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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')

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?

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