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.
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?
35
30
25
20
With dense data,
functional form is clear.
Fit serves to filter out
noise
15
10
5
0
0
5
10
15
20
25
30
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
• 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.
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.
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.
Surrogates of given functional form
y  yˆ (x, b)  
• Noisy response
• Linear approximation
• Rational
approximation
• Data from ny
experiments
• Error (fit) metrics
erms
ŷ  b1  b2 x
b1
yˆ 
x  b2
yi  yˆ (xi , b)   i
1

ny
1
eav 
ny
ny
  yi  yˆ (xi , b)
2
i 1
ny
 y  yˆ (x , b)
i 1
i
i
emax  max yi  yˆ (xi , b)
xi
Linear Regression
nb
• Functional form
yˆ   bii (x)
i 1
• For linear
1  1  2  x
approximation
• Error or difference
e  y   b  (x )
between data and
surrogate
1 T
erms 
e e
• Rms error
ny
• Minimize rms error
eTe=(y-XbT)T(y-XbT)
• Differentiate to obtain X T Xb  X T y
nb
j
j
i 1
i i
j
Beware of ill-conditioning!
e  y  Xb
Example
• Data: y(0)=0, y(1)=1, y(2)=0
• Fit linear polynomial y=b0+b1x
y (0)
 b0
y (1)  b0  b1
y (2) b0  2b1
0
1
0
• Then
3b0  3b1  1
3b0  5b1  1
• Obtain b0=1/3, b1=0, 𝑦 = 1 3.
• Surrogate preserves the average
value of the data at data points.
Other metric fits
• Assuming other fits will lead to the form 𝑦 = 𝑏,
• For average error minimize
3eav | 0  b |  |1  b |  | 0  b | Obtain b=0.
• For maximal error minimize
• emax  max | 0  b |,|1  b |,| 0  b |  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
0.5
Three lines
Original 30-point curve fit
35
• With dense data
difference due to
metrics is small
y
yav
yrms
ymax
30
25
20
15
10
5
0
.
0
5
10
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
2.934
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20
25
30
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