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Encyclopedia > Time series

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In statistics, signal processing, and econometrics, a time series is a sequence of data points, measured typically at successive times, spaced at (often uniform) time intervals. Time series analysis comprises methods that attempt to understand such time series, often either to understand the underlying theory of the data points (where did they come from? what generated them?), or to make forecasts (predictions). Time series prediction is the use of a model to predict future events based on known past events: to predict future data points before they are measured. The standard example is the opening price of a share of stock based on its past performance. A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... Econometrics literally means economic measurement. It is a combination of mathematical economics and statistics. ... In statistics, a data point is a single typed measurement. ... Prediction of future events is an ancient human wish. ... An abstract model (or conceptual model) is a theoretical construct that represents something, with a set of variables and a set of logical and quantitative relationships between them. ... This article does not cite any references or sources. ... As shown by Box and Jenkins in their book, models for time series data can have many forms and represents different stochastic processes. When modeling the mean of a process, three broad classes of practical importance are the Autoregressive (AR) models, the Integrated (I) models, and the Moving Average (MA) models (the MA process is related but not to be confused with the concept of Moving average ). These three classes depend linearly on previous data points and are treated in more detail in the articles Autoregressive Moving Average Models (ARMA) and Autoregressive Integrated Moving Average (ARIMA). There is also another kind of representation called as Autorregressive Fractionally Integrated Moving Average (ARFIMA), which is a generalisation of the former three. Non-linear dependence on previous data points is of interest because of the possibility of producing a chaotic time series. In econometrics, the Box-Jenkins methodology, named after the statisticians George Box and Gwilym Jenkins, applies autoregressive integrated moving average ARIMA models to find the best fit of a time series to past values of this time series, in order to make forecasts. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ... The term moving average is used in different contexts. ... In statistics, autoregressive moving average (ARMA) models, sometimes called Box-Jenkins models after George Box and G. M. Jenkins, are typically applied to time series data. ... In statistics, an autoregressive integrated moving average (ARIMA) model is a generalisation of an autoregressive moving average or (ARMA) model. ... A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit dynamics that are sensitive to initial conditions (popularly referred to as the butterfly effect). ... Among non-linear time series, there are models to represent the changes of variance along time (heteroskedasticity). These models are called Autoregressive Conditional Heteroskedasticity (ARCH) and the collection comprises a wide variaty of representation (GARCH, TARCH, EGARCH, FIGARCH, CGARCH, etc). In statistics, a sequence or a vector of random variables is heteroskedastic if the random variables in the sequence or vector may have different variances. ... In econometrics, an autoregressive conditional heteroskedasticity (ARCH) model considers the variance of the current error term to be a function of the variances of the previous time periods error terms. ... Contents 1 Notation 2 Assumptions 3 Related tools 4 Industry usage 5 See also 6 External links [edit] Notation A number of different notations are in use for time-series analysis: is a common notation which specifies a time series X which is indexed by the natural numbers. We also are accustomed to: In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... [edit] Assumptions There are only two assumptions from which the theory is built: Stationarity Ergodicity The general representation of a Autorregresive model well-known as AR(p) is: Stationary can mean: Look up stationary in Wiktionary, the free dictionary. ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... where the term εt is the source of randomnees and is called white noise. It is assumed to have the following characteristics: White noise spectrum White noise( ) is a random signal (or process) with a flat power spectral density. ... 1. E[εt] = 0 2. 3. If it also has a Normal distribution, it is called Normal White-Noise: [edit] Related tools Tools for investigating time-series data include: Consideration of the autocorrelation function and the spectral density function Performing a Fourier transform to investigate the series in the frequency domain. Use of a filter to remove unwanted noise. Principal components analysis (or empirical orthogonal function analysis) Artificial neural networks time-frequency analysis techniques: Continuous wavelet transform Short-time Fourier transform Chirplet transform Fractional Fourier transform Chaotic analysis Correlation dimension Recurrence plots Recurrence quantification analysis Lyapunov exponents A plot showing 100 random numbers with a hidden sine function, and an autocorrelation of the series on the bottom. ... In applied mathematics and physics, the spectral density is a general concept applied to a signal which may have any physical dimensions or none at all. