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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a time series is a series of
data point In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...
s indexed (or listed or graphed) in time order. Most commonly, a time series is a
sequence In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

sequence
taken at successive equally spaced points in time. Thus it is a sequence of
discrete-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time. Discrete time Discrete sampled signal Discrete time views values of variables as occurring at dist ...
data. Examples of time series are heights of ocean
tides (U.S.), low tide occurs roughly at moonrise and high tide with a high Moon, corresponding to the simple gravity model of two tidal bulges; at most places however, the Moon and tides have a phase shift. Tides are the rise and fall of sea leve ...

tides
, counts of
sunspots Sunspots are temporary celestial event, phenomena on the Sun's photosphere that appear as spots darker than the surrounding areas. They are regions of reduced surface temperature caused by concentrations of magnetic flux, magnetic field flux tha ...

sunspots
, and the daily closing value of the
Dow Jones Industrial Average The Dow Jones Industrial Average (DJIA), Dow Jones, or simply the Dow (), is a price-weighted measurement stock market index In finance, a stock index, or stock market index, is an Index (economics), index that measures a stock market, or a ...

Dow Jones Industrial Average
. A Time series is very frequently plotted via a
run chart A run chart, also known as a run-sequence plot is a graph that displays observed data in a . Often, the data displayed represent some aspect of the output or performance of a manufacturing or other business process. It is therefore a form of . ...
(which is a temporal
line chart A line chart or line plot or line graph or curve chart is a type of chart A chart is a graphical representation for data visualization, in which "the data Data are units of information Information can be thought of as the resol ...

line chart
). Time series are used in
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical ...

statistics
,
signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

signal processing
,
pattern recognition Pattern recognition is the automated recognition of pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of ...
,
econometrics Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of Econ ...

econometrics
,
mathematical financeMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and extend the Mathematical ...
,
weather forecasting Weather forecasting is the application of science and technology forecasting, to predict the conditions of the Earth's atmosphere, atmosphere for a given location and time. People have attempted to predict the weather informally for millennia an ...
,
earthquake prediction An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth Earth is the third planet from the Sun and the only astronomical object known t ...
,
electroencephalography Electroencephalography (EEG) is a method to record an electrogram of the electrical activity on the scalp The scalp is the anatomical area bordered by the human face The face is the front of an animal's head that features three of the he ...
,
control engineering Control engineering or control systems engineering is an engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles ...
,
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...
,
communications engineering Telecommunications Engineering is an engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The ...
, and largely in any domain of applied
science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity, awareness, or understanding of someone or something, such as facts ( descriptive knowledge), skills (procedural knowledge), or objects ...
and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more specializ ...

engineering
which involves
temporal
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 In general, a model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. ...
to predict future values based on previously observed values. While
regression analysis In statistical modelA statistical model is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unifi ...
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 Interrupted time series analysis (ITS), sometimes known as quasi-experimental time series analysis, is a method of statistical analysis involving tracking a long-term period before and after a point of intervention to assess the intervention's effe ...
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 medical research Medical research (or biomedical research), also known as experimental medicine, encompasses a wide array of research, extending from " basic research" (also called ''bench science'' or ''bench research''), – involvin ...
, 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 of London, showing cluster (epidemiology), clusters of cholera cases in the 1854 Broad Street cholera outbreak. This was one of the first uses of map-based spatial analysis. Spatial analysis or spatial statistics includes any of the formal Scienti ...
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 Stochastic () refers to the property of being well described by a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wi ...
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 A mathematical or physical process is time-reversible if the dynamics of the process remain well-defined when the sequence of time-states is reversed. A deterministic process is time-reversible if the time-reversed process satisfies the same dyna ...
). Time series analysis can be applied to
real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an in ...
, continuous data,
discrete Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual. Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic c ...
numeric data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the
English language English is a of the , originally spoken by the inhabitants of . It is named after the , one of the ancient that migrated from , a peninsula on the (not to be confused with ), to the area of later named after them: . Living languages mos ...

English language
).


