Data analysis in supersaturated designs

Transcription

1 Statistics & Probability Letters 59 (2002) Data analysis in supersaturated designs Runze Li a;b;, Dennis K.J. Lin a;b a Department of Statistics, The Pennsylvania State University, University Park, PA , USA b Department of Management Science and Information Systems, The Pennsylvania State University, University Park, PA , USA Received November 200; received in revised form March 2002 Abstract Supersaturated designs (SSDs) can save considerable cost in industrial experimentation when many potential factors are introduced in preliminary studies. Analyzing data in SSDs is challenging because the number of experiments is less than the number of candidate factors. In this paper, we introduce a variable selection approach to identifying the active eects in SSD via nonconvex penalized least squares. An iterative ridge regression is employed to nd the solution of the penalized least squares. We provide both theoretical and empirical justications for the proposed approach. Some related issues are also discussed. c 2002 Elsevier Science B.V. All rights reserved. Keywords: AIC; BIC; Penalized least squares; SCAD; Stepwise regression. Introduction Many preliminary studies in industrial experimentation contain a large number of potentially relevant factors, but actual active eects are believed to be sparse. To save experimental cost, experimenters have tried to reduce the number of experiments. An ecient use of experimental units is the employment of supersaturated design (SSD), in which the number of experiments is less than the number of candidate factors (see, for example, Lin, 99). Construction of SSDs has received increasing attention recently. For example, Lin (995) introduced an approach to generating system SSDs, Fang et al. (2000) proposed an approach for constructing multi-level SSD via a quasi-monte Carlo approach. Also, see Lin (2000) and references therein. Since the number of candidate factors is more than the number of experiments in SSD, variable selection is fundamental in analyzing SSD for identifying sparse active eects. Some traditional approaches, such as best subset variable selection, are not feasible, while stepwise variable selection may not be appropriate (see, for example, Corresponding author /02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S (02)0040-2

2 36 R. Li, D.K.J. Lin / Statistics & Probability Letters 59 (2002) Westfall et al., 998). In this paper, we introduce an approach from the viewpoint of frequentist analysis. Fan and Li (200) proposed a class of variable selection procedures via nonconcave penalized likelihood. They showed that with the proper choice of penalty function and regularization parameter, their approaches possess an oracle property. Namely, the true coecients that are zero are automatically estimated as zero, and the other coecients are estimated as if the true submodel were known in advance. However, their approach cannot be directly applied for analyzing SSD because regularity conditions imposed in their paper require the design matrix to be full rank. This cannot be satised by an SSD. In this paper, we extend the nonconcave penalized likelihood approaches to least squares, namely nonconvex penalized least squares, and focus on the situation in which the design matrix is not full rank. Theoretic properties of the proposed approach are investigated, and empirical comparisons via Monte Carlo simulation are conducted. This paper is organized as follows. In Section 2, we briey discuss nonconvex penalized least squares and introduce a variable selection procedure for SSD. Root n consistency of the resulting estimator via penalized least squares is established. We also show that the introduced procedure possesses an oracle property. An iterative ridge regression algorithm is employed to nd the solution of the penalized least squares. A choice of initial value of unknown coecients for the iterative ridge regression is suggested. In Section 3, some empirical comparisons via Monte Carlo simulation are conducted. A real data example is used for illustration. Section 4 gives the proof of the main theorem. Some conclusions are given in Section Variable selection for screening active eects 2.. Preliminary Assume that observations Y ;:::;Y n are independent samples from the linear regression model Y i = xi T + i (2.) with E( i ) = 0 and var( i )= 2, where x i is the vector of input variables. Following Fan and Li (200), a form of penalized least squares is dened as Q() n d (Y i xi T ) 2 + p n ( j ); 2n (2.2) i= j= where p n ( ) is a penalty function, and n is a tuning parameter, which can be chosen by a data-driven approach, such as cross-validation (CV) and generalized cross-validation (GCV, Craven and Wahba, 979). Many variable selection criteria are closely related to this penalized least squares. Take the penalty function to be the entropy penalty, namely, p n ( )= 2 2 ni( 0), where I( ) is an indicator function. Note that the dimension or the size of a model equals the number of nonzero regression coecients in the model. This actually equals j I( j 0). In other words, the penalized least squares (2.2) with the entropy penalty can be rewritten as n (Y i xi T ) 2 + 2n 2 2 n M ; (2.3) i=

