Sampling plans for linear regression • Given a domain, we can reduce the prediction error by good choice of the sampling points. • The choice of sampling locations is called “design of experiments” or DOE. • In this lecture we will consider DOEs for linear regression using linear and quadratic polynomials and where errors are due to noise in the data. • With a given number of points the best DOE is one that will reduce the prediction variance (reviewed in next few slides). • The simplest DOE is full factorial design where we sample each variable (factor) at a fixed number of values (levels) • Example: with four factors and three levels each we will sample 81 points • Full factorial design is not practical except for low dimensions
Linear Regression • Surrogate is linear combination of given shape functions • For linear approximation • Difference (error) between data and surrogate • Minimize square error • Differentiate to obtain
Model based error for linear regression • The common assumptions for linear regression • The true function is described by the functional form of the surrogate. • The data is contaminated with normally distributed error with the same standard deviation at every point. • The errors at different points are not correlated. • Under these assumptions, the noise standard deviation (called standard error) is estimated as • is used as estimate of the prediction error.
Prediction variance • Linear regression model • Define then • With some algebra • Standard error
Prediction variance for full factorial design • Recall that standard error (square root of prediction variance is • For full factorial design the domain is normally a box. • Cheapest full factorial design: two levels (not good for quadratic polynomials). • For a linear polynomial standard error is then • Maximum error at vertices • What does the ratio in the square root represent?
Designs for linear polynomials • Traditionally use only two levels. • Orthogonal design when XTX is diagonal. • Full factorial design is orthogonal, not so easy to produce other orthogonal designs with less points. • It is beneficial to place the points at the edges of the design domain. • Stability: Small variation of prediction variance in domain is also desirable property.
Example • Compare prediction variance for an orthogonal design based on equilateral triangle to right triangle (both are saturated) • Linear polynomial y=b1+b2x1+b3x2 • For right triangle obtain
Comparison • Prediction variances for equilateral triangle • The maximum variance at (1,1) is three times larger than the lowest one. • For right triangle Maximum variance (3) is six times the lowest, and triple that of the equilateral triangle. • A fairer comparison is when we restrict triangle to lie inside box • The prediction variance is /3 • Maximum prediction variance (1.5) and stability (ratio of 4.5) are still better than for the right triangle, but by less.
Quadratic Polynomial • A quadratic polynomial has (n+1)(n+2)/2 coefficients, so we need at least that many points. • Need at least three different values of each variable. • Simplest DOE is three-level, full factorial design • Impractical for n>5 • Also unreasonable ratio between number of points and number of coefficients • For example, for n=8 we get 6561 samples for 45 coefficients. • My rule of thumb is that you want twice as many points as coefficients
Central Composite Design • Includes 2n vertices, 2n face points plus ncrepetitions of central point • Can choose α so to • achieve spherical design • achieve rotatibility (prediction variance is spherical) • Stay in box (face centered) FCCCD • Still impractical for n>8
Repeated observations at origin • Unlike linear designs, prediction variance is high at origin. • Repetition at origin decreases variance there and improves stability. • What other rationale for choosing the origin for repetition? • Repetition also gives an independent measure of magnitude of noise. • Can be used also for lack-of-fit tests.
Without repetition (9 points) • Contours of prediction variance for spherical CCD design. • How come it is rotatable?
Center repeated 5 times (13 points) . • With five repetitions we reduce the maximum prediction variance and greatly improve the uniformity. • Five points is the optimum for uniformity.
Variance optimal designs • Full factorial and CCD are not flexible in number of points • Standard error • A key to most optimal DOE methods is moment matrix • A good design of experiments will maximize the terms in this matrix, especially the diagonal elements. • D-optimal designs maximize determinant of moment matrix. • Determinant is inversely proportional to square of volume of confidence region on coefficients.
Example • Given the model y=b1x1+b2x2, and the two data points (0,0) and (1,0), find the optimum third data point (p,q) in the unit square. • We have • So that the third point is (p,1), for any value of p • Finding D-optimal design in higher dimensions is a difficult optimization problem often solved heuristically
Matlab example >> ny=6;nbeta=6; >> [dce,x]=cordexch(2,ny,'quadratic'); >> dce' 1 1 -1 -1 0 1 -1 1 1 -1 -1 0 scatter(dce(:,1),dce(:,2),200,'filled') >> det(x'*x)/ny^nbeta ans = 0.0055 With 12 points: >> ny=12; >> [dce,x]=cordexch(2,ny,'quadratic'); >> dce' -1 1 -1 0 1 0 1 -1 1 0 -1 1 1 -1 -1 -1 1 1 -1 -1 0 0 0 1 scatter(dce(:,1),dce(:,2),200,'filled') >> det(x'*x)/ny^nbeta ans =0.0102
Other criteria • A-optimal minimizes trace of the inverse of the moment matrix. • This minimizes the sum of the variances of the coefficients. • G-optimality minimizes the maximum of the prediction variance.
Example • For the previous example, find the A-optimal design • Minimum at (0,1), so this point is both A-optimal and D-optimal.
Problems • Create a 13-point D-optimal design in two dimensional space and compare its prediction variance to that of the CCD design shown on Slide 13. • Generate noisy data for the function y=(x+y)2 and fit using the two designs and compare the accuracy of the coefficients.