PP: Multilevel wavelet decomposition network for interpretable time series analysis

Problem:

the important frequency information is lack of effective modelling.

?? what is frequency information in time series? and why other models don't model this kind of frequency information?

frequency learning

we propose two deep learning models: 1. residual classification flow for classification. 2. multi-frequency long short-term memory for forecasting

INTRODUCTION

1. two types of time series analysis methods:

与其这样说不如说time series只有两个维度，时间维度和频率维度。 

time-domain methods: analyze correlations among time series

frequency-domain methods: transform time series into a frequency spectrum, Fourier transform/ Z-transform; 

2. How to integrate wavelet transforms into the framework of deep learning models remains a great challenge.

MODEL

Multilevel Discrete Wavelet Decomposition (MDWD) [26] is a wavelet based discrete signal analysis method, which can extract multilevel time-frequency features from a time series by decomposing the series as low and high frequency sub-series level by level.

解释： 相当于将time series x分解成为i个level 的low frequency 和high frequency的子序列。而分解出来的结果则为时序数据的features；之后将feature输入到CNN和LSTM中，进行分类和预测。

该论文的唯一新意是进行了小波分解，将一个time series分解为l个high frequency and low frequency的子序列，之后将子序列feed in different neural structure. 

Residual classification flow: classification - supervised learning;

a multilayer perceptron + a residual learning method. 

?? 在这里residual learning起了什么作用呢。

Multi-frequency long short term memory:

sub-series + lstm

How to evaluate the performance of models:

MAPE: mean absolute percentage error;

RMSE: root mean square error. 

INTERPRETATION

the outputs of the middle layers in mWDN, i.e., xl (i) and xh (i), inherit the physical meanings of wavelet decompositions 

增加了可以解释性，即中间层的输出继承了小波分解的物理意义。但我十分怀疑这的意义，即使输出了中间层又怎么样，方便理解最终的classification/forecasting的结果，还是方便理解中间隐含层的特征？

SUPPLEMENTARY KNOWLEDGE

1. correlation and dependency is any statistical relationship, whether causal or not, between two random variables or bivariate data.

2. frequency-domain methods:

Fourier transform, wavelet transform

原理是把时域数据转换到频域

时间序列本身具有非线性和信噪比高的特点??待验证??，采用传统的高斯去噪、中值滤波等方法往往存在诸多缺陷。而小波理论是根据时频局部化的要求而发展起来的，具有自适应和数学显微镜性质，特别适合非平稳、非线性信号的处理。