Nonparametric Regression. Badr Missaoui

Transcription

1 Badr Missaoui

2 Outline Kernel and local polynomial regression. Penalized regression.

3 We are given n pairs of observations (X 1, Y 1 ),...,(X n, Y n ) where Y i = r(x i ) + ε i, i = 1,..., n and r(x) = E(Y X = x). If X i are deterministic, we assume that ε i N(0, σ 2 ). If X i are random variables, we assume that ε i N(0, σ 2 ) are independent of X i. In the absence of any hypothesis on the function r, we are in the nonparametric framework.

4 The simplest nonparametric estimator is the regressogram. Suppose that X i are in the interval [a, b] and denote the bins byb 1,..., B m. Let k j be the number of observations in bin B j. Define ˆr n (x) = 1 Y i = k Ȳj j i:x i B j We can rewrite the estimator as for x B j ˆr n (x) = n l i (x)y i i=1 where l i (x) = 1/k j if X j B j, and l i (x) = 0 otherwise. In other words, the estimate ˆr n is a step function obtained by averaging the Y i over each bin

5 Recall from our discussion of model selection that R(h) = E(Y ˆr n (X)) 2 = σ 2 + E(r(X) ˆr n (X)) 2 = σ 2 + MSE. We can write the MSE as MSE = bias 2 (x)p(x)dx + where and bias(x) = E(ˆr n (x) r(x)) var(x) = Variance(ˆr n (x)) var(x)p(x)dx When the data are oversmoothed, the bias term is large and the variance is small. When the data are undersmoothed, the opposite is true. This is called the bias-variance tradeoff. Minimizing risk corresponds to balancing bias and variance.

6 Ideally, we would like to choose h to minimize R(h). R(h) depends on the unknown function r(x). We use instead the average residual sums of squares to estimate R(h). 1 n n (Y i ˆr n (X i )) 2 i=1 We will estimate the risk using the leave-one-out cross validation which defined by CV = ˆR(h) = 1 n n (Y i ˆr n( i) (X i )) 2 i=1 where ˆr n( i) is the estimator obtained by omitting the i th pair (X i, Y i )

7 There is a shortcut formula for computing R just like in linear regression. Let ˆr n be a linear smoother. Then the leave-one-out cross-validation R(h) can be written as n ( Yi ˆr n( i) (X i ) ) 2 R(h) = 1 n i=1 1 L ii where L ii = l i (X i ) An alternative is to use generalized cross validation GCV (h) = 1 n n i=1 ( ) Yi ˆr n (X i ) 2 1 n 1 n i=1 L ii

8 Kernel regression Kernel refers to any smooth function K such that K (x) 0 and K (x)dx = 1, xk (x)dx = 0 and σ 2 K = x 2 K (x)dx > 0 Some commonly used kernels : the boxcar kernel : K (x) = 1 2 I x 1 the Gaussian kernel : K (x) = 1 e x 2/2 2π

9 Kernel regression Let h > 0 be a positive number (bandwidth). The Nadaraya-Watson kernel estimator is defined by where ˆr n (x) = l i (x) = n l i (x)y i i=1 ( ) K x Xi h ( n j=1 K x Xi h In R, we suggest using the loess command or the locfit library out = loess(y x,span=.25,degree=0) lines(x,fitted(out)) ) out = locfit(y x,deg=0,alpha=c(0,h))

10 Kernel regression To do the cross-validation, create a vector bandwidths h = (h 1,..., h k ) h = c(... put your values here... ) k = length(h) zero = rep(0,k) H = cbind(zero,h) out = gcvplot(yx,deg=0,alpha=h) plot(out$df,out$values)

11 Kernel regression The choice of the kernel K is not too important. What does matter much is the choice of the bandwidth which controls the amount of smoothing. In general the bandwidth depends on the sample size (h n ). We assume that f is the density of x 1,..., x n. The risk of the Nadaraya-Watson kernel estimator is R(h n ) = h4 n 4 ( + σ2 K 2 (x)dx nh n as h n 0 and nh n ) 2 ( x 2 K (x)dx r (x) + 2r (x) f ) (x) 2 dx f (x) 1 f (x) dx + o(n 1 h n ) + o(h 4 n)

12 Kernel regression What is especially notable is the presence of the term 2r (x) f (x) f (x) This means that the bias is sensitive to the position s of the X i s. If we differentiate R with respect to h n and set the result to 0, we find that the optimal h is h = ( 1 n ) 1/5 σ 2 K 2 (x)dx 1 f (x) dx ( x 2 K (x)dx ) 2 ( r (x) + 2r (x) f (x) f (x) ) 2 dx 1/5

13 Kernel regression Thus, h = n 1/5. The risk R hn decreases at rate (n 4/5 ) In practice we cannot not use the formula of h mentioned above since it depends on the unknown function r. Instead, we use leave-one-out cross-validation.