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... An FIR filter In electronics,nirali a digital filter is any electronic filter that works by performing digital mathematical operations on an intermediate form of a signal. ... In science, and especially in physics and telecommunication, noise is fluctuations in and the addition of external factors to the stream of target information (signal) being received at a detector. ... In statistics, principal components analysis (PCA) is a technique for simplifying a dataset, by reducing multidimensional datasets to lower dimensions for analysis. ... In statistics and signal processing, the method of empirical orthogonal function (EOF) analysis is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from the data. ... An artificial neural network (ANN), often just called a neural network (NN), is an interconnected group of artificial neurons that uses a mathematical model or computational model for information processing based on a connectionist approach to computation. ... A time-frequency representation (TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency. ... In mathematics and signal processing, the continuous wavelet transform (CWT) of a function is a wavelet transform defined by where represents translation, represents scale and is the mother wavelet. ... The short-time Fourier transform (STFT), or short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of a signal as it changes over time. ... Comparison of wave, wavelet, chirp, and chirplet In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets. ... The fractional Fourier transform (FRFT) is a linear transformation generalizing the continuous Fourier transform, and it can be thought of as the Fourier transform to the n-th power where n need not be an integer — thus, it can transform a function to an intermediate domain between time and... A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit dynamics that are sensitive to initial conditions (popularly referred to as the butterfly effect). ... In chaos theory the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points. ... In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for a given moment in time, the times at which a phase space trajectory visits roughly the same area in the phase space. ... Recurrence quantification analysis (RQA) is a system for analyzing nonlinear systems that works by looking at the pattern and number of recurrences present in a data signal. ... In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. ... [edit] Industry usage Any associative array of times and numbers can be viewed as a time series. The times may not necessarily be of a regular interval length. For example, the historical fluctuations in the price of a NYMEX Gold Contract can be said to be the time series for NYMEX Gold. Analysts throughout the economy will use the tools outlined here to aid in the management of their corresponding businesses. Energy traders, for example, will often attempt to forecast power consumption based upon both weather normals and short term weather forecasts. [edit] See also Analysis of rhythmic variance Anomaly time series Autocorrelation Partial autocorrelation Linear prediction Longitudinal study Moving average (finance) Prediction interval Predictive analytics Seasonal adjustment System identification Time series database Trend estimation In statistics, analysis of rhythmic variance (ANORVA) is a new simple method for detecting rhythms in biological time series, published by Peter Celec (Biol Res. ... An anomaly time series is the time series of deviations of a quantity from some mean. ... A plot showing 100 random numbers with a hidden sine function, and an autocorrelation of the series on the bottom. ... In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. ... Linear prediction is a mathematical operation where future values of a digital signal are estimated as a linear function of previous samples. ... Longitudinal studies form a class of research methods that involve observations of the same items over a longer time. ... A moving average, in finance and especially in technical analysis, is one of a family of similar statistical techniques used to analyze time series data. ... In statistics, a prediction interval bears the same relationship to a future observation that a confidence interval bears to an unobservable population parameter. ... Predictive analytics encompasses a variety of techniques from statistics and data mining that process current and historical data in order to make “predictions” about future events. ... Seasonal adjustment is a statistical method for time series analysis. ... System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data. ... A time series database server (TSDS) is a software system that is optimized for handling a time series. ... A series of measurements of a process may be treated as a time series, and then trend estimation is the application of statistical techniques to make and justify statements about trends in the data. ... [edit] External links A First Course on Time Series Analysis - an open source book on time series analysis with SAS Software for Time Series Analysis Free online chaos theory based analysis Online Tutorial 'Recurrence Plot' (Flash animation); lots of examples A Professional Environment for Time Series and Signal Analysis Categories: Statistics stubs | Time series analysis

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	Statistics Glossary - time series data (1003 words) In weekly or monthly data, the seasonal component, often referred to as seasonality, is the component of variation in a time series which is dependent on the time of year. In some time series, seasonal variation is so strong it obscures any trends or cycles which are very important for the understanding of the process being observed. Exponential smoothing is a smoothing technique used to reduce irregularities (random fluctuations) in time series data, thus providing a clearer view of the true underlying behaviour of the series. Time Series Analysis (1797 words) Time Series Analysis (TSA) is popular for estimating the process that underlies some output, or for forecasting from some observed behaviour over time. Can linkages be easily established between components in separate time series analyses, or does the method suffer from the "little room" approach - the problem has been removed from its surroundings and taken to a mathematically small room, where it has been dissected under controlled conditions in isolation from its context. The analyst can examine all of the components of the time series, make connections with other knowledge to be used in the time series analysis, or use the result of the time series analysis in some other analysis, or the whole operation can be run in batch, still with connection to other knowledge.
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