Fig 3:

An example of sequential partitioning and aggregation in a two-dimensional sample space. We partition the sample space into 64 bins to facilitate visualization. (a) Partition the two-dimensional sample space using the sequential partition strategy. Then all the bins are sequentially ordered. (b) On layer 1, the rejection set is ${ℛ}^{\left(1\right)}=\left\{\mathrm{2,10,53,55}\right\}$. (c) On layer 2, the remaining bins are aggregated with their neighbors along the order defined by sequential partition. The indices of each bin in layer 2 are relabeled and comprise the bin set $S_i^{\left(2\right)}$. After the testing procedure, null hypotheses coupled with bin set 20 and 28 are rejected. After mapping the rejections to layer 1, the rejection set is ${ℛ}^{\left(2\right)}=\left\{\mathrm{41,42,59,60}\right\}$. (d) On layer 3, we further aggregate the neighboring bin sets. After the testing procedure on layer 3, we reject the null hypotheses coupled with bin set 13. After mapping the rejections to layer 1, the rejection set is ${ℛ}^{\left(3\right)}=\left\{\mathrm{54,56,57,58}\right\}$. After three layers, the overall rejection set is $ℛ=\left\{\mathrm{2,10,53,54,55,56,57,58,59,60}\right\}$.