Locally Optimal Sequential Designs in the Dual Formulation
Abstract Generated abstract
This paper formulates a sequential experimental design problem for choosing observations that both discriminate among several linear hypotheses and refine the parameters of the true model. It introduces a duality-based functional combining a discrimination criterion with a D-optimality term, weighted by posterior hypothesis probabilities updated through Bayes’ formula. The authors derive a directional derivative and concavity result for mixtures of designs, then prove a necessary and sufficient local optimality condition expressed as an inequality over all admissible measurement points. Based on this condition, they propose an iterative procedure that updates estimates and posterior probabilities, adds measurements at maximizing points, and reduces to the standard D-optimality condition when one hypothesis has posterior probability one.
Full Text
LOCALLY OPTIMAL SEQUENTIAL DESIGNS IN THE DUAL FORMULATION
S. M. Bersenev, N. I. Sablin
(Presented by the scientific seminar of the Research Institute of Nuclear Physics, Electronics, and Automation)
In the present work a functional is introduced that reflects the duality of the experiment; necessary and sufficient conditions of local optimality are given; and an iterative procedure is proposed for constructing a locally optimal design. A similar problem was solved, for example, in [1] by means of the information method, but without deriving optimality conditions.
Let \(\eta_1(x,\Theta_1), \eta_2(x,\Theta_2), \ldots, \eta_m(x,\Theta_m)\) be continuous functions, linear in the parameters, defined on a closed set \(X\), where \(X\) is the domain of possible measurements, \(X \subseteq \bar X\); let \(\xi(x)\) be a probability measure given on the domain \(X\), and let \(y(x)\) be the results of observation, with
\[ E(y/x)= \begin{cases} \eta_1(x,\theta_1),\\ \eta_m(x,\theta_m), \end{cases} \tag{1} \]
where \(E\) is the averaging operator over the observation results.
We shall assume that the observation results are distributed according to a normal law with probability density function
\[ \varphi_i(y_n)=(2\pi(\lambda(x)^{-1}+d_i(x,\varepsilon)))^{-1/2} \exp\left\{-\frac{1}{2}\, \frac{(y_n-\hat{\eta}_i(x,\hat{\Theta}_i,\varepsilon))^2} {\lambda(x)^{-1}+d_i(x,\varepsilon)}\right\}; \]
here \(\hat{\eta}_i(x,\hat{\Theta}_i,\varepsilon)\) is the least-squares estimate of the \(i\)-th hypothesis after carrying out the experiment according to the design \(\varepsilon\), which is understood as
\[ \varepsilon= \left\{ \begin{matrix} \xi(x_1)\ \cdots\ \xi(x_n)\\ x_1\ \cdots\ x_n \end{matrix} \right\},\qquad x_i\in X,\qquad \int_X d\xi(x)=1; \]
\[
d_i(x,\varepsilon)=f_i^T(x)M_i^{-1}(\varepsilon)f_i(x)
\]
is the error corridor of the \(i\)-th hypothesis;
\[
M_i(\varepsilon)=\int_X \lambda(x) f_i^T(x)\, d\xi(x)
\]
is the information matrix;
\(\lambda(x)^{-1}\) is the variance of the observations at the point \(x\).
Introduce the vector of a priori probabilities associated with the corresponding hypotheses,
\[ P^T=\|p_{1,n},\ldots,p_{m,n}\|, \]
whose elements after each measurement at the point \(x\) can be computed by Bayes’ formula:
\[ p_{i,n}=\frac{p_{i,n-1}\varphi_i(y_n)} {\sum_{j=1}^{m}p_{j,n-1}\varphi_j(y_n)}. \tag{2} \]
Let us pose the problem of finding an experimental design, by carrying out which we shall perform the best experiment not only for discrimination, but also for refining the parameters of the true model in the sense of \(D\)-optimality. Introduce the functional:
\[ \Delta(\varepsilon)=P^T A(\varepsilon)P; \tag{3} \]
\[ A(\varepsilon)=\omega_1\int_X D(\varepsilon,x)\,d\xi(x)+\omega_2 L(\varepsilon); \]
here \(\omega_1\) and \(\omega_2\) are weighting factors;
\(D(\varepsilon,x)\), \(L(\varepsilon)\) are matrices characterizing discrimination and refinement, respectively. Define their elements as follows:
\[ d_{ij}= \begin{cases} 0, & i=j,\\ \lambda(x)\bigl(\hat{\eta}_i(\varepsilon)-\hat{\eta}_j(\varepsilon)\bigr), & i\ne j; \end{cases} \]
\[ \hat{\eta}_i(\varepsilon)=\hat{\eta}_i(x,\hat{\theta}_i,\varepsilon), \]
\[ l_{ij}= \begin{cases} \ln |M_j(\varepsilon)|, & i=j,\\ 0, & i\ne j, \end{cases} \]
In addition, introduce the matrices
\[ \overline{A}(\varepsilon,x)=\omega_1 D(\varepsilon,x)+\omega_2\overline{L}(\varepsilon,x), \]
\[ \overline{l}_{ij}= \begin{cases} 0, & i\ne j,\\ \lambda(x)d_j(x,\varepsilon), & i=j; \end{cases} \]
\[ \widetilde{A}(\varepsilon)=\omega_1\int_X D(\varepsilon,x)\,d\xi(x)+\omega_2 K, \]
\[ k_{ij}= \begin{cases} 0, & i\ne j,\\ k_j, & i=j, \end{cases} \]
\(k_j\) is the number of parameters in the \(j\)-th hypothesis.
