  Dynamic Time Warping (DTW), and its constrained (CDTW) and weighted (WDTW)
variants, are time series distances with a wide range of applications. They
minimize the cost of non-linear alignments between series. CDTW and WDTW have
been introduced because DTW is too permissive in its alignments. However, CDTW
uses a crude step function, allowing unconstrained flexibility within the
window, and none beyond it. WDTW's multiplicative weight is relative to the
distances between aligned points along a warped path, rather than being a
direct function of the amount of warping that is introduced. In this paper, we
introduce Amerced Dynamic Time Warping (ADTW), a new, intuitive, DTW variant
that penalizes the act of warping by a fixed additive cost. Like CDTW and WDTW,
ADTW constrains the amount of warping. However, it avoids both abrupt
discontinuities in the amount of warping allowed and the limitations of a
multiplicative penalty. We formally introduce ADTW, prove some of its
properties, and discuss its parameterization. We show on a simple example how
it can be parameterized to achieve an intuitive outcome, and demonstrate its
usefulness on a standard time series classification benchmark. We provide a
demonstration application in C++.
