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Moved page from wikipedia:en:Time series (history)
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{{Use American English|date = March 2019}}
{{short description|Sequence of data points over time}}
[[File:Random-data-plus-trend-r2.png|thumb|250px|Time series: random data plus trend, with best-fit line and different applied filters|alt=|right]]
In [[mathematics]], a '''time series''' is a series of [[data point]]s indexed (or listed or graphed) in time order. Most commonly, a time series is a [[sequence]] taken at successive equally spaced points in time. Thus it is a sequence of [[discrete-time]] data. Examples of time series are heights of ocean [[tides]], counts of [[sunspots]], and the daily closing value of the [[Dow Jones Industrial Average]].
thumb|250px|Time series: random data plus trend, with best-fit line and different applied filters|alt=|right
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.
拇指 | 250px | 时间序列: 随机数据加趋势，带有最佳拟合线和不同的应用过滤器 | alt = | right 在数学中，时间序列是按时间顺序索引(或列出或绘制)的一系列数据点。最常见的是，时间序列是在相继的等间隔时间点上拍摄的序列。因此，它是一个离散时间数据序列。时间序列的例子有海潮的高度、太阳黑子的数量以及道琼斯工业平均指数的每日收盘价。
A Time series is very frequently plotted via a [[run chart]] (which is a temporal [[line chart]]). Time series are used in [[statistics]], [[signal processing]], [[pattern recognition]], [[econometrics]], [[mathematical finance]], [[weather forecasting]], [[earthquake prediction]], [[electroencephalography]], [[control engineering]], [[astronomy]], [[communications engineering]], and largely in any domain of applied [[Applied science|science]] and [[engineering]] which involves [[Time|temporal]] measurements.
A Time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements.
时间序列通常是通过运行图(时间线图)绘制的。时间序列广泛应用于统计学、信号处理、模式识别、计量经济学、数学金融学、天气预报、地震预测、脑电图、控制工程、天文学、通信工程等领域，还广泛应用于涉及时间测量的任何应用科学和工程领域。
'''Time series ''analysis''''' comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. '''Time series ''forecasting''''' is the use of a [[model (abstract)|model]] to predict future values based on previously observed values. While [[regression analysis]] is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series. [[Interrupted time series]] analysis is used to detect changes in the evolution of a time series from before to after some intervention which may affect the underlying variable.
Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series. Interrupted time series analysis is used to detect changes in the evolution of a time series from before to after some intervention which may affect the underlying variable.
时间序列分析包括分析时间序列数据的方法，以提取有意义的统计数据和数据的其他特征。时间序列预测是使用一个模型来预测未来的价值基于以前观察到的价值。虽然回归分析通常用于测试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”，它特别指的是单个序列中不同时间点之间的关系。中断时间序列分析是用来检测一个时间序列从干预之前到干预之后的变化，这些变化可能会影响到基础变量。
Time series data have a natural temporal ordering. This makes time series analysis distinct from [[cross-sectional study|cross-sectional studies]], in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from [[spatial data analysis]] where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A [[stochastic]] model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see [[time reversibility]]).
Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see time reversibility).
时间序列数据具有自然的时间序列。这使得时间序列分析不同于横断面研究，横断面研究中的观察没有自然的顺序(例如:。通过参照个人的教育程度来解释人们的工资，个人的数据可以按任意顺序输入)。时间序列分析也不同于空间数据分析，因为空间数据分析的观测通常与地理位置有关(例如:。根据房屋的位置和内在特征来计算房价)。一个时间序列的随机模型通常反映了这样一个事实，即在时间上紧密相连的观察比相距较远的观察更密切相关。此外，时间序列模型往往利用时间的自然单向排序，以便某一时期的数值以某种方式从过去的数值而不是从未来的数值得出(见时间可逆性)。
Time series analysis can be applied to [[real number|real-valued]], continuous data, [[:wikt:discrete|discrete]] [[Data type#Numeric types|numeric]] data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the [[English language]]<ref>{{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>).
Time series analysis can be applied to real-valued, continuous data, discrete numeric data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the English language).
时间序列分析可以应用于实值数据、连续数据、离散数值数据或离散符号数据(即。字符序列，例如英语中的字母和单词)。
==Methods for analysis==
==Methods for analysis==
= = = 分析方法 = =
Methods for time series analysis may be divided into two classes: [[frequency-domain]] methods and [[time-domain]] methods. The former include [[frequency spectrum#Spectrum analysis|spectral analysis]] and [[wavelet analysis]]; the latter include [[auto-correlation]] and [[cross-correlation]] analysis. In the time domain, correlation and analysis can be made in a filter-like manner using [[scaled correlation]], thereby mitigating the need to operate in the frequency domain.
Methods for time series analysis may be divided into two classes: frequency-domain methods and time-domain methods. The former include spectral analysis and wavelet analysis; the latter include auto-correlation and cross-correlation analysis. In the time domain, correlation and analysis can be made in a filter-like manner using scaled correlation, thereby mitigating the need to operate in the frequency domain.
时间序列分析方法可分为两类: 频域方法和时域方法。前者包括谱分析和小波分析，后者包括自相关和互相关分析。在时域上，可以使用比例相关以类似滤波器的方式进行相关和分析，从而减少了在频域进行操作的需要。
Additionally, time series analysis techniques may be divided into [[Parametric estimation|parametric]] and [[Non-parametric statistics|non-parametric]] methods. The [[Parametric estimation|parametric approaches]] assume that the underlying [[stationary process|stationary stochastic process]] has a certain structure which can be described using a small number of parameters (for example, using an [[autoregressive]] or [[moving average model]]). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, [[Non-parametric statistics|non-parametric approaches]] explicitly estimate the [[covariance]] or the [[spectrum]] of the process without assuming that the process has any particular structure.
Additionally, time series analysis techniques may be divided into parametric and non-parametric methods. The parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters (for example, using an autoregressive or moving average model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure.
此外，时间序列分析技术可分为参数方法和非参数方法。参数方法假设潜在的平稳随机过程有一个特定的结构，这个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，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谱，而不假设过程具有任何特定的结构。
Methods of time series analysis may also be divided into [[Linear regression|linear]] and [[Nonlinear regression|non-linear]], and [[Univariate analysis|univariate]] and [[Multivariate analysis|multivariate]].
Methods of time series analysis may also be divided into linear and non-linear, and univariate and multivariate.
时间序列分析的方法也可分为线性和非线性，以及单变量和多变量。
==Panel data==
==Panel data==
= = = 面板数据 = =
A time series is one type of [[panel data]]. Panel data is the general class, a multidimensional data set, whereas a time series data set is a one-dimensional panel (as is a [[cross-sectional data]]set). A data set may exhibit characteristics of both panel data and time series data. One way to tell is to ask what makes one data record unique from the other records. If the answer is the time data field, then this is a time series data set candidate. If determining a unique record requires a time data field and an additional identifier which is unrelated to time (e.g. student ID, stock symbol, country code), then it is panel data candidate. If the differentiation lies on the non-time identifier, then the data set is a cross-sectional data set candidate.
A time series is one type of panel data. Panel data is the general class, a multidimensional data set, whereas a time series data set is a one-dimensional panel (as is a cross-sectional dataset). A data set may exhibit characteristics of both panel data and time series data. One way to tell is to ask what makes one data record unique from the other records. If the answer is the time data field, then this is a time series data set candidate. If determining a unique record requires a time data field and an additional identifier which is unrelated to time (e.g. student ID, stock symbol, country code), then it is panel data candidate. If the differentiation lies on the non-time identifier, then the data set is a cross-sectional data set candidate.
时间序列是面板数据的一种。Panel data 是一个通用类，是一个多维数据集，而时间序列数据集是一个一维面板(就像横截面数据集一样)。一个数据集可以同时显示面板数据和时间序列数据的特征。判断的一种方法是询问是什么使一个数据记录与其他记录相比是唯一的。如果答案是时间数据字段，那么这是一个候选的时间序列数据集。如果确定一个唯一的记录需要一个时间数据字段和一个与时间无关的附加标识符(例如:。学生证，股票代码，国家代码) ，然后是面板数据候选人。如果区分取决于非时间标识符，那么数据集是一个横断面数据集候选者。
==Analysis==
==Analysis==
= = 分析 = =
There are several types of motivation and data analysis available for time series which are appropriate for different purposes.
There are several types of motivation and data analysis available for time series which are appropriate for different purposes.
