Motivated by testing for pathogenic diseases we consider a new nonadaptive group testing problem for which: (1) positives occur within a burst, capturing the fact that infected test subjects often come in clusters, and (2) that the test outcomes arise from semiquantitative measurements that provide coarse information about the number of positives in any tested group. Our model generalizes prior work on detecting a single burst of positives with classical group testing[1] as well as work on semiquantitative group testing (SQGT)[2]. Specifically, we study the setting where the burst-length $\ell$ is known and the semiquantitative tests provide potentially nonuniform estimates on the number of positives in a test group. The estimates represent the index of a quantization bin containing the (exact) total number of positives, for arbitrary thresholds $\eta_1,\dots,\eta_s$. Interestingly, we show that the minimum number of tests needed for burst identification is essentially only a function of the largest threshold $\eta_s$. In this context, our main result is an order-optimal test scheme that can recover any burst of length $\ell$ using roughly $\frac{\ell}{2\eta_s}+\log_{s+1}(n)$ measurements. This suggests that a large saturation level $\eta_s$ is more important than finely quantized information when dealing with bursts. We also provide results for related modeling assumptions and specialized choices of thresholds.

翻译：受病原体检测需求的启发，我们提出了一种新的非适应性分组检测问题，其特点是：(1) 正样本呈突发性分布（反映受感染受试者通常呈集群出现的事实），(2) 测试结果来源于半定量测量，仅提供测试组中正样本数量的粗糙信息。本模型推广了经典分组检测中单次正样本突发检测[1]与半定量分组检测[2]的相关研究。具体而言，我们研究突发长度 $\ell$ 已知且半定量测试提供测试组中正样本数量非均匀估计的场景。该估计表示包含（精确）正样本总数所对应量化区间的索引，其阈值 $\eta_1,\dots,\eta_s$ 可任意设定。有趣的是，我们证明突发识别所需的最小测试数量本质上仅取决于最大阈值 $\eta_s$。在此背景下，我们的主要成果是提出了一种阶次最优的测试方案，可通过约 $\frac{\ell}{2\eta_s}+\log_{s+1}(n)$ 次测量恢复任意长度为 $\ell$ 的突发。这表明处理突发问题时，较大的饱和阈值 $\eta_s$ 比精细的量化信息更为重要。此外，我们还给出了相关建模假设及特定阈值选择下的分析结果。