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第14行： − + − 此外，时间序列分析技术可分为参数化和非参数化方法。参数方法假定基础的平稳随机过程具有某种结构，可以用少量的参数来描述（例如，使用自回归或移动平均模型）。在这些方法中，时间序列分析的任务是估计描述随机过程的模型的参数。相比之下，非参数方法明确地估计过程的协方差或频谱，而不假设过程有任何特定的结构。+ − + − + 第57行：
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第78行： − + − {{main|Statistical classification}} − Assigning time series pattern to a specific category, for example identify a word based on series of hand movements in [[sign language]]. − − − Assigning time series pattern to a specific category, for example identify a word based on series of hand movements in sign language. − − = = = 分类 = = 将时间序列模式分配到一个特定的类别，例如根据手语中的一系列动作识别一个单词。 − − ===Signal estimation=== − This approach is based on [[harmonic analysis]] and filtering of signals in the [[frequency domain]] using the [[Fourier transform]], and [[spectral density estimation]], the development of which was significantly accelerated during [[World War II]] by mathematician [[Norbert Wiener]], electrical engineers [[Rudolf E. Kálmán]], [[Dennis Gabor]] and others for filtering signals from noise and predicting signal values at a certain point in time. See [[Kalman filter]], [[Estimation theory]], and [[Digital signal processing]]+ − + − − This approach is based on harmonic analysis and filtering of signals in the frequency domain using the Fourier transform, and spectral density estimation, the development of which was significantly accelerated during World War II by mathematician Norbert Wiener, electrical engineers Rudolf E. Kálmán, Dennis Gabor and others for filtering signals from noise and predicting signal values at a certain point in time. See Kalman filter, Estimation theory, and Digital signal processing − − = = = 信号估计 = = = 这种方法是基于傅里叶分析信号和滤波的频域使用傅里叶变换和谱密度估计，其发展是显着加速二战期间由数学家诺伯特维纳，电气工程师鲁道夫·卡尔曼，丹尼斯 Gabor 和其他人从噪音信号过滤和预测信号值在一定时间点。参见卡尔曼滤波器，参数估测和数字信号处理 − − − Splitting a time-series into a sequence of segments. It is often the case that a time-series can be represented as a sequence of individual segments, each with its own characteristic properties. For example, the audio signal from a conference call can be partitioned into pieces corresponding to the times during which each person was speaking. In time-series segmentation, the goal is to identify the segment boundary points in the time-series, and to characterize the dynamical properties associated with each segment. One can approach this problem using [[Change detection|change-point detection]], or by modeling the time-series as a more sophisticated system, such as a Markov jump linear system. − − − Splitting a time-series into a sequence of segments. It is often the case that a time-series can be represented as a sequence of individual segments, each with its own characteristic properties. For example, the audio signal from a conference call can be partitioned into pieces corresponding to the times during which each person was speaking. In time-series segmentation, the goal is to identify the segment boundary points in the time-series, and to characterize the dynamical properties associated with each segment. One can approach this problem using change-point detection, or by modeling the time-series as a more sophisticated system, such as a Markov jump linear system. − − = = = 分割 = = 将一个时间序列分割成一系列的片段。