Abstract

Longitudinal trends of observations can be estimated using the generalized multivariate analysis of variance (GMANOVA) model proposed by [10]. In the present paper, we consider estimating the trends nonparametrically using known basis functions. Then, as in nonparametric regression, an overfitting problem occurs. [13] showed that the GMANOVA model is equivalent to the varying coefficient model with non-longitudinal covariates. Hence, as in the case of the ordinary linear regression model, when the number of covariates becomes large, the estimator of the varying coefficient becomes unstable. In the present paper, we avoid the overfitting problem and the instability problem by applying the concept behind penalized smoothing spline regression and multivariate generalized ridge regression. In addition, we propose two criteria to optimize hyper parameters, namely, a smoothing parameter and ridge parameters. Finally, we compare the ordinary least square estimator and the new estimator.

1. Introduction

We consider the generalized multivariate analysis of variance (GMANOVA) model withobservations of -dimensional vectors of response variables. This model was proposed by [10]. Let, , and be an matrix of response variables, an matrix of non-stochastic centerized between-individual explanatory variables (i.e.,)

of, a matrix of non-stochastic within-individual explanatory variables of , and an matrix of error variables, respectively, whereis the sample size, is an -dimensional vector of ones and is a -dimensional vector of zeros. Then, the GMANOVA model is expressed as

where is a unknown regression coefficient matrix an is -dimensional unknown vector. We assume that where is a unknown covariance matrix of. Then we can express the GMANOVA model as

Let be an unbiased estimator of the unknown covariance matrix that is given by

Then, the maximum likelihood (ML) estimators of andare given byand, respectively. The ML estimators are the unbiased and asymptotically efficiency estimators of and.

In the GMANOVA model, , is often used as the th row vector of. Then, we estimate the longitudinal trends of using -polynomial curves. However, occasionally, the polynomial curve cannot thoroughly express flexible longitudinal trends. Hence, we consider estimating the longitudinal trends nonparametrically in the same manner as [11] and [5], i.e., we use the known basis function as and assume that is large. In the present paper, we refer to the GMANOVA model with obtained from the basis function as the nonparametric GMANOVA model. In the nonparametric GMANOVA model, it is well known that the ML estimators become unstable because becomes unstable when is large. Thus, we deal with the least square (LS) estimators of and, which are obtained by minimizing . Then, the LS estimators of and are obtained by and

respectively. Note that does not depend on. The LS estimators are simple and unbiased estimators of and. However, as well as ordinary nonparametric regression model, the LS estimators cause an overfitting problem when we use basis functions to estimate the longitudinal trends nonparametrically. In order to avoid the overfitting problem, we use instead of as the penalized smoothing spline regression (see, e.g., [2]), where is a smoothing parameter and is a known penalty matrix.

Let, and let. Then, the GMANOVA model can be expressed as

where is the th element of. This expression indicates that the GMANOVA model is equivalent to the varying coefficient model with non-longitudinal covariates [13], i.e.,

(4)

where and,. Hence, estimating the longitudinal trends in the GMANOVA model nonparametrically is equivalent to estimating the varying coefficients, nonparametrically. However, when multicollinearity occurs in, the estimate of, becomes unstable, as does the ordinary LS estimator of regression coefficient, because the variance of an estimator of becomes large. Hence, we avoid the multicollinearity problem in by the ridge regression.

When and in the model (1), [4] proposed a ridge regression. This estimator is generally defined by adding to in (3), where is referred to as a ridge parameter. Since the ridge estimator changes with, optimization of is very important. One method for optimizing is minimizing the criterion proposed by [7,8] in the univariate linear regression model (for multivariate case, see e.g., [15]). For the case in which and, [17] proposed the and its bias-corrected (modified;) criteria for optimizing the ridge parameter. However, an optimal cannot be obtained without an iterative computational algorithm because an optimal cannot be obtained in closed form.

