Introduction to Regression

Transcription

1 Introduction to Regression Chad M. Schafer Carnegie Mellon University Introduction to Regression p. 1/100

2 Outline General Concepts of Regression, Bias-Variance Tradeoff Linear Regression Nonparametric Procedures Cross Validation Local Polynomial Regression Confidence Bands Basis Methods: Splines Multiple Regression Introduction to Regression p. 2/100

3 Basic Concepts in Regression The Regression Problem: Observe (X 1,Y 1 ),...,(X n,y n ), estimate f(x) = E(Y X = x). Equivalently: Estimate f(x) with where E(ǫ i ) = 0. Y i = f(x i )+ǫ i, i = 1,2,...,n Simple estimators: f(x) = β0 + β1 x parametric f(x) = mean{y i : X i x h} nonparametric Introduction to Regression p. 3/100

4 Regression Parametric Regression: Y = β 0 +β 1 X +ǫ Y = β 0 +β 1 X 1 + +β d X d +ǫ Y = β 0 e β 1X 1 + β 2 X 2 +ǫ Nonparametric Regression: Y = f(x)+ǫ Y = f 1 (X 1 )+ +f d (X d )+ǫ Y = f(x 1,X 2 )+ǫ Introduction to Regression p. 4/100

5 Next... We will first quickly look at a simple example from astronomy Introduction to Regression p. 4/100

6 Example: Type Ia Supernovae Distance Modulus Redshift From Riess, et al. (2007), 182 Type Ia Supernovae Introduction to Regression p. 5/100

7 Example: Type Ia Supernovae Assumption: Observed pairs (z i,y i ) are realizations of Y i = f(z i )+σ i ǫ i, where the ǫ i are i.i.d. standard normal with mean zero. Error standard deviations σ i assumed known. Objective: Estimate f( ). Introduction to Regression p. 6/100

8 Example: Type Ia Supernovae ΛCDM, two parameter model: ( ) c(1+z) z du f(z) = 5log 10 H 0 Ωm (1+u) 3 +(1 Ω m ) Introduction to Regression p. 7/100

9 Example: Type Ia Supernovae Distance Modulus Redshift Estimate where H 0 = and Ω m = (the MLE) Introduction to Regression p. 8/100

10 Example: Type Ia Supernovae Distance Modulus Fourth order polynomial fit. Redshift Introduction to Regression p. 9/100

11 Example: Type Ia Supernovae Distance Modulus Fifth order polynomial fit. Redshift Introduction to Regression p. 10/100

12 Next... The previous two slides illustrate the fundamental challenge of constructing a regression model: How complex should it be? Introduction to Regression p. 10/100

13 The Bias Variance Tradeoff Must choose model to achieve balance between too simple high bias, i.e., E( f(x)) not close to f(x) xxxxxx precise, but not accurate too complex high variance, i.e., Var( f(x)) large xxxxxx accurate, but not precise This is the classic bias variance tradeoff. Could be called the accuracy precision tradeoff. Introduction to Regression p. 11/100

14 The Bias Variance Tradeoff Linear Regression Model: Fit models of the form f(x) = β0 + p β i g i (x) i=1 where g i ( ) are fixed, specified functions. Increasing p yields more complex model Introduction to Regression p. 12/100

15 Next... Linear regression models are the classic approach to regression. Here, the response is modelled as a linear combination of p selected predictors. Introduction to Regression p. 12/100

16 Linear Regression Assume that p Y = β 0 + β i g i (x)+ǫ i=1 where g i ( ) are specified functions, not estimated. Assume that E(ǫ) = 0, Var(ǫ) = σ 2 Often assume ǫ is Gaussian, but this is not essential Introduction to Regression p. 13/100

17 Linear Regression The Design Matrix: D = 1 g 1 (X 1 ) g 2 (X 1 ) g p (X 1 ) 1 g 1 (X 2 ) g 2 (X 2 ) g p (X 2 ) g 1 (X n ) g 2 (X n ) g p (X n ) Then, L = D(D T D) 1 D T is the projection matrix. Note that ν = trace(l) = p+1 Introduction to Regression p. 14/100

18 Linear Regression Let Y be a n-vector filled with Y 1,Y 2,...,Y n. The Least Squares Estimates of β: β = (D T D) 1 D T Y The Fitted Values: Ŷ = LY = D β The Residuals: ǫ = Y Ŷ = (I L)Y Introduction to Regression p. 15/100

