更改

{{Use American English|date = March 2019}}

{{Use American English|date = March 2019}}

{{short description|Sequence of data points over time}}

{{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]]

[[File:Random-data-plus-trend-r2.png|thumb|250px|Time series: random data plus trend, with best-fit line and different applied filters时间序列：随机数据加趋势，带有最佳拟合线和不同的过滤器|right|链接=Special:FilePath/Random-data-plus-trend-r2.png]]

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]].

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]].

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 [[Applied science|science]] and [[engineering]] which involves [[Time|temporal]] measurements.

'''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 (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Time series analysis包含了以提取时间序列数据中有意义的统计特征和数据的其他特征为目的的方法。时间序列预测Time series forecasting是基于先前观测到的值，使用模型Model来预测未来值的方法。虽然回归分析Regression analysis经常被用于分析一个或多个不同时间序列之间的关系，但这种类型的分析通常不被称为 "时间序列分析"。时间序列分析特指的是分析单一序列中不同时间点之间的关系。中断时间序列Interrupted time series分析是用来检测时间序列在接受干预前后的变化，这种干预可能会影响基础变量。

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

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 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 name=":0">{{cite book |last1=Lin |first1=Jessica |last2=Keogh |first2=Eamonn |last3=Lonardi |first3=Stefano |last4=Chiu |first4=Bill |chapter=A symbolic representation of time series, with implications for streaming algorithms |title=Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery |pages=2–11 |year=2003 |location=New York |publisher=ACM Press |doi=10.1145/882082.882086|citeseerx=10.1.1.14.5597 |s2cid=6084733 }}</ref>).

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).

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real-valued、连续数据、离散Discrete数值Numeric数据或离散符号数据（即字符序列，如英语English language中的字母和单词<ref name=":0" />）。

 

 

 

= = = 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 [[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.

时间序列分析的方法可分为两类：频域Frequency-domain方法和时域Time-domain方法。前者包括频谱分析Spectral analysis和小波分析Wavelet analysis；后者包括自相关Auto-correlation和交叉相关Cross-correlation analysis分析。在时域中，可以用类似于滤波器的方式使用标度相关性Scaled correlation来进行关联和分析，从而减轻了在频域中操作的需要。

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 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化Parametric和非参数化Non-parametric方法。参数方法Parametric approaches假定基础的平稳随机过程Stationary stochastic process具有某种结构，可以用少量的参数来描述（例如，使用自回归Autoregressive或移动平均模型Moving average model）。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Non-parametric approaches明确地估计过程的协方差Covariance或频谱Spectrum，而不假设过程有任何特定的结构。

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 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以分为线性Linear 和非线性Non-linear，以及单变量Univariate 和多变量Multivariate。

 

 

 

 

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 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Panel data的一种类型，面板数据是更大的类别。面板数据是一个多维的数据集，而时间序列数据集是一个一维的面板（正如截面数据Cross-sectional data集一样）。一个数据集可能同时表现出面板数据和时间序列数据的特征。判断的方法之一是探究是什么使一条数据记录与其他记录不同。如果答案是时间数据字段，那么这就是一个时间序列数据集候选。如果确定一个独特的记录需要一个时间数据字段和一个与时间无关的额外标识符（如学生证、股票代码、国家代码），那么它就是面板数据的候选。如果区别在于非时间标识符，那么该数据集就是一个截面数据集候选。

 

 

 

 

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.

 

 

 

 

 

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.

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 name=":1">{{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 name=":2">{{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 name=":3">{{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 name=":4">{{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 name=":5">{{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>

===Exploratory analysis===

===Exploratory analysis探索性分析===

[[File:Tuberculosis incidence US 1953-2009.png|thumb|Tuberculosis incidence US 1953-2009]]

[[File:Tuberculosis incidence US 1953-2009.png|thumb|Tuberculosis incidence US 1953-2009美国1953-2009年结核病发病率|链接=Special:FilePath/Tuberculosis_incidence_US_1953-2009.png]]

{{further|Exploratory analysis}}

{{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.

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.

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

 

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.

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.<ref name=":6">{{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.

一项对企业数据分析师的研究发现，探索性时间序列分析有两个挑战：发现有趣模式，以及为这些模式找到解释<ref name=":6" />。将时间序列数据可视化为热力图矩阵Heat map matrices的工具可以帮助克服这些挑战。

Other techniques include:

Other techniques include:

* Separation into components representing trend, seasonality, slow and fast variation, and cyclical irregularity: see [[trend estimation]] and [[decomposition of time series]]

* 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

* 自相关分析检验序列相关性；

* 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.

