时间序列分析

时间序列分析是对按时间顺序导出的(或列出或绘制)的一系列数据点进行分析。时间序列是在连续的等距时间点上的序列。因此，这种序列上的时间是处于离散状态的。测量海洋潮汐的高度、计算太阳黑子的数量和分析道琼斯工业平均指数的每日收盘价都是时间序列在实际工作上的应用。

时间序列常通过趋势图（即时间线图Line chart）具象化。时间序列常被用于统计学、信号处理、模式识别、计量经济学、数理金融学、天气预报、地震预测、脑电图、控制工程、天文学、通信工程，以及涉及时序测量的任何科学和工程领域。

时间序列分析需要提取时间序列数据中有意义的统计特征以及数据的其他特征。时间序列分析涉及到时间序列的预测。时间序列预测是一种基于先前观测到的值去使用模型来预测未来值的方法。虽然回归分析经常被用于分析一个或多个不同时间序列之间的关系，但这种类型的分析通常不被称为 "时间序列分析"。时间序列分析特指的是分析单一序列中不同时间点之间的关系，也会分析被干预的时间序列（分析时间序列在接受干预前后的变化）。这种干预可能会影响基础变量。

时间序列数据具有自然的时间排序。这使得时间序列分析有别于截面研究。在截面研究中，观察结果没有自然排序（例如，通过参考各自的教育水平来解释人们的工资，其中个人的数据可以按任何顺序输入）。时间序列分析也有别于空间数据分析，后者的观测值通常与地理位置有关（例如，通过地点以及房屋的内在特征来说明房价）。时间序列的随机模型通常会反映这样一个事实，即在时间上相距较近的观测值会比相距较远的观测值更密切相关。此外，时间序列模型通常会利用自然的单向时间顺序，以便将给定时间段的值表示为以某种方式从过去的值而不是从未来的值中得出（参见时间可逆性）。

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）。

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。

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。

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

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。

Analysis分析

不同目的的时间序列适用的动机和数据分析方法都不同。

动机

在统计学、计量经济学、定量金融、地震学、气象学和地球物理学方面，时间序列分析的主要目标是预测。在信号处理、控制工程和通信工程方面，它被用于信号检测。在数据挖掘、模式识别和机器学习等其他应用中，时间序列分析可用于聚类、分类、按内容查询、异常检测以及预测。

探索性分析

文件:Tuberculosis incidence US 1953-2009.png

Tuberculosis incidence US 1953-2009美国1953-2009年结核病发病率

绘制折线图是分析常规时间序列的直观方法。右侧显示了一个使用电子表格程序制作的美国结核病发病率示例图表。病例的数量被标准化为每10万人的比率，并计算出该比率每年的变化百分比。几乎稳定下降的线条表明，结核病发病率在大多数年份都在下降，但该比率的变化百分比高达+/-10%，且在1975年和20世纪90年代初前后出现了 "激增"。图中应用了两个纵轴，可以在一个图表中比较两个时间序列。

一项对企业数据分析师的研究发现，探索性时间的序列分析有两个挑战：发现新模式，以及为这些模式找到解释。将时间序列数据可视化为热力图矩阵的工具可以帮助解释这些模式。

其他技巧包括:

• 通过自相关分析去检验序列相关性；
• 通过频谱分析来检查与季节性无关的周期性行为。例如，太阳黑子活动在一个周期内（11年）的变化。周期性行为常见的例子也包括天体现象、天气模式、神经活动、商品价格和经济活动；
• 将序列分离为代表趋势、季节性、慢速和快速变化以及周期性不规则的成分；

曲线拟合

Main article: Curve fitting

Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data.

正文：曲线拟合Curve fitting

曲线拟合 是构建一条曲线Curve或数学函数Mathematical function的过程，它对一系列的数据Data点具有最佳的拟合效果，可能会受到一些限制。曲线拟合包括插值Interpolation（需要精确地拟合数据）与平滑Smoothing（构造一个 "平滑 "的函数来近似地拟合数据）。与曲线拟合相近的回归分析Regression analysis更侧重于统计推断Statistical inference的问题。例如，在拟合有随机误差的数据的曲线中，有多少不确定性存在。拟合曲线可以作为数据可视化的辅助工具，在没有数据的情况下推断函数的值，并总结两个或多个变量之间的关系。外推法Extrapolation是指在观测到的数据范围Range之外使用拟合曲线，它有一定程度的不确定性Degree of uncertainty，因为它既可能是反映观测数据，也可能是反映用于构建曲线的方法。