14 Local Polynomials Kernel estimators suffer from design bias. These problem can be alleviated by using a local polynomial regression. The idea is to approximate a smooth regression function r(u) in the target value x by the polynomial : where r(u) P x (u; a) P x (u; a) = a 0 + a 1 (u x) + a 2 2! (u x)2 ) a p p! (u x)p. We estimate a = (a 0,..., a p ) T by minimizing n w i (x)(y i P x (u; a)) 2 i=1

15 Local Polynomials To find â(x), it is helpful to re-express the problem in matrix notation. Let 1 x 1 x 1 x X x = 2 x... 1 x n x and let W x = diag{w i (x)} i we can write then the problem as (x 1 x) p p! (x 2 x) p p!. (x n x) p p! (Y X x a) T W x (Y X x a) Minimizing this gives the weighted least squares estimator a(x) = (X T x W x X x ) 1 X T x W x Y

16 Local Polynomials The local polynomial regression estimate is ˆr n (x) = n l i (x)y i i=1 where l(x) T = (l 1 (x),..., l n (x)), e 1 = (1, 0,..., 0) T. l(x) T = e T 1 (X T x W x X x ) 1 X T x W x The R code is the same except we use deg=1 for local linear, deg=2 for quadratic loess(y locfit(y x,deg=1,span=h) x,deg = 1,alpha=c(0,h))

17 Local Polynomials Let Y i = r(x i ) + σ(x i )ε i for i = 1,..., n and a < X i < b. Assume that X 1,..., X n are a sample from a distribution with density f and that (i) f (x) > 0, (ii) f, r and σ 2 are continuous in a neighborhood of x, and (iii) h n 0 and nh n 0. The local linear estimator has a variance σ 2 (x) K 2 (x)dx + o(1/nh n ) f (x)nh n and has an asymptotic bias hn r (x) x 2 K (x)dx + o(h 2 ) Thus, the local linear estimator is free from design bias.

18 Penalized Regression Consider polynomial regression p Y = β j x j + ε i=0 or n ˆr(x) = ˆβ j x j j=0 We have the design matrix 1 x 1 x p 1 1 x 2 x p 2 X = x n xn p

19 Penalized Regression Least squares minimizes (Y Xβ) T (Y Xβ), which implies ˆβ = (X T X) 1 X T Y The ridge regression aims to minimize (Y Xβ) T (Y Xβ) + λβ T β then β = (X T X + λi) 1 X T Y

20 Penalized Regression An alternative way is to minimize the penalized sums of squares M(λ) = (Y i ˆr n (X i )) 2 + λj(r) i where J(r) = (r (x)) 2 dx is the roughness penalty. This penalty leads to a solution that favors smoother functions. The parameter λ controls the amount of smoothness.

21 Penalized Regression The most commonly used splines are piecewise cubic splines. Let ξ 1 <... < ξ k be a set of ordered point -called knotscontained in some interval (a, b). A cubic spline is a continuous function r such that (i) r is a cubic polynomial over (ξ 1, ξ 2 ),... and r has continuous first and second derivatives at the knots. A spline that is linear beyond the boundary knots is called a natural spline. Theorem The function ˆr n (x) that minimizes M(λ) is a natural cubic spline with knots the data points. The estimator ˆr n is called a smoothing spline.

22 Penalized Regression The theorem above does not give you an explicit form for ˆr n We will construct a basis for for the set of splines (cubic B-spline). B i,0 (t) = 1 t [ti,t i+1 ] B i,d (t) = t t i B i,d 1 (t) + t i+d+1 t B i+1,d 1 (t) t i+d t i t i+d+1 t i+1 Without the penalty, the B-spline basis interpolate the data and therefore provide a perfect fit to the data.

23 Penalized Regression Now, we can write ˆr n = n ˆβ j B j (x) j=1 where B j (x) are the basis vectors for the B-splines. We follow the pattern of polynomial regression B 1 (x 1 ) B n (x 1 ) B 1 (x 2 ) B n (x 2 ) B =... B 1 (x n ) B n (x n )

24 Penalized Regression We can rewrite the problem as follows : argmin β (Y Bβ) T (Y Bβ) + λβ T Ωβ where B ij = [B j (X i )] ij and Ω jk = B j (x)b k (x)dx. The solution is and where β = (B T B + λω) 1 B T Y Ŷ = LY L = B(B T B + λω) 1 B T We define the effective degree of freedom by df = trace(l), and we choose λ using the GCV or CV. In R, out=smooth.spline(x,y,df=10,cv=true) lines(x,out$y)

25 Penalized Regression In more general case, we will assume that r admits an expansion series wrt to a orthonormal basis ( i ) i such that r(x) = β j φ j (x) j=1 We will approximate r by r J (x) = J β j φ j (x) j=1 The number of terms J will be our smoothing parameter. Our estimate is ˆr(x) = J j=1 ˆβ j φ j (x).

26 Penalized Regression The estimate ˆβ is ˆβ = (U T U) 1 U T Y where U = [φ j (X i )] ij and Ŷ = SY where S = U(UT U) 1 U T We can choose J by cross validation. Note that trace(s) = J so the GCV takes the simple form GCV (J) = SSE n 1 (1 J/n) 2

27 Penalized Regression : Variance estimation Theorem Let ˆr n be a linear smoother. Let ˆσ 2 = n i=1 (Y i ˆr(X i )) 2 n 2ν + ˆν where ν = tr(l), ˆν = tr(l T L). If r are sufficiently smooth, ν = o(n) and ˆν = o(n) then ˆσ 2 is a consistent estimator of σ 2.