有几种类型的动机和数据分析可用于时间序列是适合不同的目的。
===Motivation===
===Motivation===
= = = 动机 = =
In the context of [[statistics]], [[econometrics]], [[quantitative finance]], [[seismology]], [[meteorology]], and [[geophysics]] the primary goal of time series analysis is [[forecasting]]. In the context of [[signal processing]], [[control engineering]] and [[communication engineering]] it is used for signal detection. Other applications are in [[data mining]], [[pattern recognition]] and [[machine learning]], where time series analysis can be used for [[cluster analysis|clustering]],<ref>{{cite journal | last1 = Liao | first1 = T. Warren | title = Clustering of time series data - a survey | journal = Pattern Recognition | volume = 38 | issue = 11 | pages = 1857–1874 | publisher = Elsevier | date = 2005 | language = en | doi = 10.1016/j.patcog.2005.01.025| bibcode = 2005PatRe..38.1857W }}{{subscription required|via=ScienceDirect }}</ref><ref>{{cite journal | last1 = Aghabozorgi | first1 = Saeed | last2 = Shirkhorshidi | first2 = Ali S. | last3 = Wah | first3 = Teh Y. | title = Time-series clustering – A decade review | journal = Information Systems | volume = 53 | pages = 16–38 | publisher = Elsevier | date = 2015 | language = en | doi = 10.1016/j.is.2015.04.007}}{{subscription required|via=ScienceDirect }}</ref> [[Statistical classification|classification]],<ref>{{cite journal | last1 = Keogh | first1 = Eamonn J. | title = On the need for time series data mining benchmarks | journal = Data Mining and Knowledge Discovery | volume = 7 | pages = 349–371 | publisher = Kluwer | date = 2003 | language = en | doi = 10.1145/775047.775062| isbn = 158113567X | s2cid = 41617550 }}{{subscription required|via=ACM Digital Library }}</ref> query by content,<ref>{{cite conference|last1=Agrawal|first1=Rakesh|last2=Faloutsos|first2=Christos|last3=Swami|first3=Arun|date=October 1993|title=Efficient Similarity Search In Sequence Databases|conference=International Conference on Foundations of Data Organization and Algorithms|volume=730|pages=69–84|book-title=Proceedings of the 4th International Conference on Foundations of Data Organization and Algorithms|doi=10.1007/3-540-57301-1_5}}{{Subscription required|via=SpringerLink}}</ref> [[anomaly detection]] as well as [[forecasting]].<ref>{{cite journal|last1=Chen|first1=Cathy W. S.|last2=Chiu|first2=L. M.|date=September 2021|title=Ordinal Time Series Forecasting of the Air Quality Index|journal=Entropy|language=en|volume=23|issue=9|pages=1167|doi=10.3390/e23091167|pmid=34573792|pmc=8469594|bibcode=2021Entrp..23.1167C|doi-access=free}}</ref>
In the context of statistics, econometrics, quantitative finance, seismology, meteorology, and geophysics the primary goal of time series analysis is forecasting. In the context of signal processing, control engineering and communication engineering it is used for signal detection. Other applications are in data mining, pattern recognition and machine learning, where time series analysis can be used for clustering, classification, query by content, anomaly detection as well as forecasting.
在统计学、计量经济学、定量金融学、地震学、气象学和地球物理学的背景下，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中，它被用于信号检测。其他应用还包括数据挖掘、模式识别和机器学习，时间序列分析可用于聚类、分类、内容查询、异常检测和预测。
===Exploratory analysis===
[[File:Tuberculosis incidence US 1953-2009.png|thumb|Tuberculosis incidence US 1953-2009]]
{{further|Exploratory analysis}}
A straightforward way to examine a regular time series is manually with a [[line chart]]. An example chart is shown on the right for tuberculosis incidence in the United States, made with a spreadsheet program. The number of cases was standardized to a rate per 100,000 and the percent change per year in this rate was calculated. The nearly steadily dropping line shows that the TB incidence was decreasing in most years, but the percent change in this rate varied by as much as +/- 10%, with 'surges' in 1975 and around the early 1990s. The use of both vertical axes allows the comparison of two time series in one graphic.
thumb|Tuberculosis incidence US 1953-2009
A straightforward way to examine a regular time series is manually with a line chart. An example chart is shown on the right for tuberculosis incidence in the United States, made with a spreadsheet program. The number of cases was standardized to a rate per 100,000 and the percent change per year in this rate was calculated. The nearly steadily dropping line shows that the TB incidence was decreasing in most years, but the percent change in this rate varied by as much as +/- 10%, with 'surges' in 1975 and around the early 1990s. The use of both vertical axes allows the comparison of two time series in one graphic.
探索性分析结核病发病率美国1953-2009一个简单的方法来检查一个定期的时间序列是手工与线图。右图显示的是美国肺结核发病率的示例图，是用电子表格程序制作的。病例数量标准化为每100000例，并计算了这一比率每年的变化百分比。几乎稳步下降的曲线表明，结核病的发病率在大多数年份都在下降，但这一比率的变化率高达 +/-10% ，在1975年和90年代初期出现了“激增”。两个垂直轴的使用允许在一个图形中比较两个时间序列。
A study of corporate data analysts found two challenges to exploratory time series analysis: discovering the shape of interesting patterns, and finding an explanation for these patterns.<ref>{{Cite journal|last=Sarkar|first=Advait|last2=Spott|first2=Martin|last3=Blackwell|first3=Alan F.|last4=Jamnik|first4=Mateja|date=2016|title=Visual discovery and model-driven explanation of time series patterns|url=https://doi.org/10.1109/VLHCC.2016.7739668|journal=2016 IEEE Symposium on Visual Languages and Human-Centric Computing (VL/HCC)|publisher=IEEE|doi=10.1109/vlhcc.2016.7739668}}</ref> Visual tools that represent time series data as [[Heat map|heat map matrices]] can help overcome these challenges.
A study of corporate data analysts found two challenges to exploratory time series analysis: discovering the shape of interesting patterns, and finding an explanation for these patterns. Visual tools that represent time series data as heat map matrices can help overcome these challenges.
一项针对企业数据分析师的研究发现，探索性时间序列分析面临两个挑战: 发现有趣模式的形状，以及为这些模式找到解释。将时间序列数据表示为热图矩阵的可视化工具可以帮助克服这些挑战。
Other techniques include:
Other techniques include:
其他技巧包括:
* [[Autocorrelation]] analysis to examine [[serial dependence]]
* [[frequency spectrum#Spectrum analysis|Spectral analysis]] to examine cyclic behavior which need not be related to [[seasonality]]. For example, sunspot activity varies over 11 year cycles.<ref>{{cite book |last=Bloomfield |first=P. |year=1976 |title=Fourier analysis of time series: An introduction |location=New York |publisher=Wiley |isbn=978-0471082569 }}</ref><ref>{{cite book |last=Shumway |first=R. H. |year=1988 |title=Applied statistical time series analysis |location=Englewood Cliffs, NJ |publisher=Prentice Hall |isbn=978-0130415004 }}</ref> Other common examples include celestial phenomena, weather patterns, neural activity, commodity prices, and economic activity.
* Separation into components representing trend, seasonality, slow and fast variation, and cyclical irregularity: see [[trend estimation]] and [[decomposition of time series]]
* Autocorrelation analysis to examine serial dependence
* Spectral analysis to examine cyclic behavior which need not be related to seasonality. For example, sunspot activity varies over 11 year cycles. Other common examples include celestial phenomena, weather patterns, neural activity, commodity prices, and economic activity.