On the other hand, [4] also proposed a generalized ridge (GR) regression in the univariate linear regression model, i.e., the model (1) with and, simultaneously with the ridge regression. The GR estimator is defined not by a single ridge parameter, but rather by multiple ridge parameters,. Then, several authors proposed a non-iterative GR estimator (see, e.g., [6]). [18] proposed a GR regression in the multivariate linear regression model, i.e., the model (1) with and. We call this generalized ridge regression the multivariate GR (MGR) regression. They also proposed the and criteria for optimizing ridge parameters in the MGR regression. They showed that the optimized by minimizing two criteria are obtained in closed form. [9] proposed non-iterative MGR estimators by extending non-iterative GR estimators. Several computational tasks are required in estimating nonparametrically because we determine the optimal and the number of basis functions simultaneously. Fortunately, [18] reported that the performance of the MGR regression is the almost same as that of the multivariate ridge regression. Hence, we use the MGR regression in order to avoid the multicollinearity problem that occurs in in order to reduce the number of computational tasks.

The remainder of the present paper is organized as follows: In Section 2, we propose new estimators using the concept of the penalized smoothing spline regression and the MGR regression. In Section 3, we show the target mean squared error (MSE) of a predicted value of. We then propose the and criteria to optimize ridge parameters and smoothing parameter in the new estimator. Using these criteria, we show that the optimized ridge parameters are obtained in closed form under the fixed. We also show the magnitude relationship between the optimized ridge parameters. In Section 4, we compare the LS estimator in (3) with the proposed estimator through numerical studies. In Section 5, we present our conclusions.

2. The New Estimators

In the model (1), we consider estimating the longitudinal trends nonparametrically by using basis functions. Then, we consider the following estimators in order to avoid the overfitting problem in the nonparametric GMANOVA model, and

(5)

where is a smoothing parameter and is a known penalty matrix. In this estimator, we must determine before using this estimator. Since is usually set as some nonnegative definite matrix, we assume thatis a nonnegative definite matrix. If, whereis a matrix of zeros, then this estimator corresponds to the LS estimators and in (1). Note that this estimator controls the smoothness of each estimated curve and, through only one parameter. When we use this estimator, we need to optimize the parameter because this estimator changes with.

If multicollinearity occurs in, then the LS estimator in (1) and the proposed estimator in (5) are not good estimators in the sense of having large variance. Note that neither the LS estimator nor the proposed estimator depend on. Hence, we avoid the multicollinearity problem for estimating. Multicollinearity often occurs when becomes large. Using the following estimator, the multicollinearity problem in can be avoided,

(6)

where is a ridge parameter. This estimator with corresponds to the estimator of [16]. If, then this estimator corresponds to the estimator in (5). Note that in this estimator corresponds to the ridge estimator for a multivariate linear model [17]. In this estimator, we need to optimize and because this estimator changes with these parameters. However, we cannot obtain the optimized and in closed form. Thus, we need to use an iterative computational algorithm to optimize two parameters. From another point of view, this estimator controls the smoothness of each estimated curve, through only one parameter. Hence, this estimator is not a well fitting curve when the smoothnesses of the true curves differ.

Hence, we apply the concept of the MGR estimator [18] to in order to obtain the optimized ridge parameter in closed form. Here, we derive the MGR estimator for the nonparametric GMANOVA model as follows:

(7)

where, is also a ridge parameter, , and is the orthogonal matrix that diagonalizes, i.e., where and  are eigenvalues of. It is clearly that,. In this estimator, since shrinks the estimators of, to 0, we can regard as controlling the smoothness of. Therefore, in this estimator, rough smoothness of the estimated curves is controlled by, and each smoothness of, is controlled by.

Clearly, and. The with for some corresponds to in (6). Thus, the estimator includes these estimators. The estimator is more flexible than these estimators and because has parameters and or has only one or two parameters. Hence, we consider and in estimating the longitudinal trends or the varying coefficient curve, while avoiding the overfitting and multicollinearity problems in the nonparametric GMANOVA model. When and, corresponds to the MGR estimator in [18].

