Direct Determination of Smoothing Parameter for Penalized Spline Regression

Penalized spline estimator is one of the useful smoothing methods. To construct the estimator, having goodness of fit and smoothness, the smoothing parameter should be appropriately selected. The purpose of this paper is to select the smoothing parameter using the asymptotic property of the penalized splines.The new smoothing parameter selectionmethod is established in the context of minimization asymptotic form of MISE of the penalized splines. The mathematical and the numerical properties of the proposed method are studied. First we organize the newmethod in univariate regression model. Next we extend to the additive models. A simulation study to confirm the efficiency of the proposed method is addressed.


Introduction
Penalized spline methods are a well-known efficient technique for nonparametric smoothing.Penalized splines were suggested by O'Sullivan [1] and Eilers and Marx [2].In O'Sullivan [1], they used a cubic -spline function and the penalty was the integrated squared second derivative of the -spline function.On the other hand, Eilers and Marx [2] use a cubic -spline function and a difference penalty on the spline coefficients.The Eilers and Marx's estimator is computationally efficient compared to smoothing splines and O'Sullivan's estimator since it removes the integration part of the penalty.Hence this paper focuses on the penalized spline estimator provided via Eilers and Marx [2].The penalized spline method is efficient for both univariate regression and multiple regressions such as the additive model (see Marx and Eilers [3]).General properties usages and a description of the flexibility of penalized splines are described in Ruppert et al. [4].
When using penalized splines, the determination of the smoothing parameter is very important since it controls the trade-off between the goodness of fit and the smoothness of the fitted curve.As the classical method for achieving this, the grid search method is often used.The grid search method is selected by minimizing one criterion from candidate points of the smoothing parameter.Criteria for grid searches include cross-validation, generalized cross-validation, Mallow's   , and so forth.Although the grid search selection generally finds one optimal smoothing parameter, it is possible that the worth curve is obtained when not all the candidates are good.This tendency is especially striking in additive models since the number of the smoothing parameter is the same as that of the covariates.Several smoothing parameter selection methods using the grid search criteria have been developed by many authors such as Krivobokova [5], Reiss and Ogden [6], Wood [7], Wood [8], and Wood [9].On the other hand, the mixed model representation of the spline smoothing has also been studied (see Lin and Zhang [10], Wand [11], and Ruppert et al. [4]).In mixed models, the grid search method is not necessary to obtain the final fit curve.The smoothing parameter in the mixed model can be written as the ratio of the variance of the random coefficient and the error.By estimating these unknown variances using a maximum likelihood method or a restricted maximum likelihood method (REML), the final fitted curves are obtained, yielding the estimated best linear unbiased predictor (EBLUP).Therefore the EBLUP does not require a grid search.However the fitted curve tends to theoretically oversmooth and the numerical stability is not guaranteed if a cubic spline is used (see Section 3).The Bayesian approach to select the smoothing parameter has been studied by Fahrmeir et al. [12], Fahrmeir and Kneib [13], and Heinzl et al. [14].Kauermann [15] compared some smoothing parameter selection methods.
In this paper, we propose a new method to determining the smoothing parameter using the asymptotic properties of the penalized splines.For the remainder of this paper, our new method will be known as the direct method.Before describing the outline of the direct method, we will briefly introduce the asymptotic studies of penalized splines.First, Hall and Opsomer [16] showed the consistency of the penalized spline estimator in white noise representation.Subsequently, Li and Ruppert [17], Claeskens et al. [18], Kauermann et al, [19], and Wang et al. [20] have developed the asymptotics for the penalized spline estimator in univariate regression.Yoshida and Naito [21] and Yoshida and Naito [22] have studied the asymptotics for penalized splines in additive regression models and generalized additive models, respectively.Xiao et al. [23] suggested a new penalized spline estimator, and developed its asymptotic properties in bivariate regression.Thus, the developments of the asymptotic theories of the penalized splines are relatively recent events.In addition, the smoothing parameter selection methods using asymptotic properties have not yet been studied.This motivates us to try to establish such methods.
The direct method is conducted by minimizing the mean integrated squared error (MISE) of the penalized spline estimator.In general, the MISE of the nonparametric estimator is divided into the integrated squared bias and the integrated variance of the estimator.Of course the penalized spline estimator is no exception and hence the direct method is stated by utilizing the expression of the asymptotic bias and variance of the penalized spline estimator, which have been derived by Claeskens et al. [18], Kauermann et al. [19], and Yoshida and Naito [22].From their result, we see that the asymptotic order of the variance of the penalized spline estimator is only dependent on the sample size and the number of knots but not the smoothing parameter.However the second term of the asymptotic variance of the penalized spline estimator contains the smoothing parameter and we can see that the variance becomes small when the smoothing parameter increases.On the other hand, the squared bias of the penalized spline estimator increases if the smoothing parameter is reduced.Therefore the minimizer of the MISE of the penalized spline estimator can be seen as one of the optimal smoothing parameters.Since the MISE is asymptotically convex with respect to the smoothing parameter, the global minimum of the MISE can be found.This detection has been sufficiently developed for bandwidth selection in kernel regression (see Ruppert et al. [24], Wand and Jones [25], etc.).First the present paper focuses on univariate regression, and we next extend the direct method to the additive models.In both models, the mathematical and the numerical properties of the direct method are studied.In additive models, we need to select a smoothing parameter of the same number as the explanatory variable, such that the computational cost of the grid search becomes large.We expect that the computational cost of the direct method is dramatically reduced compared to the grid search.
The structure of this paper is as follows.In Section 2, we introduce a penalized spline estimator in a univariate regression model.Section 3 provides the direct method and related properties.Section 4 extends the direct method to the additive model.In Section 5, we confirm the performance of the direct method using a numerical study.We provide a discussion on the outlook and further studies in Section 6.The proofs of our theorems are provided in the appendix.

