  Anomaly detection is necessary for proper and safe operation of large-scale
systems consisting of multiple devices, networks, and/or plants. Those systems
are often characterized by a pair of multivariate datasets. To detect anomaly
in such a system and localize element(s) associated with anomaly, one would
need to estimate scores that quantify anomalousness of the entire system as
well as its elements. However, it is not trivial to estimate such scores by
considering changes of relationships between the elements, which strongly
correlate with each other. Moreover, it is necessary to estimate the scores for
the entire system and its elements from a single framework, in order to
identify relationships among the scores for localizing elements associated with
anomaly. Here, we developed a new method to quantify anomalousness of an entire
system and its elements simultaneously.
  The purpose of this paper is threefold. The first one is to propose a new
anomaly detection method: Double Kernelized Scoring (DKS). DKS is a unified
framework for entire-system anomaly scoring and element-wise anomaly scoring.
Therefore, DKS allows for conducting simultaneously 1) anomaly detection for
the entire system and 2) localization for identifying faulty elements
responsible for the system anomaly. The second purpose is to propose a new
kernel function: Matrix Kernel. The Matrix Kernel is defined between general
matrices, which might have different dimensions, allowing for conducting
anomaly detection on systems where the number of elements change over time. The
third purpose is to demonstrate the effectiveness of the proposed method
experimentally. We evaluated the proposed method with synthetic and real time
series data. The results demonstrate that DKS is able to detect anomaly and
localize the elements associated with it successfully.
