summarise via summary statistics $s(\mathbf{x}^{i*})=\left( s_1, \ldots , s_{\kappa}\right) \in \mathbb{R}^{\kappa}$ and $\kappa$ is the number of summary statistics used to summarise the model output. The parameters $\mathbf{w} \in \mathbb{R}^{\kappa}$ are summary statistic weights, and these are used within a weighted Euclidan distance function $$\begin{aligned}
  d_{\mathbf{w}}\left(s(\mathbf{x}),s(\mathbf{y})\right) &= \sum_{i=1}^\kappa w_i^2 (s_i - s_{\text{obs}_i})^2 \\
  &= (s(\mathbf{x})-s(\mathbf{y}))^\top \Sigma_w^\top \Sigma_w (s(\mathbf{x})-s(\mathbf{y})) \\
  &= \left(\Sigma_w s(\mathbf{x})- \Sigma_w s(\mathbf{y})\right)^\top \left( \Sigma_w s(\mathbf{x}) - \Sigma_w s(\mathbf{y}) \right),  \end{aligned}$$ where $\Sigma_w = \text{diag}(\mathbf{w})$, to compare the pseudo data with observed data. We note that this is equivalent to stretching the space in which the pseudo data lies via the matrix $\Sigma_w$, and using the usual Euclidean distance.

Consider a small perturbation in parameter space $\mathbf{w}=\mathbf{w}_0+\epsilon$, with $||\epsilon|| \ll 1$. Then $$\Sigma_{\mathbf{w}} = \text{diag}(\mathbf{w}_0+\epsilon) = \text{diag}(\mathbf{w}_0) + \text{diag}(\epsilon) = \Sigma_{\mathbf{w}_0} + \Sigma_{\epsilon}.$$ Using this decomposition of $\Sigma_\mathbf{w}$ gives, for the weighted Euclidean distance, $$\begin{aligned}
 d_{\mathbf{w}_0+\epsilon}(s(\mathbf{x}),s(\mathbf{y})) &= \left(\left(\Sigma_{\mathbf{w}_0} + \Sigma_{\epsilon} \right) s(\mathbf{x}) - \left(\Sigma_{\mathbf{w}_0} + \Sigma_{\epsilon} \right) s(\mathbf{y})\right)^\top \\
&\, \, \, \, \, \, \, \left(\left(\Sigma_{\mathbf{w}_0} + \Sigma_{\epsilon} \right) s(\mathbf{x}) - \left(\Sigma_{\mathbf{w}_0} + \Sigma_{\epsilon} \right) s(\mathbf{y})\right) \\
&= \left( \Sigma_{\mathbf{w}_0} s(\mathbf{x})- \Sigma_{\mathbf{w}_0} s(\mathbf{y}) \right)^\top \left( \Sigma_{\mathbf{w}_0} s(\mathbf{x})- \Sigma_{\mathbf{w}_0} s(\mathbf{y}) \right) \\
&\, \, \, \, \, \, \, 