3.1.3. Distributed Computation of the D-Optimality Criterion for SOE

Let us begin first with evaluating the D-optimality criterion for the SOE case. Evaluating (37) for this data splitting is easier as compared with SOF.

Since **<sup>Π</sup>** is known to each agent, the vector *<sup>φ</sup>*(*<sup>x</sup>*, **<sup>Π</sup>**) can be evaluated without any cooperation between the agents. The covariance **Σ***<sup>w</sup>* can then be evaluated distributively using averaged consensus as **<sup>Σ</sup>***<sup>w</sup>* = (*<sup>D</sup>* <sup>+</sup> **<sup>Γ</sup>**−1)−1, where *<sup>D</sup>* is computed using networkwide averaging. To compute (46), a few more steps are needed. Specifically, in addition to **<sup>Σ</sup>***w*, we also need to compute the quantities *<sup>c</sup>*(*<sup>π</sup>*) and *<sup>b</sup>*(*<sup>π</sup>*) in (40) to evaluate the criterion. These can already be computed using averaged consensus as

$$\mathbf{c}(\tilde{\boldsymbol{\pi}}) = \boldsymbol{\Phi}^T \boldsymbol{\Lambda} \boldsymbol{\Phi}(\mathbf{X}, \tilde{\boldsymbol{\pi}}) = \boldsymbol{K} \times \frac{1}{K} \sum\_{k=1}^K \boldsymbol{\Phi}\_k^T \boldsymbol{\Lambda} \boldsymbol{\Phi}(\mathbf{X}\_k, \tilde{\boldsymbol{\pi}}), \tag{47}$$

$$b(\tilde{\pi}) = \oint (\mathbf{X}, \tilde{\pi})^T \mathbf{A} \boldsymbol{\Phi}(\mathbf{X}, \tilde{\pi}) + \tilde{\gamma}^{-1} = \boldsymbol{K} \times \frac{1}{\boldsymbol{K}} \sum\_{k=1}^{K} \boldsymbol{\Phi}(\mathbf{X}\_k, \tilde{\pi})^T \boldsymbol{\Lambda} \boldsymbol{\Phi}(\mathbf{X}\_k, \tilde{\pi}) + \tilde{\gamma}^{-1}. \tag{48}$$

Then, using (47) and (48) as well as **Σ***<sup>w</sup>* computed distributively, the criterion (46) can be easily evaluated by each agent.

It is worth noting that the choice of *<sup>γ</sup>*<sup>−</sup><sup>1</sup> in (48) is the only parameter that can be set manually in this exploration criterion. Basically, it controls how much we know about the potential measurement location. If *<sup>γ</sup>*<sup>−</sup><sup>1</sup> is large, the criterion would yield that the potential measurement location is not informative. On the other side, if *<sup>γ</sup>*<sup>−</sup><sup>1</sup> <sup>→</sup> 0, the criterion yields that the considered measurement location is potentially informative. We set *<sup>γ</sup>*<sup>−</sup><sup>1</sup> <sup>=</sup> <sup>0</sup> for all considered measurement locations, such that the current information in the model determines how informative a measurement location could be.