We shall call the design \(\varepsilon\) locally optimal if it delivers the maximum of (3) for the given \(P\).
Consider a design that is a linear combination of two designs
\[ \varepsilon=(1-\alpha)\varepsilon_1+\alpha\varepsilon_2,\quad \alpha\in[0,1]; \]
\(\varepsilon_1\) is nondegenerate, i.e. \(|M_j(\varepsilon_1)|\ne 0,\ j=1,\ldots,m\).
Lemma.
\[ \left.\frac{\partial\Delta(\varepsilon)}{\partial\alpha}\right|_{\alpha=0} = P^T\int_X \overline{A}(\varepsilon_1,x)\,d\xi_2(x)\,P - P^T\widetilde{A}(\varepsilon_1)P, \tag{4} \]
\[ \frac{\partial^2\Delta(\varepsilon)}{\partial\alpha^2}<0;\quad \alpha\in[0,1]. \tag{5} \]
Proof. Let us compute the indicated derivatives for the elements \(L(\varepsilon)\), \(\int_X D(\varepsilon,x)\,d\xi(x)\).
\[ \begin{aligned} \left.\frac{\partial}{\partial\alpha}\ln |M_j(\varepsilon)|\right|_{\alpha=0} &= \left.\frac{\partial}{\partial\alpha}\ln |(1-\alpha)M_j(\varepsilon_1)+\alpha M_j(\varepsilon_2)|\right|_{\alpha=0} \\ &=\left.\operatorname{sp} M_j^{-1}(\varepsilon)\,[M_j(\varepsilon_2)-M_j(\varepsilon_1)]\right|_{\alpha=0} \\ &=\operatorname{sp} M_j^{-1}(\varepsilon_1)M_j(\varepsilon_2) -\operatorname{sp} M_j^{-1}(\varepsilon_1)\cdot M_j(\varepsilon_1) \\ &=\int_X \lambda(x) f_j^T(x) M_j^{-1}(\varepsilon_1) f_j(x)\,d\xi(x)-k_j, \end{aligned} \]
\[ \begin{aligned} \frac{\partial^2}{\partial\alpha^2}\ln |M_j(\varepsilon)| &=\operatorname{sp}\{-M_j^{-1}(\varepsilon)[M_j(\varepsilon_2)-M_j(\varepsilon_1)] M_j^{-1}(\varepsilon)[M_j(\varepsilon_2)-M_j(\varepsilon_1)]\} \\ &=-\operatorname{sp}\{M_j^{-1}(\varepsilon)[M_j(\varepsilon_2)-M_j(\varepsilon_1)]\}^2<0. \end{aligned} \]
Let us transform the element of the matrix \(\int_X D(\varepsilon,x)\,d\xi(x)\):
\[ \int_X \lambda(x)\bigl(\hat{\eta}_i(\varepsilon)-\hat{\eta}_j(\varepsilon)\bigr)^2\,d\xi(x) = \int_X \lambda(x)\bigl(\hat{\eta}_i(\varepsilon)-\eta_u+\eta_u-\hat{\eta}_j(\varepsilon)\bigr)^2\,d\xi(x), \]
where \(\eta_u\) is the true hypothesis, equal at the point \(x\) to the result of the observation,
\(\hat{\eta}_j(\varepsilon)\) is a quantity constructed in the manner of the best linear estimate. Assuming that the system of hypotheses is independent, we obtain
\[ \begin{aligned} \int_X \lambda(x)\bigl(\hat{\eta}_i(\varepsilon)-\hat{\eta}_j(\varepsilon)\bigr)^2\,d\xi(x) &= \int_X \lambda(x)\bigl(\eta_u-\hat{\eta}_i(\varepsilon)\bigr)^2\,d\xi(x) \\ &\quad+ \int_X \lambda(x)\bigl(\eta_u-\hat{\eta}_j(\varepsilon)\bigr)^2\,d\xi(x). \end{aligned} \]
In computing the derivatives we shall use the result of [2]:
\[ \left. \frac{\partial}{\partial\alpha} \int_X \lambda(x)\bigl(\eta_u-\hat{\eta}_j(\varepsilon)\bigr)^2\,d\xi(x) \right|_{\alpha=0} = \int_X \lambda(x)\bigl(\eta_u-\hat{\eta}_j(\varepsilon_1)\bigr)^2\,d\xi_2(x) - \int_X \lambda(x)\bigl(\eta_u-\hat{\eta}_j(\varepsilon_1)\bigr)^2\,d\xi_1(x). \]
\[ \frac{\partial^2}{\partial\alpha^2} \int_X \lambda(x)\bigl(\eta_u-\hat{\eta}_j(\varepsilon)\bigr)^2\,d\xi(x)<0. \]
Then
\[ \begin{aligned} \left. \frac{\partial}{\partial\alpha} \int_X \lambda(x)\bigl(\hat{\eta}_i(\varepsilon)-\hat{\eta}_j(\varepsilon)\bigr)^2\,d\xi(x) \right|_{\alpha=0} &= \int_X \lambda(x)\bigl(\hat{\eta}_i(\varepsilon_1)-\hat{\eta}_j(\varepsilon_1)\bigr)^2\,d\xi_2(x) \\ &\quad- \int_X \lambda(x)\bigl(\hat{\eta}_i(\varepsilon_1)-\hat{\eta}_j(\varepsilon_1)\bigr)^2\,d\xi_1(x). \end{aligned} \]
Now, using the notation introduced, it is not difficult to obtain (4) and (5).
