Interpolation, extrapolation and regression Interpolation is mathematically contrasted to regression...
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- Slide 2
- Interpolation, extrapolation and regression Interpolation is
mathematically contrasted to regression or least-squares fit As
important is the contrast between interpolation and extrapolation
Extrapolation occurs when we are outside the convex hull of the
data points For high dimensional spaces we must have
extrapolation!
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- 2D example x = rand(20,1); y = rand(20,1); plot(x,y, '.'); k =
convhull(x,y) hold on, plot(x(k), y(k), '-r'), hold off Number the
points in the Figure and give a couple of Alternative sets of
alphas Approximately for the point (0.4,0.4)
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- Prediction variance Linear regression model Define then With
some algebra Standard error
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- Example 4.2.1 For a linear polynomial RS y=b 1 +b 2 x 1 +b 3 x
2 find the prediction variance in the region (a) For data at three
vertices (omitting (1,1))
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- Interpolation vs. Extrapolation At origin. At 3 vertices. At
(1,1)
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- Standard error contours Minimum error What is special about
this point x=[-1 -1 1]; y=[-1 1 -1]; [X,Y]=meshgrid(-1:0.1:1,
-1:0.1:1); Z=sqrt(0.5*(1+X+Y+X.^2+Y.^2+X.*Y));
v=linspace(0.6,1.8,7) scatter(x,y,'filled'); grid on; hold on
[C,h]=contour(X,Y,Z,v); clabel(C,h)
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- Data at four vertices Now And Error at vertices At the origin
minimum is How can we reduce error without adding points?
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- Comparison Three pointsFour points