* Separation into components representing trend, seasonality, slow and fast variation, and cyclical irregularity: see trend estimation and decomposition of time series
* 自相关分析检验序列相关性
* 谱分析检验不需要与季节性相关的循环行为。例如，太阳黑子的活动周期为11年。其他常见的例子包括天文现象、天气模式、神经活动、商品价格和经济活动。
* 分离成分代表趋势，季节性，缓慢和快速变化，周期性不规则: 见趋势估计和时间序列分解
===Curve fitting===
{{main|Curve fitting}}
Curve fitting<ref>Sandra Lach Arlinghaus, PHB Practical Handbook of Curve Fitting. CRC Press, 1994.</ref><ref>William M. Kolb. Curve Fitting for Programmable Calculators. Syntec, Incorporated, 1984.</ref> is the process of constructing a [[curve]], or [[function (mathematics)|mathematical function]], that has the best fit to a series of [[data]] points,<ref>S.S. Halli, K.V. Rao. 1992. Advanced Techniques of Population Analysis. {{isbn|0306439972}} Page 165 (''cf''. ... functions are fulfilled if we have a good to moderate fit for the observed data.)</ref> possibly subject to constraints.<ref>[https://archive.org/details/signalnoisewhymo00silv ''[[The Signal and the Noise]]]: Why So Many Predictions Fail-but Some Don't.'' By Nate Silver</ref><ref>[https://books.google.com/books?id=hhdVr9F-JfAC Data Preparation for Data Mining]: Text. By Dorian Pyle.</ref> Curve fitting can involve either [[interpolation]],<ref>Numerical Methods in Engineering with MATLAB®. By [[Jaan Kiusalaas]]. Page 24.</ref><ref>[https://books.google.com/books?id=YlkgAwAAQBAJ&printsec=frontcover#v=onepage&q=%22curve%20fitting%22&f=false Numerical Methods in Engineering with Python 3]. By Jaan Kiusalaas. Page 21.</ref> where an exact fit to the data is required, or [[smoothing]],<ref>[https://books.google.com/books?id=UjnB0FIWv_AC&printsec=frontcover#v=onepage&q&f=false Numerical Methods of Curve Fitting]. By P. G. Guest, Philip George Guest. Page 349.</ref><ref>See also: [[Mollifier]]</ref> in which a "smooth" function is constructed that approximately fits the data. A related topic is [[regression analysis]],<ref>[http://www.facm.ucl.ac.be/intranet/books/statistics/Prism-Regression-Book.unlocked.pdf Fitting Models to Biological Data Using Linear and Nonlinear Regression]. By Harvey Motulsky, Arthur Christopoulos.</ref><ref>Regression Analysis By Rudolf J. Freund, William J. Wilson, Ping Sa. Page 269.</ref> 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,<ref>Visual Informatics. Edited by Halimah Badioze Zaman, Peter Robinson, Maria Petrou, Patrick Olivier, Heiko Schröder. Page 689.</ref><ref>[https://books.google.com/books?id=rdJvXG1k3HsC&printsec=frontcover#v=onepage&q&f=false Numerical Methods for Nonlinear Engineering Models]. By John R. Hauser. Page 227.</ref> to infer values of a function where no data are available,<ref>Methods of Experimental Physics: Spectroscopy, Volume 13, Part 1. By Claire Marton. Page 150.</ref> and to summarize the relationships among two or more variables.<ref>Encyclopedia of Research Design, Volume 1. Edited by Neil J. Salkind. Page 266.</ref> [[Extrapolation]] refers to the use of a fitted curve beyond the [[range (statistics)|range]] of the observed data,<ref>[https://books.google.com/books?id=ba0hAQAAQBAJ&printsec=frontcover#v=onepage&q&f=false Community Analysis and Planning Techniques]. By Richard E. Klosterman. Page 1.</ref> and is subject to a [[Uncertainty|degree of uncertainty]]<ref>An Introduction to Risk and Uncertainty in the Evaluation of Environmental Investments. DIANE Publishing. [https://books.google.com/books?id=rJ23LWaZAqsC&pg=PA69 Pg 69]</ref> since it may reflect the method used to construct the curve as much as it reflects the observed data.
Curve fittingSandra Lach Arlinghaus, PHB Practical Handbook of Curve Fitting. CRC Press, 1994.William M. Kolb. Curve Fitting for Programmable Calculators. Syntec, Incorporated, 1984. is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points,S.S. Halli, K.V. Rao. 1992. Advanced Techniques of Population Analysis. Page 165 (cf. ... functions are fulfilled if we have a good to moderate fit for the observed data.) possibly subject to constraints.The Signal and the Noise: Why So Many Predictions Fail-but Some Don't. By Nate SilverData Preparation for Data Mining: Text. By Dorian Pyle. Curve fitting can involve either interpolation,Numerical Methods in Engineering with MATLAB®. By Jaan Kiusalaas. Page 24.Numerical Methods in Engineering with Python 3. By Jaan Kiusalaas. Page 21. where an exact fit to the data is required, or smoothing,Numerical Methods of Curve Fitting. By P. G. Guest, Philip George Guest. Page 349.See also: Mollifier in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis,Fitting Models to Biological Data Using Linear and Nonlinear Regression. By Harvey Motulsky, Arthur Christopoulos.Regression Analysis By Rudolf J. Freund, William J. Wilson, Ping Sa. Page 269. 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,Visual Informatics. Edited by Halimah Badioze Zaman, Peter Robinson, Maria Petrou, Patrick Olivier, Heiko Schröder. Page 689.Numerical Methods for Nonlinear Engineering Models. By John R. Hauser. Page 227. to infer values of a function where no data are available,Methods of Experimental Physics: Spectroscopy, Volume 13, Part 1. By Claire Marton. Page 150. and to summarize the relationships among two or more variables.Encyclopedia of Research Design, Volume 1. Edited by Neil J. Salkind. Page 266. Extrapolation refers to the use of a fitted curve beyond the range of the observed data,Community Analysis and Planning Techniques. By Richard E. Klosterman. Page 1. and is subject to a degree of uncertaintyAn Introduction to Risk and Uncertainty in the Evaluation of Environmental Investments. DIANE Publishing. Pg 69 since it may reflect the method used to construct the curve as much as it reflects the observed data.
桑德拉 · 拉赫 · 阿林豪斯，PHB 曲线拟合实用手册。1994. William m. Kolb.可编程计算器的曲线拟合。公司，1984年。是构造一条曲线或数学函数的过程，这条曲线最适合一系列数据点。哈利，k.v。男名男子名。1992.人口分析的高级技术。第165页(参考英文版)。如果我们对观察到的数据有一个良好到中等的拟合，那么功能就完成了信号和噪音: 为什么这么多预测失败——但是有些没有。数据挖掘的准备: 文本。作者: Dorian Pyle。用 MATLAB 进行曲线拟合可以包括插值、工程中的数值方法。By Jaan Kiusalaas.第24页，Python 3在工程中的数值方法。By Jaan Kiusalaas.需要对数据进行精确拟合或平滑处理的曲线拟合的数值方法。作者: p · g · 盖斯特，菲利普 · 乔治 · 盖斯特。参见: Mollifier，其中构造了一个近似适合数据的“ smooth”函数。一个相关的话题是回归分析，利用线性和非线性回归来拟合生物数据的模型。作者: Harvey Motulsky，Arthur christoph，回归分析: Rudolf j. Freund，William j. Wilson，Ping Sa。第269页。它更多地关注推论统计学的问题，比如一条曲线在多大程度上存在不确定性，而这条曲线又与随机误差观测到的数据相吻合。拟合的曲线可以用来作为一个辅助的数据可视化，视觉信息学。Edited by Halimah Badioze Zaman, Peter Robinson, Maria Petrou, Patrick Olivier, Heiko Schröder.非线性工程模型的数值方法。作者: John r. Hauser。第227页。在没有数据可用的情况下推断函数的值，实验物理学方法: 光谱学，第13卷，第1部分。作者: Claire Marton。总结两个或两个以上变量之间的关系。研究设计百科全书，第一卷。编辑: Neil j. Salkind。第266页。外推法是指使用拟合曲线超出观测数据的范围，社区分析和规划技术。作者: Richard e. Klosterman。第一页。《环境投资评估中的风险与不确定性导论》。DIANE Publishing.第69页，因为它可能反映了用来构造曲线的方法，就像它反映了观测数据一样。
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").<ref>Hamming, Richard. Numerical methods for scientists and engineers. Courier Corporation, 2012.</ref> 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.<ref>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> 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 [[Polynomial regression|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.
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").Hamming, Richard. Numerical methods for scientists and engineers. Courier Corporation, 2012. 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.Friedman, Milton. "The interpolation of time series by related series." Journal of the American Statistical Association 57.300 (1962): 729–757. 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早期和晚期数据的值(”基准”)之间的内插来估计某些数据的某些组成部分。插值是对两个已知量(历史数据)之间的未知量的估计，或者从可用信息(“字里行间的读数”)中得出缺失信息的结论。汉明，理查德。科学家和工程师的数值方法。快递公司，2012。在缺失数据周围的数据可用，其趋势、季节性和长期周期已知的情况下，插值是有用的。这通常是通过使用一个相关的系列知道所有相关的日期。按相关序列对时间序列进行插值美国统计协会杂志57.300(1962) : 729-757。或者使用多项式插值或样条插值，在这种情况下，分段多项式函数适合于时间间隔，以便它们能够平滑地组合在一起。另一个与插值密切相关的问题是用一个简单函数(也称为回归)逼近一个复杂函数。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模型的整个数据集。然而，样条插值可以产生一个由多项式组成的分段连续函数来为数据集建模。
[[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.
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.
外推法是根据一个变量与另一个变量的关系，在原始观测范围之外估计该变量的值的过程。它类似于插值，在已知的观测值之间产生估计值，但是外推法存在更大的不确定性和产生无意义结果的更高风险。
===Function approximation===
{{main|Function approximation}}
In general, a function approximation problem asks us to select a [[function (mathematics)|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 function]]s) can be approximated by a specific class of functions (for example, [[polynomial]]s or [[rational function]]s) 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 of a function|domain]] and [[codomain]] of ''g'', several techniques for approximating ''g'' may be applicable. For example, if ''g'' is an operation on the [[real number]]s, 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 [[statistical classification|classification]] problem instead. A related problem of ''online'' time series approximation<ref>Gandhi, Sorabh, Luca Foschini, and Subhash Suri. "[https://ieeexplore.ieee.org/abstract/document/5447930/ 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.</ref> 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 analysis|regression]], [[Statistical classification|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 (programming language)|Julia]], [[Python (programming language)|Python]], [[R (programming language)|R]], [[SAS (software)|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.<ref>{{cite web |title=Time Series Analysis with Spark |author=Sandy Ryza |date=2020-03-18 |access-date=2021-01-12 |url=https://databricks.com/session/time-series-analysis-with-spark |format=slides of a talk at Spark Summit East 2016 |publisher=[[Databricks]]}}</ref>
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|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|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|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 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]].