Methods for analysis

Methods for time series analysis may be divided into two classes:
frequency-domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or Signal (information theory), signals with respect to frequency, rather than time. Put simply, a time-dom ...
methods and
time-domain Time domain refers to the analysis of mathematical functions, physical signal In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio sig ...

time-domain
methods. The former include spectral analysis and
wavelet analysisA wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one recorded by a seismograph or heart monitor. Generally, wavelets a ...
; the latter include
auto-correlation File:Comparison convolution correlation.svg, 400px, Visual comparison of convolution, cross-correlation, and autocorrelation. For the operations involving function , and assuming the height of is 1.0, the value of the result at 5 different point ...
and
cross-correlation In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal ...

cross-correlation
analysis. In the time domain, correlation and analysis can be made in a filter-like manner using
scaled correlationIn statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a ...
, thereby mitigating the need to operate in the frequency domain. Additionally, time series analysis techniques may be divided into parametric and
non-parametricNonparametric statistics is the branch of statistics that is not based solely on Statistical parameter, parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on ...
methods. The parametric approaches assume that the underlying
stationary stochastic processIn mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Conse ...
has a certain structure which can be described using a small number of parameters (for example, using an
autoregressive In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...
or
moving average model In time series analysis, the moving-average model (MA model), also known as moving-average process, is a common approach for modeling univariateIn mathematics, a univariate object is an expression, equation In mathematics, an equation is a st ...
). 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 In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the less ...

covariance
or the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a Continuum (theory), continuum. The word was first used scientifically in optics to describe 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 Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...

linear
and
non-linear In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, and
univariate In mathematics, a univariate object is an expression, equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geom ...
and multivariate.


Panel data

A time series is one type of
panel data In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...
. 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 Cross-sectional data, or a cross section of a study population, in statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, ...
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.


Analysis

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


Motivation

In the context of
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical ...

statistics
,
econometrics Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of Econ ...

econometrics
,
quantitative finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, me ...
,
seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "Earthquake, earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of Linear elasticity#Elastic wave, elast ...
,
meteorology Meteorology is a branch of the (which include and ), with a major focus on . The study of meteorology dates back , though significant progress in meteorology did not begin until the 18th century. The 19th century saw modest progress in the f ...
, and
geophysics Geophysics () is a subject of concerned with the physical processes and of the and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' sometimes refers only to solid earth applic ...

geophysics
the primary goal of time series analysis is
forecasting Forecasting is the process of making predictions based on past and present data and most commonly by analysis of trends. A commonplace example might be estimation Estimation (or estimating) is the process of finding an estimate, or approximatio ...
. In the context of
signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

signal processing
,
control engineering Control engineering or control systems engineering is an engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles ...
and
communication engineering Telecommunications Engineering is an engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of eng ...
it is used for signal detection. Other applications are in
data mining Data mining is a process of extracting and discovering patterns in large data set A data set (or dataset) is a collection of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...
,
pattern recognition Pattern recognition is the automated recognition of pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of ...
and
machine learning Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data ...

machine learning
, where time series analysis can be used for clustering,
classification Classification is a process related to categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such as Object (philosophy), objects, eve ...
, query by content,
anomaly detection In data analysis, anomaly detection (also outlier detection) is the identification of rare items, events or observations which raise suspicions by differing significantly from the majority of the data. Typically the anomalous items will translate to ...
as well as
forecasting Forecasting is the process of making predictions based on past and present data and most commonly by analysis of trends. A commonplace example might be estimation Estimation (or estimating) is the process of finding an estimate, or approximatio ...
.


Exploratory analysis

A straightforward way to examine a regular time series is manually with a
line chart A line chart or line plot or line graph or curve chart is a type of chart A chart is a graphical representation for data visualization, in which "the data Data are units of information Information can be thought of as the resol ...

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. Visual tools that represent time series data as heat map matrices can help overcome these challenges. Other techniques include: *
Autocorrelation File:Comparison convolution correlation.svg, 400px, Visual comparison of convolution, cross-correlation, and autocorrelation. For the operations involving function , and assuming the height of is 1.0, the value of the result at 5 different point ...
analysis to examine serial dependence * Spectral analysis to examine cyclic behavior which need not be related to
seasonality In time series data, seasonality is the presence of variations that occur at specific regular intervals less than a year, such as weekly, monthly, or quarterly. Seasonality may be caused by various factors, such as weather, vacation, and holidays a ...

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 Linear trend estimation is a statistics, statistical technique to aid interpretation of data. When a series of measurements of a process are treated as, for example, a time series, trend estimation can be used to make and justify statements about t ...
and
decomposition of time series The decomposition of time series is a statistical task that deconstructs a time series into several components, each representing one of the underlying categories of patterns. There are two principal types of decomposition, which are outlined below. ...