3. Main Results

3.1. Target MSE

In order to define the MSE of the predicted value of, we prepare the following discrepancy function for measuring the distance between matrices and

:

Since is an unknown covariance matrix, we use the unbiased estimator in (2) instead of to estimate. Hence, we estimate using the following sample discrepancy function:

(8)

These two functions, and, correspond to the summation of the Mahalanobis distance and the sample Mahalanobis distance between the rows of and, respectively. Clearly, and. Through simple calculation, we obtain the following properties:

for any matrices, and， Using the discrepancy function, the MSE of the predicted value of is defined as

(9)

where, which is the predicted value of when we use and in (7). In the present paper, we regard and making the MSE the smallest as the principle optimum. However, we cannot use the MSE in (9) in actual application because this MSE includes unknown parameters. Hence, we must estimate (9) in order to estimate the optimum and.

3.2. The and Criteria

Letand. Note that. Hence, we obtain

From the properties of the functionand using, since is a nonstochastic variable and, and for any square matrix, we obtain

Note that.Thus, we can calculate as follows:

because, and are non-stochastic variables. For calculating the expectations in the MSE, we prove the following lemma.

Lemma 3.1. For any non-stochastic matrix, we obtain.

proof. Since, we obtain the th element of as,

. We obtain because for any and for any, where is defined as if and if. Hence we obtain . This result means that if and if. Thus, the lemma is proven.

Using this lemma, we obtain and. Hence, we obtain

By replacing with, we can propose the instinctive estimator of MSE, referred to as the criterion, as follows:

(10)

When we use this criterion, we optimize the ridge parameter and the smoothing parameter by the following algorithm:

1) We obtain, where if is given.

2) We obtain.

3) We obtain, where
, under fixed.

4) We optimize the ridge parameter and the smoothing parameter as and, respectively.

Note that this criterion corresponds to that in [18] when and.

There is some bias between the MSE in (9) and the criterion in (10) because the criterion is obtained by replacingin the MSE with. Generally, when the sample size is small or the number of explanatory variables is large, this bias becomes large. Then, we cannot obtain the higher-accuracy estimation of the optimum parameters because we cannot obtain the higher-accuracy estimation of MSE of in (9). Hence, we correct the bias between and the criterion. To correct the bias, we assume.

Letand.

Note thatand because and. Then, we obtain Since, (see, e.g., [14]) and

, we obtain

Therefore, we obtain the unbiased estimator for as, where. This implies that the bias correctedcriterion, denoted as (modified) criterion, is obtained by

(11)

As in the case of using the, we optimize and using this criterion as follows:

1) We obtain, where,

if is given.

2) We obtain.

3) We obtain, where
, under fixed.

4) We optimize the ridge parameter and the smoothing parameter as and , respectively.

Note that the criterion corresponds to that in [18] when and. The criterion completely omits the bias between the MSE of in (9) and the criterion in (10) by using a number of constant termsand. If and can be expressed in closed form for any fixed, we do not need the above iterative computational algorithm.

3.3. Optimizations using the andCriteria

Using the generalized criterion, which is given in (14), we can express the and criteria as follows:

Note that the terms with respect to in the and criteria correspond toand, respectively. Hence, we consider obtaining the optimum by minimizing the criterion. From Theorem A, the optimum is obtained in closed form as (15). Using the closed form in (15), we obtain and for each and any fixed as follows:

(12)

(13)

whereandare theth elements ofand,respectively,and. Note that and vary with. Since and are regarded as a function of, we can regard the and criteria for optimizing and in (10) and (11) as a function of. This means that we can use these criteria to optimize.

Then, we can rewrite the optimization algorithms to optimize the ridge parameter and the smoothing parameter by minimizing the and criteria in (10) and (11) as follows:

1) We obtain and.

2) We optimize the ridge parameter and the smoothing parameter as and, respectively, by using, and the closed forms in (12) and (13).

This means that we can reduce the processing time to optimize the parameters, and we need to use the optimization algorithm for only one parameter, , for any.

3.4. Magnitude Relationships between

Optimized Ridge Parameters

In this subsection, we prove the magnitude relationships between and,.

Lemma 3.2. For any, we obtain.

proof. Since we assume as a nonnegative definite matrix, there exists that satisfies (see, e.g., [3]). Then, since, we have. Hence, is a nonnegative definite matrix. This means that all of the eigenvalues of are nonnegative. Hence, all of the eigenvalues of are nonnegative. Thus, is also a nonnegative definite matrix for any. Since, we obtain as a nonnegative definite matrix for any. Thus, the lemma is proven.

