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Curve fit. noise= randn (1,30); x=1:1:30; y= x+noise 3.908 2.825 4.379 2.942 4.5314 5.7275 8.098 …………………………………25.84 27.47 27.00 30.96 [ p,s ]= polyfit (x,y,1); yfit = polyval ( p,x ); plot (x,y,\'+\',x,x,\'r\',x, yfit ,\'b\').

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curve fit
Curve fit
  • noise=randn(1,30); x=1:1:30; y=x+noise
  • 3.908 2.825 4.379 2.942 4.5314 5.7275 8.098 …………………………………25.84 27.47 27.00 30.96
  • [p,s]=polyfit(x,y,1); yfit=polyval(p,x); plot(x,y,\'+\',x,x,\'r\',x,yfit,\'b\')

With dense data, functional form is clear. Fit serves to filter out noise

regression
Regression
  • The process of fitting data with a curve by minimizing root mean square error is known as regression
  • Term originated from first paper to use regression “regression of heights to the mean” http://www.jcu.edu.au/cgc/RegMean.html
  • Can get the same curve from a lot of data or very little. So confidence in fit is major concern.
surrogate approximations
Surrogate (approximations)
  • Originated from experimental optimization where measurements are very noisy
  • In the 1920s it was used to maximize crop yields by changing inputs such as water and fertilizer
  • With a lot of data, can use curve fit to filter out noise
  • “Approximation” can be then more accurate than data!
  • The term “surrogate” captures the purpose of the fit: using it instead of the data for prediction.
  • Most important when data is expensive
surrogates for simulation based optimization
Surrogates for Simulation based optimization
  • Great interest now in applying these techniques to computer simulations
  • Computer simulations are also subject to noise (numerical)
  • However, simulations are exactly repeatable, and if noise is small may be viewed as exact.
  • Some surrogates (e.g. polynomial response surfaces) cater mostly to noisy data. Some (e.g. Kriging) to exact data.
polynomial response surface approximations
Polynomial response surface approximations
  • Data is assumed to be “contaminated” with normally distributed error of zero mean and standard deviation 
  • Response surface approximation has no bias error, and by having more points than polynomial coefficients it filters out some of the noise.
  • Consequently, approximation may be more accurate than data
fitting approximation to given data
Fitting approximation to given data
  • Noisy response model
  • Data from ny experiments
  • Linear approximation
  • Rational approximation
  • Error measures
linear regression
Linear Regression
  • Functional form
  • For linear approximation
  • Estimate of coefficient vector denoted as b
  • Rms error
  • Minimize rms error

eTe=(y-XbT)T(y-XbT)

  • Differentiate to obtain

Beware of ill-conditioning!

example 3 1 1
Example 3.1.1
  • Data: y(0)=0, y(1)=1, y(2)=0
  • Fit linear polynomial y=b0+b1x
  • Then
  • Obtain b0=1/3, b1=0.
comparison with alternate fits
Comparison with alternate fits
  • Errors for regression fit
  • To minimize maximum error obviously y=0.5. Then eav=erms=emax=0.5
  • To minimize average error, y=0 eav=1/3, emax=1, erms=0.577
  • What should be the order of the progression from low to high?
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