19 Linear Regression The estimates of β given above minimize the sum of squared errors: n n ) 2 SSE = ǫ 2 i = (Y i Ŷ i i=1 i=1 If ǫ i are Gaussian, then β is Gaussian, leading to confidence intervals, hypothesis tests Introduction to Regression p. 16/100

20 Linear Regression in R Linear regression in R is done via the command lm() > holdlm = lm(y x) Default is to include intercept term. If want to exclude, use > holdlm = lm(y x - 1) Introduction to Regression p. 17/100

21 Linear Regression in R y x Introduction to Regression p. 18/100

22 Linear Regression in R Fitting the model Y = β 0 +β 1 x+ǫ The summary() shows the key output > holdlm = lm(y x) > summary(holdlm) < CUT SOME HERE > Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) * x e-10 *** < CUT SOME HERE > Note that β 1 = 2.07, the SE for β 1 is approximately Introduction to Regression p. 19/100

23 Linear Regression in R Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) * x e-10 *** If the ǫ are Gaussian, then the last column gives the p-value for the test of H 0 : β i = 0 versus H 1 : β i 0. Also, if the ǫ are Gaussian, β i ±2ŜE( β i ) is an approximate 95% confidence interval for β i. Introduction to Regression p. 20/100

24 Linear Regression in R Residuals Fitted Values Always a good idea to look at plot of residuals versus fitted values. There should be no apparent pattern. > plot(holdlm$residuals,holdlm$fitted.values) Introduction to Regression p. 21/100

25 Linear Regression in R Sample Quantiles Theoretical Quantiles Is it reasonable to assume ǫ are Gaussian? Check the normal probability plot of the residuals. > qqplot(holdlm$residuals) > qqline(holdlm$residuals) Introduction to Regression p. 22/100

26 Linear Regression in R Simple to include more terms in the model: > holdlm = lm(y x + z) Unfortunately, this does not work: > holdlm = lm(y x + xˆ2) Instead, create a new variable and use that: > x2 = xˆ2 > holdlm = lm(y x + x2) Introduction to Regression p. 23/100

27 The Correlation Coefficient A standard way of quantifying the strength of the linear relationship between two variables is via the (Pearson) correlation coefficient. Sample Version: r = 1 n 1 n i=1 X i Y i where X i and Y i are the standardized versions of X i and Y i, i.e., X i = X i X s X. Introduction to Regression p. 24/100

28 The Correlation Coefficient Note that 1 r 1. r = 1 indicates perfect linear, increasing relationship r = 1 indicates perfect linear, decreasing relationship Also, note that In R: use cor() β 1 = r ( sy s X ) Introduction to Regression p. 25/100

29 The Correlation Coefficient Examples of different scatter plots and values for r (Wikipedia) Introduction to Regression p. 26/100

30 The Coefficient of Determination A very popular, but overused, regression diagnostic is the coefficient of determination, denoted R 2 n i=1(y i Ŷ i ) 2 R 2 = 1 n i=1( Yi Y ) 2 The proportion of the variation explained by the model. Note that 0 R 2 1, and, R 2 = r 2. But: Large R 2 does not imply that the model is a good fit. Introduction to Regression p. 27/100

31 Anscombe s Quartet (1973) Each of these has the same value for the marginal means and variances, same r, same R 2, same β0 and β1. Introduction to Regression p. 28/100

32 Next... Now we will begin our discussion of nonparametric regression. Reconsider the SNe example from the start of the lecture. Introduction to Regression p. 28/100

33 Example: Type Ia Supernovae Distance Modulus Nonparametric estimate. Redshift Introduction to Regression p. 29/100

34 The Bias Variance Tradeoff Nonparametric Case: Every nonparametric procedure requires a smoothing parameter h. Consider the regression estimator based on local averaging: f(x) = mean{y i : X i x h}. Increasing h yields smoother estimate f Introduction to Regression p. 30/100

35 What is nonparametric? Data Estimate Assumptions In the parametric case, the influence of assumptions is fixed Introduction to Regression p. 31/100

36 What is nonparametric? Data Estimate Assumptions h In the nonparametric case, the influence of assumptions is controlled by h, where h = O ( 1 n 1/5 ) Introduction to Regression p. 32/100