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 number]]s is written
:''X'' = (''X''<sub>1</sub>, ''X''<sub>2</sub>, ...).
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<sub>t</sub>'': ''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 [[Stationary process#Weaker forms of stationarity|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 [[Cyclostationary process|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.<ref>Boashash, B. (ed.), (2003) ''Time-Frequency Signal Analysis and Processing: A Comprehensive Reference'', Elsevier Science, Oxford, 2003 {{isbn|0-08-044335-4}}</ref>
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|autocorrelation function]] and the [[Spectral density|spectral density function]] (also [[cross-correlation function]]s and cross-spectral density functions)
* [[Scaled correlation|Scaled]] cross- and auto-correlation functions to remove contributions of slow components<ref name="Nikolicetal">{{cite journal |last1=Nikolić |first1=D. |last2=Muresan |first2=R. C. |last3=Feng |first3=W. |last4=Singer |first4=W. |year=2012 |title=Scaled correlation analysis: a better way to compute a cross-correlogram |journal=European Journal of Neuroscience |volume=35 |issue=5 |pages=742–762 |doi=10.1111/j.1460-9568.2011.07987.x |pmid=22324876 |s2cid=4694570 |url=https://semanticscholar.org/paper/caa784fc3c22656413143559c402b54d0567f4d1 }}</ref>
* Performing a [[Fourier transform]] to investigate the series in the [[frequency domain]]
* Use of a [[digital filter|filter]] to remove unwanted [[noise (physics)|noise]]
* [[Principal component analysis]] (or [[empirical orthogonal function]] analysis)
* [[Singular spectrum analysis]]
* "Structural" models:
** General [[State Space Model]]s
** Unobserved Components Models
* [[Machine Learning]]
** [[Artificial neural network]]s
** [[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 model|Nonlinear mixed-effects modeling]]
* [[Dynamic time warping]]<ref name="Sakoe 1978">{{cite book |last1=Sakoe |first1=Hiroaki |last2=Chiba |first2=Seibi |year=1978 |chapter=Dynamic programming algorithm optimization for spoken word recognition |volume=26 |pages=43–49 |doi=10.1109/TASSP.1978.1163055 |journal=IEEE Transactions on Acoustics, Speech, and Signal Processing |s2cid=17900407 |chapter-url=https://semanticscholar.org/paper/18f355d7ef4aa9f82bf5c00f84e46714efa5fd77 }}</ref>
* [[Cross-correlation]]<ref>{{cite book |last1=Goutte |first1=Cyril |last2=Toft |first2=Peter |last3=Rostrup |first3=Egill |last4=Nielsen |first4=Finn Å. |last5=Hansen |first5=Lars Kai |year=1999 |chapter=On Clustering fMRI Time Series |volume=9 |issue=3 |pages=298–310 |doi=10.1006/nimg.1998.0391 |pmid=10075900 |journal=NeuroImage |s2cid=14147564 |chapter-url=https://semanticscholar.org/paper/2d5c663fb53d8348bdf3c4df0f881b5db2dcf5e3 }}</ref>
* [[Dynamic Bayesian network]]
* [[Time-frequency representation|Time-frequency analysis techniques:]]
** [[Fast Fourier transform]]
** [[Continuous wavelet transform]]
** [[Short-time Fourier transform]]
** [[Chirplet transform]]
** [[Fractional Fourier transform]]
* [[Chaos theory|Chaotic analysis]]
** [[Correlation dimension]]
** [[Recurrence plot]]s
** [[Recurrence quantification analysis]]
** [[Lyapunov exponent]]s
** [[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
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*CUSUM 图
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* EWMA 图
* 非趋势波动分析
* 非线性混合效应建模
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* 动态时间规整互相关
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* 动态贝氏网路
* 时频分析技术:
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* 快速傅里叶变换连续小波转换
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* 短时距傅里叶变换
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===Measures===
Time series metrics or [[Features (pattern recognition)|features]] that can be used for time series [[Classification (machine learning)|classification]] or [[regression analysis]]:<ref>{{cite journal |last1=Mormann |first1=Florian |last2=Andrzejak |first2=Ralph G. |last3=Elger |first3=Christian E. |last4=Lehnertz |first4=Klaus |title=Seizure prediction: the long and winding road |journal=[[Brain (journal)|Brain]] |year=2007 |volume=130 |issue=2 |pages=314–333 |doi=10.1093/brain/awl241 |pmid=17008335|doi-access=free }}</ref>
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]]
** [[Fast Fourier transform|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]]<ref>{{cite web |last1=Land |first1=Bruce |last2=Elias |first2=Damian |title=Measuring the 'Complexity' of a time series |url=http://www.nbb.cornell.edu/neurobio/land/PROJECTS/Complexity/ }}</ref>
** [[Sample entropy]]
** {{iw2|Fourier entropy||uk|Ентропія Фур'є}}
** 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 information theory|Algorithmic complexity]]
** [[Kolmogorov complexity]] estimates
** [[Hidden Markov Model]] states
** [[Rough path#Signature|Rough path signature]]<ref>[1] Chevyrev, I., Kormilitzin, A. (2016) "[https://arxiv.org/abs/1603.03788 A Primer on the Signature Method in Machine Learning], arXiv:1603.03788v1"</ref>
** 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''':<ref>{{cite journal |last1=Ropella |first1=G. E. P. |last2=Nag |first2=D. A. |last3=Hunt |first3=C. A. |title=Similarity measures for automated comparison of in silico and in vitro experimental results |journal=Engineering in Medicine and Biology Society |year=2003 |volume=3 |pages=2933–2936 |doi=10.1109/IEMBS.2003.1280532 |isbn=978-0-7803-7789-9 |s2cid=17798157 }}</ref>
** [[Cross-correlation]]
** [[Dynamic Time Warping]]<ref name="Sakoe 1978"/>
** [[Hidden Markov Models]]
** [[Edit distance]]
** [[Total correlation]]
** [[Newey–West estimator]]
** [[Prais–Winsten estimation|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
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* 矩(数学)
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* 谱带功率
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* 谱边缘频率
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* 累积能量(信号处理)
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* 自相关函数特性
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* Hjorth 参数
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* FFT 参数
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* 自回归模型参数
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* 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
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* 本地流
* 其他单变量度量
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* 算法复杂度
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* 柯氏复杂性估计
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* 隐马尔可夫模型状态
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* 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”
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* 替代时间序列和替代校正
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* 递归损失(非平稳度)
* 双变量线性度量
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* 最大线性互相关
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* 线性相干性(信号处理)
* 双变量非线性度量
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* 非线性相互依赖
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* 动态卷吸(物理学)
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* 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
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* Prais-Winsten 变换
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* 数据作为向量在乌雷松度量化定理
*
*
* 明氏距离
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* 马氏距离
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* 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)<ref>{{cite web|last1=Tominski|first1=Christian|last2= Aigner|first2=Wolfgang|title=The TimeViz Browser:A Visual Survey of Visualization Techniques for Time-Oriented Data|url=http://survey.timeviz.net/|access-date=1 June 2014}}</ref>
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
* {{iw2|GapChart||fr}}
* 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==
{{Columns-list|colwidth=30em|
* [[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]]
}}
==References==
{{Reflist|2}}
==Further reading==
* {{Citation
| author-link = George E. P. Box
| last1 = Box | first1 = George
| last2 = Jenkins | first2 = Gwilym
| title = Time Series Analysis: forecasting and control, rev. ed.
| publisher = Holden-Day
| location = Oakland, California
| year = 1976
}}
* [[James Durbin|Durbin J.]], Koopman S.J. (2001), ''Time Series Analysis by State Space Methods'', [[Oxford University Press]].
* {{Citation
| last = Gershenfeld | first = Neil
| year = 2000
| title = The Nature of Mathematical Modeling
| isbn = 978-0-521-57095-4
| publisher = [[Cambridge University Press]]
| oclc = 174825352
}}
* {{Citation
| author-link = James D. Hamilton
| last = Hamilton | first = James
| year = 1994
| title = Time Series Analysis
| isbn = 978-0-691-04289-3
| publisher = [[Princeton University Press]]
}}
* [[Maurice Priestley|Priestley, M. B.]] (1981), ''Spectral Analysis and Time Series'', [[Academic Press]]. {{ISBN|978-0-12-564901-8}}
* {{Citation | last = Shasha | first = D. | title = High Performance Discovery in Time Series | publisher = [[Springer Science+Business Media|Springer]] | year = 2004 | 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, {{ISBN|978-3-319-52451-1}}
* 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]].
* [[Norbert Wiener|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]].
