Curve fit

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# Curve fit - PowerPoint PPT Presentation

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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Presentation Transcript
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
• 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)
• 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
• 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
• 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
• Noisy response model
• Data from ny experiments
• Linear approximation
• Rational approximation
• Error measures
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
• 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
• 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?