3 R. Li, D.K.J. Lin / Statistics & Probability Letters 59 (2002) where M = j I( j 0), the size of the underlying candidate model. Hence, many popular variable selection criteria can be derived from the penalized least squares (2.3) by taking dierent values of n. For instance, the AIC and BIC correspond to n = 2(= n) and log n(= n), respectively, although these two criteria were motivated from dierent principles. The entropy penalty is not continuous, and can be improved by its smooth version, namely a hard thresholding penalty function, p ( )= 2 ( ) 2 I( ); proposed by Fan (997). Other penalty functions have been used in the literature. The L 2 penalty p n ( )=2 n 2 yields a ridge regression, and the L penalty p n ( )= n results in LASSO (Tibshirani, 996). More generally, the L q penalty leads to a bridge regression (see Frank and Friedman, 993; Fu, 998; Knight and Fu, 2000). All aforementioned penalties do not satisfy the conditions for desired properties in terms of continuity, sparsity and unbiasedness, advocated by Fan and Li (200). They suggested using the smoothly clipped absolute deviation (SCAD) penalty, proposed by Fan (997). The rst-order derivative of SCAD is dened by { p ()= I( 6 )+ (a ) } + (a ) I( ) for 0 with a =3:7, and p (0) = 0. For simplicity of presentation, we will use SCAD for all procedures using the SCAD penalty. As recommended by Fan and Li, we employ penalized least squares with the SCAD penalty to identify the sparse active eects in the analysis stage of SSD in this paper Asymptotic properties When (x i ;Y i );i=;:::;n, are independent and identically distributed, Fan and Li have established an oracle property for their penalized likelihood estimator. In what follows, we will show that the oracle property still holds for the penalized least-squares estimator when the predictor variable x is a xed design even if design matrix is singular. We next introduce some related notations. Denote by 0 the true value of, and let 0 =( 0 ;:::; d0 ) T =( T 0 ; T 20 )T. Without loss of generality, assume that 20 = 0, and all components of 0 are not equal to 0. Denote by s the number of nonzero components of. Let X consist of the rst s columns of X, the design matrix of the linear regression model (2.), and x k consist of the rst s components of x k ;k=;:::;n. Dene V = lim n (=n)x T X and let V consist of the rst s columns and rows of V. To establish the oracle property for the penalized least-squares estimate, we need the following two conditions: (C) max 6k6n x T k (XT X ) x k 0; as n. (C2) V is nite and V 0. Conditions (C) and (C2) guarantee that the asymptotic normality holds for the least-squares estimator of ˆ. We show in Theorem that the penalized least-squares estimate is root n consistent and possesses the oracle property for the penalized least-squares estimator with the SCAD penalty.