Penalized Spline Estimator
Consider the regression problem with  observations, where   is the response variable,   is the explanatory variable,  is the true regression function, and   is the random error which is assumed to be independently distributed with mean 0 and variance  2 .Throughout the paper we assume the explanatory variable   ∈ [, ] ( = 1, . . ., ) is not a random variable from which the expectation of   can be expressed as [  |   ] = (  ).The support of the explanatory   can be relaxed as the real space R. In order to simplify the way of locating the knots in the following, the support of the explanatory is assumed to be with compact space.We aim to estimate  via a nonparametric penalized spline method.We consider the knots  =  0 <  1 < ⋅ ⋅ ⋅ <   <  +1 =  and, for  = −, . . .,  − 1, let []  () be the th degree -spline basis function defined as  () is a piecewise th degree polynomial on an interval [  ,  ++1 ].The details of the -spline basis functions are described in de Boor [26].For simplicity, we write   () = []  () since we do not specify the  in the following sentence.
The penalized spline estimator b = ( b− ⋅ ⋅ ⋅ b−1 ) of b is defined as the minimizer of where  is the smoothing parameter and Δ is the backward difference operator defined as Δ  =   −  −1 and Let   = ( ()  )  be a ( +  − ) × ( + ) matrix, where can then be expressed as The minimum of ( 7) is obtained when The penalized spline estimator f() of () for  ∈ [, ] is defined as where If  ≡ 0, f() is reduced to the regression spline estimator, which is the spline estimator obtained via the least squares method.The regression spline estimator will lead to an oscillatory fit if the number of knots  is large.However the determination of  and the location of knots are very difficult problems.The advantage of the penalized spline smoothing is that the good smoothing parameter brings the estimator to the curve with the fitness and smoothness simultaneously without choosing the number and location of knots precisely.In the present paper, we use equidistant knots   =  + / and focus on the determination of the smoothing parameter.As the location of knots other than above, the quantiles of the data points { 1 , . . .,   } are often used (see Ruppert [27]).However it is known that the penalized spline estimator is hardly affected by the location of knots if  is not too small.Therefore we do not discuss the location of knots.We suggest the direct method for this in the next section.

