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Function in probability theory
theory and statistics, the covariance function describes how much two random variables change together (their covariance) with varying spatial or temporal
Covariance_function
Tool in multivariate statistical analysis
In statistics, the Matérn covariance, also called the Matérn kernel, is a covariance function used in spatial statistics, geostatistics, machine learning
Matérn_covariance_function
Measure of covariance of components of a random vector
and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square
Covariance_matrix
Measure of the joint variability
calculating covariance Analysis of covariance Autocovariance Covariance function Covariance matrix Covariance operator Distance covariance, or Brownian
Covariance
Statistical model
we might choose a rougher covariance function. Extreme examples of the behaviour is the Ornstein–Uhlenbeck covariance function and the squared exponential
Gaussian_process
Correlation of a signal with a time-shifted copy of itself, as a function of shift
t} . Subtracting the mean before multiplication yields the auto-covariance function between times t 1 {\displaystyle t_{1}} and t 2 {\displaystyle t_{2}}
Autocorrelation
Concept in probability and statistics
statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance
Autocovariance
Concepts in probability and statistics
In probability theory and statistics, the mathematical concepts of covariance and correlation are very similar. Both describe the degree to which two random
Covariance_and_correlation
Measure of joint variability in statistics
{\displaystyle \left\{Y_{t}\right\}} , the cross-covariance is a function that gives the covariance of one process with the other at pairs of time points
Cross-covariance
Statistics concept
statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals
Estimation of covariance matrices
Estimation_of_covariance_matrices
In statistics, the rational quadratic covariance function is used in spatial statistics, geostatistics, machine learning, image analysis, and other fields
Rational quadratic covariance function
Rational_quadratic_covariance_function
Type of matrix in probability theory and statistics
probability theory and statistics, a cross-covariance matrix is a matrix whose element in the i, j position is the covariance between the i-th element of a random
Cross-covariance_matrix
Operator in probability theory
linear functional x on the element z. Quite similarly, the covariance function of a function-valued random element (in special cases is called random process
Covariance_operator
Theory of stochastic processes
process. In fact, the orthogonal basis functions used in this representation are determined by the covariance function of the process. One can think that
Kosambi–Karhunen–Loève theorem
Kosambi–Karhunen–Loève_theorem
Covariance and correlation
jointly wide-sense stationary. Then the cross-covariance function and the cross-correlation function are a function only of time lag τ = t 2 − t 1 {\displaystyle
Cross-correlation
Method of interpolation
function will be normally distributed, where the covariance between any two samples is the covariance function (or kernel) of the Gaussian process evaluated
Kriging
Concept in digital signal processing
imply causation Covariance function Pearson product-moment correlation coefficient Correlation function (astronomy) Correlation function (statistical mechanics)
Cross-correlation_matrix
Type of mathematical function
Basis Functions were developed thereafter. Some methods are the RBF-FD method, the RBF-QR method and the RBF-PUM method. Matérn covariance function Radial
Radial_basis_function
Swedish statistician and mathematician (1917–2007)
May 1917 – 6 November 2007) was a Swedish statistician. The Matérn covariance function is named after him. Bertil Matérn was born on 18 May 1917 in Gothenburg
Bertil_Matérn
Probability theory concept
{\displaystyle t} in [ 0 , T ] {\textstyle [0,T]} , and has the following covariance function: E [ B H ( t ) , B H ( s ) ] = 1 2 ( | t | 2 H + | s | 2 H − | t
Fractional_Brownian_motion
Method in signal processing and statistics
of the cross-covariance function. This means it takes the relationship between the two signals over time and represents it as a function of frequency
Cross-spectrum
Evolutionary algorithm
Covariance matrix adaptation evolution strategy (CMA-ES) is a particular kind of strategy for numerical optimization. Evolution strategies (ES) are stochastic
CMA-ES
Mathematical function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f ( x ) = exp ( − x 2 ) {\displaystyle f(x)=\exp(-x^{2})}
Gaussian_function
Collection of random variables
cloud computing infrastructures. List of stochastic processes topics Covariance function Deterministic system Dynamics of Markovian particles Entropy rate
Stochastic_process
Part of signal processing in time-frequency analysis
time series x [ t ] {\displaystyle x[t]} , its non-stationary auto-covariance function is given by C x ( t 1 , t 2 ) = ⟨ ( x [ t 1 ] − μ [ t 1 ] ) ( x [
Wigner_distribution_function
Subset of artificial intelligence