4 38 R. Li, D.K.J. Lin / Statistics & Probability Letters 59 (2002) Theorem. Consider model (2.) and suppose that the random errors are independent and identically distributed with zero mean and nite positive variance 2. Suppose that conditions (C) and (C2) hold. For the SCAD penalty; if n 0 and n n ; then (a) (Root n consistency). With probability tendingto one; there exists a local minimizer ˆ of Q(); dened in (2.2); such that ˆ is a root n consistent estimator of ; (b) (Oracle property). With probability tendingto one; the root n consistent estimator in Part (a) satises ˆ 2 = 0 and n( ˆ 0 ) N(0; 2 V ): The proof of Theorem is given in Section 4. From Theorem, with proper rate of n, the SCAD results in a root n consistency estimator. Moreover, the resulting estimator correctly identies the inactive eects and estimates the active eects as if we knew the true submodel in advance Iterative ridge regression Local quadratic approximation Note that the SCAD penalty function is singular at the origin and may not have the second derivative at some points. In order to apply the Newton Raphson algorithm to the penalized least squares, Fan and Li (200) locally approximate the SCAD penalty function by a quadratic function as follows. Given an initial value (0) that is close to the true value of, when (0) j is not very close to 0, the penalty p ( j ) can be locally approximated by the quadratic function as [p ( j )] = p ( j ) sgn( j ) {p ( (0) j )= (0) j } j and set ˆj =0 if (0) j is very close to 0. With the local quadratic approximation, the solution for the penalized least squares can be found by iteratively computing the following ridge regression with an initial value (0) : where () = {X T X + n ( (0) )} X T y; ( (0) ) = diag{p ( (0) )= (0) ;:::;p ( (0) d )= (0) d }: This can be easily implemented in many statistical packages Initial value The local quadratic approximation requires a good initial value which is close to the true value 0. When the design matrix is full rank, the least-squares estimate can serve as the initial value of. When the sample size is relatively large, the least-squares estimate possesses root n consistency, and hence it is very close to the true value of. In the SSD settings, however, the number of experiments may be less than the number of potential candidates, and therefore the design matrix is not full rank. In fact, the regression coecients are not identiable without further assumptions, for example, the eect sparsity assumption. Here we use stepwise variable selection to nd an initial

5 R. Li, D.K.J. Lin / Statistics & Probability Letters 59 (2002) value of. In other words, we rst apply stepwise variable selection to the full model with small thresholding values (i.e. large value of signicance level ) such that all active factors are included in the selected model. Of course, some insignicant factors may still stay in the resulting model at this step. 3. Simulation study and example In this section we compare the performance of the SCAD and stepwise variable selection procedure for analyzing SSD. All simulations are conducted using MATLAB codes. GCV (Craven and Wahba, 979) was used for determining the regularization parameter in the SCAD. In the following examples, the level of signicance is set to be 0. for both F-enter and F-remove in the stepwise variable selection procedure for choosing an initial value for the SCAD. The comparison is made in terms of the ability of identifying true model and the size of resulting model. Example. Given the design matrix constructed in Lin (993); we simulated 000 data sets from the linear model Y = x T + ; where the random error is N(0; ). We generate data from the following three models: Model I: =8, 2 = 5 and all other components of are equal to 0. Model II: =0; 2 =9, 3 = 2 and all other components of are equal to 0. Model III: = 20; 3 =2; 5 =0; 7 =5; 6 = 2 and all other components of are equal to 0. In Model I, there are two large active eects; while in Models II and III, there are some large eects, some moderate eects and a small eect. In our simulations, we also employ stepwise regression with three dierent criteria. One selects signicant eects by setting the level of signicance to be 0.05, the other two select a subset with the best AIC and BIC score in the stepwise regression, respectively. Table summarizes the simulation results. From Table, it is clear that SCAD outperforms all three stepwise regression methods in terms of rate of identifying true models and size of selected models. Example 2 (Williams Rubber Experiment). The proposed procedure is applied to the SSD constructed by Lin (993). Stepwise variable selection selects the following factors: X 5 ; X 2 ; X 20 ; X 4 ;X 0 ;X ;X 7 ;X ;X 4 ;X 7 and X 22. Then the SCAD procedure is applied. The selected tuning parameter is ˆ =6:5673. The nal model selected by the SCAD identies X 4 ;X 2 ;X 5 ;X 20 as the active eects. The result is consistent with the conclusion of Williams (968). 4. Proofs To prove Theorem, we prove the following two lemmas rst.

