A-optimal designs for non-parametric symmetrical global sensitivity analysis

Abstract

In the early stage of exploring a complex system, a preliminary experiment is used to capture the key characteristics of the model. Symmetrical global sensitivity analysis (SGSA) is one such experiment that explores the symmetrical structure of model by decomposing the model into independent symmetric functions. However, the existing experimental plans for SGSA rely on deterministic computational models that produce unique values of outputs when executed for specific values of inputs. In this paper, the problem of designing experiments for non-parametric SGSA is considered. Here the phrase “non-parametric” refers to model outputs containing random errors. The main result in the paper shows that a symmetrical design with certain constraints achieves A-optimum for the estimation of each output element function, and guarantees the superiority of the SGSA result. The statistical properties of non-parametric SGSA based on the optimal designs are further discussed, showing that the non-influential sensitivity indices can be estimated with low bias and volatility. Two explicit structures of the optimal designs are obtained. The optimality of the derived design is validated by simulation in the end.

Appendix

1.1 Proof of Property 2.2

We first choose a ${\varvec{\gamma _{1}}}$ from H. Then from the definition of symmetrical design, each ${\varvec{\gamma _{1}}}\sigma$ ($\sigma \in G$) is in H.

Denote the matrix that merges the same rows of

$$\begin{aligned} \left( \begin{array}{cccc} {\varvec{\gamma _{1}}}^{'} &{} {\varvec{\gamma _{1}}}\sigma _{1}^{'} &{} \ldots &{} {\varvec{\gamma _{1}}}\sigma _{g}^{'} \\ \end{array} \right) ^{'} \end{aligned}$$

as $H^{{\varvec{\gamma _{1}}}},$ and secondly choose a ${\varvec{\gamma _{2}}}$ from H that is not in $H^{{\varvec{\gamma _{1}}}}.$ Then each ${\varvec{\gamma _{2}}}\sigma ,$ $\sigma \in G$ is in H but not in $H^{{\varvec{\gamma _{1}}}}.$ Otherwise, suppose ${\varvec{\gamma _{2}}}\pi \in H^{{\varvec{\gamma _{1}}}},$ then $({\varvec{\gamma _{2}}}\pi )\pi ^{-1}={\varvec{\gamma _{2}}}\in H^{{\varvec{\gamma _{1}}}}.$

And so on, H can be expressed as the form of Property 2.2. $\square$

1.2 Proof of Lemma 3.1

Denote the independent parameters of $f_l({\varvec{X}})$ as ${\varvec{\beta _l}},$ then based on the restrictions of $f_l({\varvec{X}})$, a matrix $C_l$ with rank $d_{l}$ can be obtained such that

$$\begin{aligned} f_{l}(\varvec{X})=C_{l} \varvec{\beta }_{l}. \end{aligned}$$

It follows that Model (2) can be transformed into the linear model:

$$\begin{aligned} {\varvec{Y}}=\sum _{l=1}^{k} C_{l} \varvec{\beta }_{l}+\varvec{\varepsilon }=C \varvec{\beta }+\varvec{\varepsilon } \end{aligned}$$

where $C=\left( C_{0}, C_{1}, \ldots , C_{k}\right)$ and $\varvec{\beta }=\left( \varvec{\beta }_{0}^{\prime }, \varvec{\beta }_{1}^{\prime }, \ldots , \varvec{\beta }_{\varvec{k}}^{\prime }\right) ^{\prime }.$

If H is feasible, then ${\varvec{\beta }}$ is estimable, and C has full column rank. The least-square estimation of $\beta$ is $\widehat{\beta }=\left( C^{\prime } C\right) ^{-1} C^{\prime } {\varvec{Y}}.$ Let

$$\begin{aligned} E_{l}=\left( \begin{array}{ccccccc}{0}&{\ldots }&{0}&{I_{d_{l}}}&{0}&{\ldots }&{0}\end{array}\right) ^{\prime } \end{aligned}$$

where $I_{d_l}$ is the identity matrix of order $d_l.$ Then by Gauss-Markov theorem,

$$\begin{aligned} \widehat{f_{l}(\varvec{X})}=C E_{l} E_{l}^{\prime } \widehat{\varvec{\beta }} \end{aligned}$$