* {{cite book|last1=Auffarth|first1=Ben|year=2021|title= Machine Learning for Time-Series with Python: Forecast, predict, and detect anomalies with state-of-the-art machine learning methods|publisher=Packt Publishing|edition=1st|isbn=978-1801819626|url=https://www.packtpub.com/product/machine-learning-for-time-series-with-python/9781801819626|access-date=5 November 2021}}
*
* 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==
{{Commons category}}
*[http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4.htm 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。
{{Statistics}}
{{Portal bar|Mathematics}}
{{Authority control}}
{{DEFAULTSORT:Time Series}}
[[Category:Time series| ]]
[[Category:Statistical data types]]
[[Category:Mathematical and quantitative methods (economics)]]
[[Category:Machine learning]]
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
<noinclude>
<small>This page was moved from [[wikipedia:en:Time series]]. Its edit history can be viewed at [[时间序列分析/edithistory]]</small></noinclude>
[[Category:待整理页面]]
{{Use American English|date = March 2019}}
{{short description|Sequence of data points over time}}
[[File:Random-data-plus-trend-r2.png|thumb|250px|Time series: random data plus trend, with best-fit line and different applied filters|alt=|right]]
In [[mathematics]], a '''time series''' is a series of [[data point]]s indexed (or listed or graphed) in time order. Most commonly, a time series is a [[sequence]] taken at successive equally spaced points in time. Thus it is a sequence of [[discrete-time]] data. Examples of time series are heights of ocean [[tides]], counts of [[sunspots]], and the daily closing value of the [[Dow Jones Industrial Average]].
thumb|250px|Time series: random data plus trend, with best-fit line and different applied filters|alt=|right
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.
拇指 | 250px | 时间序列: 随机数据加趋势，带有最佳拟合线和不同的应用过滤器 | alt = | right 在数学中，时间序列是按时间顺序索引(或列出或绘制)的一系列数据点。最常见的是，时间序列是在相继的等间隔时间点上拍摄的序列。因此，它是一个离散时间数据序列。时间序列的例子有海潮的高度、太阳黑子的数量以及道琼斯工业平均指数的每日收盘价。
A Time series is very frequently plotted via a [[run chart]] (which is a temporal [[line chart]]). Time series are used in [[statistics]], [[signal processing]], [[pattern recognition]], [[econometrics]], [[mathematical finance]], [[weather forecasting]], [[earthquake prediction]], [[electroencephalography]], [[control engineering]], [[astronomy]], [[communications engineering]], and largely in any domain of applied [[Applied science|science]] and [[engineering]] which involves [[Time|temporal]] measurements.
A Time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements.
时间序列通常是通过运行图(时间线图)绘制的。时间序列广泛应用于统计学、信号处理、模式识别、计量经济学、数学金融学、天气预报、地震预测、脑电图、控制工程、天文学、通信工程等领域，还广泛应用于涉及时间测量的任何应用科学和工程领域。
'''Time series ''analysis''''' comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. '''Time series ''forecasting''''' is the use of a [[model (abstract)|model]] to predict future values based on previously observed values. While [[regression analysis]] is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series. [[Interrupted time series]] analysis is used to detect changes in the evolution of a time series from before to after some intervention which may affect the underlying variable.
Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series. Interrupted time series analysis is used to detect changes in the evolution of a time series from before to after some intervention which may affect the underlying variable.
时间序列分析包括分析时间序列数据的方法，以提取有意义的统计数据和数据的其他特征。时间序列预测是使用一个模型来预测未来的价值基于以前观察到的价值。虽然回归分析通常用于测试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”，它特别指的是单个序列中不同时间点之间的关系。中断时间序列分析是用来检测一个时间序列从干预之前到干预之后的变化，这些变化可能会影响到基础变量。
Time series data have a natural temporal ordering. This makes time series analysis distinct from [[cross-sectional study|cross-sectional studies]], in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from [[spatial data analysis]] where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A [[stochastic]] model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see [[time reversibility]]).
Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see time reversibility).
时间序列数据具有自然的时间序列。这使得时间序列分析不同于横断面研究，横断面研究中的观察没有自然的顺序(例如:。通过参照个人的教育程度来解释人们的工资，个人的数据可以按任意顺序输入)。时间序列分析也不同于空间数据分析，因为空间数据分析的观测通常与地理位置有关(例如:。根据房屋的位置和内在特征来计算房价)。一个时间序列的随机模型通常反映了这样一个事实，即在时间上紧密相连的观察比相距较远的观察更密切相关。此外，时间序列模型往往利用时间的自然单向排序，以便某一时期的数值以某种方式从过去的数值而不是从未来的数值得出(见时间可逆性)。
Time series analysis can be applied to [[real number|real-valued]], continuous data, [[:wikt:discrete|discrete]] [[Data type#Numeric types|numeric]] data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the [[English language]]<ref>{{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>).
Time series analysis can be applied to real-valued, continuous data, discrete numeric data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the English language).
时间序列分析可以应用于实值数据、连续数据、离散数值数据或离散符号数据(即。字符序列，例如英语中的字母和单词)。
==Methods for analysis==
==Methods for analysis==
= = = 分析方法 = =
Methods for time series analysis may be divided into two classes: [[frequency-domain]] methods and [[time-domain]] methods. The former include [[frequency spectrum#Spectrum analysis|spectral analysis]] and [[wavelet analysis]]; the latter include [[auto-correlation]] and [[cross-correlation]] analysis. In the time domain, correlation and analysis can be made in a filter-like manner using [[scaled correlation]], thereby mitigating the need to operate in the frequency domain.
Methods for time series analysis may be divided into two classes: frequency-domain methods and time-domain methods. The former include spectral analysis and wavelet analysis; the latter include auto-correlation and cross-correlation analysis. In the time domain, correlation and analysis can be made in a filter-like manner using scaled correlation, thereby mitigating the need to operate in the frequency domain.
时间序列分析方法可分为两类: 频域方法和时域方法。前者包括谱分析和小波分析，后者包括自相关和互相关分析。在时域上，可以使用比例相关以类似滤波器的方式进行相关和分析，从而减少了在频域进行操作的需要。
Additionally, time series analysis techniques may be divided into [[Parametric estimation|parametric]] and [[Non-parametric statistics|non-parametric]] methods. The [[Parametric estimation|parametric approaches]] assume that the underlying [[stationary process|stationary stochastic process]] has a certain structure which can be described using a small number of parameters (for example, using an [[autoregressive]] or [[moving average model]]). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, [[Non-parametric statistics|non-parametric approaches]] explicitly estimate the [[covariance]] or the [[spectrum]] of the process without assuming that the process has any particular structure.
Additionally, time series analysis techniques may be divided into parametric and non-parametric methods. The parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters (for example, using an autoregressive or moving average model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure.
此外，时间序列分析技术可分为参数方法和非参数方法。参数方法假设潜在的平稳随机过程有一个特定的结构，这个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，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谱，而不假设过程具有任何特定的结构。
Methods of time series analysis may also be divided into [[Linear regression|linear]] and [[Nonlinear regression|non-linear]], and [[Univariate analysis|univariate]] and [[Multivariate analysis|multivariate]].
Methods of time series analysis may also be divided into linear and non-linear, and univariate and multivariate.
时间序列分析的方法也可分为线性和非线性，以及单变量和多变量。
==Panel data==
==Panel data==
= = = 面板数据 = =
A time series is one type of [[panel data]]. Panel data is the general class, a multidimensional data set, whereas a time series data set is a one-dimensional panel (as is a [[cross-sectional data]]set). A data set may exhibit characteristics of both panel data and time series data. One way to tell is to ask what makes one data record unique from the other records. If the answer is the time data field, then this is a time series data set candidate. If determining a unique record requires a time data field and an additional identifier which is unrelated to time (e.g. student ID, stock symbol, country code), then it is panel data candidate. If the differentiation lies on the non-time identifier, then the data set is a cross-sectional data set candidate.
A time series is one type of panel data. Panel data is the general class, a multidimensional data set, whereas a time series data set is a one-dimensional panel (as is a cross-sectional dataset). A data set may exhibit characteristics of both panel data and time series data. One way to tell is to ask what makes one data record unique from the other records. If the answer is the time data field, then this is a time series data set candidate. If determining a unique record requires a time data field and an additional identifier which is unrelated to time (e.g. student ID, stock symbol, country code), then it is panel data candidate. If the differentiation lies on the non-time identifier, then the data set is a cross-sectional data set candidate.
时间序列是面板数据的一种。Panel data 是一个通用类，是一个多维数据集，而时间序列数据集是一个一维面板(就像横截面数据集一样)。一个数据集可以同时显示面板数据和时间序列数据的特征。判断的一种方法是询问是什么使一个数据记录与其他记录相比是唯一的。如果答案是时间数据字段，那么这是一个候选的时间序列数据集。如果确定一个唯一的记录需要一个时间数据字段和一个与时间无关的附加标识符(例如:。学生证，股票代码，国家代码) ，然后是面板数据候选人。如果区分取决于非时间标识符，那么数据集是一个横断面数据集候选者。
==Analysis==
==Analysis==
= = 分析 = =
There are several types of motivation and data analysis available for time series which are appropriate for different purposes.
There are several types of motivation and data analysis available for time series which are appropriate for different purposes.