Using the same idea, we have for any,. Therefore, the final terms of the and criteria in (10) and (11) are always greater than. In order to prove the magnitude relationship between and, we consider two situations in which is satisfied and is satisfied.

First, we consider to be satisfied. Let. Using, we obtain the following corollary:

Corollary 3.1. For any, we obtain .

proof. Through simple calculation, we obtain

Since, and from lemma 3.1, the corollary is proven.

This corollary indicates that is satisfied when is satisfied because, and is satisfied when is satisfied because and. Using these relationships, we obtain the following theorem.

Theorem 3.1. For any, we obtain .

proof. We consider the following situations:

1) is satisfied

2) is satisfied

3) is satisfied.

In (1), , because. In (3), , because becomes. Hence, we only consider situation (2). Note that, because and. This means that does not become. This theorem holds when, because, in this case, and. We also consider to be satisfied. Then, we obtain

Since is a positive definite matrix, for any. From corollary 3.1, we have for any. Hence we obtain for anysince, and. Thus, this theorem is proven.

This theorem corresponds to that in [9] when and.

From Theorem 3.1, we obtained the relationships between and for the case in which the optimized smoothing parameters and are the same. However, and are optimized by minimizing the and criteria in (10) and (11). Hence, and are generally different. Thus, we consider the relationship between and when. Since is regarded as a function of, we write as and for each optimized smoothing parameter.

Theorem 3.2. We consider the following situations:

1) or is satisfied

2) and are satisfied

3) is satisfied

4) is satisfied

5) oris satisfied.

For any and, we obtain the following relationships based on the above situations:

1) If (1), then

2) If (2) and (3), then

3) If (2) and (4), then

4) If (5), then.

proof. In (1) and (5), the relationships (i) and (iv) are true. Hence we need only prove relationships (ii) and (iii). Then we obtain and using the closed forms of (12) and (13). Through simple calculation, we obtain Since and the denominator is positive, the sign of is the same as the sign of. Hence we obtain relationships (ii) and (iii). Thus, the theorem is proven.

4. Numerical Studies

In this section, we compare the LS estimator and in (3) with the proposed estimator and in (7) through a numerical study. Let, and let be an matrix as follows:

The explanatory matrix is given by where, is an matrix and each row vector of is generated from the independent -dimensional normal distribution with mean and covariance matrix. Let, be a -dimensional vector. We set each as follows:

where and theth element of is. Each element of is Richard's growth curve model [12]. We set the longitudinal trends using these as. Note that, , which indicates that the last six rows of are obtained by changing the scale of. The response matrix is generated by where. Then, we standardized. Let, and. We set each element of as a cubic -spline basis function. Since is set using the cubic -spline, we note that. Additional details concerning and are reported in [2]. We simulate repetitions for each, , , and. In each repetition, we fixed, but varies. We searchandusing fminsearch, which is a program in the software Matlab used to search for a minimum value, because and cannot be obtained in closed form. In searching and, we transformand search optimizedby each criterion becauseand. In the search algorithm, the starting point for the search is set as. Then, we obtain the optimized ridge parameters andusing the closed forms of (12) and (13) in each repetition. In each repetition, we need to optimizebecauseandvary with. We calculateand for each in each repetition. Then, we adopt the optimized by minimizing each criterion in each repetition. After that, we calculate for each criterion, where, which is obtained using and for each criterion and the optimized in each repetition. The average of over repetitions is regarded as the MSE of. We compare the values predictedusing the estimators and with those using the LS estimators and, and the estimators and in (5). When we use, we obtain by minimizing and. As in the case of using, we adopt by using each criterion in each repetition for and. Some of the results are shown in Tables 1 and 2. The values in the tables are obtained by,

where,and

where.