The construction of economic time series involves the estimation of some components for some dates by interpolation between values ("benchmarks") for earlier and later dates. Interpolation is estimation of an unknown quantity between two known quantities (historical data), or drawing conclusions about missing information from the available information ("reading between the lines"). Interpolation is useful where the data surrounding the missing data is available and its trend, seasonality, and longer-term cycles are known. This is often done by using a related series known for all relevant dates. Alternatively polynomial interpolation or spline interpolation is used where piecewise polynomial functions are fit into time intervals such that they fit smoothly together. A different problem which is closely related to interpolation is the approximation of a complicated function by a simple function (also called regression).The main difference between regression and interpolation is that polynomial regression gives a single polynomial that models the entire data set. Spline interpolation, however, yield a piecewise continuous function composed of many polynomials to model the data set.

经济时间序列的构建涉及通过在早期和晚期的值（“基准”）之间进行插值Interpolation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"）。如果围绕缺失数据的数据是可用的，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。或者使用多项式插值Polynomial interpolation或样条插值Spline interpolation，将分段多项式Polynomial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Regression）。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。而样条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。

Extrapolation is the process of estimating, beyond the original observation range, the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a higher risk of producing meaningless results.

外推法Extrapolation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Interpolation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Uncertainty更大，产生无意义结果的风险也更大。

Function approximation

Main article: Function approximation

In general, a function approximation problem asks us to select a function among a well-defined class that closely matches ("approximates") a target function in a task-specific way. One can distinguish two major classes of function approximation problems: First, for known target functions, approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).

In general, a function approximation problem asks us to select a function among a well-defined class that closely matches ("approximates") a target function in a task-specific way. One can distinguish two major classes of function approximation problems: First, for known target functions, approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).

= = = = = = = = 一般来说，一个函数逼近问题要求我们在一个定义良好的类中选择一个函数，这个类以一种特定于任务的方式与目标函数非常匹配(“近似”)。人们可以区分两类主要的函数逼近问题: 首先，对于已知的目标函数，逼近理论是数值分析的一个分支，研究某些已知函数(例如，特殊函数)如何可以用一类特定的函数(例如，多项式或有理函数)来近似，这类函数通常具有理想的性质(廉价计算、连续性、积分和极限值等等)。).

Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of points (a time series) of the form (x, g(x)) is provided. Depending on the structure of the domain and codomain of g, several techniques for approximating g may be applicable. For example, if g is an operation on the real numbers, techniques of interpolation, extrapolation, regression analysis, and curve fitting can be used. If the codomain (range or target set) of g is a finite set, one is dealing with a classification problem instead. A related problem of online time series approximation is to summarize the data in one-pass and construct an approximate representation that can support a variety of time series queries with bounds on worst-case error.

Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of points (a time series) of the form (x, g(x)) is provided. Depending on the structure of the domain and codomain of g, several techniques for approximating g may be applicable. For example, if g is an operation on the real numbers, techniques of interpolation, extrapolation, regression analysis, and curve fitting can be used. If the codomain (range or target set) of g is a finite set, one is dealing with a classification problem instead. A related problem of online time series approximationGandhi, Sorabh, Luca Foschini, and Subhash Suri. "Space-efficient online approximation of time series data: Streams, amnesia, and out-of-order." Data Engineering (ICDE), 2010 IEEE 26th International Conference on. IEEE, 2010. is to summarize the data in one-pass and construct an approximate representation that can support a variety of time series queries with bounds on worst-case error.

其次，目标函数，称为 g，可能是未知的; 代替黎曼显式公式，只有一组形式(x，g (x))的点(时间序列)被提供。根据 g 的畴和余畴的结构，几种近似 g 的方法可能是适用的。例如，如果 g 是对实数的运算，可以使用插值、外推、回归分析和曲线拟合等技术。如果 g 的余域(范围或目标集)是一个有限集，那么我们就是在处理一个分类问题。在线时间序列的一个相关问题接近于 andhi，Sorabh，Luca Foschini，和 Subhash Suri。时间序列数据的空间有效在线近似: 数据流、失忆和无序数据工程，2010年 IEEE 第26届国际会议。2010.是对数据进行一次总结，构造一个近似表示，可以支持各种时间序列查询，最坏情况下的错误界限。

To some extent, the different problems (regression, classification, fitness approximation) have received a unified treatment in statistical learning theory, where they are viewed as supervised learning problems.

