Curve fit
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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

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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?


Three lines

Three lines


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