# 时间序列分析

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

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.

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.

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.

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).

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[1]).

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).

## 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 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.

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.

Methods of time series analysis may also be divided into linear and non-linear, and univariate and multivariate.

## 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 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.

## Analysis分析

There are several types of motivation and data analysis available for time series which are appropriate for different purposes.

### 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 clustering,[2][3] classification,[4] query by content,[5] anomaly detection as well as forecasting.[6]

### Exploratory analysis探索性分析

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

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.[7] Visual tools that represent time series data as heat map matrices can help overcome these challenges.

Other techniques include:

• 自相关分析检验序列相关性；
• 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；
• 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[10][11] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points,[12] possibly subject to constraints.[13][14] Curve fitting can involve either interpolation,[15][16] where an exact fit to the data is required, or smoothing,[17][18] in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis,[19][20] 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,[21][22] to infer values of a function where no data are available,[23] and to summarize the relationships among two or more variables.[24] Extrapolation refers to the use of a fitted curve beyond the range of the observed data,[25] and is subject to a degree of uncertainty[26] since it may reflect the method used to construct the curve as much as it reflects the observed data.

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").[27] 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.[28] 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.

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

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

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

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

Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of points (a time series) of the form (x, g(x)) is provided. Depending on the structure of the domain and codomain of g, several techniques for approximating g may be applicable. For example, if g is an operation on the real numbers, techniques of interpolation, extrapolation, regression analysis, and curve fitting can be used. If the codomain (range or target set) of g is a finite set, one is dealing with a classification problem instead. A related problem of online time series approximation[29] 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.

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

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

### Prediction and forecasting

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

• Fully formed statistical models for stochastic simulation purposes, so as to generate alternative versions of the time series, representing what might happen over non-specific time-periods in the future
• Simple or fully formed statistical models to describe the likely outcome of the time series in the immediate future, given knowledge of the most recent outcomes (forecasting).
• Forecasting on time series is usually done using automated statistical software packages and programming languages, such as Julia, Python, R, SAS, SPSS and many others.
• Forecasting on large scale data can be done with Apache Spark using the Spark-TS library, a third-party package.[30]

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.

• 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

Assigning time series pattern to a specific category, for example identify a word based on series of hand movements in sign language.

Assigning time series pattern to a specific category, for example identify a word based on series of hand movements in sign language.

= = = 分类 = = 将时间序列模式分配到一个特定的类别，例如根据手语中的一系列动作识别一个单词。

### Signal estimation

This approach is based on harmonic analysis and filtering of signals in the frequency domain using the Fourier transform, and spectral density estimation, the development of which was significantly accelerated during World War II by mathematician Norbert Wiener, electrical engineers Rudolf E. Kálmán, Dennis Gabor and others for filtering signals from noise and predicting signal values at a certain point in time. See Kalman filter, Estimation theory, and Digital signal processing

This approach is based on harmonic analysis and filtering of signals in the frequency domain using the Fourier transform, and spectral density estimation, the development of which was significantly accelerated during World War II by mathematician Norbert Wiener, electrical engineers Rudolf E. Kálmán, Dennis Gabor and others for filtering signals from noise and predicting signal values at a certain point in time. See Kalman filter, Estimation theory, and Digital signal processing

= = = 信号估计 = = = 这种方法是基于傅里叶分析信号和滤波的频域使用傅里叶变换和谱密度估计，其发展是显着加速二战期间由数学家诺伯特维纳，电气工程师鲁道夫·卡尔曼，丹尼斯 Gabor 和其他人从噪音信号过滤和预测信号值在一定时间点。参见卡尔曼滤波器，参数估测和数字信号处理

### Segmentation

Splitting a time-series into a sequence of segments. It is often the case that a time-series can be represented as a sequence of individual segments, each with its own characteristic properties. For example, the audio signal from a conference call can be partitioned into pieces corresponding to the times during which each person was speaking. In time-series segmentation, the goal is to identify the segment boundary points in the time-series, and to characterize the dynamical properties associated with each segment. One can approach this problem using change-point detection, or by modeling the time-series as a more sophisticated system, such as a Markov jump linear system.