multivariate normal distribution, and it relies on a pre-defined covariance function, or kernel, that models how pairs of points relate to each other
Machine_learning
Correlation as a function of distance
a function of shift Correlation does not imply causation – Refutation of a logical fallacy Correlogram – Chart of correlation statistics Covariance function –
Correlation_function
fundamental component of Gaussian processes, where the kernel function operates as a covariance function that defines relationships between inputs. Traditionally
Bayesian interpretation of kernel regularization
Bayesian_interpretation_of_kernel_regularization
Statistical measure
For an implementation see dcov.test function in the energy package for R. The population value of distance covariance can be defined along the same lines
Distance_correlation
Formula in probability theory
In probability theory, the law of total covariance, covariance decomposition formula, or conditional covariance formula states that if X, Y, and Z are
Law_of_total_covariance
Concept in probability theory and statistics
pseudo-covariance (also called complementary variance): The second order statistics are fully characterized by the covariance and the pseudo-covariance. Properties
Complex_random_variable
{X}}_{d}} which has mean function μ : X → Y {\displaystyle \mu :{\mathcal {X}}\rightarrow {\mathcal {Y}}} and covariance function K : X × X → R {\displaystyle
Gaussian process approximations
Gaussian_process_approximations
covariance function. Regarding the analysis of loads, it involves time series, in which case the covariance function becomes the autocovariance function. In
Rotational sampling in wind turbines
Rotational_sampling_in_wind_turbines
Type of stochastic process
stationary if they are both wide-sense stationary and their cross-covariance function K X Y ( t 1 , t 2 ) = E [ ( X t 1 − m X ( t 1 ) ) ( Y t 2 − m Y
Stationary_process
Method of data analysis
calculation of the covariance matrix is avoided, just as in the matrix-free implementation of the power iterations to XTX, based on the function evaluating the
Principal_component_analysis
Algorithm that estimates unknowns from a series of measurements over time
sigma points are then propagated through the nonlinear functions, from which a new mean and covariance estimate are formed. The resulting filter depends on
Kalman_filter
Class of algorithms for pattern analysis
still be referred to as a "kernel". If the kernel function k {\displaystyle k} is also a covariance function as used in Gaussian processes, then the Gram matrix
Kernel_method
Generalization of a positive-definite matrix
high-performance computing environments. Covariance function Integral equation Integral transform Positive-definite function on a group Reproducing kernel Hilbert
Positive-definite_kernel
Type of probability distribution
measure of the relationship between two random variables is the covariance. Covariance is a measure of linear relationship between the random variables
Joint probability distribution
Joint_probability_distribution
Spatial prediction technique
deterministic part of variation is estimated using OLS, then the covariance function of the residuals is used to obtain the GLS coefficients. Next, these
Regression-kriging
Techniques and methods in signal processing
of x(t). The value of x(t) is expressed as a probability function. Auto-covariance function (ACF) R x ( t , τ ) {\displaystyle R_{x}(t,\tau )} R x ( t
Time–frequency_analysis
Branch of statistics mathematics
{\displaystyle \Sigma } are continuous functions and then the covariance function Σ {\displaystyle \Sigma } defines a covariance operator C : H → H {\displaystyle
Functional_data_analysis
Machine learning technique
classification. It is actually equivalent to a Gaussian process model with covariance function: k ( x , x ′ ) = ∑ j = 1 N 1 α j φ ( x , x j ) φ ( x ′ , x j ) {\displaystyle
Relevance_vector_machine
Programming language concept
to write this kind of polymorphic function without relying on covariance. The array comparison and shuffling functions can be given the parameterized types
Type_variance
Statistical method for investigating the dominant modes of variation of functional data
local linear smoothing or spline smoothing. Then the estimate of the covariance function G ^ ( s , t ) {\displaystyle {\hat {G}}(s,t)} is obtained by averaging
Functional principal component analysis
Functional_principal_component_analysis
Mathematical function characterizing set membership
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all
Indicator_function
Counternull Counting process Covariance Covariance and correlation Covariance intersection Covariance matrix Covariance function Covariate Cover's theorem
List_of_statistics_articles
classes. In Gaussian processes, kernels are called covariance functions. Multiple-output functions correspond to considering multiple processes. See Bayesian
Kernel methods for vector output
Kernel_methods_for_vector_output
Method used in statistics, pattern recognition, and other fields
that the class covariances are identical, so Σ 0 = Σ 1 = Σ {\displaystyle \Sigma _{0}=\Sigma _{1}=\Sigma } ) and that the covariances have full rank.