37 The Bias Variance Tradeoff Can formalize this via appropriate measure of error in estimator Squared error loss: L(f(x), f(x)) = (f(x) f(x)) 2. Mean squared error MSE (risk) MSE = R(f(x), f(x)) = E(L(f(x), f(x))). Introduction to Regression p. 33/100

38 The Bias Variance Tradeoff The key result: where R(f(x), f(x)) = bias 2 x +variance x bias x = E( f(x)) f(x) variance x = Var( f(x)) Often written as MSE = BIAS 2 + VARIANCE. Introduction to Regression p. 34/100

39 The Bias Variance Tradeoff Mean Integrated Squared Error: MISE = = R(f(x), f(x))dx bias 2 x dx+ variance x dx Empirical version: 1 n n i=1 R(f(x i ), f(xi )). Introduction to Regression p. 35/100

40 The Bias Variance Tradeoff Risk Bias squared Variance Less Optimal smoothing More Introduction to Regression p. 36/100

41 The Bias Variance Tradeoff For nonparametric procedures, when x is one-dimensional, which is minimized at Hence, MISE c 1 h 4 + c 2 nh h = O MISE = O ( 1 n 1/5 ( 1 ) n 4/5 whereas, for parametric problems, when the model is correct, ( ) 1 MISE = O n ) Introduction to Regression p. 37/100

42 Next... Now we will begin our discussion of nonparametric regression procedures. But, these nonparametric procedures share a common structure with linear regression. Namely, they are all linear smoothers. Introduction to Regression p. 37/100

43 Linear Smoothers For a linear smoother: f(x) = i Y i l i (x) for some weights: l 1 (x),...,l n (x) The vector of weights depends on the target point x. Introduction to Regression p. 38/100

44 Linear Smoothers If f(x) = n i=1 l i(x)y i then Ŷ 1... Ŷn }{{} Ŷ f(x 1 )... f(xn) = l 1 (X 1 ) l 2 (X 1 ) ln(x 1 ) Y l 1 (X 2 ) l 2 (X 2 ) ln(x 2 ) Yn l 1 (Xn) l 2 (Xn) ln(xn) }{{}}{{} Y L Ŷ = L Y The effective degrees of freedom is: ν = trace(l) = n L ii. i=1 Can be interpreted as the number of parameters in the model. Recall that ν = p+1 for linear regression. Introduction to Regression p. 39/100

45 Linear Smoothers Consider the Extreme Cases: When l i (X j ) = 1/n for all i,j, and ν = 1. f(x j ) = Y for all j When l i (X j ) = δ ij for all i,j, and ν = n. f(x j ) = Y j Introduction to Regression p. 40/100

46 Next... To begin the discussion of nonparametric procedures, we start with a simple, but naive, approach: binning. Introduction to Regression p. 40/100

47 Binning Divide the x-axis into bins B 1,B 2,... of width h. f is a step function based on averaging the Y i s in each bin: for x B j : f(x) = mean { Y i : X i B j }. The (arbitrary) choice of the boundaries of the bins can affect inference, especially when h large. Introduction to Regression p. 41/100

48 Binning WMAP data, binned estimate with h = 15. Cl l Introduction to Regression p. 42/100

49 Binning WMAP data, binned estimate with h = 50. Cl l Introduction to Regression p. 43/100

50 Binning WMAP data, binned estimate with h = 75. Cl l Introduction to Regression p. 44/100

51 Next... A step beyond binning is to take local averages to estimate the regression function. Introduction to Regression p. 44/100

52 Local Averaging For each x let N x = [ x h 2, x+ h 2 ]. This is a moving window of length h, centered at x. Define } f(x) = mean {Y i : X i N x. This is like binning but removes the arbitrary boundaries. Introduction to Regression p. 45/100

53 Local Averaging WMAP data, local average estimate with h = 15. Cl Binned Local Averaging l Introduction to Regression p. 46/100

54 Local Averaging WMAP data, local average estimate with h = 50. Cl Binned Local Averaging l Introduction to Regression p. 47/100

55 Local Averaging WMAP data, local average estimate with h = 75. Cl Binned Local Averaging l Introduction to Regression p. 48/100

56 Next... Local averaging is a special case of kernel regression. Introduction to Regression p. 48/100

57 Kernel Regression The local average estimator can be written: n i=1 f(x) = Y i K ( ) x X i h n i=1 K( ) x X i h where K(x) = { 1 x < 1/2 0 otherwise. Can improve this by using a function K which is smoother. Introduction to Regression p. 49/100