(9)

is the best linear unbiased estimation of $f_{l}(\varvec{X})$ and

$$\begin{aligned} \begin{aligned} {\text {Var}}\left( \widehat{f_{l}(\varvec{X})}\right)&=C E_{l} E_{l}^{\prime }\left( C^{\prime } C\right) ^{-1} C^{\prime } C\left( C^{\prime } C^{\prime } C^{-1} E_{l} E_{l}^{\prime } C^{\prime } \sigma ^{2}\right. \\&=C E_{l} E_{l}^{\prime }\left( C^{\prime } C\right) ^{-1} E_{l} E_{l}^{\prime } C^{\prime } \sigma ^{2} \end{aligned} \end{aligned}$$

From Lemma 2.2 of Wang et al. (2012), we can be obtain that

$$\begin{aligned} E_{l}^{\prime }\left( C^{\prime } C\right) ^{-1} E_{l} \ge \left( E_{l}^{\prime } C^{\prime } C E_{l}\right) ^{-1} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {\text {Var}}\left( \widehat{f_{l}(\varvec{X})}\right)&\ge C E_{l}\left( E_{l}^{\prime }\left( C^{\prime } C\right) E_{l}\right) ^{-1} E_{l}^{-1} C^{-1} \sigma ^{2} \\&=C_{l}\left( C_{l}^{\prime } C_{l}\right) ^{-1} C_{l}^{\prime } \sigma ^{2} \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} {\text {tr}}\left( {\text {Var}}(\widehat{f_{l}(\varvec{X})})\right) \ge \sigma ^{2} {\text {tr}}\left( P_{C_{l}}\right) =\sigma ^{2} r k\left( P_{C_{l}}\right) =\sigma ^{2} r k\left( C_{l}\right) =\sigma ^{2} d_{l} \end{aligned}$$

$\square$

1.3 Proof of Theorem 3.1

First, we introduce Lemma 5.1 to show the Theorem 3.1.

Lemma 5.1

For symmetrical designs, if $\chi _{1}(\sigma )\equiv 1$ and $\sum \nolimits _{\sigma \in G}\chi _{l}(\sigma )=0$ for $l=2,\ldots ,k,$ then we have ${\varvec{M_l}}f_l({\varvec{X}})=f_l({\varvec{X}})$ for $l=1,\ldots ,k.$

Proof

From the definition of ${\varvec{M_l}},$ we have

$$\begin{aligned} {\varvec{M_{l}}} \cdot f_{l}(\varvec{X})= & {} \left[ \frac{1}{|G|} \sum _{\sigma \in G} \chi _{l}(\sigma )\left( P(\sigma )-P_{n}\right) \right] \left( \begin{array}{c}{f_{l}\left( \varvec{X}_{1}\right) } \\ {f_{l}\left( \varvec{X}_{2}\right) } \\ {\vdots } \\ {f_{l}\left( \varvec{X}_{\varvec{n}}\right) }\end{array}\right) \\= & {} \frac{1}{|G|} \sum _{\sigma \in G} \chi _{l}(\sigma )\left[ \left( \begin{array}{c}{f_{l}\left( \varvec{X}_{\mathbf {1} \sigma }\right) } \\ {f_{l}\left( \varvec{X}_{2} \sigma \right) } \\ {\vdots } \\ {f_{l}\left( \varvec{X}_{n} \sigma \right) }\end{array}\right) -\frac{1}{n} \left( \begin{array}{c}{\sum \limits _{i=1}^{n} f_{l}\left( \varvec{X}_{i}\right) } \\ {\sum \limits _{i=1}^{n} f_{l}\left( \varvec{X}_{i}\right) } \\ {\vdots } \\ {\sum \limits _{i=1}^{n} f_{l}\left( \varvec{X}_{i}\right) }\end{array}\right) \right] \end{aligned}$$