To some extent, the different problems (regression, classification, fitness approximation) have received a unified treatment in statistical learning theory, where they are viewed as supervised learning problems.

在某种程度上，不同的问题(回归、分类、适应度逼近)在统计学习理论中得到了统一的处理，它们被视为监督式学习问题。

Prediction and forecasting

In statistics, prediction is a part of statistical inference. One particular approach to such inference is known as predictive inference, but the prediction can be undertaken within any of the several approaches to statistical inference. Indeed, one description of statistics is that it provides a means of transferring knowledge about a sample of a population to the whole population, and to other related populations, which is not necessarily the same as prediction over time. When information is transferred across time, often to specific points in time, the process is known as forecasting.

• Fully formed statistical models for stochastic simulation purposes, so as to generate alternative versions of the time series, representing what might happen over non-specific time-periods in the future
• Simple or fully formed statistical models to describe the likely outcome of the time series in the immediate future, given knowledge of the most recent outcomes (forecasting).
• Forecasting on time series is usually done using automated statistical software packages and programming languages, such as Julia, Python, R, SAS, SPSS and many others.
• Forecasting on large scale data can be done with Apache Spark using the Spark-TS library, a third-party package.

In statistics, prediction is a part of statistical inference. One particular approach to such inference is known as predictive inference, but the prediction can be undertaken within any of the several approaches to statistical inference. Indeed, one description of statistics is that it provides a means of transferring knowledge about a sample of a population to the whole population, and to other related populations, which is not necessarily the same as prediction over time. When information is transferred across time, often to specific points in time, the process is known as forecasting.

• Fully formed statistical models for stochastic simulation purposes, so as to generate alternative versions of the time series, representing what might happen over non-specific time-periods in the future
• Simple or fully formed statistical models to describe the likely outcome of the time series in the immediate future, given knowledge of the most recent outcomes (forecasting).
• Forecasting on time series is usually done using automated statistical software packages and programming languages, such as Julia, Python, R, SAS, SPSS and many others.
• Forecasting on large scale data can be done with Apache Spark using the Spark-TS library, a third-party package.

在统计学中，预测是推论统计学的一部分。一种特殊的推理方法被称为预测推理，但是这种预测可以在几种推论统计学推理方法中的任何一种中进行。事实上，对统计的一种描述是，它提供了一种将关于某一人口样本的知识转移给整个人口和其他相关人口的手段，这不一定与随着时间的推移所作的预测相同。当信息跨越时间传递，通常是传递到特定的时间点，这个过程就被称为预测。

• 为随机模拟目的而建立完整的统计模型，以产生时间序列的替代版本，反映未来在非特定时间段内可能发生的情况
• 简单或完整的统计模型，以描述时间序列在最近期间可能产生的结果(预测)。
• 时间序列预测通常使用自动化的统计软件包和编程语言，例如 Julia、 Python、 r、 SAS、 SPSS 等。
• 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

Main article: 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

See also: Signal processing and Estimation theory

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

Segmentation

Main article: Time-series segmentation

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.

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。

= 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 (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".

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 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)

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.

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。

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.

在最近的无模型分析工作中，基于小波变换的方法(如局部平稳小波和小波分解神经网络)得到了广泛的关注。多尺度(通常称为多分辨率)技术分解给定的时间序列，试图说明在多个尺度上的时间依赖。参见马尔可夫切换多重分形(MSMF)建模波动演化技术。

A Hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved (hidden) states. An HMM can be considered as the simplest dynamic Bayesian network. HMM models are widely used in speech recognition, for translating a time series of spoken words into text.

A Hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved (hidden) states. An HMM can be considered as the simplest dynamic Bayesian network. HMM models are widely used in speech recognition, for translating a time series of spoken words into text.

隐马尔可夫模型模型是一个统计马尔可夫模型，其中被建模的系统被假定为一个具有不可观测(隐藏)状态的马尔可夫过程。隐马尔科姆可以被认为是最简单的动态贝氏网路。隐马尔可夫模型广泛应用于语音识别中，用于将语音序列转换成文本。

Notation

A number of different notations are in use for time-series analysis. A common notation specifying a time series X that is indexed by the natural numbers is written

X = (X₁, X₂, ...).