6 40 R. Li, D.K.J. Lin / Statistics & Probability Letters 59 (2002) Table Percent of 000 simulations in Example Model Method Rate of the true model being identied Avg. size of tted model Median Mean I II III True model: Y =8x +5x 2 + Stepwise (p = 0:05) 27.00% Stepwise (AIC) 2.20% Stepwise (BIC).0% SCAD 82.70% True model: Y =0x +9x 2 +2x 3 + Stepwise (p = 0:05) 32.30% Stepwise (AIC) 3.00% Stepwise (BIC) 3.50% SCAD 74.70% True model: Y = 20x +2x 3 +0x 5 +5x 7 +2x 6 + Stepwise (p = 0:05) 38.90% Stepwise (AIC) 7.00% Stepwise (BIC) 24.70% SCAD 7.90% Lemma. Under the conditions of Theorem ; there exists a large constant C such that { } P inf Q( 0 + n =2 u) Q( 0 ) ; (4.) u =C where u =(u T ; 0) and the dimension of u is the same as that of 0. Proof. Dene D n (u) Q( 0 + n u) Q( 0 ): Note that p n (0) = 0 and 20 =0; D n (u) n { y X ( 2n 0 + u ) 2 y X 0 2 } + i= s {p n ( j0 + n u j ) p n ( j0 )}; (4.2) j= where y =(Y ;:::;Y n ) T. The rst term in (4.2) can be simplied ( ) n 2 n u T n XT X u n =2 u T x i (Y i xi T 0 ): (4.3) i=

7 R. Li, D.K.J. Lin / Statistics & Probability Letters 59 (2002) By the assumption of (C2); the rst term in (4.3) equals n 2 ut V u { + o()}. Using R = E(R)+ O P { var(r)} for any random variable with nite second moment; it follows that the second term in (4.3) equals to n u T V u O P (). Note that V is nite and positive denite. By choosing a suciently large C; the rst term will dominate the second term; uniformly in u = C. By Taylor s expansion, the second term on the right-hand side of (4.2) becomes s j= [n =2 p n ( j0 ) sgn( j0 )u j + n p n ( j0 )u 2 j ( + o())] which is bounded by sn =2 a n u + n b n u 2 ; where a n = max{ p n ( j0 ) : j0 0} and b n = max{ p n ( j0 ) : j0 0}: For the SCAD penalty, if n 0, then a n = 0 and b n = 0 when n is large enough. Therefore, the second term in (4.2) is dominated by the rst term of (4.3). Hence, by choosing suciently large C, (4.) holds. This completes the proof. Lemma 2. Under the conditions of Theorem ; for any given satisfyingthat 0 = O P (n =2 ) and any constant C; the followingequation holds with probability tendingto one; {( )} {( )} Q = min Q : 0 2 6Cn =2 2 Proof. It is sucient to show that; with probability tending to as n ; for any satisfying 0 =O P (n =2 ) and for some small n = Cn =2 and j = s 0 for 0 j n j 0 for n j 0: (4.5) By some straightforward computations; it follows j n xt (j)(y X 0 )+ n xt (j)x( 0 )+p n ( j ) sgn( j ); where x (j) is the jth column of X. Since V is nite, var{(=n)x(j) T (y XT 0 )} =O(n ), and it follows that n xt (j)(y X 0 )=O P (n =2 ):

