Predictor Selection Algorithm for Bayesian Lasso

Transcription

1 Predictor Selection Algorithm for Baesian Lasso Quan Zhang Ma 16, Introduction The Lasso [1] is a method in regression model for coefficients shrinkage and model selection. It is often used in the linear regression model = µ1 n + Xβ + ε where is the response vector with the length of n, µ is the overall mean, X is the n p standardized design matri, β = (β 1,, β p ), and ε is the n 1 vector of iid normal errors with zero mean and unknown variance σ 2. The Lasso estimates, ˆβ, minimizes the sum of the squared residuals subject to β t. It can be described as argmin β p (ỹ Xβ) (ỹ Xβ) + λ β i i=1 where ỹ = ȳ1 n, and λ 0 is the tuning parameter relating to the bound t. It is indicated that ˆβ can be interpreted as a Baesian mode estimate when the parameters β i s have iid double eponential priors, i.e., π(β) = The double eponential distribution can be writen as a scale miture of normals: p i=1 λ 2 e λ β i. (1) α 1 2 e a z = e z2 /(2s) a2 s/2 0 2πs 2 e a2 ds, a > 0. (2) The detailed hierarchical model formulation and the Gibbs sampler implementation can been seen in Park and Casella s Baesian Lasso paper [2]. Note that λ can be not onl pre-specified, but also estimated if a hperprior is placed. A slightl more complicated Baesian hierarchical model and its Gibbs sampler implementation will be given in Section 4. However, as Park and Cassella said [2], this method cannot automaticall perform predictor selection, in other words, it cannot force some of the regression coefficients to zero. We propose an algorithm to automaticall select predictors in the process of the Gibbs sampler. We applied this method to ordinar linear regression problem with variable selection. Another important application is the penalized spline using the l 1 penalt. Using the Baesian Lasso with predictor selection, we can not onl fit the spline model, but also automaticall select knots. Division of Biostatistics, School of Public Health, Universit of Minnesota. 1

2 2 Algorithm for predictor selection As described above, we have p predictors, X 1,, X p and denote the corresponding regression parameters of the predictors as β 1,, β p. The main idea of the algorithm is to throw out one predictor whose coefficient is the most insignificant after some iterations of the Gibbs sampler till all the remaining predictors coefficients are significant based on a specific Baesian credible interval (BCI). The algorithm can be fulfilled through the following steps: 1. Run N 1 iterations of the Gibbs sampler, where N 1 is a large number, sa 10, 000. Note that the N 1 iterations are just burn-in and we found it neccessar especiall when p is large. 2. Run another M iterations where M is relativel small, like 1, 000. For a specific p, for eample, 0.05, if the (1 p ) Baesian credible interval based on the last M iterations of all the p β s eclude zero, then continue running the Gibbs sampler till convergence. Otherwise, go to Step Define N1+M k drop = argma k min( g=n 1 +1 I(β (g) k > 0), N 1 +M g=n 1 +1 I(β (g) k < 0)) where β (g) k is the sample value of β k in the gth iteration of the Gibbs sampler for k = 1,, p. Throw out X kdrop. That is to sa, we throw out the β that is most centered to 0 based on the last M iterations of the Gibbs sampler. 4. Replace N 1 b N 1 + M and p b p 1. Let k = 1,, k drop 1, k drop + 1,, p and go to step 2. This algorithm will select ver few predictors if X i s are highl correlated, even though there are a large number of predictors. We can define the number of predictors to be selected in the algorithm rather than defining p. This ma result in some of the selected predictors coefficient of insignificance. 3 Linear regression We applied our algorithm to a linear regression problem on the diabetes data [3], which has n = 442 and p = 10. The predictors are 10 baseline variables, age, se, bod mass inde, average blood pressure, and si blood serum measurements were obtained for each of 442 diabetes patients, and the response of interest is a quantitative measure of disease progression one ear after baseline. We performed a Baesian Lasso with the predictor selection algorithm described in Section 2, setting p = 0.05 and placing a hperprior on λ to estimate it simultaneousl. Then we perform the Baesian Lasso without selecting predictors and the ordinar Lasso both of which have the tuning parameter matching the value of λ estimated b the algorithm. The results are summarized in Table 1 along with the ordinar Lasso whose λ is chosen b n-fold cross-validation. Comparing the Baesian Lasso using the algorithm with the one not using the algorithm, the algorithm threw out all predictors that have an insignificant coefficient based on the 95% credible interval, ecept one predictor, (7) hdl. hdl has a marginall significant coefficient in the Baesian Lasso without the algorithm, but has a highl significant coefficient if using the algorithm. If we compare the Baesian Lasso with the algorithm and the ordinar Lasso with λ matched, we find the algorithm tends to select fewer predictors: (5) tc and (10) glu are non-zero in the ordinar Lasso but are thrown out b the algorithm. If we look at the last two columns, we find the Lasso with λ matched shinks β more than that b the cross-validation. Furthermore, we performed the Baesian Lasso without the predictor selection algorithm and estimate λ simultaneousl b placing a hperprior. We also fit the model b the Baesian Lasso using the algorithm and the ordinar Lasso 2