58 Kernels A kernel is any smooth function K such that K(x) 0 and K(x)dx = 1, xk(x)dx = 0 and σk 2 x 2 K(x)dx > 0. Some commonly used kernels are the following: the boxcar kernel : K(x) = 1 I( x < 1), 2 the Gaussian kernel : K(x) = 1 e x2 /2, 2π the Epanechnikov kernel : K(x) = 3 4 (1 x2 )I( x < 1) the tricube kernel : K(x) = (1 x 3 ) 3 I( x < 1). Introduction to Regression p. 50/100

59 Kernels Introduction to Regression p. 51/100

60 Kernel Regression We can write this as f(x) = n i=1 Y i K ( ) x X i h n i=1 K( ) x X i h f(x) = i Y i l i (x) where l i (x) = K ( x X i ) h n i=1 K( x X i ). h Introduction to Regression p. 52/100

61 Kernel Regression WMAP data, kernel regression estimates, h = 15. Cl boxcar kernel Gaussian kernel tricube kernel l Introduction to Regression p. 53/100

62 Kernel Regression WMAP data, kernel regression estimates, h = 50. Cl boxcar kernel Gaussian kernel tricube kernel l Introduction to Regression p. 54/100

63 Kernel Regression WMAP data, kernel regression estimates, h = 75. Cl boxcar kernel Gaussian kernel tricube kernel l Introduction to Regression p. 55/100

64 Kernel Regression MISE h4 4 ( x 2 K(x)dx + σ2 K 2 (x)dx nh ) 2 ( 1 g(x) dx. f (x)+2f (x) g (x) g(x) where g(x) is the density for X. What is especially notable is the presence of the term 2f (x) g (x) g(x) = design bias. ) 2 dx Also, bias is large near the boundary. We can reduce these biases using local polynomials. Introduction to Regression p. 56/100

65 Kernel Regression WMAP data, h = 50. Note the boundary bias. Cl boxcar kernel Gaussian kernel tricube kernel l Introduction to Regression p. 57/100

66 Next... The deficiencies of kernel regression can be addressed by considering models which, on small scales, model the regression function as a low order polynomial. Introduction to Regression p. 57/100

67 Local Polynomial Regression Recall polynomial regression: f(x) = β0 + β1 x+ β2 x βp x p where β = ( β0,..., βp ) are obtained by least squares: n ( minimize Y i [β 0 +β 1 x+β 2 x 2 + +β p x p ] i=1 ) 2 Introduction to Regression p. 58/100

68 Local Polynomial Regression Local polynomial regression: approximate f(x) locally by a different polynomial for every x: f(u) β 0 (x)+β 1 (x)(u x)+β 2 (x)(u x) 2 + +β p (x)(u x) p for u near x. Estimate ( β0 (x),..., βp (x)) by local least squares: minimize n ( ) (Y i [β 0 (x)+β 1 (x)x+β 2 (x)x 2 + +β p (x)x p ]) 2 x Xi K h i=1 }{{} kernel f(x) = β0 (x) Introduction to Regression p. 59/100

69 Local Polynomial Regression Taking p = 0 yields the kernel regression estimator: n f n (x) = l i (x)y i l i (x) = i=1 K ( ) x x i h n j=1 K( x x j ). h Taking p = 1 yields the local linear estimator. This is the best, all-purpose smoother. Choice of Kernel K: not important Choice of bandwidth h: crucial Introduction to Regression p. 60/100

70 Local Polynomial Regression The local polynomial regression estimate is n f n (x) = l i (x)y i i=1 where l(x) T = (l 1 (x),...,l n (x)), l(x) T = e T 1 (XT x W xx x ) 1 Xx T W x, e 1 = (1,0,...,0) T and X x and W x are defined by Xx = 1 X 1 x 1 X 2 x Xn x (X 1 x) p p! (X 2 x) p p! (Xn x) p p! Wx = K ( x X1 h.. ) K ( x X n h.... ). Introduction to Regression p. 61/100

71 Local Polynomial Regression Note that f(x) = n i=1 l i(x)y i is a linear smoother. Define Ŷ = (Ŷ 1,...,Ŷ n ) where Ŷ i = f(xi ). Then Ŷ = LY where L is the smoothing matrix: L = l 1 (X 1 ) l 2 (X 1 ) ln(x 1 ) l 1 (X 2 ) l 2 (X 2 ) ln(x 2 ).... l 1 (Xn) l 2 (Xn) ln(xn)..... The effective degrees of freedom is: ν = trace(l) = n i=1 L ii. Introduction to Regression p. 62/100