Noting that $(\sum \nolimits _{\sigma \in G}\chi _l(\sigma ))(\sum \nolimits _{i=1}^{n}f_l({\varvec{X_i}}))=0$ based on the constrain of the model and the condition of $\chi _l$, the above expression equals

$$\begin{aligned} \frac{1}{|G|} \sum _{\sigma \in G} \chi _{l}(\sigma ) \left( \begin{array}{c}{f_{l}\left( \varvec{X}_{\mathbf {1}} \sigma \right) } \\ {f_{l}\left( \varvec{X}_{\mathbf {2}} \sigma \right) } \\ {\vdots } \\ {f_{l}\left( \varvec{X}_{\varvec{n}} \sigma \right) }\end{array}\right) =\left( \begin{array}{c}{\frac{1}{|G|} \sum \limits _{\sigma \in G} \chi _{l}(\sigma ) f_{l}\left( \varvec{X}_{\mathbf {1}} \sigma \right) } \\ {\frac{1}{|G|} \sum \limits _{\sigma \in G} \chi _{l}(\sigma ) f_{l}\left( \varvec{X}_{2} \sigma \right) } \\ {\vdots } \\ {\frac{1}{|G|} \sum \limits _{\sigma \in G} \chi _{l}(\sigma ) f_{l}\left( \varvec{X}_{\varvec{n}} \sigma \right) }\end{array}\right) . \end{aligned}$$

From the symmetry of $f_l,$ we have

$$\begin{aligned} f_l({\varvec{X_i}})=\frac{1}{|G|}\sum \limits _{\sigma \in G}\chi _l(\sigma )f_l({\varvec{X_i}}\sigma ).\end{aligned}$$

Thus

$$\begin{aligned} {\varvec{M_{l}}} \cdot f_{l}(\varvec{X})=\left( \begin{array}{c}{f_{l}\left( \varvec{X}_{1}\right) } \\ {f_{l}\left( \varvec{X}_{2}\right) } \\ {\vdots } \\ {f_{l}\left( \varvec{X}_{n}\right) }\end{array}\right) =f_{l}(\varvec{X}). \end{aligned}$$

$\square$

Then we give the proof of Theorem 3.1.

From Lemma 5.1 and Property 2.1, we have

$$\begin{aligned} E(\widehat{f_{l}({\varvec{X}})})= & {} {\varvec{M_l}}E({\varvec{Y}})={\varvec{M_l}}({\varvec{f_0}}+f_{l}({\varvec{X}})+\ldots +f_{l}({\varvec{X}}))\\= & {} {\varvec{M_l}}({\varvec{M_0}}{\varvec{f_0}}+{\varvec{M_1}}f_{l}({\varvec{X}})+\ldots +{\varvec{M_k}}f_{k}({\varvec{X}}))\\= & {} {\varvec{M_lM_l}}f_{l}({\varvec{X}})={\varvec{M_l}}f_{l}({\varvec{X}})=f_{l}({\varvec{X}}) \end{aligned}$$

Thus $\widehat{f_{l}({\varvec{X}})}$ is unbiased.

Furthermore, we have from Lemma 3.1 that,

$$\begin{aligned}{\varvec{M_l}}f_{l}({\varvec{X}})={\varvec{M_l}}C_l{\varvec{\beta _l}}=C_l{\varvec{\beta _l}}\end{aligned}$$

for any ${\varvec{\beta _l}}.$ Thus $R(C_l)\subset R({\varvec{M_l}}),$ and

$$\begin{aligned}d_l=rk(C_l)\le rk({\varvec{M_l}}).\end{aligned}$$

Given $rk({\varvec{M_l}})\le d_l,$ we obtain $rk({\varvec{M_l}})=d_l,$ and

$$\begin{aligned} Var(\widehat{f_{l}({\varvec{X}})})= & {} Var({\varvec{M_lY}})={\varvec{M_l}}Var({\varvec{Y}}){\varvec{M_l}}^{'}\\= & {} \sigma ^{2}{\varvec{M_l}}{\varvec{M_l}}^{'}=\sigma ^{2}{\varvec{M_l}}. \end{aligned}$$