3 predictor Baesian Lasso (predictor selection) 95% BCI Baesian Lasso 95% BCI Lasso (λ matched) Lasso (n-fold c.v.) (1) age 0 NA ( , ) 0 0 (2) se ( , ) ( , ) (3) bmi (396.68, ) (391.97, ) (4) map (181.85, ) (176.65, ) (5) tc 0 NA ( , ) (6) ldl 0 NA ( , ) 0 0 (7) hdl ( , ) ( , 64.75) (8) tch 0 NA ( , ) 0 0 (9) ltg (335.30, ) (325.79, ) (10) glu 0 NA (-52.77, ) Table 1: Comparison of linear regression parameters for the diabetes data using the λ estimated b the Baesian lasso with the predictor selection algorithm. both of which match the tuning parameter. The results are given in Table 2. We found the Baesian Lasso using the algorithm selects 5 predictors while the ordinar Lasso selects 7. From the 5th and 10th rows in both Table 1 and 2 we can see, given a tuning paramter, for a predictor that the Baesian Lasso using the algorithm throws out whereas the ordinar Lasso selects, the estimate of its coefficent b the ordinar Lasso alwas has a larger absolute value than b the Baesian Lasso without the algorithm. predictor Baesian Lasso (predictor selection) 95% BCI Baesian Lasso 95% BCI Lasso (λ matched) (1) age 0 NA ( , ) 0 (2) se ( , ) ( , ) (3) bmi (396.80, ) (395.94, ) (4) map (186.78, ) (183.42, ) (5) tc 0 NA ( , ) (6) ldl 0 NA ( , ) 0 (7) hdl ( , ) ( , 92.73) (8) tch 0 NA ( , ) 0 (9) ltg (349.14, ) (335.42, ) (10) glu 0 NA (-49.00, ) Table 2: Comparison of linear regression parameters for the diabetes data using the λ estimated b the Baesian Lasso without predictor selection. 4 Penalized Spline using l 1 penalt The Baesian Lasso with the predictor selection algorithm can be applied to penalized spline models using l 1 penalt so that we can fit the model and select knots from a set of candidates simultaneousl. The model is constructed as = µ1 n + Xβ + Zu + ε (3) 3

4 where Z is the n K standardized design matri, u = (u 1,, u K ) and other notations are defined as in Section 1. Then (β, u) are estimated as ( ˆβ, û) = argmin (β,u) where ỹ and λ has the same meaning as in Section 1. K (ỹ Xβ Zu) (ỹ Xβ Zu) + λ u i i=1 4.1 Baesian hierarchical model and Gibbs sampler implementation Analougous to Baesian Lasso for linear regression model [2], the hierarchical representation of the full model is ỹ X, β, Z, u, σ 2 N n (Xβ + Zu, σ 2 I n ) β π(β)dβ, π(β) 1 u τ 2 1,, τ 2 K, λ, σ 2 N K (0 K, σ 2 D τ ), D τ = diag(τ 2 1,, τ 2 K) τ 2 1,, τ 2 K λ Gamma(1, λ2 2 ) λ 2 Gamma(r, δ) σ 2 π(σ 2 )dσ 2, π(σ 2 ) 1 σ 2 with τ and σ 2 independent. According to (2), integrating out τ, the conditional prior on u will be π(u λ, σ 2 ) = K i=1 λ 2σ e λ u i /σ. (4) Park and Casella [2] eplained in detail wh the prior in (4) is preferred to the one in (1). As for the Gamma prior on λ 2, we need the prior variance of λ large enough to make the prior relativel flat. So set r = 1 and δ = 0.1 so that the prior mean of λ 2 is much larger than ˆλ (ˆλ s are less than 1 throughout all models in this paper). The full conditional distributions of the parameters are ( β N p (X X) 1 X (ỹ Zu), σ 2 (X X) 1) ( u N K (Z Z + Dτ 1 ) 1 Z (ỹ Xβ), σ 2 (Z Z + Dτ 1 ) 1) ( p + n 1 σ 2 InvGamma, ỹ Xβ Zu 2 + u D 1 ) τ u λ Inverse Gaussian with mean parameter as 2 σ 2 τ j u 2 and scale parameter as λ 2, j = 1,, K j ( Kj=1 λ 2 τ 2 ) j Gamma K + r, + δ 2 The Gibbs sampler samples parameters cclicall from their full conditionals above. We construct the hierarchical model using ỹ instead of since µ is often of secondar interest. But we make inference of µ, we can still draw samples from its full conditional distribution, N( 1 ni=1 n, σ2 n ), if each column of X and Z is centered. 4

