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,
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.
Exploratory analysis A simple way to examine a regular time series is manually with a
line chart. The datagraphic shows tuberculosis deaths in the United States, along with the yearly change and the percentage change from year to year. The total number of deaths declined in every year until the mid-1980s, after which there were occasional increases, often proportionately - but not absolutely - quite large. 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.
Estimation, filtering, and smoothing This approach may be based on
harmonic analysis and filtering of signals in the
frequency domain using the
Fourier transform, and
spectral density estimation. Its development 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. An equivalent effect may be achieved in the time domain, as in a
Kalman filter; see
filtering and
smoothing for more techniques. Other related techniques include: •
Autocorrelation analysis to examine
serial dependence •
Spectral analysis to examine cyclic behavior which need not be related to
seasonality. For example, sunspot activity varies over 11 year cycles. Other common examples include celestial phenomena, weather patterns, neural activity, commodity prices, and economic activity. • Separation into components representing trend, seasonality, slow and fast variation, and cyclical irregularity: see
trend estimation and
decomposition of time series Curve fitting Curve fitting is the process of constructing a
curve, or
mathematical function, that has the best fit to a series of
data points, possibly subject to constraints. Curve fitting can involve either
interpolation, where an exact fit to the data is required, or
smoothing, in which a "smooth" function is constructed that approximately fits the data. A related topic is
regression analysis, which focuses more on questions of
statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables.
Extrapolation refers to the use of a fitted curve beyond the
range of the observed data, and is subject to a
degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data. For processes that are expected to generally grow in magnitude one of the curves in the graphic (and many others) can be fitted by estimating their parameters. The construction of economic time series involves the estimation of some components for some dates by
interpolation between values ("benchmarks") for earlier and later dates. Interpolation is estimation of an unknown quantity between two known quantities (historical data), or drawing conclusions about missing information from the available information ("reading between the lines"). Interpolation is useful where the data surrounding the missing data is available and its trend, seasonality, and longer-term cycles are known. This is often done by using a related series known for all relevant dates. Alternatively
polynomial interpolation or
spline interpolation is used where piecewise
polynomial functions are fitted in 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.). Second, the target function, call it
g, may be unknown; instead of an explicit formula, only a set of points (a time series) of the form (
x,
g(
x)) is provided. Depending on the structure of the
domain and
codomain of
g, several techniques for approximating
g may be applicable. For example, if
g is an operation on the
real numbers, techniques of
interpolation,
extrapolation,
regression analysis, and
curve fitting can be used. If the
codomain (range or target set) of
g is a finite set, one is dealing with a
classification problem instead. A related problem of
online time series approximation is to summarize the data in one-pass and construct an approximate representation that can support a variety of time series queries with bounds on worst-case error. 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.
Classification Assigning time series pattern to a specific category, for example identify a word based on series of hand movements in
sign language.
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.
Clustering Time series data may be clustered, however special care has to be taken when considering subsequence clustering. Time series clustering may be split into • whole time series clustering (multiple time series for which to find a cluster) • subsequence time series clustering (single timeseries, split into chunks using sliding windows) • time point clustering
Subsequence time series clustering Subsequence time series clustering resulted in unstable (random) clusters
induced by the feature extraction using chunking with sliding windows. It was found that the cluster centers (the average of the time series in a cluster - also a time series) follow an arbitrarily shifted sine pattern (regardless of the dataset, even on realizations of a
random walk). This means that the found cluster centers are non-descriptive for the dataset because the cluster centers are always nonrepresentative sine waves. ==Models==