有几种类型的动机和数据分析可用于时间序列是适合不同的目的。
===Motivation===
===Motivation===
= = = 动机 = =
In the context of [[statistics]], [[econometrics]], [[quantitative finance]], [[seismology]], [[meteorology]], and [[geophysics]] the primary goal of time series analysis is [[forecasting]]. In the context of [[signal processing]], [[control engineering]] and [[communication engineering]] it is used for signal detection. Other applications are in [[data mining]], [[pattern recognition]] and [[machine learning]], where time series analysis can be used for [[cluster analysis|clustering]],<ref>{{cite journal | last1 = Liao | first1 = T. Warren | title = Clustering of time series data - a survey | journal = Pattern Recognition | volume = 38 | issue = 11 | pages = 1857–1874 | publisher = Elsevier | date = 2005 | language = en | doi = 10.1016/j.patcog.2005.01.025| bibcode = 2005PatRe..38.1857W }}{{subscription required|via=ScienceDirect }}</ref><ref>{{cite journal | last1 = Aghabozorgi | first1 = Saeed | last2 = Shirkhorshidi | first2 = Ali S. | last3 = Wah | first3 = Teh Y. | title = Time-series clustering – A decade review | journal = Information Systems | volume = 53 | pages = 16–38 | publisher = Elsevier | date = 2015 | language = en | doi = 10.1016/j.is.2015.04.007}}{{subscription required|via=ScienceDirect }}</ref> [[Statistical classification|classification]],<ref>{{cite journal | last1 = Keogh | first1 = Eamonn J. | title = On the need for time series data mining benchmarks | journal = Data Mining and Knowledge Discovery | volume = 7 | pages = 349–371 | publisher = Kluwer | date = 2003 | language = en | doi = 10.1145/775047.775062| isbn = 158113567X | s2cid = 41617550 }}{{subscription required|via=ACM Digital Library }}</ref> query by content,<ref>{{cite conference|last1=Agrawal|first1=Rakesh|last2=Faloutsos|first2=Christos|last3=Swami|first3=Arun|date=October 1993|title=Efficient Similarity Search In Sequence Databases|conference=International Conference on Foundations of Data Organization and Algorithms|volume=730|pages=69–84|book-title=Proceedings of the 4th International Conference on Foundations of Data Organization and Algorithms|doi=10.1007/3-540-57301-1_5}}{{Subscription required|via=SpringerLink}}</ref> [[anomaly detection]] as well as [[forecasting]].<ref>{{cite journal|last1=Chen|first1=Cathy W. S.|last2=Chiu|first2=L. M.|date=September 2021|title=Ordinal Time Series Forecasting of the Air Quality Index|journal=Entropy|language=en|volume=23|issue=9|pages=1167|doi=10.3390/e23091167|pmid=34573792|pmc=8469594|bibcode=2021Entrp..23.1167C|doi-access=free}}</ref>
In the context of statistics, econometrics, quantitative finance, seismology, meteorology, and geophysics the primary goal of time series analysis is forecasting. In the context of signal processing, control engineering and communication engineering it is used for signal detection. Other applications are in data mining, pattern recognition and machine learning, where time series analysis can be used for clustering, classification, query by content, anomaly detection as well as forecasting.
在统计学、计量经济学、定量金融学、地震学、气象学和地球物理学的背景下，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中，它被用于信号检测。其他应用还包括数据挖掘、模式识别和机器学习，时间序列分析可用于聚类、分类、内容查询、异常检测和预测。
===Exploratory analysis===
[[File:Tuberculosis incidence US 1953-2009.png|thumb|Tuberculosis incidence US 1953-2009]]
{{further|Exploratory analysis}}
A straightforward way to examine a regular time series is manually with a [[line chart]]. An example chart is shown on the right for tuberculosis incidence in the United States, made with a spreadsheet program. The number of cases was standardized to a rate per 100,000 and the percent change per year in this rate was calculated. The nearly steadily dropping line shows that the TB incidence was decreasing in most years, but the percent change in this rate varied by as much as +/- 10%, with 'surges' in 1975 and around the early 1990s. The use of both vertical axes allows the comparison of two time series in one graphic.
thumb|Tuberculosis incidence US 1953-2009
A straightforward way to examine a regular time series is manually with a line chart. An example chart is shown on the right for tuberculosis incidence in the United States, made with a spreadsheet program. The number of cases was standardized to a rate per 100,000 and the percent change per year in this rate was calculated. The nearly steadily dropping line shows that the TB incidence was decreasing in most years, but the percent change in this rate varied by as much as +/- 10%, with 'surges' in 1975 and around the early 1990s. The use of both vertical axes allows the comparison of two time series in one graphic.
探索性分析结核病发病率美国1953-2009一个简单的方法来检查一个定期的时间序列是手工与线图。右图显示的是美国肺结核发病率的示例图，是用电子表格程序制作的。病例数量标准化为每100000例，并计算了这一比率每年的变化百分比。几乎稳步下降的曲线表明，结核病的发病率在大多数年份都在下降，但这一比率的变化率高达 +/-10% ，在1975年和90年代初期出现了“激增”。两个垂直轴的使用允许在一个图形中比较两个时间序列。
A study of corporate data analysts found two challenges to exploratory time series analysis: discovering the shape of interesting patterns, and finding an explanation for these patterns.<ref>{{Cite journal|last=Sarkar|first=Advait|last2=Spott|first2=Martin|last3=Blackwell|first3=Alan F.|last4=Jamnik|first4=Mateja|date=2016|title=Visual discovery and model-driven explanation of time series patterns|url=https://doi.org/10.1109/VLHCC.2016.7739668|journal=2016 IEEE Symposium on Visual Languages and Human-Centric Computing (VL/HCC)|publisher=IEEE|doi=10.1109/vlhcc.2016.7739668}}</ref> Visual tools that represent time series data as [[Heat map|heat map matrices]] can help overcome these challenges.
A study of corporate data analysts found two challenges to exploratory time series analysis: discovering the shape of interesting patterns, and finding an explanation for these patterns. Visual tools that represent time series data as heat map matrices can help overcome these challenges.
一项针对企业数据分析师的研究发现，探索性时间序列分析面临两个挑战: 发现有趣模式的形状，以及为这些模式找到解释。将时间序列数据表示为热图矩阵的可视化工具可以帮助克服这些挑战。
Other techniques include:
Other techniques include:
其他技巧包括:
* [[Autocorrelation]] analysis to examine [[serial dependence]]
* [[frequency spectrum#Spectrum analysis|Spectral analysis]] to examine cyclic behavior which need not be related to [[seasonality]]. For example, sunspot activity varies over 11 year cycles.<ref>{{cite book |last=Bloomfield |first=P. |year=1976 |title=Fourier analysis of time series: An introduction |location=New York |publisher=Wiley |isbn=978-0471082569 }}</ref><ref>{{cite book |last=Shumway |first=R. H. |year=1988 |title=Applied statistical time series analysis |location=Englewood Cliffs, NJ |publisher=Prentice Hall |isbn=978-0130415004 }}</ref> Other common examples include celestial phenomena, weather patterns, neural activity, commodity prices, and economic activity.
* Separation into components representing trend, seasonality, slow and fast variation, and cyclical irregularity: see [[trend estimation]] and [[decomposition of time series]]
* Autocorrelation analysis to examine serial dependence
* Spectral analysis to examine cyclic behavior which need not be related to seasonality. For example, sunspot activity varies over 11 year cycles. Other common examples include celestial phenomena, weather patterns, neural activity, commodity prices, and economic activity.