Linear_discriminant_analysis
Measure of linear correlation
the covariance of two variables and the product of their standard deviations; thus, it is essentially a normalized measurement of the covariance, such
Pearson correlation coefficient
Pearson_correlation_coefficient
Spatial statistics function
define the weights of the kriging function. Note that the experimental variogram is an empirical estimate of the covariance of a Gaussian process. As such
Variogram
Experiment used to study computer simulation
mean function and C {\displaystyle C} is the covariance function. Popular mean functions are low order polynomials and a popular covariance function is
Computer_experiment
Classification algorithm
transforms a vector of random variables with a known covariance matrix into a set of new variables whose covariance is the identity matrix, meaning that they are
Whitening_transformation
Stochastic process modeling random walk with friction
constant. Moreover, the Itō isometry can be used to calculate the covariance function by cov ( x s , x t ) = E [ ( x s − E [ x s ] ) ( x t − E
Ornstein–Uhlenbeck_process
Statistical measure of how far values spread from their average
also used in sample covariance and the sample standard deviation (the square root of variance). The square root is a concave function and thus introduces
Variance
Mathematical relation assigning a probability event to a cost
optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one
Loss_function
denote the mean function by μ ( t ) = E ( X ( t ) ) {\displaystyle \mu (t)=\operatorname {E} (X(t))} , and the covariance function by G ( s , t ) =
Modes_of_variation
Method in statistics
Matérn covariance function of smoothness 3 / 2 {\displaystyle 3/2} and correlation length ρ = 1 / 5 {\displaystyle \rho =1/5} . This covariance function is
Bayesian_quadrature
Mathematical concept
realization problem where we have knowledge only of the Auto-correlation (covariance) function of the output of an LTI system driven by white noise, was derived
Subspace identification method
Subspace_identification_method
Statistical method
distribution. A GP is defined by a mean function and a covariance function, which specify the mean vectors and covariance matrices for each finite collection
Bootstrapping_(statistics)
Brownian surface represents a Gaussian process with a nonstationary covariance function, one can use the Cholesky decomposition method. A more efficient
Brownian_surface
Bimodal function
upon the distance between them (via function f), then function f must be positive-definite to ensure the covariance matrix A is positive-definite. See
Positive-definite_function
Topics referred to by the same term
Autocovariance, the covariance of a signal with a time-shifted version of itself Covariance function, a function giving the covariance of a random field
Covariance_(disambiguation)
Effect of variables' uncertainties on the uncertainty of a function based on them
^{\top }.} That is, the Jacobian of the function is used to transform the rows and columns of the variance-covariance matrix of the argument. Note this is
Propagation_of_uncertainty
Generalization of the one-dimensional normal distribution to higher dimensions
such that the covariance matrix for this subset is positive definite; then the other coordinates may be thought of as an affine function of these selected
Multivariate normal distribution
Multivariate_normal_distribution
Prediction model used in Engineering
prior covariance matrix P {\displaystyle P} is generated from a covariance function. One example of a covariance function is the Gaussian covariance: P i
Gradient-enhanced_kriging
by S {\displaystyle {\mathcal {S}}} with mean function μ {\displaystyle \mu } and covariance function K {\displaystyle K} . Assume that S = { s 1 , …
Vecchia_approximation
Measure of the shape of a function
Moments of a function in mathematics are certain quantitative measures related to the shape of the function's graph. For example, if the function represents
Moment_(mathematics)
independence Pairwise independence Covariance Covariance matrix De Finetti's theorem Correlation Uncorrelated Correlation function Canonical correlation Convergence
List_of_probability_topics
Function related to statistics and probability theory
A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability
Likelihood_function
Representation of a type of random process
and the covariance function of the process, and this correspondence can be inverted to determine the parameters from the autocorrelation function (which
Autoregressive_model
Vector behavior under coordinate changes
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Filter for nonlinear state estimation
mean multivariate Gaussian noises with covariance Qk and Rk respectively. uk is the control vector. The function f can be used to compute the predicted
Extended_Kalman_filter
General linear model that blends ANOVA and regression
Analysis of covariance (ANCOVA) is a general linear model that blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable
Analysis_of_covariance
Probability distribution
possess a 2k-dimensional multivariate normal distribution. The variance-covariance structure of X is described by two matrices: the variance matrix Γ, and
Normal_distribution
Mapping between categories