Direct Determination of Smoothing Parameter
In this section, we provide the direct method for determining the smoothing parameter without a grid search.This direct method is given theoretical justification by asymptotic theory of the penalized spline estimator.To investigate the asymptotic property of the penalized spline estimator, we assume that  ∈  +1 ,  = ( 1/2 ), and  = (/ 1− ).
For convenience we first give some notation.Let   =  −1    and Λ  =   + (/)     .Let b * be a best  ∞ approximation to the true function .This means that b * satisfies sup where ( <  < ) is the indicator function of an interval (, ), and Br  () is the th Bernoulli polynomial (see Zhou et al. [28]).It can be easily shown that   () = (1) as  → ∞.
The penalized spline estimator can be written as The first term of the right hand side of ( 12) is equal to the regression spline estimator denoted by frs ().The asymptotics for the regression spline estimator has been developed by Zhou et al. [28] and can be expressed as From Theorem 2(a) of Claeskens et al. [18], we have where is the covariance of frs () and the second term of the right hand side of (12).The variance of the second term of ( 12) can be shown to be negligible (see the appendix).The following theorem leads to  controlling the trade-off between the squared bias and variance of the penalized spline estimator.From the asymptotic form of [ f()] and [ f()] and Theorem 1, we see that, for small , the bias of f() is small and the variance becomes large.On the other hand, the large  indicates the bias of f() increases and the variance decreases.From Theorem 1, the MISE of f() can be expressed as where  1 () and  2 (, ) are of negligible order, respectively, compared to the regression spline and penalized spline of the second term of the right hand side of ( 12).Actually we have  1 () = (/) and The MISE of f() is asymptotically quadratic and a global minimum exists.Let And let  opt be the minimizer of MISE().We suggest the use of  opt as the optimal smoothing parameter, where However it is easy to see that MISE() and  opt contain an unknown function and parameters, and hence these must be estimated.We construct the estimator of  opt by using the consistent estimator of  (+1) and b * .We can use the penalized spline estimator and its derivative as the pilot estimator of  (+1) and b * .If we then use another smoothing parameter  0 , it should be chosen appropriately.Therefore we use the regression spline estimator as the pilot estimator of  (+1) () and b * .First we establish b = (  ) −1   .Next we construct the pilot estimator with  (+1) () by using the ( + 2)th degree -spline basis.Let B [] () = ( []  − () ⋅ ⋅ ⋅ [] −1 ()) .Using the fundamental property of the -spline function,  () () can be written as f() () =  [−] ()    b * asymptotically.Hence the regression spline estimator f(+1) can be constructed as f(+1) () = B [+1] ()   +1 b (+2) , where b −+−1 (  ))  .Since the regression spline estimator tends to be oscillatory with a higher order th spline function, the fewer knots are used to construct f(+1) ().The variance  2 included in ( | , , 0) is estimated via Using the above pilot estimators, with some finite grid points {  }  1 on [, ].Consequently the final penalized spline estimator is defined as where bopt = (   + λopt      ) It is known that the optimal order of  of the penalized spline estimator is the same as that of the regression spline estimator,  = ( 1/(2+3) ) (see Kauermann et al. [19] and Zhou et al. [28]).Using this, we show the asymptotic property of  opt in following theorem.Theorem 2. Let  ∈  +1 .Suppose that  = ( 1/2 ) and  = (/ 1− ).Then   given in (21) exists, and   = ( −−2 + 2 ) as  → ∞.Furthermore  = ( 1/(2+3) ) leads to the optimal order and the rate of convergence of MISE of f() becomes ( −2(+1)/(2+3) ).
The proof of Theorem 2 is given in the appendix.At the end of this section, we give a few remarks.
Remark 5.The penalized spline regression is often considered as the mixed model representation (see Ruppert et al. [4]).In this frame work, we use the th truncated spline model where ( − ) + = max{ − , 0} and the 's are unknown parameters.Each   is independently distributed as   ∼ (0,  2  ), where  2  is an unknown variance parameter of   .The penalized spline fit of () is defined as the estimated BLUP (see Robinson [29]).The smoothing parameter in the ordinal spline regression model corresponds to  2 / 2  in the spline mixed model.Since the  2 and  2  are estimated using the ML or REML method, we do not need to choose the smoothing parameter via a grid search.It is known that the estimated BLUP fit is linked to the penalized spline estimator (9) with  =  + 1 (see Kauermann et al. [19]).Hence the estimated BLUP tends to have a theoretically underfit (see Remark 4).Remark 6.From Lyapunov's theorem, the asymptotic normality of the penalized spline estimator f() with λopt can be derived under the same assumption as Theorem 2 and some additional mild conditions.Although the proof is omitted, it is straightforward since λopt is the consistent estimator of  opt and the asymptotic order of  opt satisfies Lyapunov's condition.