通常情况下，时间序列可以表示为一系列单独的片段，每个片段都有自己的特征属性。例如，来自电话会议的音频信号可以根据每个人发言的时间分割成相应的部分。在时间序列分割中，目标是识别时间序列中的分段边界点，并刻画每个分段的动态特性。人们可以通过变点检测来解决这个问题，或者将时间序列建模为一个更复杂的系统，如马尔可夫跳跃线性系统。 − − ==Models== − − ==Models== − − = = 模型 = = − Models for time series data can have many forms and represent different [[stochastic processes]]. When modeling variations in the level of a process, three broad classes of practical importance are the ''[[autoregressive]]'' (AR) models, the ''integrated'' (I) models, and the ''[[moving average model|moving average]]'' (MA) models. These three classes depend linearly on previous data points.<ref name="linear time series">{{cite book |author-link=Neil Gershenfeld |last=Gershenfeld |first=N. |year=1999 |title=The Nature of Mathematical Modeling |url=https://archive.org/details/naturemathematic00gers_334 |url-access=limited |location=New York |publisher=Cambridge University Press |pages=[https://archive.org/details/naturemathematic00gers_334/page/n206 205]–208 |isbn=978-0521570954 }}</ref> Combinations of these ideas produce [[autoregressive moving average]] (ARMA) and [[autoregressive integrated moving average]] (ARIMA) models. The [[autoregressive fractionally integrated moving average]] (ARFIMA) model generalizes the former three. Extensions of these classes to deal with vector-valued data are available under the heading of multivariate time-series models and sometimes the preceding acronyms are extended by including an initial "V" for "vector", as in VAR for [[vector autoregression]]. An additional set of extensions of these models is available for use where the observed time-series is driven by some "forcing" time-series (which may not have a causal effect on the observed series): the distinction from the multivariate case is that the forcing series may be deterministic or under the experimenter's control. For these models, the acronyms are extended with a final "X" for "exogenous". − Models for time series data can have many forms and represent different stochastic processes. When modeling variations in the level of a process, three broad classes of practical importance are the autoregressive (AR) models, the integrated (I) models, and the moving average (MA) models. These three classes depend linearly on previous data points. Combinations of these ideas produce autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) models. The autoregressive fractionally integrated moving average (ARFIMA) model generalizes the former three. Extensions of these classes to deal with vector-valued data are available under the heading of multivariate time-series models and sometimes the preceding acronyms are extended by including an initial "V" for "vector", as in VAR for vector autoregression. An additional set of extensions of these models is available for use where the observed time-series is driven by some "forcing" time-series (which may not have a causal effect on the observed series): the distinction from the multivariate case is that the forcing series may be deterministic or under the experimenter's control. For these models, the acronyms are extended with a final "X" for "exogenous".