A number of different notations are in use for time-series analysis. A common notation specifying a time series X that is indexed by the natural numbers is written

X = (X1, X2, ...).

= = = 表示法 = = 用于时间序列分析的许多不同的表示法。一个用于指定时间序列 x 的通用符号是: x = (X1，X2，...)。

Another common notation is

Y = (Y_t: t ∈ T),

where T is the index set.

Another common notation is

Y = (Yt: t ∈ T),

where T is the index set.

另一种常用的表示法是: y = (Yt: t ∈ t) ，其中 t 是索引集。

Conditions

There are two sets of conditions under which much of the theory is built:

• Stationary process
• Ergodic process

There are two sets of conditions under which much of the theory is built:

• Stationary process
• Ergodic process

这个理论的大部分建立在两个条件之下:

• 平稳过程遍历过程

However, ideas of stationarity must be expanded to consider two important ideas: strict stationarity and second-order stationarity. Both models and applications can be developed under each of these conditions, although the models in the latter case might be considered as only partly specified.

However, ideas of stationarity must be expanded to consider two important ideas: strict stationarity and second-order stationarity. Both models and applications can be developed under each of these conditions, although the models in the latter case might be considered as only partly specified.

然而，平稳性的概念必须扩展到考虑两个重要的概念: 严格平稳性和二阶平稳性。模型和应用程序都可以在这些条件中的每一种情况下开发，尽管后一种情况下的模型可能被认为只是部分具体说明。

In addition, time-series analysis can be applied where the series are seasonally stationary or non-stationary. Situations where the amplitudes of frequency components change with time can be dealt with in time-frequency analysis which makes use of a time–frequency representation of a time-series or signal.

In addition, time-series analysis can be applied where the series are seasonally stationary or non-stationary. Situations where the amplitudes of frequency components change with time can be dealt with in time-frequency analysis which makes use of a time–frequency representation of a time-series or signal.Boashash, B. (ed.), (2003) Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, Elsevier Science, Oxford, 2003

此外，时间序列分析可以应用于季节性平稳或非平稳的序列。时频分析利用时间序列或信号的时频表示，可以处理频率分量振幅随时间变化的情况。波阿什，b。我不知道你在说什么。) ，(2003)《时频信号分析与处理: 综合参考》 ，爱思唯尔科学出版社，牛津，2003

Tools

Tools for investigating time-series data include:

Tools for investigating time-series data include:

= = = 调查时间序列数据的工具包括:

• Consideration of the autocorrelation function and the spectral density function (also cross-correlation functions and cross-spectral density functions)
• Scaled cross- and auto-correlation functions to remove contributions of slow components
• Performing a Fourier transform to investigate the series in the frequency domain
• Use of a filter to remove unwanted noise
• Principal component analysis (or empirical orthogonal function analysis)
• Singular spectrum analysis
• "Structural" models:
  • General State Space Models
  • Unobserved Components Models
• Machine Learning
  • Artificial neural networks
  • Support vector machine
  • Fuzzy logic
  • Gaussian process
  • Genetic Programming
  • Gene expression programming
  • Hidden Markov model
  • Multi expression programming
• Queueing theory analysis
• Control chart
  • Shewhart individuals control chart
  • CUSUM chart
  • EWMA chart
• Detrended fluctuation analysis
• Nonlinear mixed-effects modeling
• Dynamic time warping
• Cross-correlation
• Dynamic Bayesian network
• Time-frequency analysis techniques:
  • Fast Fourier transform
  • Continuous wavelet transform
  • Short-time Fourier transform
  • Chirplet transform
  • Fractional Fourier transform
• Chaotic analysis
  • Correlation dimension
  • Recurrence plots
  • Recurrence quantification analysis
  • Lyapunov exponents
  • Entropy encoding

