  Sequentially obtained dataset usually exhibits different behavior at
different data resolutions/scales. Instead of inferring from data at each scale
individually, it is often more informative to interpret the data as an ensemble
of time series from different scales. This naturally motivated us to propose a
new concept referred to as the scale-based inference. The basic idea is that
more accurate prediction can be made by exploiting scale information of a time
series. We first propose a nonparametric predictor based on $k$-nearest
neighbors with an optimally chosen $k$ for a single time series. Based on that,
we focus on a specific but important type of scale information, the
resolution/sampling rate of time series data. We then propose an algorithm to
sequentially predict time series using past data at various resolutions. We
prove that asymptotically the algorithm produces the mean prediction error that
is no larger than the best possible algorithm at any single resolution, under
some optimally chosen parameters. Finally, we establish the general
formulations for scale inference, and provide further motivating examples.
Experiments on both synthetic and real data illustrate the potential
applicability of our approaches to a wide range of time series models.
