用扫描检测和平均似然比
Detection with the scan and the average likelihood ratio
摘要：
研究了扫描（极大似然比统计量）和平均似然比统计量在具有高斯白噪声的典型单变量采样数据模型中检测空间范围未知的确定性信号的性能.我们的研究结果表明，扫描统计，一个流行的工具检测问题，是最佳的信号检测与最小的空间范围。对于具有较大空间范围的信号，扫描是次优的，并且功率损耗可能是相当大的。相比之下，平均似然比统计量对于除最小尺度之外的所有尺度上的信号检测都是最优的，在最小尺度上，其性能仅略次优。我们给出了严格的数学陈述，这些结果以及启发式的解释表明，这些发现的本质适用于检测问题相当普遍，如检测集群的模型，涉及密度或强度或检测的多元信号。我们提出了一个修改的平均似然比，产生最佳的检测信号与任意的空间范围，并具有额外的好处，允许快速计算的统计。相比之下，扫描的最佳检测似乎需要使用依赖于尺度的临界值。
英文摘要：
We investigate the performance of the scan (maximum likelihood ratio statistic) and of the average likelihood ratio statistic in the problem of detecting a deterministic signal with unknown spatial extent in the prototypical univariate sampled data model with white Gaussian noise. Our results show that the scan statistic, a popular tool for detection problems, is optimal only for the detection of signals with the smallest spatial extent. For signals with larger spatial extent the scan is suboptimal, and the power loss can be considerable. In contrast, the average likelihood ratio statistic is optimal for the detection of signals on all scales except the smallest ones, where its performance is only slightly suboptimal. We give rigorous mathematical statements of these results as well as heuristic explanations which suggest that the essence of these findings applies to detection problems quite generally, such as the detection of clusters in models involving densities or intensities or the detection of multivariate signals. We present a modification of the average likelihood ratio that yields optimal detection of signals with arbitrary spatial extent and which has the additional benefit of allowing for a fast computation of the statistic. In contrast, optimal detection with the scan seems to require the use of scale-dependent critical values.
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