72 Theoretical Aside Why local linear (p = 1) is better than kernel (p = 0). Both have (approximate) variance σ 2 (x) g(x)nh The kernel estimator has bias h 2 ( 1 2 f (x)+ f (x)g (x) g(x) K 2 (u)du ) u 2 K(u)du whereas the local linear estimator has asymptotic bias h f (x) u 2 K(u)du The local linear estimator is free from design bias. At the boundary points, the kernel estimator has asymptotic bias of O(h) while the local linear estimator has bias O(h 2 ). Introduction to Regression p. 63/100

73 Next... Parametric proceudres force the choice p, the number of predictors. Nonparametric procedures force the choice of the smoothing parameter h. Next we will discuss how this can be selected in a data-driven manner via cross-validation. Introduction to Regression p. 63/100

74 Choosingpandh Estimate the risk 1 n n E( f(xi ) f(x i )) 2 i=1 with the leave-one-out cross-validation score: CV = 1 n (Y i f( i) (X i )) 2 n i=1 where f( i) is the estimator obtained by omitting the i th pair (X i,y i ). Introduction to Regression p. 64/100

75 Choosingpandh Amazing shortcut formula: ( CV = 1 n n i=1 ) 2 Y i fn (x i ). 1 L ii An commonly used approximation is GCV (generalized cross-validation): GCV = 1 n ( 1 ν n n ) 2 (Y i f(xi )) 2 i=1 ν = trace(l). Introduction to Regression p. 65/100

76 Next... Next, we will go over a bit of how to perform local polynomial regression in R. Introduction to Regression p. 65/100

77 Using locfit() Need to include the locfit library: > install.packages("locfit") > library(locfit) > result = locfit(y x, alpha=c(0, 1.5), deg=1) y and x are vectors the second argument to alpha gives the bandwidth (h) the first argument to alpha specifies the nearest neighbor fraction, an alternative to the bandwidth fitted(result) gives the fitted values, f(xi ) residuals(result) gives the residuals, Y i f(xi ) Introduction to Regression p. 66/100

78 Using locfit() degree=1, n=1000, sigma=0.1 y x(1 x)sin((2.1π) (x + 0.2)) estimate using h = estimate using gcv optimal h = x Introduction to Regression p. 67/100

79 Using locfit() degree=1, n=1000, sigma=0.1 y x(1 x)sin((2.1π) (x + 0.2)) estimate using h = 0.1 estimate using gcv optimal h = x Introduction to Regression p. 68/100

80 Using locfit() degree=1, n=1000, sigma=0.5 y (x + 5exp( 4x 2 )) estimate using h = 0.05 estimate using gcv optimal h = x Introduction to Regression p. 69/100

81 Using locfit() degree=1, n=1000, sigma=0.5 y (x + 5exp( 4x 2 )) estimate using h = 0.5 estimate using gcv optimal h = x Introduction to Regression p. 70/100

82 Example WMAP data, local linear fit, h = 15. y estimate using h = 15 estimate using gcv optimal h = x Introduction to Regression p. 71/100

83 Example WMAP data, local linear fit, h = 125. y estimate using h = 125 estimate using gcv optimal h = x Introduction to Regression p. 72/100

84 Next... We should quantify the amount of error in our estimator for the regression function. Next we will look at how this can be done using local polynomial regression. Introduction to Regression p. 72/100

85 Variance Estimation Let where σ 2 = n i=1 (Y i f(xi )) 2 n 2ν + ν ν = tr(l), ν = tr(l T L) = n l(x i ) 2. If f is sufficiently smooth, then σ 2 is a consistent estimator of σ 2. i=1 Introduction to Regression p. 73/100

86 Variance Estimation For the WMAP data, using local linear fit. > wmap = read.table("wmap.dat",header=t) > opth = 63.0 > locfitwmap = locfit(wmap$cl[1:700] wmap$ell[1:700], alpha=c(0,opth),deg=1) > nu = as.numeric(locfitwmap$dp[6]) > nutilde = as.numeric(locfitwmap$dp[7]) > sigmasqrhat = sum(residuals(locfitwmap)ˆ2)/(700-2*nu+nutilde) > sigmasqrhat [1] But, does not seem reasonable to assume homoscedasticity... Introduction to Regression p. 74/100

