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Autoregressive Process
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 nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation (or recurrence relation which should not be confused with differential equation). Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random vari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 univariate time series. The moving-average model specifies that the output variable is Cross-correlation, cross-correlated with a non-identical to itself random-variable. Together with the Autoregressive model, autoregressive (AR) model, the moving-average model is a special case and key component of the more general Autoregressive–moving-average model, ARMA and Autoregressive integrated moving average, ARIMA models of time series, which have a more complicated stochastic structure. The moving-average model should not be confused with the moving average, a distinct concept despite some similarities. Contrary to the AR model, the finite MA model is always Stationary process, stationary. Definition The notation MA(''q'') refers to the moving average model of order ''q'': : X_t = \mu + \varepsilon_t + \theta_1 \varepsilon_ + \cdots + \theta_q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Constant
In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a system that can be modeled by a single first order differential equation in time. Examples include the simplest single-stage electrical RC circuits and RL circuits. The time constant is the main characteristic unit of a first-order LTI system. In the time domain, the usual choice to explore the time response is through the step response to a step input, or the impulse response to a Dirac delta function input. In the frequency domain (for example, looking at the Fourier transform of the step response, or using an input that is a simple sinusoidal function of time) the time constant also determines the bandwidth of a first-order time-invariant system, that is, the frequency at which the output signal power drops to half the value it has at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophical Transactions Of The Royal Society
''Philosophical Transactions of the Royal Society'' is a scientific journal published by the Royal Society. In its earliest days, it was a private venture of the Royal Society's secretary. It was established in 1665, making it the first journal in the world exclusively devoted to science, and therefore also the world's longest-running scientific journal. It became an official society publication in 1752. The use of the word ''philosophical'' in the title refers to natural philosophy, which was the equivalent of what would now be generally called ''science''. Current publication In 1887 the journal expanded and divided into two separate publications, one serving the physical sciences ('' Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences'') and the other focusing on the life sciences ('' Philosophical Transactions of the Royal Society B: Biological Sciences''). Both journals now publish themed issues and issues resulting from pap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gilbert Walker (physicist)
Sir Gilbert Thomas Walker (14 June 1868 – 4 November 1958) was an English physicist and statistician of the 20th century. Walker studied mathematics and applied it to a variety of fields including aerodynamics, electromagnetism and the analysis of time-series data before taking up a teaching position at the University of Cambridge. Although he had no experience in meteorology, he was recruited for a post in the Indian Meteorological Department where he worked on statistical approaches to predict the monsoons. He developed the methods in the analysis of time-series data that are now called the Yule-Walker equations. He is known for his groundbreaking description of the Southern Oscillation, a major phenomenon of global climate, and for discovering what is named after him as the Walker circulation, and for greatly advancing the study of climate in general. He was also instrumental in aiding the early career of the Indian mathematical prodigy, Srinivasa Ramanujan. Early ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Udny Yule
George Udny Yule FRS (18 February 1871 – 26 June 1951), usually known as Udny Yule, was a British statistician, particularly known for the Yule distribution. Personal life Yule was born at Beech Hill, a house in Morham near Haddington, Scotland and died in Cambridge, England. He came from an established Scottish family composed of army officers, civil servants, scholars, and administrators. His father, Sir George Udny Yule (1813–1886) was a brother of the noted orientalist Sir Henry Yule (1820–1889). His great uncle was the botanist John Yule. In 1899, Yule married May Winifred Cummings. The marriage was annulled in 1912, producing no children.annulment: Yates, 1952 Education and teaching Udny Yule was educated at Winchester College and at the age of 16 at University College London where he read engineering. After a year in Bonn doing research in experimental physics under Heinrich Rudolf Hertz, Yule returned to University College in 1893 to work as a demonstra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Method Of Moments (statistics)
In statistics, the method of moments is a method of estimation of population parameters. The same principle is used to derive higher moments like skewness and kurtosis. It starts by expressing the population moments (i.e., the expected values of powers of the random variable under consideration) as functions of the parameters of interest. Those expressions are then set equal to the sample moments. The number of such equations is the same as the number of parameters to be estimated. Those equations are then solved for the parameters of interest. The solutions are estimates of those parameters. The method of moments was introduced by Pafnuty Chebyshev in 1887 in the proof of the central limit theorem. The idea of matching empirical moments of a distribution to the population moments dates back at least to Pearson. Method Suppose that the problem is to estimate k unknown parameters \theta_, \theta_2, \dots, \theta_k characterizing the distribution f_W(w; \theta) of the random va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Least Squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values of the variable being observed) in the input dataset and the output of the (linear) function of the independent variable. Geometrically, this is seen as the sum of the squared distances, parallel to the axis of the dependent variable, between each data point in the set and the corresponding point on the regression surface—the smaller the differences, the better the model fits the data. The resulting estimator can be expressed by a simple formula, especially in the case of a simple linear regression, in which there is a single regressor on the right side of the regression equation. The OLS estimator is consiste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Condition
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For a system of order ''k'' (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension ''n'' (that is, with ''n'' different evolving variables, which together can be denoted by an ''n''-dimensional coordinate vector), generally ''nk'' initial conditions are needed in order to trace the system's variables forward through time. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. Examples of a geometric sequence are powers ''r''''k'' of a fixed non-zero number ''r'', such as 2''k'' and 3''k''. The general form of a geometric sequence is :a,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \ldots where ''r'' ≠ 0 is the common ratio and ''a'' ≠ 0 is a scale factor, equal to the sequence's start value. The sum of a geometric progression terms is called a ''geometric series''. Elementary properties The ''n''-th term of a geometric sequence with initial value ''a'' = ''a''1 and common ratio ''r'' is given by :a_n = a\,r^, and in general :a_n = a_m\,r^. Such a geometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Limit Theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory. If X_1, X_2, \dots, X_n, \dots are random samples drawn from a population with overall mean \mu and finite variance and if \bar_n is the sample mean of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distribution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