• Consideration of the autocorrelation function and the spectral density function (also cross-correlation functions and cross-spectral density functions)
• Scaled cross- and auto-correlation functions to remove contributions of slow components
• Performing a Fourier transform to investigate the series in the frequency domain
• Use of a filter to remove unwanted noise
• Principal component analysis (or empirical orthogonal function analysis)
• Singular spectrum analysis
• "Structural" models:
  • General State Space Models
  • Unobserved Components Models
• Machine Learning
  • Artificial neural networks
  • Support vector machine
  • Fuzzy logic
  • Gaussian process
  • Genetic Programming
  • Gene expression programming
  • Hidden Markov model
  • Multi expression programming
• Queueing theory analysis
• Control chart
  • Shewhart individuals control chart
  • CUSUM chart
  • EWMA chart
• Detrended fluctuation analysis
• Nonlinear mixed-effects modeling
• Dynamic time warping
• Cross-correlation
• Dynamic Bayesian network
• Time-frequency analysis techniques:
  • Fast Fourier transform
  • Continuous wavelet transform
  • Short-time Fourier transform
  • Chirplet transform
  • Fractional Fourier transform
• Chaotic analysis
  • Correlation dimension
  • Recurrence plots
  • Recurrence quantification analysis
  • Lyapunov exponents
  • Entropy encoding

• 考虑自相关函数和谱密度函数(也包括互相关函数和互谱密度函数)
• 调整互相关函数和自相关函数以去除慢分量的贡献
• 在频域中执行一个傅里叶变换来调查这个序列
• 使用
• CUSUM 图
• EWMA 图
• 非趋势波动分析
• 非线性混合效应建模
• 动态时间规整互相关
• 动态贝氏网路
• 时频分析技术:
• 快速傅里叶变换连续小波转换
• 短时距傅里叶变换
• 混沌分析
• 复发图
• 递归量化分析
• Lyapunov 指数

Measures

Time series metrics or features that can be used for time series classification or regression analysis:

Time series metrics or features that can be used for time series classification or regression analysis:

= = = = 可用于时间序列分类或回归分析的时间序列度量或特征:

• Univariate linear measures
  • Moment (mathematics)
  • Spectral band power
  • Spectral edge frequency
  • Accumulated Energy (signal processing)
  • Characteristics of the autocorrelation function
  • Hjorth parameters
  • FFT parameters
  • Autoregressive model parameters
  • Mann–Kendall test
• Univariate non-linear measures
  • Measures based on the correlation sum
  • Correlation dimension
  • Correlation integral
  • Correlation density
  • Correlation entropy
  • Approximate entropy
  • Sample entropy
  • 模板:Iw2
  • Wavelet entropy
  • Dispersion entropy
  • Fluctuation dispersion entropy
  • Rényi entropy
  • Higher-order methods
  • Marginal predictability
  • Dynamical similarity index
  • State space dissimilarity measures
  • Lyapunov exponent
  • Permutation methods
  • Local flow
• Other univariate measures
  • Algorithmic complexity
  • Kolmogorov complexity estimates
  • Hidden Markov Model states
  • Rough path signature
  • Surrogate time series and surrogate correction
  • Loss of recurrence (degree of non-stationarity)
• Bivariate linear measures
  • Maximum linear cross-correlation
  • Linear Coherence (signal processing)
• Bivariate non-linear measures
  • Non-linear interdependence
  • Dynamical Entrainment (physics)
  • Measures for Phase synchronization
  • Measures for Phase locking
• Similarity measures:
  • Cross-correlation
  • Dynamic Time Warping
  • Hidden Markov Models
  • Edit distance
  • Total correlation
  • Newey–West estimator
  • Prais–Winsten transformation
  • Data as Vectors in a Metrizable Space
    • Minkowski distance
    • Mahalanobis distance
  • Data as time series with envelopes
    • Global standard deviation
    • Local standard deviation
    • Windowed standard deviation
  • Data interpreted as stochastic series
    • Pearson product-moment correlation coefficient
    • Spearman's rank correlation coefficient
  • Data interpreted as a probability distribution function
    • Kolmogorov–Smirnov test
    • Cramér–von Mises criterion