Curve fitting

Curve fitting is the process of constructing a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
, or
mathematical function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, that has the best fit to a series of
data Data (; ) are individual facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to correspond to experience. Standard reference works are often used ...

data
points, possibly subject to constraints. Curve fitting can involve either
interpolation In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantitie ...

interpolation
, where an exact fit to the data is required, or
smoothing In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...

smoothing
, in which a "smooth" function is constructed that approximately fits the data. A related topic is
regression analysis In statistical modelA statistical model is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unifi ...
, which focuses more on questions of
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables.
Extrapolation In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known ...

Extrapolation
refers to the use of a fitted curve beyond the
range Range may refer to: Geography * Range (geographic)A range, in geography, is a chain of hill A hill is a landform A landform is a natural or artificial feature of the solid surface of the Earth or other planetary body. Landforms together ...
of the observed data, and is subject to a
degree of uncertainty
degree of uncertainty
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 In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantitie ...

interpolation
between values ("benchmarks") for earlier and later dates. Interpolation is estimation of an unknown quantity between two known quantities (historical data), or drawing conclusions about missing information from the available information ("reading between the lines"). Interpolation is useful where the data surrounding the missing data is available and its trend, seasonality, and longer-term cycles are known. This is often done by using a related series known for all relevant dates. Alternatively
polynomial interpolation In numerical analysis (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... Numerical analysis is the study of ...
or
spline interpolation In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...

spline interpolation
is used where piecewise
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known ...

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 In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantitie ...

interpolation
, which produces estimates between known observations, but extrapolation is subject to greater
uncertainty Uncertainty refers to epistemic Epistemology (; ) is the branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, ...

uncertainty
and a higher risk of producing meaningless results.


Function approximation

In general, a function approximation problem asks us to select a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler function (mathematics), functions, and with Quantitative property, quantitatively characterization (ma ...
is the branch of
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
that investigates how certain known functions (for example,
special function Special functions are particular mathematical function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), ...
s) can be approximated by a specific class of functions (for example,
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
s or
rational function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

rational function
s) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.). 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 Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
and
codomain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

codomain
of ''g'', several techniques for approximating ''g'' may be applicable. For example, if ''g'' is an operation on the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, techniques of
interpolation In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantitie ...

interpolation
,
extrapolation In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known ...

extrapolation
,
regression analysis In statistical modelA statistical model is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unifi ...
, and
curve fitting Curve fitting is the process of constructing a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve m ...

curve fitting
can be used. If the
codomain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

codomain
(range or target set) of ''g'' is a finite set, one is dealing with a
classification Classification is a process related to categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience Experience refers to conscious , an English Paracels ...
problem instead. A related problem of ''online'' time series approximation is to summarize the data in one-pass and construct an approximate representation that can support a variety of time series queries with bounds on worst-case error. To some extent, the different problems ( regression,
classification Classification is a process related to categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such as Object (philosophy), objects, eve ...
,
fitness approximation Fitness approximation Y. JinA comprehensive survey of fitness approximation in evolutionary computation Soft Computing, 9:3–12, 2005 aims to approximate the objective or fitness functions in evolutionary optimization by building up machine learn ...
) have received a unified treatment in
statistical learning theory Statistical learning theory is a framework for machine learning drawing from the fields of statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying s ...
, where they are viewed as
supervised learning Supervised learning (SL) is the machine learning task of learning a function that Map (mathematics), maps an input to an output based on example input-output pairs. It infers a function from ' consisting of a set of ''training examples''. In supe ...
problems.


Prediction and forecasting

In
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical ...

statistics
,
prediction Image:Old Farmer's Almanac 1793 cover.jpg, frame, ''The Old Farmer's Almanac'' is famous in the US for its (not necessarily accurate) long-range weather predictions. A prediction (Latin ''præ-'', "before," and ''dicere'', "to say"), or forecas ...
is a part of
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
. One particular approach to such inference is known as
predictive inference Predictive inference is an approach to statistical inference that emphasizes the prediction of future observations based on past observations. Initially, predictive inference was based on ''observable'' parameters and it was the main purpose of stu ...
, 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 Forecasting is the process of making predictions based on past and present data and most commonly by analysis of trends. A commonplace example might be estimation Estimation (or estimating) is the process of finding an estimate, or approximatio ...
. * Fully formed statistical models for
stochastic simulationA stochastic simulation is a simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model , models; the model represents the key characteristics or behavi ...
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 Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio (given name), Julio and Julius. (For further details on etymology, see wikt:Iulius#Latin, Wiktionary entry “Julius”.) The given name ''Julia'' had been ...
,
Python PYTHON was a Cold War contingency plan of the Government of the United Kingdom, British Government for the continuity of government in the event of Nuclear warfare, nuclear war. Background Following the report of the Strath Committee in 1955, the ...
, R, SAS,
SPSS SPSS Statistics is a statistical software Statistical software are specialized computer program A computer program is a collection of instructions that can be executed by a computer to perform a specific task. A computer program is usual ...

