连续观测的通用离群序列检测
Universal Outlying sequence detection For Continuous Observations
摘要：
研究了存在样本的$M$序列，需要从其中检测出一个离群序列的检测问题。每个典型序列包含独立同分布（i.i.d.）的$n$。来自已知分布$π$的连续观测，并且离群值序列包含$n$ i.i.d.来自离群值分布$μ$的观测，其不同于$π$，但在其他方面未知。建立了一种基于KL散度的通用检验方法，用于近似最大似然检验，其中包含已知的$π$和未知的$μ$ .基于数据依赖的分区KL散度估计。当密度比满足$0<K_1\leq \frac{dμ}{dπ}\leq K_2$，其中$K_1$和$K_2$是正常数时，这种KL散度估计量以指数形式快速收敛到其真值，这进一步表明检验是指数一致的.测试的性能进行了比较，最近推出的测试这个问题的基础上的最大均值差异（MMD）的机器学习方法。我们确定的KL分歧为基础的测试制度是优于MMD为基础的测试。
英文摘要：
The following detection problem is studied, in which there are $M$ sequences of samples out of which one outlier sequence needs to be detected. Each typical sequence contains $n$ independent and identically distributed (i.i.d.) continuous observations from a known distribution $π$, and the outlier sequence contains $n$ i.i.d. observations from an outlier distribution $μ$, which is distinct from $π$, but otherwise unknown. A universal test based on KL divergence is built to approximate the maximum likelihood test, with known $π$ and unknown $μ$. A data-dependent partitions based KL divergence estimator is employed. Such a KL divergence estimator is further shown to converge to its true value exponentially fast when the density ratio satisfies $0<K_1\leq \frac{dμ}{dπ}\leq K_2$, where $K_1$ and $K_2$ are positive constants, and this further implies that the test is exponentially consistent. The performance of the test is compared with that of a recently introduced test for this problem based on the machine learning approach of maximum mean discrepancy (MMD). We identify regimes in which the KL divergence based test is better than the MMD based test.
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