Time Series Decomposition using the Fréchet Distance

In this paper, we introduce a new data analysis problem that aims to decompose a set of univariate time series into a small set of $k$ base curves of length at most $l$ such that the sum of Fr\'echet distances of the time series to a ``Fr\'echet combination'' of the base curves is minimized. Here, a Fr\'echet combination allows to combine individually scaled base curves using a $k$-dimensional traversal. We call the problem of finding a set of optimal base curves the Fr\'echet decomposition problem and we consider two variants: (a) the base curves can be arbitrary curves of bounded length and (b) the curves come from a given finite set of candidate curves. We think of the Fr\'echet decomposition problem as a Fr\'echet variant of principal component analysis. For the case of a single base curve we develop a $(1+\varepsilon)$-approximation algorithm for the Fr\'echet decomposition problem. Additionally we give an exact algorithm for the projection distance problem that asks to compute the distance of one given time series to a given set of $k$ base curves. This allows us to design an exact algorithm for the Fr\'echet decomposition problem for general $k$ when curves come from a fixed candidate set.