When we fit a curve to data we ask What is the error metric for the best fit What is more accurate the data or the fit This lecture deals with the following case The data is noisy The functional form of the true function is known ID: 538966
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Slide1
Curve fit metrics
When we fit a curve to data we ask:What is the error metric for the best fit?What is more accurate, the data or the fit?This lecture deals with the following case:The data is noisy.The functional form of the true function is known.The data is dense enough to allow us some noise filtering.The objective is to answer the two questions.Slide2
Curve fit
We sample the function y=x (in red) at x=1,2,…,30, add noise with standard deviation 1 and fit a linear polynomial (blue).How would you check the statement that fit is more accurate than the data?With dense data, functional form is clear. Fit serves to filter out noiseSlide3
Regression
The process of fitting data with a curve by minimizing the mean square difference from the data is known as regressionTerm originated from first paper to use regression dealt with a phenomenon called regression to the mean http://www.jcu.edu.au/cgc/RegMean.html The polynomial regression on the previous slide is a simple regression, where we know or assume the functional shape and need to determine only the coefficients.Slide4
Surrogate (
metamodel)The algebraic function we fit to data is called surrogate, metamodel or approximation.Polynomial surrogates were invented in the 1920s to characterize crop yields in terms of inputs such as water and fertilizer.They were called then “response surface approximations.”
The term “surrogate” captures the purpose of the fit: using it instead of the data for prediction.
Most important when data is expensive and noisy, especially for optimization.Slide5
Surrogates for fitting simulations
Great interest now in fitting computer simulationsComputer simulations are also subject to noise (numerical)
Simulations
are
exactly repeatable
, so noise is hidden.
Some surrogates
(
e.g. polynomial response surfaces) cater mostly to noisy data.
Some
(e.g.
Kriging
)
interpolate data. Slide6
Surrogates of given functional form
Noisy response Linear approximationRational approximationData from ny experimentsError (fit) metricsSlide7
Linear Regression
Functional formFor linear approximationError or difference between data and surrogateRms errorMinimize
rms
error
e
T
e=(y-XbT)T(y-X
bT)Differentiate to obtain
Beware of ill-conditioning
!Slide8
Example
Data: y(0)=0, y(1)=1, y(2)=0Fit linear polynomial y=b0+b1xThen
Obtain
b
0
=1/3,
b1=0,
.Surrogate preserves the average value of the data at data points.
Slide9
Other metric fits
Assuming other fits will lead to the form ,For average error minimize
Obtain b=0.
For maximal error minimize
obtain b=0.5
Rms
fit
Av. Err. fit
Max
err. fit
RMS error
0.471
0.577
0.5
Av
. error
0.444
0.333
0.5
Max
error
0.667
1
0.5Slide10
Three linesSlide11
Original 30-point curve fit
With dense data difference due to metrics is small
.
Rms
fit
Av. Err. fit
Max
err. fit
RMS error
1.278
1.283
1.536
Av
. error
0.958
0.951
1.234
Max
error
3.007
2.987
2.934Slide12
problemsFind other metrics for a fit beside the three discussed in this lecture.
Redo the 30-point example with the surrogate y=bx. Use the same data.
3.
Redo the 30-point example using only every third point (x=3,6,…). You can consider the other 20 points as test points used to check the fit. Compare the difference between the fit and the data points to the difference between the fit and the test points. It is sufficient to do it for one
fit metric.
Source: Smithsonian Institution
Number: 2004-57325