+ − 时间序列数据的模型可以有多种形式，表示不同的随机过程。在对过程层次的变化进行建模时，三大类实际的重要性是自回归(AR)模型、综合(i)模型和移动平均(MA)模型。这三个类线性地依赖于以前的数据点。这些想法的结合产生了自回归移动平均(ARMA)和 ARIMA模型移动平均(ARIMA)模型。自回归分数积分移动平均(ARFIMA)模型对前三种模型进行了推广。处理矢量值数据的这些类的扩展可以在多元时间序列模型的标题下得到，有时前面的首字母缩略词被扩展，包括一个初始的“ v”代表“矢量”，如在 VAR 代表向量自回归模型。这些模型的另外一组扩展可用于观测到的时间序列是由某种“强迫”的时间序列驱动的(这种时间序列可能对观测到的序列没有因果效应) : 与多变量情况的区别在于强迫序列可能是确定的或者在实验者的控制之下。对于这些模型，首字母缩略词被扩展成最后一个“ x”，表示“外生的”。+ − Non-linear dependence of the level of a series on previous data points is of interest, partly because of the possibility of producing a [[chaos theory|chaotic]] time series. However, more importantly, empirical investigations can indicate the advantage of using predictions derived from non-linear models, over those from linear models, as for example in [[nonlinear autoregressive exogenous model]]s. Further references on nonlinear time series analysis: (Kantz and Schreiber),<ref>{{cite book|last1=Kantz|first1=Holger|last2=Thomas|first2=Schreiber|title=Nonlinear Time Series Analysis|date=2004|publisher=Cambridge University Press|location=London|isbn=978-0521529020}}</ref> and (Abarbanel)<ref>{{cite book|last1=Abarbanel|first1=Henry|title=Analysis of Observed Chaotic Data|date=Nov 25, 1997|publisher=Springer|location=New York|isbn=978-0387983721}}</ref> − Non-linear dependence of the level of a series on previous data points is of interest, partly because of the possibility of producing a chaotic time series. However, more importantly, empirical investigations can indicate the advantage of using predictions derived from non-linear models, over those from linear models, as for example in nonlinear autoregressive exogenous models. Further references on nonlinear time series analysis: (Kantz and Schreiber), and (Abarbanel)+ − 一系列数据的水平对以前的数据点的非线性依赖是有趣的，部分是因为产生混沌时间序列的可能性。然而，更重要的是，经验调查可以表明使用来自非线性模型的预测优于来自线性模型的预测，例如在非线性自回归外生模型中。非线性时间序列分析的进一步参考文献: (Kantz 和 Schreiber)和(Abarbanel) − Among other types of non-linear time series models, there are models to represent the changes of variance over time ([[heteroskedasticity]]). These models represent [[autoregressive conditional heteroskedasticity]] (ARCH) and the collection comprises a wide variety of representation ([[GARCH]], TARCH, EGARCH, FIGARCH, CGARCH, etc.). Here changes in variability are related to, or predicted by, recent past values of the observed series. This is in contrast to other possible representations of locally varying variability, where the variability might be modelled as being driven by a separate time-varying process, as in a [[doubly stochastic model]].+ − 在其他类型的非线性时间序列模型中，有一些模型可以表示方差随时间的变化(异方差)。这些模型代表了 ARCH模型(ARCH) ，收藏包括各种各样的代表(GARCH，TARCH，EGARCH，FIGARCH，CGARCH 等等)。在这里，变异性的变化与观测系列的最近过去的值有关，或者是预测的。这与局部变化的其他可能表现形式形成对比，在这种情况下，变化可能被模拟为由一个单独的时变过程驱动，如双重随机模型。+
无编辑摘要
时间序列分析可以应用于实值、连续数据、离散数值Numeric数据或离散符号数据（即字符序列，如英语中的字母和单词<ref name=":0">{{cite book |last1=Lin |first1=Jessica |last2=Keogh |first2=Eamonn |last3=Lonardi |first3=Stefano |last4=Chiu |first4=Bill |chapter=A symbolic representation of time series, with implications for streaming algorithms |title=Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery |pages=2–11 |year=2003 |location=New York |publisher=ACM Press |doi=10.1145/882082.882086|citeseerx=10.1.1.14.5597 |s2cid=6084733 }}</ref>）。