5 4.2 Penalized spline model for the global temperature data using B-spline Basis The Global temperature data [4] recorded 125 ears () of global temperature (Y ) from ear 1881 to The design matri X is obtained b standardizing (, 2, 3 ) so that each column of X has mean zero and l 2 norm one. If Z is a polnomial basis matri, its columns will be highl correlated. It is known that Lasso is not good at dealing with problems when the predictors are highl correlated [5], so we use a B-spline basis matri generated b the R function splinedesign: set the option order = 4 and knots as 23 equall spaced points on the interval [min()+2, ma() 2] such that the function returns a design matri with 21 columns. We obtain Z b standardizing this matri B-spline and truncated cubic spline using l 2 penalt This section is to show how B-spline fits (3) using the l 2 penalt and compare it to the truncated cubic spline. Define W = ( ) ( κ 1 ) 3 +,, ( κ 21 ) 3 + where the knots κ 1,, κ 21 are equall spaced points on the interval [min()+2, ma() 2]. Then we obatin Z b standardizing W. We use Z to fit (3) using l 2 penalt on u b the R function lme, and compare the fitted curve with the one using Z. The curves are plotted in Figure 1 and we can see the curve fitted using B-spline are more bump. B spline truncated cubic spline Figure 1: B-spline and truncated cubic spline using l 2 penalt. 5

6 4.2.2 B-spline knots selection We can appl the predictor selection algorithm to the penalized spline model to select knots automaticall. We use the design matri X and the B-spline basis matri Z using l 1 penalt to fit (3). Running the Gibbs sampler with the algorithm, 3, 5, 6 and 6 knots are selected corresponding to p equal to 0.01, 0.05, 0.1 and 0.2, respectivel. The fitted curves are plotted in Figure 2. We can see the smoothed curves do not fit ver well; the curve tends to fit better as p increases. We can also specif the number of knots to select in the algorithm, rather then setting p. Figure 3 plotted p * = p * = p * = p * = Figure 2: B-spline using l 1 penalt and knots selection for p = 0.01, 0.05, 0.1, 0.2. the fitted curve with 10 knots. It fits much better than all in Figure 2. 5 Discussion Like ordinar Lasso, the Baesian Lasso with the predictor selection algorithm does not select predictors stabl; even if we run the Gibbs sampler several times with all the same settings, the results of predictor selection and parameter estimation ma differ a lot especiall when the predictors are highl correlated. Another problem of the algorithm is the computing speed. First, we need a burn-in of N 1 iterations and N 1 ma be a ver big number, sa , when we initiall have a lot of predictors. Furthermore, M should also be large, sa 10000, because the coefficient of a predictor ma change a lot after throwing out a correlated predictor. When we stop throwing out predictors, we need to run the Gibbs sampler till convergence. In this paper, we just run a big number of iterations and use the trace plot to check convergence. The method of Monte Carlo standard errors 6

7 Figure 3: B-spline using l 1 penalt with 10 knots. can be used to determine when to stop the sampler [6]. 7

8 References [1] Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Roal Statistical Societ. Series B (Methodological), pages , [2] Trevor Park and George Casella. The baesian lasso. Journal of the American Statistical Association, 103(482): , [3] Bradle Efron, Trevor Hastie, Iain Johnstone, Robert Tibshirani, et al. Least angle regression. The Annals of statistics, 32(2): , [4] James S Hodges. Richl Parameterized Linear Models: Additive, Time Series, and Spatial Models Using Random Effects. CRC Press, [5] Hui Zou and Trevor Hastie. Regularization and variable selection via the elastic net. Journal of the Roal Statistical Societ: Series B (Statistical Methodolog), 67(2): , [6] James M Flegal, Murali Haran, Galin L Jones, et al. Markov chain monte carlo: Can we trust the third significant figure? Statistical Science, 23(2): ,