* 频谱分析来检查与季节性无关的周期性行为。例如，太阳黑子活动在一个周期内（11年）的变化。其他常见的例子包括天体现象、天气模式、神经活动、商品价格和经济活动；

* Separation into components representing trend, seasonality, slow and fast variation, and cyclical irregularity: see trend estimation and decomposition of time series

* 将序列分离为代表趋势、季节性、慢速和快速变化以及周期性不规则的成分：见趋势估计和时间序列的分。

 

===Curve fitting曲线拟合===

 

===Curve fitting===

{{main|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 fitting<ref name=":7">Sandra Lach Arlinghaus, PHB Practical Handbook of Curve Fitting. CRC Press, 1994.</ref><ref name=":8">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 name=":9">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 name=":10">[https://archive.org/details/signalnoisewhymo00silv]''[[The Signal and the Noise]]'': Why So Many Predictions Fail-but Some Don't.'' By Nate Silver''</ref><ref name=":11">[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 name=":12">Numerical Methods in Engineering with MATLAB®. By [[Jaan Kiusalaas]]. Page 24.</ref><ref name=":13">[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 name=":14">[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 name=":15">See also: [[Mollifier]]</ref> in which a "smooth" function is constructed that approximately fits the data.  A related topic is [[regression analysis]],<ref name=":16">[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 name=":17">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 name=":18">Visual Informatics. Edited by Halimah Badioze Zaman, Peter Robinson, Maria Petrou, Patrick Olivier, Heiko Schröder. Page 689.</ref><ref name=":19">[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 name=":20">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 name=":21">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 name=":22">[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 name=":23">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.

正文：曲线拟合Curve fitting

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。

曲线拟合<ref name=":7" /><ref name=":8" /> 是构建一条曲线Curve或数学函数Mathematical function的过程，它对一系列的数据Data点具有最佳的拟合效果<ref name=":9" />，可能会受到一些限制<ref name=":10" /><ref name=":11" />。曲线拟合包括插值Interpolation<ref name=":12" /><ref name=":13" />（需要精确地拟合数据）与平滑Smoothing<ref name=":14" /><ref name=":15" />（构造一个 "平滑 "的函数来近似地拟合数据）。与曲线拟合相近的回归分析Regression analysis<ref name=":16" /><ref name=":17" />更侧重于统计推断Statistical inference的问题。例如，在拟合有随机误差的数据的曲线中，有多少不确定性存在。拟合曲线可以作为数据可视化的辅助工具<ref name=":18" /><ref name=":19" />，在没有数据的情况下推断函数的值<ref name=":20" />，并总结两个或多个变量之间的关系<ref name=":21" />。外推法Extrapolation是指在观测到的数据范围Range之外使用拟合曲线<ref name=":22" />，它有一定程度的不确定性Degree of uncertainty<ref name=":23" />，因为它既可能是反映观测数据，也可能是反映用于构建曲线的方法。

The construction of economic time series involves the estimation of some components for some dates by interpolation between values ("benchmarks") for earlier and later dates. Interpolation is estimation of an unknown quantity between two known quantities (historical data), or drawing conclusions about missing information from the available information ("reading between the lines").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.

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 name=":24">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 name=":25">Friedman, Milton. "[http://www.nber.org/chapters/c2062.pdf The interpolation of time series by related series]." Journal of the American Statistical Association 57.300 (1962): 729–757.</ref> 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.

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。

经济时间序列的构建涉及通过在早期和晚期的值（“基准”）之间进行插值Interpolation来估计某些日期的某些组成部分。插值法是在两个已知量（历史数据）之间估计一个未知量，或从现有信息中得出关于缺失信息的结论（"从字里行间阅读"）<ref name=":24" />。如果围绕缺失数据的数据是可用的，并且其趋势、季节性和长期周期是已知的，那么插值法就很有用。插值法通常是通过使用已知所有相关日期的相关序列来实现的<ref name=":25" />。或者使用多项式插值Polynomial interpolation或样条插值Spline interpolation，将分段多项式Polynomial函数拟合到时间间隔中，使其平滑地拟合在一起。一个与插值密切相关的问题是用一个简单的函数来逼近一个复杂的函数（也称为回归Regression）。回归和插值的主要区别是，多项式回归给出一个单一的多项式来模拟整个数据集。而样条插值则产生一个由许多多项式组成的分段连续函数来模拟数据集。

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

[[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.

 

===Function approximation===

===Function approximation===