It follows that

$$\begin{aligned} tr(Var(\widehat{f_{l}({\varvec{X}})}))= & {} tr(\sigma ^{2}{\varvec{M_l}})=\sigma ^{2}rk({\varvec{M_l}})=\sigma ^{2}d_{l}. \end{aligned}$$

$\square$

1.4 Proof of Corollary 3.1

From the proof of Theorem 3.1, we just need to prove that $rk({\varvec{M_l}})=d_l.$

Since ${\varvec{M_l}}$ is a projection matrix, we have

$$\begin{aligned} rk({\varvec{M_l}})= & {} tr({\varvec{M_l}})=tr\left[ \frac{1}{|G|}\sum \limits _{\sigma \in G}\chi _{l}(\sigma )(P(\sigma )-P_n)\right] \\= & {} \frac{1}{|G|}\sum \limits _{\sigma \in G}\chi _{l}(\sigma )tr(P(\sigma )-P_n)=\frac{1}{|G|}\sum \limits _{\sigma \in G}\chi _{l}(\sigma )(trP(\sigma )-1)\\= & {} \frac{1}{|G|}\sum \limits _{\sigma \in G}\chi _{l}(\sigma )trP(\sigma )-\frac{1}{|G|}\sum \limits _{\sigma \in G}\chi _{l}(\sigma ). \end{aligned}$$

Suppose there are t rows of H satisfying (ii) of the corollary and $n-t$ rows satisfying (i) with w pathways. Obviously, each of the w pathways constitute a block of run size |G|.

Then from the definition of $P(\sigma ),$ we obtain that $P(e)=I_n$ and the diagonal elements of $P(\sigma )$ have t numbers of 1 and $n-t$ numbers of 0. It follows that $trP(e)=n=w|G|+t$ and $trP(\sigma )=t$ for $\sigma \ne e.$

${\varvec{(i).}}$ For $l=1,$ we obtain from $\chi _{1}(\sigma )\equiv 1$ that

$$\begin{aligned} rk({\varvec{M_l}})= & {} \frac{1}{|G|}\sum \limits _{\sigma \in G}trP(\sigma )-1=\frac{1}{|G|}(n+(|G|-1)t)-1\\= & {} \frac{1}{|G|}(w|G|+t+(|G|-1)t)-1=w+t-1. \end{aligned}$$

On the other hand, based on constrain (4), each $\{f_1({\varvec{X_i}}), f_1({\varvec{X_i}}\sigma _1),\ldots ,f_1({\varvec{X_i}}\sigma _{g})\}$ has one independent parameter, i.e., there are $w+t$ independent parameters. Furthermore, given constrain (3)

$$\begin{aligned} \sum \limits _{i=1}^{n}f_1({\varvec{X_i}})=0,\end{aligned}$$

there are $w+t-1$ independent parameters in $f_1({\varvec{X}}),$ and

$$\begin{aligned} rk({\varvec{M_1}})=w+t-1=d_1.\end{aligned}$$

${\varvec{(ii).}}$ For $l=2,\ldots ,k,$ we obtain from the conditions $\sum \nolimits _{\sigma \in G}\chi _{l}(\sigma )=0$ and $\chi _{l}(e)=1$ that

$$\begin{aligned} rk({\varvec{M_l}})=\frac{1}{|G|}\sum \limits _{\sigma \in G}\chi _{l}(\sigma )trP(\sigma )=\frac{1}{|G|}tr\left( \sum \limits _{\sigma \in G}\chi _{l}(\sigma )P(\sigma )\right) ,\end{aligned}$$

