  Signatures, one of the key concepts of rough path theory, have recently
gained prominence as a means to find appropriate feature sets in machine
learning systems. In this paper, in order to compute signatures directly from
discrete data without going through the transformation to continuous data, we
introduced a discretized version of signatures, called "flat discrete
signatures". We showed that the flat discrete signatures can represent the
quadratic variation that has a high relevance in financial applications. We
also introduced the concept of "discrete signatures" that is a generalization
of "flat discrete signatures". This concept is defined to reflect the fact that
data closer to the current time is more important than older data, and is
expected to be applied to time series analysis. As an application of discrete
signatures, we took up a stock market related problem and succeeded in
performing a good estimation with fewer data points than before.