87 Variance Estimation Allow σ to be a function of x: Y i = f(x i )+σ(x i )ǫ i. Let Z i = log(y i f(x i )) 2 and δ i = logǫ 2 i. Then, Z i = log(σ 2 (x i ))+δ i. This suggests estimating logσ 2 (x) by regressing the log squared residuals on x. Introduction to Regression p. 75/100

88 Variance Estimation 1. Estimate f(x) with any nonparametric method to get an estimate fn (x). 2. Define Z i = log(y i fn (x i )) Regress the Z i s on the x i s (again using any nonparametric method) to get an estimate q(x) of logσ 2 (x) and let σ 2 (x) = e q(x). Introduction to Regression p. 76/100

89 Example WMAP data, log squared residuals, local linear fit, h = log residual squared l Introduction to Regression p. 77/100

90 Example With local linear fit, h = 130, chosen via GCV log residual squared l Introduction to Regression p. 78/100

91 Example Estimating σ(x): sigma hat l Introduction to Regression p. 79/100

92 Confidence Bands Recall that f(x) = i Y i l i (x) so Var( f(x)) = σ 2 (X i )l 2 i(x). An approximate 1 α confidence interval for f(x) is i f(x)±z α/2 i σ 2 (X i )l 2 i (x). When σ(x) is smooth, we can approximate σ 2 (X i )l 2 i (x) σ(x) l(x). i Introduction to Regression p. 80/100

93 Confidence Bands Two caveats: 1. f is biased so this is realy an interval for E( f(x)). Result: bands can miss sharp peaks in the function. 2. Pointwise coverage does not imply simultaneous coverage for all x. Solution for 2 is to replace z α/2 with a larger number (Sun and Loader 1994). locfit() does this for you. Introduction to Regression p. 81/100

94 More on locfit() > diaghat = predict.locfit(locfitwmap,where="data",what="infl") > normell = predict.locfit(locfitwmap,where="data",what="vari") diaghat will be L ii, i = 1,2,...,n. normell will be l(x i ), i = 1,2,...,n The Sun and Loader replacement for z α/2 is found using kappa0(locfitwmap)$crit.val. Introduction to Regression p. 82/100

95 Example 95% pointwise confidence bands: Cl estimate 95% pointwise band assuming constant variance 95% pointwise band using nonconstant variance l Introduction to Regression p. 83/100

96 Example 95% simultaneous confidence bands: Cl estimate 95% simultaneous band assuming constant variance 95% simultaneous band using nonconstant variance l Introduction to Regression p. 84/100

97 Next... We discuss the use of regression splines. These are a well-motivated choice of basis for constructing regression functions. Introduction to Regression p. 84/100

98 Basis Methods Idea: expand f as f(x) = j β j ψ j (x) where ψ 1 (x),ψ 2 (x),... are specially chosen, known functions. Then estimate β j and set f(x) = j β j ψ j (x). Here we consider a popular version, splines Introduction to Regression p. 85/100

99 Splines and Penalization Define fn to be the function that minimizes M(λ) = (Y i fn (x i )) 2 +λ (f (x)) 2 dx. i λ = 0 = fn (X i ) = Y i (no smoothing) λ = = fn (x) = β0 + β1 x (linear) 0 < λ < = fn (x) = cubic spline with knots at X i. A cubic spline is a continuous function f such that 1. f is a cubic polynomial between the X i s 2. f has continuous first and second derivatives at the X i s. Introduction to Regression p. 86/100

100 Basis For Splines Define (z) + = max{z,0}, N = n+4, ψ 1 (x) = 1 ψ 2 (x) = x ψ 3 (x) = x 2 ψ 4 (x) = x 3 ψ 5 (x) = (x X 1 ) 3 + ψ 6 (x) = (x X 2 ) 3 + ψ N (x) = (x X n ) 3 +. These functions form a basis for the splines: we can write f(x) = N j=1 β j ψ j (x). (For numerical calculations it is actually more efficient to use other spline bases.) We can thus write (1) f n (x) = N β j ψ j (x), j=1 Introduction to Regression p. 87/100

101 Basis For Splines We can now rewrite the minimization as follows: minimize : (Y Ψβ) T (Y Ψβ)+λβ T Ωβ where Ψ ij = ψ j (X i ) and Ω jk = ψ j (x)ψ k (x)dx. The value of β that minimizes this is β = (Ψ T Ψ+λΩ) 1 Ψ T Y. The smoothing spline fn (x) is a linear smoother, that is, there exist weights l(x) such that fn (x) = n i=1 Y il i (x). Introduction to Regression p. 88/100