* Separation into components representing trend, seasonality, slow and fast variation, and cyclical irregularity: see trend estimation and decomposition of time series
* 自相关分析检验序列相关性
* 谱分析检验不需要与季节性相关的循环行为。例如，太阳黑子的活动周期为11年。其他常见的例子包括天文现象、天气模式、神经活动、商品价格和经济活动。
* 分离成分代表趋势，季节性，缓慢和快速变化，周期性不规则: 见趋势估计和时间序列分解
===Curve fitting===
{{main|Curve fitting}}
Curve fitting<ref>Sandra Lach Arlinghaus, PHB Practical Handbook of Curve Fitting. CRC Press, 1994.</ref><ref>William M. Kolb. Curve Fitting for Programmable Calculators. Syntec, Incorporated, 1984.</ref> is the process of constructing a [[curve]], or [[function (mathematics)|mathematical function]], that has the best fit to a series of [[data]] points,<ref>S.S. Halli, K.V. Rao. 1992. Advanced Techniques of Population Analysis. {{isbn|0306439972}} Page 165 (''cf''. ... functions are fulfilled if we have a good to moderate fit for the observed data.)</ref> possibly subject to constraints.<ref>[https://archive.org/details/signalnoisewhymo00silv ''[[The Signal and the Noise]]]: Why So Many Predictions Fail-but Some Don't.'' By Nate Silver</ref><ref>[https://books.google.com/books?id=hhdVr9F-JfAC Data Preparation for Data Mining]: Text. By Dorian Pyle.</ref> Curve fitting can involve either [[interpolation]],<ref>Numerical Methods in Engineering with MATLAB®. By [[Jaan Kiusalaas]]. Page 24.</ref><ref>[https://books.google.com/books?id=YlkgAwAAQBAJ&printsec=frontcover#v=onepage&q=%22curve%20fitting%22&f=false Numerical Methods in Engineering with Python 3]. By Jaan Kiusalaas. Page 21.</ref> where an exact fit to the data is required, or [[smoothing]],<ref>[https://books.google.com/books?id=UjnB0FIWv_AC&printsec=frontcover#v=onepage&q&f=false Numerical Methods of Curve Fitting]. By P. G. Guest, Philip George Guest. Page 349.</ref><ref>See also: [[Mollifier]]</ref> in which a "smooth" function is constructed that approximately fits the data. A related topic is [[regression analysis]],<ref>[http://www.facm.ucl.ac.be/intranet/books/statistics/Prism-Regression-Book.unlocked.pdf Fitting Models to Biological Data Using Linear and Nonlinear Regression]. By Harvey Motulsky, Arthur Christopoulos.</ref><ref>Regression Analysis By Rudolf J. Freund, William J. Wilson, Ping Sa. Page 269.</ref> 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,<ref>Visual Informatics. Edited by Halimah Badioze Zaman, Peter Robinson, Maria Petrou, Patrick Olivier, Heiko Schröder. Page 689.</ref><ref>[https://books.google.com/books?id=rdJvXG1k3HsC&printsec=frontcover#v=onepage&q&f=false Numerical Methods for Nonlinear Engineering Models]. By John R. Hauser. Page 227.</ref> to infer values of a function where no data are available,<ref>Methods of Experimental Physics: Spectroscopy, Volume 13, Part 1. By Claire Marton. Page 150.</ref> and to summarize the relationships among two or more variables.<ref>Encyclopedia of Research Design, Volume 1. Edited by Neil J. Salkind. Page 266.</ref> [[Extrapolation]] refers to the use of a fitted curve beyond the [[range (statistics)|range]] of the observed data,<ref>[https://books.google.com/books?id=ba0hAQAAQBAJ&printsec=frontcover#v=onepage&q&f=false Community Analysis and Planning Techniques]. By Richard E. Klosterman. Page 1.</ref> and is subject to a [[Uncertainty|degree of uncertainty]]<ref>An Introduction to Risk and Uncertainty in the Evaluation of Environmental Investments. DIANE Publishing. [https://books.google.com/books?id=rJ23LWaZAqsC&pg=PA69 Pg 69]</ref> since it may reflect the method used to construct the curve as much as it reflects the observed data.
Curve fittingSandra Lach Arlinghaus, PHB Practical Handbook of Curve Fitting. CRC Press, 1994.William M. Kolb. Curve Fitting for Programmable Calculators. Syntec, Incorporated, 1984. is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points,S.S. Halli, K.V. Rao. 1992. Advanced Techniques of Population Analysis. Page 165 (cf. ... functions are fulfilled if we have a good to moderate fit for the observed data.) possibly subject to constraints.The Signal and the Noise: Why So Many Predictions Fail-but Some Don't. By Nate SilverData Preparation for Data Mining: Text. By Dorian Pyle. Curve fitting can involve either interpolation,Numerical Methods in Engineering with MATLAB®. By Jaan Kiusalaas. Page 24.Numerical Methods in Engineering with Python 3. By Jaan Kiusalaas. Page 21. where an exact fit to the data is required, or smoothing,Numerical Methods of Curve Fitting. By P. G. Guest, Philip George Guest. Page 349.See also: Mollifier in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis,Fitting Models to Biological Data Using Linear and Nonlinear Regression. By Harvey Motulsky, Arthur Christopoulos.Regression Analysis By Rudolf J. Freund, William J. Wilson, Ping Sa. Page 269. 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,Visual Informatics. Edited by Halimah Badioze Zaman, Peter Robinson, Maria Petrou, Patrick Olivier, Heiko Schröder. Page 689.Numerical Methods for Nonlinear Engineering Models. By John R. Hauser. Page 227. to infer values of a function where no data are available,Methods of Experimental Physics: Spectroscopy, Volume 13, Part 1. By Claire Marton. Page 150. and to summarize the relationships among two or more variables.Encyclopedia of Research Design, Volume 1. Edited by Neil J. Salkind. Page 266. Extrapolation refers to the use of a fitted curve beyond the range of the observed data,Community Analysis and Planning Techniques. By Richard E. Klosterman. Page 1. and is subject to a degree of uncertaintyAn Introduction to Risk and Uncertainty in the Evaluation of Environmental Investments. DIANE Publishing. Pg 69 since it may reflect the method used to construct the curve as much as it reflects the observed data.
桑德拉 · 拉赫 · 阿林豪斯，PHB 曲线拟合实用手册。1994. William m. Kolb.可编程计算器的曲线拟合。公司，1984年。是构造一条曲线或数学函数的过程，这条曲线最适合一系列数据点。哈利，k.v。男名男子名。1992.人口分析的高级技术。第165页(参考英文版)。如果我们对观察到的数据有一个良好到中等的拟合，那么功能就完成了信号和噪音: 为什么这么多预测失败——但是有些没有。数据挖掘的准备: 文本。作者: Dorian Pyle。用 MATLAB 进行曲线拟合可以包括插值、工程中的数值方法。By Jaan Kiusalaas.第24页，Python 3在工程中的数值方法。By Jaan Kiusalaas.需要对数据进行精确拟合或平滑处理的曲线拟合的数值方法。作者: p · g · 盖斯特，菲利普 · 乔治 · 盖斯特。参见: Mollifier，其中构造了一个近似适合数据的“ smooth”函数。一个相关的话题是回归分析，利用线性和非线性回归来拟合生物数据的模型。作者: Harvey Motulsky，Arthur christoph，回归分析: Rudolf j. Freund，William j. Wilson，Ping Sa。第269页。它更多地关注推论统计学的问题，比如一条曲线在多大程度上存在不确定性，而这条曲线又与随机误差观测到的数据相吻合。拟合的曲线可以用来作为一个辅助的数据可视化，视觉信息学。Edited by Halimah Badioze Zaman, Peter Robinson, Maria Petrou, Patrick Olivier, Heiko Schröder.非线性工程模型的数值方法。作者: John r. Hauser。第227页。在没有数据可用的情况下推断函数的值，实验物理学方法: 光谱学，第13卷，第1部分。作者: Claire Marton。总结两个或两个以上变量之间的关系。研究设计百科全书，第一卷。编辑: Neil j. Salkind。第266页。外推法是指使用拟合曲线超出观测数据的范围，社区分析和规划技术。作者: Richard e. Klosterman。第一页。《环境投资评估中的风险与不确定性导论》。DIANE Publishing.第69页，因为它可能反映了用来构造曲线的方法，就像它反映了观测数据一样。
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").<ref>Hamming, Richard. Numerical methods for scientists and engineers. Courier Corporation, 2012.</ref> 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.<ref>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> 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 [[Polynomial regression|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.
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").Hamming, Richard. Numerical methods for scientists and engineers. Courier Corporation, 2012. 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.Friedman, Milton. "The interpolation of time series by related series." Journal of the American Statistical Association 57.300 (1962): 729–757. 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早期和晚期数据的值(”基准”)之间的内插来估计某些数据的某些组成部分。插值是对两个已知量(历史数据)之间的未知量的估计，或者从可用信息(“字里行间的读数”)中得出缺失信息的结论。汉明，理查德。科学家和工程师的数值方法。快递公司，2012。在缺失数据周围的数据可用，其趋势、季节性和长期周期已知的情况下，插值是有用的。这通常是通过使用一个相关的系列知道所有相关的日期。按相关序列对时间序列进行插值美国统计协会杂志57.300(1962) : 729-757。或者使用多项式插值或样条插值，在这种情况下，分段多项式函数适合于时间间隔，以便它们能够平滑地组合在一起。另一个与插值密切相关的问题是用一个简单函数(也称为回归)逼近一个复杂函数。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模型的整个数据集。然而，样条插值可以产生一个由多项式组成的分段连续函数来为数据集建模。
[[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.
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.