Extension to Additive Models
We extend the direct method to the regression model with multidimensional explanatory variables.In particular, we consider additive models in this section.For the dataset {(  , x  ) :  = 1, . . ., } with 1-dimensional response   and -dimensional explanatory x  = ( 1 , . . .,   ), the additive models are connected via unknown regression functions   ( = 1, . . ., ) and the mean parameter  such that We assume that   is located on an interval [  ,   ] and  1 , . . .,   are normalized as to ensure the identifiability of   .Then the intercept  is typically estimated via Hence we replace   with   − μ in (26) and set  = 0, redefining the additive model as where each   is centered.We aim to estimate   via a penalized spline method.Let and estimate   and  , .For  = 1, . . ., , the penalized spline estimator b = ( b,− ⋅ ⋅ ⋅ b,−1 ) where   are the smoothing parameters and   is the th order difference matrix with size (  +  − ) × (  + ) for  = 1, . . ., .By using b , the penalized spline estimator of   (  ) is defined as The asymptotics for f (  ) have been studied by Yoshida and Naito [22] who derived the asymptotic bias and variance to be where The above asymptotic bias and variance of f (  ) are similar to that of the penalized spline estimator in univariate regression with {(  ,   ) :  = 1, . . ., }.Furthermore the asymptotic normality of [ f1 ( 1 ) ⋅ ⋅ ⋅ f (  )]  has been shown by Yoshida and Naito [22].From their paper, we find that f (  ) and f (  ) are then asymptotically independent for  ̸ = .This indicates some theoretical justification to select   in minimizing the MISE of f .Similar to the discussion in Section 3, the minimizer  ,opt of the MISE of f (  ) can be obtained for  = 1, . . ., , where Since   , b *  , and  2 are unknown, they should be estimated.The pilot estimators of   , b *  , and  2 are constructed based on the regression spline method.By using the pilot estimators f , b of   , b *  and the estimator of  2 , we construct the estimator of  ,opt : where {  }  1 is some finite grid point sequence on [  ,   ].We obtain for  = 1, . . ., , the penalized spline estimator f (  ) of   (  ), where b,opt is the penalized spline estimator of b  using λ,opt .From Theorem 2 and the proof of Theorem 3.4 of Yoshida and Naito [22], the asymptotic normality of [ f1 ( 1 ) ⋅ ⋅ ⋅ f (  )]  using ( 1,opt , . . .,  ,opt ) can be shown.
Remark 7. Since the true regression functions are normalized, the estimator f should be also centered as Remark 8.The penalized spline estimator of b 1 , . . ., b  can be obtained using a backfitting algorithm (Hastie and Tibshirani [30]).The backfitting algorithm for the penalized splines in additive regression is detailed in Marx and Eilers [3] and Yoshida and Naito [21].
Remark 9.Although we focus on the nonparametric additive regression in this paper, the direct method can be also applied to the generalized additive models.However we omit this discussion because the procedure is similar to the case of the additive models discussed in this section.
Remark 10.The direct method is quite computationally efficient when compared to the grid search method in additive models.In grid searches, we prepare the candidate of   .Let   be the set of all possible candidate grid value of   for  = 1, . . ., .Then we need to compute the backfitting algorithm { 1 × ⋅ ⋅ ⋅ ×   } times.On the other hand, it is sufficient to perform the backfitting algorithm for only two steps for the pilot estimator and the final penalized spline estimator.Thus, compared with the conventional grid search method, the direct method can drastically reduce computation time.