8 42 R. Li, D.K.J. Lin / Statistics & Probability Letters 59 (2002) By condition (C2) n xt (j)x = V j ( + o()); where V j is the jth column of V. By the assumption that 0 =O P (n =2 ), j = n { n p n ( j ) sgn( j )+O P (n =2 = n )}: Since lim inf 0 + n p n ()= and n =2 = n 0asn, the sign of the derivative is completely determined by that of j. Hence, (4.4) and (4.5) follows. This completes the proof. Proof of Theorem.. To show Part (a); it suces to show that there exists a large constant C such that { } P inf Q( 0 + n =2 u) Q( 0 ) : (4.6) u =C This implies that with probability tending to one there exists a local minimum in the ball { 0 + n =2 u : u 6 C}: Hence; there exists a local minimizer ˆ such that ˆ 0 =O P (n =2 ). Since Q( 0 + n =2 u) Q( 0 ) [ {( 0 + n =2 u = Q 20 + n =2 u 2 )} Q {( 0 + n =2 u 20 )}] [ + Q {( 0 + n =2 u 20 )} ] Q( 0 ) : By Lemma 2, with probability tending to the rst term is positive as 20 = 0. By Lemma, (4.6) holds. This completes the proof of Part (a). Now we show Part (b). It follows by Lemma 2 that with probability tending to one, ˆ 2 = 0. It can be easily shown that there exists a ˆ in Part (a) that is a root n consistent local minimizer of {( )} Q ; 0 regarded as a function of, and satisfying the j ( ) ˆ = 0 for j =;:::;s: = 0 Note that ˆ is a consistent estimator and 20 ˆ) j n xt (j)(y X 0 )+ n xt (j)x () ( ˆ 0 ) p n ( ˆ j ) = n xt (j)(y X 0 )+V T (j)( ˆ 0 )(+o P ()) (p n ( 0 j ) sgn( j0 )+{p n ( j0 )+o P ()}( ˆ j j0 ));

9 R. Li, D.K.J. Lin / Statistics & Probability Letters 59 (2002) where X () consists of the rst s columns of X and V (j) is the jth-column of V. It follows by Slutsky s Theorem and the Hajek Sidak Central Limit Theorem (CLT) that n(v + ){ ˆ 0 +(V + ) b} N(0; 2 V ) in distribution, where = diag{p n ( 0 );:::;p n ( s0 )}; b =(p n ( 0 ) sgn( 0 );:::;p n ( s0 ) sgn( s0 )) T : Note that for the SCAD penalty, = 0 and b = 0 as n 0. Thus, n( ˆ 0 ) N(0; 2 V ) in distribution. This completes the proof of Part (b). 5. Conclusion While the construction of SSDs has been paid increasing attention, the analysis of SSDs deserves special attention. In this paper, we proposed a variable selection approach for analyzing experiments when the full model contains many potential candidate eects. It has been shown that the proposed approach possesses an oracle property. The proposed approach has been empirically tested. A real data example was used to illustrate the eectiveness of the proposed approach. Acknowledgements The authors thank the referee and Dr. H. McGrath for their valuable comments and suggestions which led to a great improvement in the presentation of this paper. Li s research was supported by an NSF Grant DMS References Craven, P., Wahba, G., 979. Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation Numer. Math. 3, Fan, J., 997. Comments on Wavelets in statistics: a review by A. Antoniadis. J. Italian Statist. Assoc. 6, Fan, J., Li, R., 200. Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assos. 96, Fang, K.T., Lin, D.K.J., Ma, C.X., On the construction of multi-level supersaturated designs. J. Statist. Plan. Inference 86, Frank, I.E., Friedman, J.H., 993. A statistical view of some chemometrics regression tools. Technometrics 35, Fu, W.J., 998. Penalized regression: the bridge versus the LASSO. J. Comput. Graphical Statist. 7, Knight, K., Fu, W., Asymptotic for Lasso-type estimators. Ann. Statist. 28, Lin, D.K.J., 99. Systematic supersaturated designs. Working Paper, No. 264, Department of Statistics, University of Tennessee. Lin, D.K.J., 993. A new class of supersaturated designs. Technometrics 35, Lin, D.K.J., 995. Generating system supersaturated designs. Technometrics 37,

10 44 R. Li, D.K.J. Lin / Statistics & Probability Letters 59 (2002) Lin, D.K.J., Supersaturated design: theory and application. In: Park, S.H., Vining, G.G. (Eds.), Statistical Process Monitoring and Optimization. Marcel Dekker, New York (Chapter 8). Tibshirani, R.J., 996. Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. B 58, Westfall, P.H., Young, S.S., Lin, D.K.J., 998. Forward selection error control in analysis of supersaturated designs. Statist. Sinica 8, 0 7. Williams, K.R., 968. Designed experiments. Rubber Age 00, 65 7.