Theorem. In order that the design \(\varepsilon\) be locally optimal, it is necessary and sufficient that
\[ P^{T}\widetilde A(\varepsilon)P \geq P^{T}\overline A(\varepsilon,x)P,\quad \forall x\in X . \tag{6} \]
Proof.
- Necessity. Let \(\varepsilon^{*}\) be a locally optimal design and suppose that at the point \(\dot x\) condition (6) is not satisfied. Then, considering the design
\[ \varepsilon=(1-\alpha)\varepsilon^{*}+\alpha\varepsilon(\dot x), \]
where \(\varepsilon(\dot x)\) is a design concentrated at a single point, and choosing sufficiently small \(\alpha\), we, using (4), obtain
\[ \Delta(\varepsilon)>\Delta(\varepsilon^{*}), \]
which contradicts the definition of \(\varepsilon^{*}\).
- Sufficiency. Suppose that (6) is satisfied for \(\varepsilon\), which is not locally optimal, i.e., there exists \(\widetilde\varepsilon\) for which
\[ \Delta(\widetilde\varepsilon)>\Delta(\varepsilon^{*}). \]
Then, taking the linear combination
\[ \varepsilon=(1-\alpha)\varepsilon^{*}+\alpha\widetilde\varepsilon, \]
and by virtue of concavity (condition (5)), we obtain
\[ \left.\frac{\partial\Delta(\varepsilon)}{\partial\alpha}\right|_{\alpha=0} = P^{T}\left\{\int_{\dot X}\overline A(\varepsilon^{*},x)\,d\widetilde\xi(x)\right\}P - P^{T}\widetilde A(\varepsilon^{*})P >0, \]
i.e.,
\[ \max_{x\in X} P^{T}\overline A(\varepsilon^{*},x)P > P^{T}\widetilde A(\varepsilon^{*})P, \]
which contradicts (6). The theorem is proved.
Using the theorem just proved, one may propose the following iterative procedure for constructing a locally optimal experimental-design strategy.
A. An initial experiment is carried out with design \(\varepsilon^{s}\), nondegenerate for all hypotheses; estimates \(\hat\eta_i(\varepsilon^{s})\), \(i=1,\ldots,m\), are obtained.
B. The vector of prior probabilities is specified:
\[ P_s^{T}=\|p_{1,s},\ldots,p_{m,s}\|. \]
C. A point \(x^{s}\) is found satisfying
\[ \max_{x\in X} P_s^{T}\overline A(\varepsilon^{s},x)P_s. \]
D. At this point an additional measurement is made, the probability vector is recalculated, new estimates \(\hat\eta_i(\varepsilon^{s+1})\), \(i=1,\ldots,m\), are constructed, and then we return to C,
The procedure \((B,\Gamma)\) is carried out until (6) is satisfied. At each step of the iterative procedure we have the inequality
\[ \Delta\!\left(\varepsilon^{s+1}\right)>\Delta(\varepsilon), \]
(\(s\) is the iteration number), since for small \(\alpha_s\) the following relations hold:
\[ \varepsilon^{s+1}=(1-\alpha)\varepsilon^s+\alpha\,\varepsilon(x^s), \]
\[ \frac{\Delta\!\left(\varepsilon^{s+1}\right)-\Delta(\varepsilon^s)}{\alpha_s} \simeq P_s^T \widetilde{A}(\varepsilon^s,x^s)P_s - P_s^T \widetilde{A}(\varepsilon^s)P_s . \]
Condition (6), when one of the \(p_{j,s}\) is equal to one, becomes the condition of \(D\)-optimality for the \(j\)-th hypothesis.
REFERENCES
- G. S. Fedorova. Zavodskaya laboratoriya, 1970, No. 5.
- V. V. Fedorov. Locally Optimal Sequential Designs of Discriminating Experiments, Moscow, Moscow State University Press, 1970.