Interpolation extrapolation and regression Interpolation is mathematically contrasted

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!

2 D 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)

Prediction variance • Linear regression model • Define • With some algebra • Standard error then

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))

Interpolation vs. Extrapolation • At origin . At 3 vertices . At (1, 1)

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)

Data at four vertices • Now • And • Error at vertices • At the origin minimum is • How can we reduce error without adding points?

Comparison Three points Four points