vectors in general are covariant since they can be pushed forward. See also covariance and contravariance of vectors. Every functor F : C → D {\displaystyle
Functor
Estimation method
mean and covariance equal to the given mean and covariance. This distribution can be propagated exactly by applying the nonlinear function to each point
Unscented_transform
{\displaystyle t=(t_{1},\dots t_{n})\in \mathbb {R} _{+}^{n}} for the covariance function cov ( B s ( i ) , B t ( j ) ) = { ∏ l = 1 n min ( s l , t l )
Brownian_sheet
Function valued in a vector space; typically a real or complex one
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional
Vector-valued_function
Measure of variation in statistics
variables can be related to their individual standard deviations and the covariance between them: σ ( X + Y ) = var ( X ) + var ( Y ) + 2 cov ( X ,
Standard_deviation
Statistical relationship
correlation Correlation disattenuation Correlation function Correlation gap Covariance Covariance and correlation Cross-correlation Ecological correlation
Correlation
Random variable with multiple component dimensions
the respective random variables. The covariance matrix (also called second central moment or variance-covariance matrix) of an n × 1 {\displaystyle n\times
Multivariate_random_variable
Statistical distribution of complex random variables
The complex normal family has three parameters: location parameter μ, covariance matrix Γ {\displaystyle \Gamma } , and the relation matrix C {\displaystyle
Complex_normal_distribution
(1:R) Correlation / (2:R) Correlation function / (U:R) Covariance / (2F:R) (1:G) Covariance function / (U:R) Covariance matrix / (F:R) Cumulant / (12F:DCR)
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Important algorithms in numerical statistics
mean1 # covariance = sum((i1 - mean1) * (i2 - mean2) for i1, i2 in zip(data1, data2)) b = i2 - mean2 covariance += a * b / n return covariance A stable
Algorithms for calculating variance
Algorithms_for_calculating_variance
Matrix of second derivatives
partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix
Hessian_matrix
Integral of the Gaussian function, equal to sqrt(π)
also known as the Euler–Poisson integral, is the integral of the Gaussian function f ( x ) = e − x 2 {\displaystyle f(x)=e^{-x^{2}}} over the entire real
Gaussian_integral
element is the covariance between the i th and the j th random variables. Unlike in the case of real random variables, the covariance between two random
Complex_random_vector
Branch of nonparametric statistics
the mean density function as μ ( t ) = E [ f ( t ) ] {\displaystyle \mu (t)=\mathbb {E} \left[f(t)\right]} and the covariance function as G ( s , t ) =
Distributional_data_analysis
Branch of statistics focusing on spatial data sets
automata Multiple-Point Geostatistics Regionalized variable theory Covariance function Semi-variance Variogram Kriging Range (geostatistics) Sill (geostatistics)
Geostatistics
Concept in statistics
random variables, the covariance between the random variables is equal to the variance of the mean of the underlying distribution function. For finite exchangeable
Exchangeable_random_variables
Numerical measure of a statistical relationship between variables
the linear relationship between two variables that is defined as the covariance of the variables divided by the product of their standard deviations.
Correlation_coefficient
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
delta (named after Leopold Kronecker) is a function of two variables, usually non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
Kronecker_delta
Statistical estimator
the precision matrix (also called the concentration matrix or inverse covariance matrix) of a multivariate elliptical distribution. Through the use of
Graphical_lasso
Extension to cover cases with multiple dependent variables
Multivariate analysis of covariance (MANCOVA) is an extension of analysis of covariance (ANCOVA) methods to cover cases where there is more than one dependent
Multivariate analysis of covariance
Multivariate_analysis_of_covariance
Recursive filter
Kalman filter for large problems (essentially, the covariance matrix is replaced by the sample covariance), and it is now an important data assimilation component
Ensemble_Kalman_filter
Type of signal in signal processing
they be statistically uncorrelated; that is, their covariance is zero. Therefore, the covariance matrix R of the components of a white noise vector w
White_noise
Distribution function associated with the empirical measure of a sample
an empirical distribution function (a.k.a. an empirical cumulative distribution function, eCDF) is the distribution function associated with the empirical
Empirical distribution function
Empirical_distribution_function
Method of frequency estimation
Processing", 2000. Pisarenko, V. F. The retrieval of harmonics from a covariance function Geophysics, J. Roy. Astron. Soc., vol. 33, pp. 347-366, 1973.
Pisarenko harmonic decomposition
Pisarenko_harmonic_decomposition
Probability distribution
{\displaystyle \Sigma } is a p × p {\displaystyle p\times p} positive-semidefinite covariance matrix with strictly positive diagonal entries, then for independent and
Cauchy_distribution
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