102 Basis For Splines Basis functions with 5 knots. Psi x Introduction to Regression p. 89/100

103 Basis For Splines Same span as previous slide, the B-spline basis, 5 knots: x Introduction to Regression p. 90/100

104 Smoothing Splines in R > smosplresult = smooth.spline(x,y, cv=false, all.knots=true) If cv=true, then cross-validiation used to choose λ; if cv=false, gcv is used If all.knots=true, then knots are placed at all data points; if all.knots=false, then set nknots to specify the number of knots to be used. Using fewer knots eases the computational cost. predict(smosplresult) gives fitted values. Introduction to Regression p. 91/100

105 Example WMAP data, λ chosen using GCV. Cl l Introduction to Regression p. 92/100

106 Next... Finally, we address the challenges of fitting regression models when the predictor x is of large dimension. Introduction to Regression p. 92/100

107 Recall: The Bias Variance Tradeoff For nonparametric procedures, when x is one-dimensional, which is minimized at Hence, MISE c 1 h 4 + c 2 nh h = O MISE = O ( 1 n 1/5 ( 1 ) n 4/5 whereas, for parametric problems, when the model is correct, ( ) 1 MISE = O n ) Introduction to Regression p. 93/100

108 The Curse of Dimensionality For nonparametric procedures, when x is d-dimensional, which is minimized at Hence, MISE c 1 h 4 + c 2 nh d h = O ( MISE = O 1 n 1/(4+d) ( 1 ) n 4/(4+d) whereas, for parametric problems, when the model is correct, still, ( ) 1 MISE = O n ) Introduction to Regression p. 94/100

109 Multiple Regression Y = f(x 1,X 2,...,X d )+ǫ The curse of dimensionality: Optimal rate of convergence for d = 1 is n 4/5. In d dimensions the optimal rate of convergence is is n 4/(4+d). Thus, the sample size m required for a d-dimensional problem to have the same accuracy as a sample size n in a one-dimensional problem is m n cd where c = (4+d)/(5d) > 0. To maintain a given degree of accuracy of an estimator, the sample size must increase exponentially with the dimension d. Put another way, confidence bands get very large as the dimension d increases. Introduction to Regression p. 95/100

110 Multiple Local Linear Given a nonsingular positive definite d d bandwidth matrix H, we define K H (x) = 1 H 1/2K(H 1/2 x). Often, one scales each covariate to have the same mean and variance and then we use the kernel h d K( x /h) where K is any one-dimensional kernel. Then there is a single bandwidth parameter h. Introduction to Regression p. 96/100

111 Multiple Local Linear At a target value x = (x 1,...,x d ) T, the local sum of squares is given by n ( d ) 2 w i (x) Y i a 0 a j (x ij x j ) i=1 j=1 where w i (x) = K( x i x /h). The estimator is fn (x) = â 0 â = (â 0,...,â d ) T is the value of a = (a 0,...,a d ) T that minimizes the weighted sums of squares. Introduction to Regression p. 97/100

112 Multiple Local Linear The solution â is â = (X T xw x X x ) 1 X T xw x Y where X x = 1 (x 11 x 1 ) (x 1d x d ) 1 (x 21 x 1 ) (x 2d x d ) (x n1 x 1 ) (x nd x d ) and W x is the diagonal matrix whose (i,i) element is w i (x). Introduction to Regression p. 98/100

113 A Compromise: Additive Models Y = α+ d j=1 f j (x j )+ǫ Usually take α = Y. Then estimate the f j s by backfitting. 1. set α = Y, f1 = fd = Iterate until convergence: for j = 1,...,d: Compute Ỹ i = Y i α k j f k (X i ), i = 1,...,n. Apply a smoother to Ỹ i on X j to obtain fj. Set fj (x) equal to fj (x) n 1 n i=1 f j (x i ). The last step ensures that i f j (X i ) = 0 (identifiability). Introduction to Regression p. 99/100

114 To Sum Up... Classic linear regression is computationally and theoretically simple Nonparametric procedures for regression offer increased flexibility Careful consideration of MISE suggests use of local polynomial fits Smoothing parameters can be chosen objectively High-dimensional data present a challenge Introduction to Regression p. 100/100