时间序列分析可以应用于实值、连续数据、离散数值Numeric数据或离散符号数据（即字符序列，如英语中的字母和单词<ref name=":0">{{cite book |last1=Lin |first1=Jessica |last2=Keogh |first2=Eamonn |last3=Lonardi |first3=Stefano |last4=Chiu |first4=Bill |chapter=A symbolic representation of time series, with implications for streaming algorithms |title=Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery |pages=2–11 |year=2003 |location=New York |publisher=ACM Press |doi=10.1145/882082.882086|citeseerx=10.1.1.14.5597 |s2cid=6084733 }}</ref>）。
==分析方法==
===分析方法===
时间序列分析的方法可分为两类：频域方法和时域方法。前者包括频谱分析和小波分析；后者包括自相关和交叉相关分析。在时域中，可以用类似于滤波器的方式使用标度相关性来进行关联和分析。
时间序列分析的方法可分为两类：频域方法和时域方法。前者包括频谱分析和小波分析；后者包括自相关和交叉相关分析。在时域中，可以用类似于滤波器的方式使用标度相关性来进行关联和分析。
此外，时间序列分析技术可分为参数化和非参数化方法。参数方法假定基础的平稳随机过程具有某种结构，可以用少量的参数来描述（例如，使用自回归或移动平均模型）。在这些模型中，时间序列分析的任务是估计描述随机过程的模型的参数。相比之下，非参数方法明确地估计过程的协方差或频谱，而不假设过程有任何特定的结构。
时间序列分析的方法也可以分为线性和非线性，以及单变量 和多变量。
时间序列分析的方法也可以分为线性和非线性，以及单变量 和多变量。
==面板数据==
===面板数据===
时间序列是面板数据的一种类型，面板数据是更大的类别。面板数据是一个多维的数据集，而时间序列数据集是一个一维的面板（正如截面数据集一样）。一个数据集可能同时表现出面板数据和时间序列数据的特征。判断是面板数据还是时间序列的方法之一是探究使一条数据记录与其他记录不同的因素。如果答案是时间数据字段，那么这就是一个时间序列数据集候选。如果确定一个独特的记录需要一个时间数据字段和一个与时间无关的额外标识符（如学生证、股票代码、国家代码），那么它就是面板数据的候选。如果区别在于非时间标识符，那么该数据集就是一个截面数据集候选。
时间序列是面板数据的一种类型，面板数据是更大的类别。面板数据是一个多维的数据集，而时间序列数据集是一个一维的面板（正如截面数据集一样）。一个数据集可能同时表现出面板数据和时间序列数据的特征。判断是面板数据还是时间序列的方法之一是探究使一条数据记录与其他记录不同的因素。如果答案是时间数据字段，那么这就是一个时间序列数据集候选。如果确定一个独特的记录需要一个时间数据字段和一个与时间无关的额外标识符（如学生证、股票代码、国家代码），那么它就是面板数据的候选。如果区别在于非时间标识符，那么该数据集就是一个截面数据集候选。
==分析==
==='''分析'''===
经济时间序列的构建涉及通过在早期和晚期的基准值之间进行插值来估计某些日期的某些组成部分。插值法是在两个已知量（历史数据）之间估计一个未知量，或从现有信息中得出关于缺失信息的结论（"从字里行间阅读"）<ref name=":24">Hamming, Richard. Numerical methods for scientists and engineers. Courier Corporation, 2012.</ref>。如果与缺失数据相关的数据是可用的，并且其趋势、季节性和长期周期是已知的，那么插值法就很有用。插值法通常是通过使用已知所有相关日期的相关序列来实现的<ref name=":25">Friedman, Milton. "[http://www.nber.org/chapters/c2062.pdf The interpolation of time series by related series]." Journal of the American Statistical Association 57.300 (1962): 729–757.</ref>。或者使用多项式插值或样条插值，将分段多项式函数拟合到时间间隔中，使其平滑地拟合在一起。一个与插值密切相关的问题是用一个简单的函数来逼近一个复杂的函数（也称为回归）。回归和插值的主要区别是，多项式回归给出一个单一的多项式来模拟整个数据集。而插值则产生一个由许多多项式组成的分段连续函数来模拟数据集。
经济时间序列的构建涉及通过在早期和晚期的基准值之间进行插值来估计某些日期的某些组成部分。插值法是在两个已知量（历史数据）之间估计一个未知量，或从现有信息中得出关于缺失信息的结论（"从字里行间阅读"）<ref name=":24">Hamming, Richard. Numerical methods for scientists and engineers. Courier Corporation, 2012.</ref>。如果与缺失数据相关的数据是可用的，并且其趋势、季节性和长期周期是已知的，那么插值法就很有用。插值法通常是通过使用已知所有相关日期的相关序列来实现的<ref name=":25">Friedman, Milton. "[http://www.nber.org/chapters/c2062.pdf The interpolation of time series by related series]." Journal of the American Statistical Association 57.300 (1962): 729–757.</ref>。或者使用多项式插值或样条插值，将分段多项式函数拟合到时间间隔中，使其平滑地拟合在一起。一个与插值密切相关的问题是用一个简单的函数来逼近一个复杂的函数（也称为回归）。回归和插值的主要区别是，多项式回归给出一个单一的多项式来模拟整个数据集。而插值则产生一个由许多多项式组成的分段连续函数来模拟数据集。
外推法是指在原始观察范围之外，根据一个变量与另一个变量的关系来估计其数值的过程。它与插值I类似，插值在已知的观测值之间产生估计值，但外推法的不确定性更大，产生无意义结果的风险也更大。
外推法是指在原始观察范围之外，根据一个变量与另一个变量的关系来估计其数值的过程。它与插值I类似，插值在已知的观测值之间产生估计值，但外推法的不确定性更大，产生无意义结果的风险也更大。
* 使用第三方软件包 Spark-TS 库，Apache Spark 可以对大规模数据进行预测。
* 使用第三方软件包 Spark-TS 库，Apache Spark 可以对大规模数据进行预测。
===Classification===
===时间序列分析在信号处理上的应用===
{{see also|Signal processing|Estimation theory}}
{{see also|Signal processing|Estimation theory}}