Numerical Study
In this section, we investigate the finite sample performance of the proposed direct method in a Monte Carlo simulation.Let us first consider the univariate regression model (1) for the data {(  ,   ) :  = 1, . . ., }.Then we use the three  which are labeled by F1, F2, and F3, respectively.Here () is the density function of the normal distribution.The explanatory   and error   are independently generated from uniform distribution on [0, 1] and (0, 0.5 2 ), respectively.We estimate each true regression function via the penalized spline method.We then use the linear and cubic -spline bases with equidistant knots and the second order difference penalty.In addition we set  = 5 2/5 equidistant knots and the smoothing parameter is determined by the direct method.The penalized spline estimator with the linear spline and cubic spline are denoted as L-Direct and C-Direct, respectively.For comparison with L-Direct, the same studies are also implemented for the penalized spline estimator with a linear spline and the smoothing parameter selected via GCV and restricted maximum likelihood method (REML) in mixed model representation, and the local polynomial estimator with normal kernel and Plug-in bandwidth (see Ruppert et al. [24]).In GCV, we set the candidate values of  as {/10 :  = 0, 1, . . ., 99}.The above three estimators are denoted by L-GCV, L-REML, and local linear, respectively.Furthermore we compare C-Direct with C-GCV and C-REML, which are the penalized spline estimator with the cubic spline and the smoothing parameter determined by GCV and REML.Let be the sample MISE of any estimator f() of (), where f () is f() with th replication and   = /.We calculate the sample MISE of the penalized spline estimator with the direct method, GCV, and REML and the local linear estimator.In this simulation, we use  = 100 and  = 1000.
We have simulated  = 50 and 200.
The sMISE of all estimators for each model and  are given in Table 1.The penalized spline estimator using the direct method shows good performance in each setting.In comparison with other smoothing parameter methods, the direct method is a little better than the GCV as a whole.However for  = 200, C-GCV is better than C-Direct in F1 and F3 though its difference is very small.We see that the sMISE of C-Direct is smaller than local linear, whereas local linear behaves better than L-Direct in some case.In F2, C-Direct is the smallest of all estimators for  = 50 and 200.Although the performance totally seems to depend on situation in which data sets are generated, we believe that the proposed method is one of the efficient methods.
Next the difference between λopt and  opt is investigated empirically.Let λopt, be the λopt with th replications for  = 1, . . ., 1000.Then we calculate the sample MSE of λopt : In Table 2, the sMSE of λopt for each true function are described.For comparison, the sMSE of the smoothing be the sample PSE for any estimator f(), where   ( = 1, . . ., ) is independently generated from   ∼ ((  ), (0.5) 2 ) for all .
In Table 3, the modified sPSE, |sPSE − (0.5) 2 |, of all estimators for each model and  are described.In Remark 11, we discuss the modified sPSE.From the result, we can confirm that the direct method has good predictability.GCV can be regarded as the estimator of sPSE.Therefore in some case, sPSE with GCV is smaller than that with the direct method.It seems that the cause is the accuracy of the estimates of variance (see Remark 11).However its difference is very small.We admit that the computational time (in second) taken to obtain f() for F1,  = 3, and  = 200.The fits with the direct method, GCV and REML took 0.04, 1.22, and 0.34.Although the difference is small, the computational time of the direct method was faster than that of GCV and REML.
Next we confirm the behavior of the penalized spline estimator with the direct method in the additive model.For the data {(  ,  1 ,  2 ,  3 ) :  = 1, . . ., }, we assume the additive model with true functions  1 ( 1 ) = sin(2 1 ),  2 ( 2 ) = (( 2 − 0.5)/0.2),and  3 ( 3 ) = 0.4(( 3 − 0.1)/0.2) + 0.6(( 2 − 0.8)/0.2).The error is similar to the first simulation.The design points ( 1 ,  2 ,  3 ) are generated by where   ( = 1, . . ., ,  = 1, 2, 3) are generated independently from the uniform distribution on [0, 1].In this simulation, we adopt  = 0 and  = 0.5.We then corrected to satisfy ()  = 0,  = 1, 2, 3 in each .We construct the penalized spline estimator via the backfitting algorithm with the cubic spline and second order difference penalty.The number of equidistant knots is   = 5 2/5 and the smoothing parameters are determined using the direct method.The pilot estimator to construct λ,opt is the regression spline estimator with fifth spline and   =  2/5 .We calculate the sample MISE of each f for  = 200 and 1000 Monte Carlo iterations.Then we calculate the sample MISE of f1 , f2 , and f3 .In order to compare with the direct method, we also conduct the same simulation with GCV and REML.In GCV, we set the candidate values of   as {/10 :  = 0, 1, . . ., 9} for  = 1, 2, 3. Table 4 summarizes the sample MISE of f ( = 1, 2, 3) denoted by MISE for direct method, GCV, and REML.The penalized spline estimator with the direct method performs like that with the GCV in both the uncorrelated and correlated design cases.For  = 0, the behaviors of MISE1, MISE2, and MISE3 with the direct method are similar.On the other hand, for GCV, the MISE1 is slightly larger than MISE2 and MISE3.The direct method leads to an efficient estimate of all covariates.On the whole, the direct method is better than REML.From above, we believe that the direct method is preferable in practice.

Remark 4 .
O'Sullivan[1] used  ∫   { () ()} 2  as the penalty term, where  is the smoothing parameter.When equidistant knots are used, the penalty ∫   { () ()} 2  can be expressed as  2 b       b, where  = (∫ )  .The penalty b       b provided by Eilers and Marx [2] can be seen as the simple version of  2 b       b by replacing  with  −1  and  =  2−1 , where  is the identity matrix.

Table 1 :
Results of sample MISE for  = 50 and  = 200.All entries for MISE are 10 2 times their actual values.

Table 2 :
Results of sample MSE of smoothing parameter obtained by direct method, GCV, and REML for  = 50 and  = 200.All entries for MSE are 10 2 times their actual values.

Table 3 :
Results of sample PSE for  = 50 and  = 200.All entries for PSE are 10 2 times their actual values.Direct and the C-Direct is small even in  = 50 for all true functions.Therefore it seems that the accuracy of λopt is guaranteed.It indicates that the pilot estimator constructed via least squares method is not bad.The sMSE with the direct method are smaller than that with GCV and REML.This result is not surprising since GCV and REML are not concerned with  opt .However together with Table1, it seems that the sample MSE of the smoothing parameter is reflected in the sample MISE of the estimator.The proposed direct method was derived based on the MISE.On the other hand the GCV and the REML are obtained in context of minimizing prediction squared error (PSE) and prediction error.Hence we compare the sample PSE of the penalized spline estimator with the direct method, GCV and REML, and the local linear estimator.Since the prediction error is almost the same as MISE (seeSection 4 of