SPSS
and many others. * Forecasting on large scale data can be done with
Apache Spark Apache Spark is an open-source unified analytics engine for large-scale data processing. Spark provides an interface Interface or interfacing may refer to: Academic journals * Interface (journal), ''Interface'' (journal), by the Electrochemi ...
using the Spark-TS library, a third-party package.


Classification

Assigning time series pattern to a specific category, for example identify a word based on series of hand movements in
sign language Sign languages (also known as signed languages) are languages that use the visual-manual modality to convey meaning. Sign languages are expressed through manual articulations in combination with non-manual elements. Sign languages are full-fled ...

sign language
.


Signal estimation

This approach is based on
harmonic analysis Harmonic analysis is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...
and filtering of signals in the
frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or Signal (information theory), signals with respect to frequency, rather than time. Put simply, a time-dom ...
using the
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
, and
spectral density estimation In statistical signal processing, the goal of spectral density estimation (SDE) is to estimate the spectral density (also known as the power spectral density) of a random signal from a sequence of time samples of the signal. Intuitively speaki ...
, the development of which was significantly accelerated during
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by mathematician
Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

Norbert Wiener
, electrical engineers
Rudolf E. Kálmán Rudolf Emil Kálmán (May 19, 1930 – July 2, 2016) was an Hungarian-American electrical engineer, mathematician, and inventor. He is most noted for his co-invention and development of the Kalman filter In statistics Statistics is the d ...
,
Dennis Gabor Dennis Gabor ( hu, Gábor Dénes; , ; 5 June 1900 – 9 February 1979) was a Hungarian people, Hungarian-British people, British Electrical engineering, electrical engineer and physicist, most notable for inventing holography, for which he la ...
and others for filtering signals from noise and predicting signal values at a certain point in time. See
Kalman filter For statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin w ...
,
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, and
Digital signal processing Digital signal processing (DSP) is the use of digital processing Digital data, in information theory and information systems, is information represented as a string of discrete symbols each of which can take on one of only a finite number of ...


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.


Models

Models for time series data can have many forms and represent different
stochastic processes In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
. When modeling variations in the level of a process, three broad classes of practical importance are the ''
autoregressive In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...
'' (AR) models, the ''integrated'' (I) models, and the ''
moving average In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a moving mean (MM) or rolling mean and is ...
'' (MA) models. These three classes depend linearly on previous data points. Combinations of these ideas produce
autoregressive moving average In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in natural science, nature, economics, etc. The autoregr ...
(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 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 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. 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.


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'' = (''X''1, ''X''2, ...). 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: * Stationary process * Ergodic process 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. 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.


Tools

Tools for investigating time-series data include: * Consideration of the autocorrelation, autocorrelation function and the Spectral density, spectral density function (also cross-correlation functions and cross-spectral density functions) * Scaled correlation, Scaled cross- and auto-correlation functions to remove contributions of slow components * Performing a
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
to investigate the series in the
frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or Signal (information theory), signals with respect to frequency, rather than time. Put simply, a time-dom ...
* 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 Models ** Unobserved Components Models * Machine Learning ** Artificial neural networks ** Support vector machine ** Fuzzy logic ** Gaussian process ** Hidden Markov model * 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 * Cross-correlation * 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 plots ** Recurrence quantification analysis ** Lyapunov exponents ** Entropy encoding


Measures

Time series metrics or Features (pattern recognition), features that can be used for time series Classification (machine learning), classification or
regression analysis In statistical modelA statistical model is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unifi ...
: * 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 ** Sample entropy ** ** Wavelet 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 ** Surrogate time series and surrogate correction ** Loss of recurrence (degree of non-stationarity) * Bivariate linear measures ** Maximum linear
cross-correlation In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal ...

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 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


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)


Overlapping charts

* Braided graphs * Line charts * Slope graphs *


Separated charts

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


See also


References


Further reading

* * James Durbin, Durbin J., Koopman S.J. (2001), ''Time Series Analysis by State Space Methods'', Oxford University Press. * * * Maurice Priestley, 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. * 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. *


External links


Introduction to Time series Analysis (Engineering Statistics Handbook)
— A practical guide to Time series analysis. {{DEFAULTSORT:Time Series Time series, Statistical data types Mathematical and quantitative methods (economics) Machine learning