大规模时间序列数据挖掘方法：现状与挑战

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"时间序列数据挖掘方法综述" 时间序列分析是一种专门研究数据随时间变化趋势的统计学方法，它在各种领域，如金融、气象学、经济学、生物医学和工程等，有着广泛的应用。随着大数据时代的到来，时间序列数据的规模已经达到了前所未有的程度，甚至可以包含万亿级的观测值。这种大规模的数据为数据挖掘提供了新的挑战和机遇。本篇硕士论文，由Caroline Kleist撰写，对时间序列数据挖掘方法进行了全面的回顾。论文指出，虽然在横截面数据分析方面已发展出成熟的技术，但针对时间序列数据的方法尚未达到相同的复杂度和成熟度。大时间序列数据带来的问题包括极高的维度以及至今仍未达成共识的最佳实践。论文中，作者首先探讨了大型时间序列数据所面临的挑战，如高维性问题。在高维数据中，特征之间可能存在的多重共线性、噪声增加以及计算复杂性等问题都显著影响了数据挖掘的效率和准确性。为解决这些问题，时间序列数据挖掘社区提出了一系列方法，如降维技术（如主成分分析PCA）、特征选择策略以及适应时间序列特性的建模方法。作者通过谷歌趋势数据来展示这些重要技术的实际应用，谷歌趋势数据是一个典型的时间序列数据源，反映了用户对特定搜索词的兴趣随时间的变化。论文中可能涉及的技术包括时间序列分割、自回归模型（ARIMA）、状态空间模型、季节性分解趋势（STL）以及基于机器学习的时间序列预测方法，如支持向量机（SVM）和神经网络。此外，论文还对当前的研究方向进行了审查，强调了深度学习在时间序列分析中的应用，如长短时记忆网络（LSTM）和门控循环单元（GRU），这些技术在处理序列数据的长期依赖性和动态模式识别上表现突出。同时，论文还指出了未来可能的研究热点，如集成学习在时间序列预测中的应用、时空数据的联合分析以及如何利用元学习和迁移学习来提升时间序列模型的泛化能力。这篇综述论文为读者提供了关于时间序列数据挖掘的全面概述，不仅介绍了现有的技术，还展望了该领域的未来趋势，对于研究人员和从业者来说，是一份宝贵的参考资料。

(PAA) since the research by Keogh et al. (2001a). Originally, Keogh and Pazzani (2000b)

called the PAA method Piecewise C onstant Approximation (PCA) and the name was after-

wards changed to PAA as the abbreviation PCA is already reserved for Principal Component

Analysis. The PAA coeﬃcients are generated by dividing the time series into ω equi-sized

windows and calculating averages of the subsequences in the corresponding windows. The

averages of each window are stacked into a vector and called “PAA coeﬃcients”. Hence, the

dimension reduction of a time series X of length n into a string

X = ˆx

, . . . , ˆx

of arbitrary

length ω with ω  n is performed by the following transformation:

ˆx

= d

−1

d i

j=d (i−1)+1

, i = 1, . . . , ω, j = 1, . . . , n, d = n w

−1

(1)

Figure 1 shows the PAA dimension reduction of the Google search volume index for the

query term “Data Mining” with ω = 40. Although the PAA dimension reduction approach is

very simple, it is well comparable to more complex competitors (see for example Ding et al.

(2008)). Furthermore, a very strong advantage is the fact that each PAA window is of the

same length. This facilitates the indexing technique enormously (see Subsection 3.2). An

extension of the PAA approach is Adaptive Piecewise C onstant Approximation (APCA) by

Keogh et al. (2001b). APCA aims to approximate a time series by PAA segments of varying

length, i.e. it allows the diﬀerent windows to have an arbitrary size. In this way, we try to

minimize the individual reconstruction error of the reduced time series. For the APCA, we

do not only store the mean for each window but its length as well.

Principal C omponent Analysis (PCA) is a further dimension reduction technique adopted

from static data approaches (see for instance Yang and Shahabi (2005b) or Raychaudhuri et al.

(2000)). PCA disregards less signiﬁcant components and therefore gives a reduced represen-

tation of the data. PCA computations are based on orthogonal transformations with the goal

to obtain linearly uncorrelated variables, the so called principal components. So, in order to

reduce dimensionality, PCA requires the covariance matrix of the corresponding time series.

However, for the covariance calculations it is ignored whether the considered time series are

similar or correlated at diﬀerent points in time. If two time series are evolving very similar

and are only shifted in time, the traditional PCA approach would lead to false conclusions.

To avoid this ineﬀectiveness, Li (2014) present an asynchronism-based principal component

analysis. In order to improve PCA, Li (2014) embed Dynamic T ime W arping (DTW, see

Section 3.5.1) before applying PCA. By doing that, they employ asynchronous and not only

synchronous covariances. Furthermore, DTW does not require the time series to be of the

from the SVD representation, the error we make is relatively low. Korn et al. (1997) even

aim to further minimize the reconstruction error and further improve the SVD representation.

For this purpose they introduce S ingular V alue Decomposition with Deltas (SVDD).

A very frequently used and probably the most popular dimensional reduction and indexing

technique is based on a symbolic representation and called SAX. It was introduced by Lin et al.

(2003). The S ymbolic Aggregate Approximation (SAX) represenation is said to outperform

all other dimensionality reduction techniques (see for instance Ratanamahatana et al. (2010)

or Kadam and Thakore (2012)). In order to illustrate how the SAX algorithm works, Figure

2 shows the SAX representation including the PAA representation which is the preceding

interim dimension reducing stage of the SAX algorithm. The illustrating example is computed

for the trajectories of the Google SVI for the query terms “Data Mining” and “Google”. As

already indicated, the transformation of a time series X of length n into a string X =

, . . . , x

of arbitrary length ω with ω  n is performed in two steps. In a ﬁrst step, the

z-normalized time series is converted to a PAA representation, i.e. the Piecewise Aggregate

Approximation. As a reminder, the PAA coeﬃcients are derived by slicing the data at hand

along the temporal axis into ω equidistant windows and thereupon calculating sample means

for all observations falling into one window (see Equation 1). Hereafter, the PAA coeﬃcients

are mapped to symbols (mostly letters). For doing so, the cutoﬀ lines dividing the distribution

space into α equiprobable segments need to be speciﬁed. Assuming that the z-normalized

time series are standard normally distributed, these cutoﬀ lines are tabulated for α diﬀerent

symbols as reported in Table 1. Hence, α is a hyperparameter that has to be chosen a priori.

Equation 2 depicts how the PAA coeﬃcients from the vector

X = ˆx

, . . . , ˆx

are mapped

into α diﬀerent symbols yielding the SAX string X = x

, . . . , x

= α

if ˆx

∈ [β

j−1

, β

) (2)

with β

, . . . , β

being the corresponding tabulated breakpoints as reported in Table 1 for

i, j = 1, . . . , α − 1. Finally, as symbols require fewer bits than real valued numbers, symbols

are stored instead of the original time series. If we chose the hyperparameter to be α = 3,

the three used symbols are “a”, “b” and “c”. Instead of the PAA coeﬃcients or even the real

valued time series objects, the symbols are stored in the corresponding order as a “word”.

In order to speed the SAX algorithm dramatically up, Shieh and Keogh (2008) propose a

modiﬁcation of the SAX representation called indexable Symbolic Aggregate Approximation

(iSAX). This multi-resolution symbolic representation allows extensible hashing and hence

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