• Univariate linear measures
  • Moment (mathematics)
  • Spectral band power
  • Spectral edge frequency
  • Accumulated Energy (signal processing)
  • Characteristics of the autocorrelation function
  • Hjorth parameters
  • FFT parameters
  • Autoregressive model parameters
  • Mann–Kendall test
• Univariate non-linear measures
  • Measures based on the correlation sum
  • Correlation dimension
  • Correlation integral
  • Correlation density
  • Correlation entropy
  • Approximate entropy
  • Sample entropy
  • Wavelet entropy
  • Dispersion entropy
  • Fluctuation dispersion entropy
  • Rényi entropy
  • Higher-order methods
  • Marginal predictability
  • Dynamical similarity index
  • State space dissimilarity measures
  • Lyapunov exponent
  • Permutation methods
  • Local flow
• Other univariate measures
  • Algorithmic complexity
  • Kolmogorov complexity estimates
  • Hidden Markov Model states
  • Rough path signature[1] Chevyrev, I., Kormilitzin, A. (2016) "A Primer on the Signature Method in Machine Learning, arXiv:1603.03788v1"
  • Surrogate time series and surrogate correction
  • Loss of recurrence (degree of non-stationarity)
• Bivariate linear measures
  • Maximum linear cross-correlation
  • Linear Coherence (signal processing)
• Bivariate non-linear measures
  • Non-linear interdependence
  • Dynamical Entrainment (physics)
  • Measures for Phase synchronization
  • Measures for Phase locking
• Similarity measures:
  • Cross-correlation
  • Dynamic Time Warping
  • Hidden Markov Models
  • Edit distance
  • Total correlation
  • Newey–West estimator
  • Prais–Winsten transformation
  • Data as Vectors in a Metrizable Space
    • Minkowski distance
    • Mahalanobis distance
  • Data as time series with envelopes
    • Global standard deviation
    • Local standard deviation
    • Windowed standard deviation
  • Data interpreted as stochastic series
    • Pearson product-moment correlation coefficient
    • Spearman's rank correlation coefficient
  • Data interpreted as a probability distribution function
    • Kolmogorov–Smirnov test
    • Cramér–von Mises criterion

• 单变量线性测量
• 矩(数学)
• 谱带功率
• 谱边缘频率
• 累积能量(信号处理)
• 自相关函数特性
• Hjorth 参数
• FFT 参数
• 自回归模型参数
• 相关积分相关密度相关熵近似熵小波熵色散熵涨落色散熵高阶方法边际可预测动力学相似性指数状态空间相异性度量李亚普诺夫指数排列方法
• 本地流
• 其他单变量度量
• 算法复杂度
• 柯氏复杂性估计
• 隐马尔可夫模型状态
• 粗糙路径签名[1] Chevyrev，i. ，Kormilitzin，a。(2016)“ a Primer on the Signature Method in Machine Learning，arXiv: 1603.03788 v1”
• 替代时间序列和替代校正
• 递归损失(非平稳度)
• 双变量线性度量
• 最大线性互相关
• 线性相干性(信号处理)
• 双变量非线性度量
• 非线性相互依赖
• 动态卷吸(物理学)
• 相位同步的度量
• 相位锁定的度量
• 相似度量:
• 互相关
• 动态时间规整
• 隐马尔可夫模型
• 编辑距离
• 总相关性
• Newey-West 估计
• Prais-Winsten 变换
• 数据作为向量在乌雷松度量化定理
• 明氏距离
• 马氏距离
• 数据作为时间序列与信封
• 局部标准差标准差
• 窗口标准差
• 数据解释为随机序列
• 斯皮尔曼的秩相关系数
• 数据解释为概率分布函数
• Kolmogorov-Smirnov 检验
• Cramér-von Mises 准则

Visualization

Time series can be visualized with two categories of chart: Overlapping Charts and Separated Charts. Overlapping Charts display all-time series on the same layout while Separated Charts presents them on different layouts (but aligned for comparison purpose)

Time series can be visualized with two categories of chart: Overlapping Charts and Separated Charts. Overlapping Charts display all-time series on the same layout while Separated Charts presents them on different layouts (but aligned for comparison purpose)

= = 可视化 = = 时间序列可以用两类图表进行可视化: 重叠图表和分离图表。重叠图表显示同一布局的所有时间序列，而分离图表显示不同的布局(但对齐用于比较)

Overlapping charts

• Braided graphs
• Line charts
• Slope graphs
• 模板:Iw2

• Braided graphs
• Line charts
• Slope graphs

= = = 重叠图 =

• 编织图
• 线图
• 斜率图

Separated charts

• Horizon graphs
• Reduced line chart (small multiples)
• Silhouette graph
• Circular silhouette graph

• Horizon graphs
• Reduced line chart (small multiples)
• Silhouette graph
• Circular silhouette graph

