  We propose an anomaly detection method for multi-variate scientific data
based on analysis of high-order joint moments. Using kurtosis as a reliable
measure of outliers, we suggest that principal kurtosis vectors, by analogy to
principal component analysis (PCA) vectors, signify the principal directions
along which outliers appear. The inception of an anomaly, then, manifests as a
change in the principal values and vectors of kurtosis. Obtaining the principal
kurtosis vectors requires decomposing a fourth order joint cumulant tensor for
which we use a simple, computationally less expensive approach that involves
performing a singular value decomposition (SVD) over the matricized tensor. We
demonstrate the efficacy of this approach on synthetic data, and develop an
algorithm to identify the occurrence of a spatial and/or temporal anomalous
event in scientific phenomena. The algorithm decomposes the data into several
spatial sub-domains and time steps to identify regions with such events.
Feature moment metrics, based on the alignments of the principal kurtosis
vectors, are computed at each sub-domain and time step for all features to
quantify their relative importance towards the overall kurtosis in the data.
Accordingly, spatial and temporal anomaly metrics for each sub-domain are
proposed using the Hellinger distance of the feature moment metric distribution
from a suitable nominal distribution. We apply the algorithm to a
one-dimensional turbulent auto-ignition combustion case and demonstrate that
the anomaly metrics reliably capture the occurrence of auto-ignition in
relevant spatial sub-domains at the right time steps.
