interpolation, extrapolation and regression interpolation is mathematically contrasted to regression...

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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! 1 1 1 1 , 1, 0, n n i i i i i i x x

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  • Slide 1
  • 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!
  • Slide 3
  • 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)
  • Slide 4
  • Prediction variance Linear regression model Define then With some algebra Standard error
  • Slide 5
  • 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))
  • Slide 6
  • Interpolation vs. Extrapolation At origin. At 3 vertices. At (1,1)
  • Slide 7
  • 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)
  • Slide 8
  • Data at four vertices Now And Error at vertices At the origin minimum is How can we reduce error without adding points?
  • Slide 9
  • Comparison Three pointsFour points