这种方法是基于傅里叶分析信号和滤波的频域使用傅里叶变换和谱密度估计，该方法在二战期间迅速得以推广。数学家诺伯特维纳，电气工程师鲁道夫·卡尔曼，丹尼斯和其他学者完成了信号的滤波处理，且预测了在特定时间段中的信号值。相关知识参见卡尔曼滤波器，参数估测和数字信号处理的介绍。
===时间序列的分割处理===
===Segmentation===
{{main|Time-series segmentation}}
{{main|Time-series segmentation}}
时间序列的分割处理是将一个时间序列分割成一系列的片段。通常情况下，时间序列可以表示为一系列单独的片段，每个片段都有自己的特征属性。例如，来自电话会议的音频信号可以根据每个人发言的时间分割成相应的部分。时间序列分割的目标是识别时间序列中的分段临界点，并分析每个分段的动态特性。此外，时间序列可以建模成一个更复杂的系统，如马尔可夫跳跃线性系统。
===模型 ===
时间序列数据的模型可以用多种形式来表示不同的随机过程。在对过程层次的变化进行建模时，三大重要模型是自回归(AR)模型、综合(i)模型和移动平均(MA)模型。这三个模型线性地依赖于以前的数据点。这些模型的结合产生了自回归移动平均(ARMA)和 模型移动平均(ARIMA)模型。自回归分数积分移动平均(ARFIMA)模型对三大重要模型进行了推广。处理矢量值数据的模型的扩展可以在多元时间序列模型下继续完成，有一些是前面的首字母缩略词被扩展，例如一个初始的“ v”代表“矢量”； VAR 代表向量自回归模型。从这些模型的另外一组扩展可以发现观测到的时间序列是由某种“强迫”的时间序列驱动的(这种时间序列可能对观测到的序列没有因果效应)；与多变量情况的区别在于这种强迫序列可能是在实验者的控制之下所得到的。对于这些模型，首字母缩略词被扩展成最后一个“ x”，表示“外生的”。
研究数据对过往的数据点的非线性依赖关系是有趣的，主要是它有产生混沌时间序列的可能性。然而，更重要的是，使用来自非线性模型的预测优于来自线性模型的预测。例如非线性自回归外生模型的预测准确度优于线性的回顾模型。
Among other types of non-linear time series models, there are models to represent the changes of variance over time (heteroskedasticity). These models represent autoregressive conditional heteroskedasticity (ARCH) and the collection comprises a wide variety of representation (GARCH, TARCH, EGARCH, FIGARCH, CGARCH, etc.). Here changes in variability are related to, or predicted by, recent past values of the observed series. This is in contrast to other possible representations of locally varying variability, where the variability might be modelled as being driven by a separate time-varying process, as in a doubly stochastic model.
Among other types of non-linear time series models, there are models to represent the changes of variance over time (heteroskedasticity). These models represent autoregressive conditional heteroskedasticity (ARCH) and the collection comprises a wide variety of representation (GARCH, TARCH, EGARCH, FIGARCH, CGARCH, etc.). Here changes in variability are related to, or predicted by, recent past values of the observed series. This is in contrast to other possible representations of locally varying variability, where the variability might be modelled as being driven by a separate time-varying process, as in a doubly stochastic model.
在其他的非线性时间序列模型中，有一些模型可以表示方差随时间的变化(异方差)。这些模型有ARCH模型，其中包括GARCH，TARCH，EGARCH，FIGARCH，CGARCH 等等。在这里，变异性的变化与观测系列的最近过去的值有关，或者是预测的。这与局部变化的其他可能表现形式形成对比，在这种情况下，变化可能被模拟为由一个单独的时变过程驱动，如双重随机模型。
In recent work on model-free analyses, wavelet transform based methods (for example locally stationary wavelets and wavelet decomposed neural networks) have gained favor. Multiscale (often referred to as multiresolution) techniques decompose a given time series, attempting to illustrate time dependence at multiple scales. See also [[Markov switching multifractal]] (MSMF) techniques for modeling volatility evolution.
In recent work on model-free analyses, wavelet transform based methods (for example locally stationary wavelets and wavelet decomposed neural networks) have gained favor. Multiscale (often referred to as multiresolution) techniques decompose a given time series, attempting to illustrate time dependence at multiple scales. See also [[Markov switching multifractal]] (MSMF) techniques for modeling volatility evolution.