Lecture 21 Approximation Methods

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Lecture 21 - Approximation Methods

• CVEN 302
• October 15, 2001

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Lectures Goals

• Discrete Least Square Approximation
• Linear
• Quadratic
• Higher order Polynomials
• Nonlinear
• Continuous Least Square
• Orthogonal Polynomials
• Gram Schmidt -Legendre Polynomial

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Approximation Methods
What is the difference between approximation and
interpolation?

• Interpolation matches the data points exactly.
  In case of experimental data, this assumption is
  not often true.
• Approximation - we want to consider the curve
  that will fit the data with the smallest error.

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Least Square Fit Approximations
Suppose we want to fit the data set.
We would like to find the best straight line to
fit the data?
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Least Square Fit Approximations
The problem is how to minimize the error. We can
use the error defined as
However, the errors can cancel one another and
still be wrong.
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Least Square Fit Approximations
We could minimax the error, defined as
The error minimization is going to have problems.
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Least Square Fit Approximations
The solution is the minimization of the sum of
squares. This will give a least square solution.
This is known as the maximum likelihood principle.
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Least Square Approximations
Assume
The error is defined as
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Least Square Error
The sum of the errors
Substitute for the error
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Least Square Error
How do you minimize the error?
Take the derivative with the coefficients and set
it equal to zero.
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Least Square Error
The first component, a
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Least Square Error
The second component, b
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Least Square Coefficients
The equations can be rewritten
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Least Square Coefficients
The equations can be rewritten
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Least Square Coefficients
The coefficients are defined as
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Least Square Example
Using the results into table of the values
Given the data
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Least Square Example
The equation is
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Least Square Error
How do you minimize the error for a quadratic fit?
Take the derivative with the coefficients and set
it equal to zero.
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Least Square Coefficients for Quadratic fit
The equations can be written as
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Least Square of Quadratic Fit
The matrix can be solved using a Gaussian
elimination and the coefficients can be found.
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Quadratic Least Square Example
The linear results

• Given a set of data

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Quadratic Least Square Example
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Quadratic Least Square Example

• The results are

a 0.225, b -1.018 , c 0.998
y 0.225x2 -1.018x 0.998
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Polynomial Least Square
The technique can be used to all forms of
polynomials of the form
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Polynomial Least Square
Solving large sets of linear equations are not a
simple task. They can have the undesirable
property known as ill-conditioning. The results
of this method is that round-off errors in
solving for the coefficients cause unusually
large errors in the curve fits.
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Polynomial Least Square
How do you measure the error of higher order
polynomials?
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Polynomial Least Square
Or measure of the variance of the problem
Where, n is the degree polynomial and N is the
number of elements and Yk are the data points
and,
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Polynomial Least Square Example
Example 2 can be fitted with cubic equation and
the coefficients are a0 1.004 a1 -1.079 a2
0.351 a3 - 0.069
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Polynomial Least Square Example
However, if we were to look at the higher order
polynomials such the sixth and seventh
order. The results are not all that promising.
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Polynomial Least Square Example
The standard deviation of the polynomial fit
shows that the best fit for the data is the
second order polynomial.
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Summary

• The linear least squared method is straight
  forward to determine the coefficients of the line.

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Summary

• The quadratic and higher order polynomial curve
  fits use a similar technique and involve solving
  a matrix of (n1) x (n1).

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Summary

• The higher order polynomials fit required that
  one selects the best fit for the data and a means
  of measuring the fit is the standard deviation of
  the results as a function of the degree of the
  polynomial.

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Homework

• Check the Homework webpage