Each estimator optimized by using the criterion for, , and is more improve than that by using the criterion for each estimator in almost all situations. This indicates that the criterion is a better estimator of the MSE of each predicted value of than the criterion. The reasons for this are that the criterion is an unbiased estimator of MSE

and each of the parameters in each estimator is optimized by minimizing the criterion. When, provides a greater improvement than either or in all situations. The estimator, which is optimized using the criterion, has the smallest MSE among these estimators for almost situations when. Here, provides a greater improvement than when in all situations. When is large, the estimator provides a greater improvement than in most situations when. On the other hand, provides a greater improvement than in most situations when is small, and. If, then and improve the LS estimator. Comparing the results for with the results for reveals that these estimators become poor esti mators when becomes large. The reasons for this are thought to be that and become unstable and the has some curves that are in a different scale. Each MSE using each method and the criterion is similar to that using the criterion if becomes large because is close to 1. When becomes large, improves the LS estimator more than when is small. Since controls the correlation in, the multicollinearity in becomes large when becomes large. Then, is not a good estimator because is unstable. Hence, we can avoid the multicollinearity problem in by using, which is one of the purposes of the present study. In all situations, the new estimators improve the LS estimator. In addition, is better than in most situations, especially when is small or is large. In general, optimized using is the best method.

5. Conclusions

In the present paper, we estimate the longitudinal trends nonparametrically by using the nonparametric GMANOVA model in (1), which is defined using basis functions as in the GMANOVA model. When we use basis functions as, the LS estimators and incur overfitting. In order to avoid this problem, we proposed and in (5) using the smoothing parameter and the known penalty non-negative definite matrix. However, if multicollinearity occurs in, and are not good estimators due to large variance. In the present paper, we also proposed in (7) in order to avoid the multicollinearity problem that occurs in and the overfitting problem by using basis functions as. The estimator controls the smoothness of each estimated longitudinal curve using only one parameter. On the other hand, in the estimator, the rough smoothness of estimated longitudinal curves is controlled using, and each smoothness of in the varying coefficient model (4) is controlled by.

We also proposed the and criteria in (10) and (11) for optimizing the ridge parameter and the smoothing parameter. Then, using the criterion in (14) and minimizing this criterion in Theorem A, we obtain the optimized using the and criteria in closed form as (12) and (13) for any. Thus, we can regard the and criteria as a function of.

Hence, we need to optimize only one parameter in order to optimize parameters in using these criteria. On the other hand, we must optimize two parameters when we use in (6). This optimization is difficult and requires a complicated program and a long processing time for simulation or analysis of real data because the optimized cannot be obtained in closed form even if is fixed. This is the advantage of using. This advantage does not appear to be important because of the high calculation power of CPUs. However, this advantage is made clear when we use together with variable selection. Even if becomes large, then this advantage remains when is used because the optimized obtained using each criterion is always obtained as (12) and (13) for any. Furthermore, we must optimize if we use model (1) to estimate the longitudinal trends. This means that we optimize the parameters in the estimators and calculate the valuation of the estimator for each, and then we compare these values in order to optimize. Since this optimization requires an iterative computational algorithm, we must reduce the processing time for estimating the parameters in the estimator. Hence, the advantage of using is very important. This optimized ridge parameter in (12) and (13) corresponds to that in [18] when and

.

Using some matrix properties, we showed that and in the and criteria are always nonnegative. From for any in lemma 3.1, we also established the relationship between and for any in corollary 3.1. Then, in Theorem 3.1, we established the relationship between and if and are the same, where and are obtained by minimizing the and criteria. Note that this relationship corresponds to that in [9] when and. In Theorem 3.2, we also established the relationships between and for the more general case, in which and are different. The reason of the relationship in Theorem 3.2 is occurred is that and for each can be regarded as a function of.

The numerical results reveal that and have some following properties. These estimation methods and improve the LS estimator in all situations, especially when is large. This indicates that the proposed estimators are better than the LS estimator. Even if becomes large, we note that is stable because we add the ridge parameter to in the LS estimator. This result indicates that the multicollinearity problem in can be avoided by using the estimator in (7). These estimators can be used to estimate the true longitudinal trends nonparametrically using basis functions as without overfitting. The LS estimator and the proposed estimators and optimized using the criterion provide a greater improvement than the estimators optimized using the criterion in most situations. The reason for this is that the criterion is the unbiased estimator of MSE of the predicted value of. Based on the present numerical study, and can be used to estimate the longitudinal trends in most situations. In addition, the can be used to optimize the smoothing parameter and the number of basis functions. Hence, we can use and, the parameters, , and of which are optimized by the criterion for estimating the longitudinal trends.

