Transcript of Curve fit metrics When we fit a curve to data we ask: –What is the error metric for the best fit?...
- Slide 1
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- 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.
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- 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 noise
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- Regression The process of fitting data with a curve by
minimizing the mean square difference from the data is known as
regression Term originated from first paper to use regression dealt
with a phenomenon called regression to the mean
http://www.jcu.edu.au/cgc/RegMean.html
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.
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- 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.
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- Surrogates for fitting simulations Great interest now in
fitting computer simulations Computer 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.
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- Surrogates of given functional form Noisy response Linear
approximation Rational approximation Data from n y experiments
Error (fit) metrics
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- Linear Regression Functional form For linear approximation
Error or difference between data and surrogate Rms error Minimize
rms error e T e=(y-Xb T ) T (y-Xb T ) Differentiate to obtain
Beware of ill-conditioning !
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- Example
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- Other metric fits Rms fitAv. Err. fitMax err. fit RMS
error0.4710.5770.5 Av. error0.4440.3330.5 Max error0.66710.5
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- Three lines
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- Original 30-point curve fit With dense data difference due to
metrics is small. Rms fitAv. Err. fitMax err. fit RMS
error1.2781.2831.536 Av. error0.9580.9511.234 Max
error3.0072.9872.934
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- problems 1.Find other metrics for a fit beside the three
discussed in this lecture. 2.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