F8 : 11' Curve Fitting

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F8 11. Curve Fitting

• Curvilinear Regression
• Nonlinear Regression (NLR)

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Drying time of a varnish
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Drying time of a varnish

• Independent variable X Amount of varnish
  additive in grams x1,...,xn
• Dependent variable Y Drying time in hours
  y1,...yn
• Wanted value for X , which gives the minimal
  drying time
• Y is random (varibility of material, procedure,
  measurement errors).
• X is fixed and known

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SPSS-Result linear regression
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SPSS-Result linear regression

• H Beta 0 cannot rejected!
• No significant relation between x and y was
  founded.
• No result.

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SPSS-Result Quadratic regression
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SPSS-Result quadratic regression

• Model is accepted.
• All components have significant influence.
• fitted curve 12.2 - 1.85 x 0.183 xx
• minimum at x0 5.05 gram of additive

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Curvilinear Regressions model

• regressions function f is not linear
• but the unknown parameters are linear parameters
  f(x) betah(x)
• Can apply linear estimation methods after
  transformation of x by h.

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Transformation method

• Find a transformation for the independent
  variable h(X)
• Find a transformation for the dependent variable
  g(Y)
• Apply simple linear regression on the transformed
  variables!

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How to apply the Transformation Method?

• Plot a Scatter Plot.
• Gives an idea for the type of curve!
• Check usual transformations for X and Y.
• Scatter Plot of the transformed variables h(X)
  and g(Y).
• If it looks like a line, apply linear regression
  to the transformed variables.

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Classic Transformations
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Example Radial tires

• Independent Variable Miles driven
• Response Percentage usable
• Life time gtgtgtgtgt exponential relation
• f(x) beta1exp(beta2x)
• ln(y)

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Nonlinear Regression NLR

• Regressions function f is not linear in x
• Regression function f is in beta nonlinear also!
• known the form of the regression function f.
• Physical interpretation of the parameters.

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Idea of Nonlinear Least Squares

• Take the curve with minimal average of squared
  differences between curve and observations !
• Apply a numerical procedure for determining the
  minimum sum of squares.

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Method of Least Squares in simple linear
regression
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Residual Analysis
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Application to Soya beans
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Result for Soya beans
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SPSS-Result Soya beans
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Properties of LSE
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Inference based on LSE
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SPSS-Result Soya beans
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Confidence and Prediction
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