= = = = 分离图表 = =

• 地平线图
• 简化线图(小倍数)
• 轮廓线图
• 圆形轮廓线图

See also

• Anomaly time series
• Chirp
• Decomposition of time series
• Detrended fluctuation analysis
• Digital signal processing
• Distributed lag
• Estimation theory
• Forecasting
• Frequency spectrum
• Hurst exponent
• Least-squares spectral analysis
• Monte Carlo method
• Panel analysis
• Random walk
• Scaled correlation
• Seasonal adjustment
• Sequence analysis
• Signal processing
• Time series database (TSDB)
• Trend estimation
• Unevenly spaced time series

Further reading

• Box, George; Jenkins, Gwilym (1976), Time Series Analysis: forecasting and control, rev. ed., Oakland, California: Holden-Day
• Durbin J., Koopman S.J. (2001), Time Series Analysis by State Space Methods, Oxford University Press.
• Gershenfeld, Neil (2000), The Nature of Mathematical Modeling, Cambridge University Press, ISBN 978-0-521-57095-4, OCLC 174825352
• Hamilton, James (1994), Time Series Analysis, Princeton University Press, ISBN 978-0-691-04289-3
• Priestley, M. B. (1981), Spectral Analysis and Time Series, Academic Press.
  • Shasha, D. (2004), High Performance Discovery in Time Series, Springer, ISBN 978-0-387-00857-8
  • Shumway R. H., Stoffer D. S. (2017), Time Series Analysis and its Applications: With R Examples (ed. 4), Springer,
    • Weigend A. S., Gershenfeld N. A. (Eds.) (1994), Time Series Prediction: Forecasting the Future and Understanding the Past. Proceedings of the NATO Advanced Research Workshop on Comparative Time Series Analysis (Santa Fe, May 1992), Addison-Wesley.
    • Wiener, N. (1949), Extrapolation, Interpolation, and Smoothing of Stationary Time Series, MIT Press.
    • Woodward, W. A., Gray, H. L. & Elliott, A. C. (2012), Applied Time Series Analysis, CRC Press.
    • Auffarth, Ben (2021). Machine Learning for Time-Series with Python: Forecast, predict, and detect anomalies with state-of-the-art machine learning methods (1st ed.). Packt Publishing. ISBN 978-1801819626. https://www.packtpub.com/product/machine-learning-for-time-series-with-python/9781801819626. 
    • Durbin J., Koopman S.J. (2001), Time Series Analysis by State Space Methods, Oxford University Press.
    • Priestley, M. B. (1981), Spectral Analysis and Time Series, Academic Press.
    • Shumway R. H., Stoffer D. S. (2017), Time Series Analysis and its Applications: With R Examples (ed. 4), Springer,
    • Weigend A. S., Gershenfeld N. A. (Eds.) (1994), Time Series Prediction: Forecasting the Future and Understanding the Past. Proceedings of the NATO Advanced Research Workshop on Comparative Time Series Analysis (Santa Fe, May 1992), Addison-Wesley.
    • Wiener, N. (1949), Extrapolation, Interpolation, and Smoothing of Stationary Time Series, MIT Press.
    • Woodward, W. A., Gray, H. L. & Elliott, A. C. (2012), Applied Time Series Analysis, CRC Press.

    = 进一步阅读 =

    • Durbin j，Koopman s.j。(2001) ，《状态空间法时间序列分析》 ，牛津大学出版社。
    • Priestley, M. B.(1981) ，光谱分析与时间序列，学术出版社。
    • Shumway r. h. ，Stoffer d. s. (2017) ，时间序列分析及其应用: 与 R.示例(ed。4), Springer,
    • Weigend A. S., Gershenfeld N. A.(Eds.)(1994) ，时间序列预测: 预测未来和了解过去。北约比较时间序列分析高级研究讲习班论文集(圣达菲，1992年5月) ，Addison-Wesley。《平稳时间序列的外推、插值和平滑》 ，麻省理工学院出版社。
    • 伍德沃德，w. a. ，格雷，h. l. & 埃利奥特，a. c. (2012) ，应用时间序列分析，CRC 出版社。

    External links

    • Introduction to Time series Analysis (Engineering Statistics Handbook) — A practical guide to Time series analysis.

    
    

    • Introduction to Time series Analysis (Engineering Statistics Handbook) — A practical guide to Time series analysis.

    = 外部链接 =

    • 时间序列分析导论(工程统计手册)ー时间序列分析实用指南。

    
    

    Category:Statistical data types Category:Mathematical and quantitative methods (economics) Category:Machine learning

    类别: 统计数据类型类别: 数学和定量方法(经济学)类别: 机器学习

    
    

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