外推法是根据一个变量与另一个变量的关系，在原始观测范围之外估计该变量的值的过程。它类似于插值，在已知的观测值之间产生估计值，但是外推法存在更大的不确定性和产生无意义结果的更高风险。
===Function approximation===
{{main|Function approximation}}
In general, a function approximation problem asks us to select a [[function (mathematics)|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 function]]s) can be approximated by a specific class of functions (for example, [[polynomial]]s or [[rational function]]s) 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 of a function|domain]] and [[codomain]] of ''g'', several techniques for approximating ''g'' may be applicable. For example, if ''g'' is an operation on the [[real number]]s, 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 [[statistical classification|classification]] problem instead. A related problem of ''online'' time series approximation<ref>Gandhi, Sorabh, Luca Foschini, and Subhash Suri. "[https://ieeexplore.ieee.org/abstract/document/5447930/ 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.</ref> 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 analysis|regression]], [[Statistical classification|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 (programming language)|Julia]], [[Python (programming language)|Python]], [[R (programming language)|R]], [[SAS (software)|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.<ref>{{cite web |title=Time Series Analysis with Spark |author=Sandy Ryza |date=2020-03-18 |access-date=2021-01-12 |url=https://databricks.com/session/time-series-analysis-with-spark |format=slides of a talk at Spark Summit East 2016 |publisher=[[Databricks]]}}</ref>
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|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|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|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 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]].
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 number]]s is written
:''X'' = (''X''<sub>1</sub>, ''X''<sub>2</sub>, ...).
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<sub>t</sub>'': ''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 [[Stationary process#Weaker forms of stationarity|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 [[Cyclostationary process|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.<ref>Boashash, B. (ed.), (2003) ''Time-Frequency Signal Analysis and Processing: A Comprehensive Reference'', Elsevier Science, Oxford, 2003 {{isbn|0-08-044335-4}}</ref>
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|autocorrelation function]] and the [[Spectral density|spectral density function]] (also [[cross-correlation function]]s and cross-spectral density functions)
* [[Scaled correlation|Scaled]] cross- and auto-correlation functions to remove contributions of slow components<ref name="Nikolicetal">{{cite journal |last1=Nikolić |first1=D. |last2=Muresan |first2=R. C. |last3=Feng |first3=W. |last4=Singer |first4=W. |year=2012 |title=Scaled correlation analysis: a better way to compute a cross-correlogram |journal=European Journal of Neuroscience |volume=35 |issue=5 |pages=742–762 |doi=10.1111/j.1460-9568.2011.07987.x |pmid=22324876 |s2cid=4694570 |url=https://semanticscholar.org/paper/caa784fc3c22656413143559c402b54d0567f4d1 }}</ref>
* Performing a [[Fourier transform]] to investigate the series in the [[frequency domain]]
* Use of a [[digital filter|filter]] to remove unwanted [[noise (physics)|noise]]
* [[Principal component analysis]] (or [[empirical orthogonal function]] analysis)
* [[Singular spectrum analysis]]
* "Structural" models:
** General [[State Space Model]]s
** Unobserved Components Models
* [[Machine Learning]]
** [[Artificial neural network]]s
** [[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 model|Nonlinear mixed-effects modeling]]
* [[Dynamic time warping]]<ref name="Sakoe 1978">{{cite book |last1=Sakoe |first1=Hiroaki |last2=Chiba |first2=Seibi |year=1978 |chapter=Dynamic programming algorithm optimization for spoken word recognition |volume=26 |pages=43–49 |doi=10.1109/TASSP.1978.1163055 |journal=IEEE Transactions on Acoustics, Speech, and Signal Processing |s2cid=17900407 |chapter-url=https://semanticscholar.org/paper/18f355d7ef4aa9f82bf5c00f84e46714efa5fd77 }}</ref>
* [[Cross-correlation]]<ref>{{cite book |last1=Goutte |first1=Cyril |last2=Toft |first2=Peter |last3=Rostrup |first3=Egill |last4=Nielsen |first4=Finn Å. |last5=Hansen |first5=Lars Kai |year=1999 |chapter=On Clustering fMRI Time Series |volume=9 |issue=3 |pages=298–310 |doi=10.1006/nimg.1998.0391 |pmid=10075900 |journal=NeuroImage |s2cid=14147564 |chapter-url=https://semanticscholar.org/paper/2d5c663fb53d8348bdf3c4df0f881b5db2dcf5e3 }}</ref>
* [[Dynamic Bayesian network]]
* [[Time-frequency representation|Time-frequency analysis techniques:]]
** [[Fast Fourier transform]]
** [[Continuous wavelet transform]]
** [[Short-time Fourier transform]]
** [[Chirplet transform]]
** [[Fractional Fourier transform]]
* [[Chaos theory|Chaotic analysis]]
** [[Correlation dimension]]
** [[Recurrence plot]]s
** [[Recurrence quantification analysis]]
** [[Lyapunov exponent]]s
** [[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
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*CUSUM 图
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* EWMA 图
* 非趋势波动分析
* 非线性混合效应建模
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* 动态时间规整互相关
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* 动态贝氏网路
* 时频分析技术:
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* 快速傅里叶变换连续小波转换
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* 短时距傅里叶变换
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* 混沌分析
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* Lyapunov 指数
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===Measures===
Time series metrics or [[Features (pattern recognition)|features]] that can be used for time series [[Classification (machine learning)|classification]] or [[regression analysis]]:<ref>{{cite journal |last1=Mormann |first1=Florian |last2=Andrzejak |first2=Ralph G. |last3=Elger |first3=Christian E. |last4=Lehnertz |first4=Klaus |title=Seizure prediction: the long and winding road |journal=[[Brain (journal)|Brain]] |year=2007 |volume=130 |issue=2 |pages=314–333 |doi=10.1093/brain/awl241 |pmid=17008335|doi-access=free }}</ref>
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]]
** [[Fast Fourier transform|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]]<ref>{{cite web |last1=Land |first1=Bruce |last2=Elias |first2=Damian |title=Measuring the 'Complexity' of a time series |url=http://www.nbb.cornell.edu/neurobio/land/PROJECTS/Complexity/ }}</ref>
** [[Sample entropy]]
** {{iw2|Fourier entropy||uk|Ентропія Фур'є}}
** 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 information theory|Algorithmic complexity]]
** [[Kolmogorov complexity]] estimates
** [[Hidden Markov Model]] states
** [[Rough path#Signature|Rough path signature]]<ref>[1] Chevyrev, I., Kormilitzin, A. (2016) "[https://arxiv.org/abs/1603.03788 A Primer on the Signature Method in Machine Learning], arXiv:1603.03788v1"</ref>
** 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''':<ref>{{cite journal |last1=Ropella |first1=G. E. P. |last2=Nag |first2=D. A. |last3=Hunt |first3=C. A. |title=Similarity measures for automated comparison of in silico and in vitro experimental results |journal=Engineering in Medicine and Biology Society |year=2003 |volume=3 |pages=2933–2936 |doi=10.1109/IEMBS.2003.1280532 |isbn=978-0-7803-7789-9 |s2cid=17798157 }}</ref>
** [[Cross-correlation]]
** [[Dynamic Time Warping]]<ref name="Sakoe 1978"/>
** [[Hidden Markov Models]]
** [[Edit distance]]
** [[Total correlation]]
** [[Newey–West estimator]]
** [[Prais–Winsten estimation|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)<ref>{{cite web|last1=Tominski|first1=Christian|last2= Aigner|first2=Wolfgang|title=The TimeViz Browser:A Visual Survey of Visualization Techniques for Time-Oriented Data|url=http://survey.timeviz.net/|access-date=1 June 2014}}</ref>
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
* {{iw2|GapChart||fr}}
* 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==
{{Columns-list|colwidth=30em|
* [[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]]
}}
==References==
{{Reflist|2}}
==Further reading==
* {{Citation
| author-link = George E. P. Box
| last1 = Box | first1 = George
| last2 = Jenkins | first2 = Gwilym
| title = Time Series Analysis: forecasting and control, rev. ed.
| publisher = Holden-Day
| location = Oakland, California
| year = 1976
}}
* [[James Durbin|Durbin J.]], Koopman S.J. (2001), ''Time Series Analysis by State Space Methods'', [[Oxford University Press]].
* {{Citation
| last = Gershenfeld | first = Neil
| year = 2000
| title = The Nature of Mathematical Modeling
| isbn = 978-0-521-57095-4
| publisher = [[Cambridge University Press]]
| oclc = 174825352
}}
* {{Citation
| author-link = James D. Hamilton
| last = Hamilton | first = James
| year = 1994
| title = Time Series Analysis
| isbn = 978-0-691-04289-3
| publisher = [[Princeton University Press]]
}}
* [[Maurice Priestley|Priestley, M. B.]] (1981), ''Spectral Analysis and Time Series'', [[Academic Press]]. {{ISBN|978-0-12-564901-8}}
* {{Citation | last = Shasha | first = D. | title = High Performance Discovery in Time Series | publisher = [[Springer Science+Business Media|Springer]] | year = 2004 | 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, {{ISBN|978-3-319-52451-1}}
* 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]].
* [[Norbert Wiener|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]].
* {{cite book|last1=Auffarth|first1=Ben|year=2021|title= Machine Learning for Time-Series with Python: Forecast, predict, and detect anomalies with state-of-the-art machine learning methods|publisher=Packt Publishing|edition=1st|isbn=978-1801819626|url=https://www.packtpub.com/product/machine-learning-for-time-series-with-python/9781801819626|access-date=5 November 2021}}
*
* 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==
{{Commons category}}
*[http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4.htm 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。
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