For $\sigma \ne e,$ suppose the first t diagonal elements of $P(\sigma )$ are non-zero, then

$$\begin{aligned} \sum \limits _{\sigma \in G}\chi _{l}(\sigma )P(\sigma )= & {} \chi _{l}(e)\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 1 &{} \star &{} \star &{} \star \\ \star &{} 1 &{} \star &{} \star \\ \star &{} \star &{} \ddots &{} \star \\ \star &{} \star &{} \star &{} 1 \\ \end{array}\right) +\sum \limits _{\sigma \ne e}\chi _{l}(\sigma )\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1 &{} \star &{} \star &{} \star &{} \star &{}\star \\ \star &{} \ddots &{} \star &{} \star &{}\star &{}\star \\ \star &{} \star &{} 1 &{} \star &{}\star &{}\star \\ \star &{} \star &{} \star &{} 0 &{} \star &{}\star \\ \star &{} \star &{} \star &{} \star &{}\ddots &{}\star \\ \star &{} \star &{} \star &{} \star &{} \star &{} 0\\ \end{array} \right) \\= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \sum \limits _{\sigma \in G}\chi _{l}(\sigma ) &{} \star &{} \star &{} \star &{} \star &{}\star \\ \star &{} \ddots &{} \star &{} \star &{}\star &{}\star \\ \star &{} \star &{} \sum \limits _{\sigma \in G}\chi _{l}(\sigma ) &{} \star &{}\star &{}\star \\ \star &{} \star &{} \star &{} \chi _{l}(e) &{} \star &{} \star \\ \star &{} \star &{} \star &{} \star &{}\ddots &{}\star \\ \star &{} \star &{} \star &{} \star &{} \star &{} \chi _{l}(e)\\ \end{array} \right) \\= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0 &{} \star &{} \star &{} \star &{} \star &{}\star \\ \star &{} \ddots &{} \star &{} \star &{}\star &{}\star \\ \star &{} \star &{} 0 &{} \star &{}\star &{}\star \\ \star &{} \star &{} \star &{} 1 &{} \star &{} \star \\ \star &{} \star &{} \star &{} \star &{}\ddots &{}\star \\ \star &{} \star &{} \star &{} \star &{} \star &{} 1\\ \end{array} \right) \end{aligned}$$

It follows that

$$\begin{aligned} rk({\varvec{M_l}})=\frac{1}{|G|}(n-t)=\frac{1}{|G|}(|G|w)=w. \end{aligned}$$

Furthermore, for the ${\varvec{X_t}}$ whose corresponding ${\varvec{\alpha }}$ satisfies (ii) of the corollary, we have

$$\begin{aligned} f_l({\varvec{X_t}})=\frac{1}{|G|}\sum \limits _{\sigma \in G}\chi _{l}(\sigma )f({\varvec{X_t}}\sigma )=\left( \frac{1}{|G|}\sum \limits _{\sigma \in G}\chi _{l}(\sigma )\right) f({\varvec{X_t}})=0.\end{aligned}$$

For the remaining ${\varvec{X_t}}$s, the corresponding ${\varvec{\alpha }}$ satisfy (i) of the corollary, and compose w pathways. Suppose each pathway corresponds to ${\varvec{F}}=(f_l({\varvec{X_i}}), f_l({\varvec{X_i}}\sigma _1),\ldots ,f_l({\varvec{X_i}}\sigma _{g}))^{'},$ then it has ${\varvec{F}}={\varvec{A_lF}}$ from constrain (4) of the model. Given the condition $rk(I_{|G|}-{\varvec{A_l}})=g$ in (II), there is one independent parameter in each $\{f_l({\varvec{X_i}}), f_l({\varvec{X_i}}\sigma _1),\ldots ,f_l({\varvec{X_i}}\sigma _{g})\}.$ Then there are w independent parameters in $f_l({\varvec{X}}).$

Therefore,

$$\begin{aligned} rk({\varvec{M_l}})=w=d_l \end{aligned}$$

for $l=2,\ldots ,k.$ $\square$