6. Acknowledgments

I would like to express my deepest gratitude to Dr. Hirokazu Yanagihara of Hiroshima University for his valuable ideas and useful discussions. In addition, I would like to thank Prof. Yasunori Fujikoshi, Prof. Hirofumi Wakaki, Dr. Kenichi Satoh and Dr. Kengo Kato of Hiroshima University for their useful suggestions and comments. Finally, I would like to thank Dr. Tomoyuki Akita of Hiroshima University for his advice with regard to programming.

7. Appendix

7.1. Minimization of the Criterion

In this appendix, we show that the optimizations using the and criteria in (10) and (11) are obtained in closed form as (12) and (13) for any. [9] proposed the generalized criterion for the MGR regression (originally the criterion for selection variables in the univariate regression was proposed by [1]). Similar to their idea, we proposed the criterion for the nonparametric GMANOVA model.

By omitting constant terms and some terms with respect to in the and criteria in (10) and (11), these criteria are included in a class of criteria specified by. This class is expressed by thecriterion as

(14)

where the function is given by (8). Note that and correspond to the terms with respect to in the and criteria. Using this criterion, we can deal systematically with the and criteria for optimizing. Let, which minimize the criterion for any. Then, and are obtained asand, respectively. Thus, we can deal systematically with the optimizations of when we use the and criteria. This means that we need only obtain in order to obtain and for any and some. If is obtained in closed form for any fixed, we do not need to use the iterative computational algorithm for optimizing the ridge parameter. In order to obtain, we obtain, in closed form, as shown in the following theorem.

Theorem A. For any and, is obtained as

(15)

where.

proof. Since and we use the

properties of the function in Section 3.1, we can calculate  in the criterion in (14) as follows:

Sincefor any, and for any, the second term in the right-hand side of the above equation can be calculated as Note that  because is an orthogonal matrix and. Hence, we obtain the following results:

Since and are diagonal matrices, we obtain. Hence is calculated as

whereand. Clearly, and change with. Based on this result and, we can calculate the criterion in (14) as follows:

Then, we calculate the second and third terms in the right-hand side of the above equation as follows:

where and are the th element of and, respectively. Clearly, and also vary with. Note that, for any because is a positive definite matrix (see, e.g., [3]). Let, be as follows:

(16)

Using, we can express

Since does not depend on, we can obtain by minimizing for each and any . In order to obtain, we consider the following function for:

(17)

If we restrict to be greater than or equal to 0, then this function is equivalent to the function in (16), which must be minimized. Note that and. Letting, we obtain

Let satisfy and, then is obtained by

where. Note that in (17) has a minimum value at, which is and. Note that the sign of is the same as the sign of. In order to obtain, we consider the following situations:

1) is satisfied

2) and are satisfied

3) and are satisfied.

In (1), , because and. In addition, for any, because, and indicates that the sign of is nonnegative. This means that the minimum value of in is obtained when in situation (1). In (2), , and then the minimum value of in is obtained when. In (3), since and, we obtain for any. Hence, is minimized when in. From the above results, we obtain as follows:

Thus, the theorem is proven.

Note that corresponds to that in [9] when and. Since we obtain and in closed form as (15) for any, we must optimize only one parameter in order to optimize parameters. The use of is advantageous because only an iterative computational algorithm is required for optimizing only one parameter for any. This means that we can reduce the processing time required to optimize the parameters in the estimator which is defined by (7). When we use in (5), we also need the same iterative computational algorithm to optimize only one parameter.

On the other hand, when we use in (6), the criterion for optimizing for any fixed is obtained as

Since we need to minimize in order to optimize, we cannot obtain that minimizes this criterion for in closed form, even if is fixed. Thus, we use an iterative computational algorithm to optimize the parameters and simultaneously. This iterative computational algorithm for optimizing two parameters is difficult and requires a longer processing time than the optimization of a single parameter

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