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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. Decomposition based on rates of change This is an important technique for all types of time series analysis, especially for seasonal adjustment. It seeks to construct, from an observed time series, a number of component series (that could be used to reconstruct the original by additions or multiplications) where each of these has a certain characteristic or type of behavior. For example, time series are usually decomposed into: *T_t, the trend component at time ''t'', which reflects the long-term progression of the series (secular variation). A trend exists when there is a persistent increasing or decreasing direction in the data. The trend component does not have to be linear. *C_t, the cyclical component at time ''t'', which refle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical
Statistics (from German: '' Statistik'', "description of a state, a country") 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 population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Airlines Of The United Kingdom
The following is a list of operational airlines in the United Kingdom. For British Overseas Territories, see the sections for List of airlines of the Americas# Anguilla, Anguilla, List of airlines of Bermuda, Bermuda, List of airlines of the British Virgin Islands, British Virgin Islands, List of airlines of the Cayman Islands, Cayman Islands, List of airlines of the Falkland Islands, Falkland Islands, List of airlines of Montserrat, Montserrat, List of airlines of the Turks and Caicos Islands, Turks and Caicos Islands. Scheduled airlines Charter airlines Cargo airlines Helicopter airlines/general aviation Channel Islands and the Isle of Man See also * Lists of airlines * List of defunct airlines of the United Kingdom References {{List of airlines Airlines of the United Kingdom, * Lists of airlines by country, United Kingdom Lists of companies of the United Kingdom by industry, Airlines Lists of airlines of the United Kingdom, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Drift
In probability theory, stochastic drift is the change of the average value of a stochastic (random) process. A related concept is the drift rate, which is the rate at which the average changes. For example, a process that counts the number of heads in a series of n fair coin tosses has a drift rate of 1/2 per toss. This is in contrast to the random fluctuations about this average value. The stochastic mean of that coin-toss process is 1/2 and the drift rate of the stochastic mean is 0, assuming 1 = heads and 0 = tails. Stochastic drifts in population studies Longitudinal studies of secular events are frequently conceptualized as consisting of a trend component fitted by a polynomial, a cyclical component often fitted by an analysis based on autocorrelations or on a Fourier series, and a random component (stochastic drift) to be removed. In the course of the time series analysis, identification of cyclical and stochastic drift components is often attempted by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least-squares Spectral Analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in long gapped records; LSSA mitigates such problems. Unlike with Fourier analysis, data need not be equally spaced to use LSSA. LSSA is also known as the Vaníček method or the Gauss-Vaniček method after Petr Vaníček, and as the Lomb method or the Lomb–Scargle periodogram, based on the contributions of Nicholas R. Lomb and, independently, Jeffrey D. Scargle. Historical background The close connections between Fourier analysis, the periodogram, and least-squares fitting of sinusoids have long been known. Most developments, however, are restricted to complete data sets of equally spaced samples. In 1963, Freek J. M. Barning of Mathematisch Centrum, Amsterdam, handled unequally spaced data by similar t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each individual equation. The most important application is in data fitting. When the problem has substantial uncertainties in the independent variable (the ''x'' variable), then simple regression and least-squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares. Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regress ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Huang Transform
The Hilbert–Huang transform (HHT) is a way to decompose a Signal processing, signal into so-called intrinsic mode functions (IMF) along with a trend, and obtain instantaneous frequency data. It is designed to work well for data that is Stationary process, nonstationary and nonlinear. In contrast to other common transforms like the Fourier transform, the HHT is an algorithm that can be applied to a data set, rather than a theoretical tool. The Hilbert–Huang transform (HHT), a NASA designated name, was proposed by Norden E. Huang et al. (1996, 1998, 1999, 2003, 2012). It is the result of the empirical mode decomposition (EMD) and the Hilbert spectral analysis (HSA). The HHT uses the EMD method to decompose a Signal processing, signal into so-called intrinsic mode functions (IMF) with a trend, and applies the HSA method to the IMFs to obtain instantaneous frequency data. Since the signal is decomposed in time domain and the length of the IMFs is the same as the original signal, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequency Spectrum
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum. When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (or simply power spectrum), which applies to signals existing over ''all'' time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The power spectral density (PSD) then refers to the spectral energy distribution that would b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berlin Procedure
The Berlin procedure (BV) is a mathematical procedure for time series decomposition and seasonal adjustment of monthly and quarterly economic time series. The mathematical foundations of the procedure were developed in 1960's at the Technical University of Berlin and the German Institute for Economic Research (DIW). The most important user of the procedure is the Federal Statistical Office of Germany. For the latest version 4.1 of BV a BV4.1 software is available as freeware Freeware is software, most often proprietary, that is distributed at no monetary cost to the end user. There is no agreed-upon set of rights, license, or EULA that defines ''freeware'' unambiguously; every publisher defines its own rules for the ... for non-commercial purposes. Specific features of the procedure The latest version 4.1 of the Berlin procedure is distinguished from other commonly used decomposition and seasonal adjustment methods (i.e. X-12-ARIMA) by the following characteristic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wold Decomposition
In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator. In time series analysis, the theorem implies that any stationary discrete-time stochastic process can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a moving average process. Details Let ''H'' be a Hilbert space, ''L''(''H'') be the bounded operators on ''H'', and ''V'' ∈ ''L''(''H'') be an isometry. The Wold decomposition states that every isometry ''V'' takes the form :V = (\oplus_ S) \oplus U for some index set ''A'', where ''S'' is the unilateral shift on a Hilbert space ''Hα'', and ''U'' is a unitary operator (possible vacuous). The family consists of isomorphic Hilbert spaces. A proo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Series
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 ''fore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wold's Theorem
In statistics, Wold's decomposition or the Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the Wiener–Khinchin theorem), named after Herman Wold, says that every covariance-stationary time series Y_ can be written as the sum of two time series, one ''deterministic'' and one ''stochastic''. Formally :Y_t=\sum_^\infty b_j \varepsilon_+\eta_t, where: :*Y_t is the time series being considered, :*\varepsilon_t is an uncorrelated sequence which is the innovation process to the process Y_t – that is, a white noise process that is input to the linear filter \ . :*b is the ''possibly'' infinite vector of moving average weights (coefficients or parameters) :*\eta_t is a deterministic time series, such as one represented by a sine wave. The moving average coefficients have these properties: # Stable, that is square summable \sum_^, b_, ^2 < # Causal (i.e. there are no terms with ''j'' < 0) # Minim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
