm of each side
• ln y ln a bx
• By finding ln yi for each point in a data set, we
  can solve for a and b using the least squares
  method.