SE 301: Numerical Methods Topic 4: Least Squares Curve Fitting

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SE 301 Numerical MethodsTopic 4Least Squares
Curve Fitting

• Dr. Samir Al-Amer
• (Term 062)

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SE 301 Numerical MethodsIntroduction to Least
Squares

• Dr. Samir Al-Amer
• (Term 061)

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Motivation

• Given a set of experimental data

• The relationship between x and y may not be clear
  
• we want to find an expression for f(x)

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Curve Fitting

• Given a set of tabulated data, find a curve
  or a function that best represents the data.
• Given
• The tabulated data
• The form of the function
• The curve fitting criteria
• Find the unknown coefficients

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Selection of the functions
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Decide on the criterion
Chapter 17
Chapter 18
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Least Squares
Given
The form of the function is assumed to be known
but the coefficients are unknown
The difference is assumed to be the result of
experimental error
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Determine the Unknowns
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Determine the Unknowns
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Example 1
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Remember
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Example 1
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Example 1
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Example 1
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Example 1
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Example 2
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Example 2
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Example 2
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How do you judge performance?
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Multiple Regression

• Example
• Given the following data
• It is required to determine a function of two
  variables
• f(x,t) a b x c t
• to explain the data that is best in the
  least square sense.

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Solution of Multiple Regression

• Construct , the sum of the square of the
  error and derive the necessary conditions by
  equating the partial derivatives with respect to
  the unknown parameters to zero then solve the
  equations.

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Solution of Multiple Regression
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SE301 Numerical MethodsLecture 13 Nonlinear
least squares problems More

• Examples of nonlinear least squares
• Solution of inconsistent equations
• Continuous least square problems

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Outlines

• Examples of nonlinear least squares
• Solution of inconsistent equations
• Continuous least square problems

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Nonlinear Problem
Given
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Alternative Solution(Linearization Method)
Given
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Inconsistent System of Equations
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Inconsistent System of EquationsReasons
Inconsistent equations may occur because of
errors in formulating the problem, errors in
collecting the data or computational errors.
Solution if all lines intersect at one point
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Inconsistent System of EquationsFormulation as a
least squares problem
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Solution
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Solution
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Examples (Linearization Method)
Given
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HW

• Check WebCT for HW problems and due date