  Recent years have witnessed the unprecedented rising of time series from
almost all kindes of academic and industrial fields. Various types of deep
neural network models have been introduced to time series analysis, but the
important frequency information is yet lack of effective modeling. In light of
this, in this paper we propose a wavelet-based neural network structure called
multilevel Wavelet Decomposition Network (mWDN) for building frequency-aware
deep learning models for time series analysis. mWDN preserves the advantage of
multilevel discrete wavelet decomposition in frequency learning while enables
the fine-tuning of all parameters under a deep neural network framework. Based
on mWDN, we further propose two deep learning models called Residual
Classification Flow (RCF) and multi-frequecy Long Short-Term Memory (mLSTM) for
time series classification and forecasting, respectively. The two models take
all or partial mWDN decomposed sub-series in different frequencies as input,
and resort to the back propagation algorithm to learn all the parameters
globally, which enables seamless embedding of wavelet-based frequency analysis
into deep learning frameworks. Extensive experiments on 40 UCR datasets and a
real-world user volume dataset demonstrate the excellent performance of our
time series models based on mWDN. In particular, we propose an importance
analysis method to mWDN based models, which successfully identifies those
time-series elements and mWDN layers that are crucially important to time
series analysis. This indeed indicates the interpretability advantage of mWDN,
and can be viewed as an indepth exploration to interpretable deep learning.