Splitting a time-series into a sequence of segments. It is often the case that a time-series can be represented as a sequence of individual segments, each with its own characteristic properties. For example, the audio signal from a conference call can be partitioned into pieces corresponding to the times during which each person was speaking. In time-series segmentation, the goal is to identify the segment boundary points in the time-series, and to characterize the dynamical properties associated with each segment. One can approach this problem using change-point detection, or by modeling the time-series as a more sophisticated system, such as a Markov jump linear system.

= = = 分割 = = 将一个时间序列分割成一系列的片段。通常情况下，时间序列可以表示为一系列单独的片段，每个片段都有自己的特征属性。例如，来自电话会议的音频信号可以根据每个人发言的时间分割成相应的部分。在时间序列分割中，目标是识别时间序列中的分段边界点，并刻画每个分段的动态特性。人们可以通过变点检测来解决这个问题，或者将时间序列建模为一个更复杂的系统，如马尔可夫跳跃线性系统。

# = 模型 =

Models for time series data can have many forms and represent different stochastic processes. When modeling variations in the level of a process, three broad classes of practical importance are the autoregressive (AR) models, the integrated (I) models, and the moving average (MA) models. These three classes depend linearly on previous data points.[31] 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".

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),[32] and (Abarbanel)[33]

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)

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.

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.

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.

### Notation

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

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

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

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

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

Another common notation is

Y = (Yt: tT),

where T is the index set.

Another common notation is

Y = (Yt: t ∈ T),

where T is the index set.

### Conditions

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

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

• Stationary process
• Ergodic process

• 平稳过程遍历过程

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

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

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.[34]

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

### Tools

Tools for investigating time-series data include:

Tools for investigating time-series data include:

= = = 调查时间序列数据的工具包括:

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

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

### Measures

Time series metrics or features that can be used for time series classification or regression analysis:[38]

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

= = = = 可用于时间序列分类或回归分析的时间序列度量或特征:

• Univariate linear measures
• Moment (mathematics)
• Spectral band power
• Spectral edge frequency
• Accumulated Energy (signal processing)
• Characteristics of the autocorrelation function
• Hjorth parameters
• FFT parameters
• Autoregressive model parameters
• Mann–Kendall test
• Univariate non-linear measures
• Measures based on the correlation sum
• Correlation dimension
• Correlation integral
• Correlation density
• Correlation entropy
• Approximate entropy
• Sample entropy
• 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)[42]

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

# = = = 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

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• Box, George; Jenkins, Gwilym (1976), Time Series Analysis: forecasting and control, rev. ed., Oakland, California: Holden-Day
• Durbin J., Koopman S.J. (2001), Time Series Analysis by State Space Methods, Oxford University Press.
• Gershenfeld, Neil (2000), The Nature of Mathematical Modeling, Cambridge University Press, ISBN 978-0-521-57095-4, OCLC 174825352
• Hamilton, James (1994), Time Series Analysis, Princeton University Press, ISBN 978-0-691-04289-3
• Priestley, M. B. (1981), Spectral Analysis and Time Series, Academic Press.
• Shasha, D. (2004), High Performance Discovery in Time Series, Springer, ISBN 978-0-387-00857-8
• Shumway R. H., Stoffer D. S. (2017), Time Series Analysis and its Applications: With R Examples (ed. 4), Springer,
• Weigend A. S., Gershenfeld N. A. (Eds.) (1994), Time Series Prediction: Forecasting the Future and Understanding the Past. Proceedings of the NATO Advanced Research Workshop on Comparative Time Series Analysis (Santa Fe, May 1992), Addison-Wesley.
• Wiener, N. (1949), Extrapolation, Interpolation, and Smoothing of Stationary Time Series, MIT Press.
• Woodward, W. A., Gray, H. L. & Elliott, A. C. (2012), Applied Time Series Analysis, CRC Press.
•
• 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。