D-optimal designs for full and reduced Fourier regression models

Abstract

The optimal designs for Fourier regression models under the D-optimality criterion are discussed in this article. First, we investigate the D-optimal designs for estimating two coefficients corresponding to either sine or cosine terms in a full Fourier regression model. In many biological applications, estimating such specific pairs of coefficients is of interest. As a result of this article, the D-optimal designs for estimating these “coefficient pairs” can be constructed either explicitly or numerically for Fourier regression models with any order. Our resulting designs are provided for Fourier regression models with order less than 6. Secondly, we discuss the sensitivity of our resulting optimal designs for a full Fourier regression model when the true model is actually a reduced version of the assumed one. Lastly, we provide the algorithm for obtaining the D-optimal designs for a reduced Fourier regression model and the D-optimal designs for a useful reduced Fourier model are constructed. The comparison study shows that the constructed designs incorporating the reduced model are efficient.

Appendix: Proof of Lemma 1

Let $\mathbf {x}_{\mathbf {\tau }}=(\cos ( a\tau _{1}) ,\ldots ,\cos ( a\tau _{m-1})) ^{T}.$ The following expansion

$$\begin{aligned} \cos ( t) =1-\frac{t^{2}}{2}+o\left( t^{2}\right) , \end{aligned}$$

implies that as $a\rightarrow 0,$

$$\begin{aligned} \varphi \left( \mathbf {x}_{\mathbf {\tau }},\,a\right) =\frac{a^{2m( m+1) }}{2^{m( m+1) }}\prod \limits _{i=1}^{m}\left( \tau _{i}^{2}\right) ^{2}\prod \limits _{1\le i<j\le m}\left( \tau _{j} ^{2}-\tau _{i}^{2}\right) ^{2}( 1+o(a)); \end{aligned}$$

then, $\lim _{a\rightarrow 0}\mathbf {\tau }^{*}( a)$ exists and can be obtained by maximizing

$$\begin{aligned} \tilde{\varphi }( \mathbf {\tau }) =\prod \limits _{i=1} ^{m-1}\left( \tau _{i}^{2}\right) ^{2}\left( 1-\tau _{i}^{2}\right) ^{2}\prod \limits _{1\le i<j\le m-1}\left( \tau _{j}^{2}-\tau _{i} ^{2}\right) ^{2} \end{aligned}$$

over T defined in Sect. 4.2. Let $y_{i}=\tau _{i}^{2}\in (0,\,1).$ In order to maximize the following quantity,

$$\begin{aligned} \hat{\varphi }( y)=\prod \limits _{i=1}^{m-1}y_{i}^{2}\left( 1-y_{i}\right) ^{2}\prod \limits _{1\le i<j\le m-1}\left( y_{j} -y_{i}\right) ^{2}, \end{aligned}$$

the conditions in (11)

$$\begin{aligned} \frac{\partial \log \hat{\varphi }( y) }{\partial y_{i}}=\frac{4}{y_{i}}+\sum \limits _{j=1,j\ne i}^{m-1}\frac{4}{y_{i}-y_{j}}=0,\quad i=1,\ldots ,m-1, \end{aligned}$$

(11)

must be satisfied. Similar arguments as given in Fedorov (1972) show that the polynomial $\phi (y)=( y-y_{1})(y-y_{2}), \ldots ,( y-y_{m-1})$ satisfies the differential equation

$$\begin{aligned} y( 1-y) \phi ^{\prime \prime }(y)( 2-4y) \phi ^{\prime }(y)+( m-1)( m+2) \phi (y)=0. \end{aligned}$$

(12)

It is well known that Eq. (12) has a unique solution given by the Jacobi polynomial $P_{m-1}^{(1,1)}(1-2y),$ and the lemma is now proved by transformation $y=\tau ^{2}.$ $\square$