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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 the joint variability
calculating covariance Analysis of covariance Autocovariance Covariance function Covariance matrix Covariance operator Distance covariance, or Brownian
Covariance
Statistical model
defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral
Gaussian_process
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
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
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
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
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
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
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
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
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
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
imply causation Covariance function Pearson product-moment correlation coefficient Correlation function (astronomy) Correlation function (statistical mechanics)
Cross-correlation_matrix
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
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
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
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
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
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
{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
Collection of random variables
cloud computing infrastructures. List of stochastic processes topics Covariance function Deterministic system Dynamics of Markovian particles Entropy rate
Stochastic_process
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
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
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
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
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
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
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
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
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
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
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
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
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
Counternull Counting process Covariance Covariance and correlation Covariance intersection Covariance matrix Covariance function Covariate Cover's theorem
List_of_statistics_articles
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
In mathematics, a quantitative measure of the shape of a set of points
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
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)
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
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
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
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
Matrix of second derivatives
Notably regarding Randomized Search Heuristics, the evolution strategy's covariance matrix adapts to the inverse of the Hessian matrix, up to a scalar factor
Hessian_matrix
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
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
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
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
Brownian surface represents a Gaussian process with a nonstationary covariance function, one can use the Cholesky decomposition method. A more efficient
Brownian_surface
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
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)
by S {\displaystyle {\mathcal {S}}} with mean function μ {\displaystyle \mu } and covariance function K {\displaystyle K} . Assume that S = { s 1 , …
Vecchia_approximation
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
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
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
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
Form of causal modeling that fit networks of constructs to data
other fit measures. It is a function of the discrepancy between the observed covariance matrix and the model-implied covariance matrix. Chi-square increases
Structural_equation_modeling
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
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
Set of statistical processes for estimating the relationships among variables
e_{i}} are uncorrelated with one another. Mathematically, the variance–covariance matrix of the errors is diagonal. A handful of conditions are sufficient
Regression_analysis
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
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
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
Statistical property
a scedastic function), so the variance is proportional to the value of x {\displaystyle x} . More generally, if the variance-covariance matrix of disturbance
Homoscedasticity and heteroscedasticity
Homoscedasticity_and_heteroscedasticity
Statistical relationship
correlation Correlation disattenuation Correlation function Correlation gap Covariance Covariance and correlation Cross-correlation Ecological correlation
Correlation
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
Probability distribution
quadratic forms in a normal system, with applications to the analysis of covariance". Mathematical Proceedings of the Cambridge Philosophical Society. 30
Student's_t-distribution
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
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
{\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
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
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
Special mathematical functions defined on the surface of a sphere
then Sff(ℓ) and Sfg(ℓ) represent the contributions to the function's variance and covariance for degree ℓ, respectively. It is common that the (cross-)power
Spherical_harmonics
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
Method for model fitting in statistics
of generalized least squares, when all the off-diagonal entries of the covariance matrix of the errors are null. The fit of a model to a data point is measured
Weighted_least_squares
Notion in statistics
as the covariance of the fitted Gaussian. Statistical systems of a scientific nature (physical, biological, etc.) whose likelihood functions obey shift
Fisher_information
Method in computer modeling
distributions of covariance functions of the errors of the "first guess" field (previous forecast) and "true field". These functions are never known.
Data_assimilation
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
COVARIANCE FUNCTION
COVARIANCE FUNCTION
Male
Egyptian
, the son of the functionary Heknofre.
Male
Egyptian
, an Egyptian functionary.
Male
Celtic
, great justiciary, or functionary.
Boy/Male
Shakespearean
The Tragedy of Romeo And Juliet' Juliet's Father, head of the Capulet house, at variance with the...
Male
Egyptian
, a great functionary.
Biblical
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Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Male
Egyptian
, a high Egyptian functionary.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, Functionary of the Interior.
COVARIANCE FUNCTION
COVARIANCE FUNCTION
Boy/Male
Biblical
The cloak, glory, grandeur or power of the king.
Boy/Male
Sikh
Ceaselessly remembers the Lord
Girl/Female
Indian, Sikh
Precious to the World
Girl/Female
Hindu
The heavens
Boy/Male
Hindu
Lord/protector
Girl/Female
Hindu, Indian
Jewel Among the Leader
Boy/Male
German Teutonic
Resolute spirit.
Female
Romanian
Romanian form of Greek Elisabet, ELISABETA means "God is my oath."
Boy/Male
Hindu, Indian, Telugu
Strong; Love
Boy/Male
Tamil
The intelligent one, Name of Brihaspati, Planet jupiter, Spiritual preceptor, Epithet of Narayan
COVARIANCE FUNCTION
COVARIANCE FUNCTION
COVARIANCE FUNCTION
COVARIANCE FUNCTION
COVARIANCE FUNCTION
n.
One who attempts to unite principles or parties which are irreconcilably at variance;
n.
The property of remaining invariable under prescribed or implied conditions.
n.
Quarrel; strife; cause of variance; difference.
n.
The state or quality of being discrepant; disagreement; variance; discordance; dissimilarity; contrariety.
a.
Discordant; at variance; disagreeing; contrary; different.
v. t.
To cause to be different or unlike; to set at variance.
v. t. & i.
To unite again; to join after separation or variance.
n.
A disagreement or difference between two parts of the same legal proceeding, which, to be effectual, ought to agree, -- as between the writ and the declaration, or between the allegation and the proof.
n.
Disunion; difference in opinion or feeling; discord; variance; alienation.
n.
The quality or state of being variant; change of condition; variation.
n.
A function involving the coefficients and the variables of a quantic, and such that when the quantic is lineally transformed the same function of the new variables and coefficients shall be equal to the old function multiplied by a factor. An invariant is a like function involving only the coefficients of the quantic.
n.
The state of disagreeing; a being at variance; dissimilitude; diversity.
n.
A withholding of intimacy; alienation; coldness; disagreement; variance; restraint; reserve.
n.
Recovery.
n.
Disagreeing; incongruous; being at variance; clashing; opposing; not harmonious.
v.
Reconciliation; agreement after variance; harmony; concord.
n.
Difference that produce dispute or controversy; disagreement; dissension; discord; dispute; quarrel.
n.
Attempted union of principles or parties irreconcilably at variance with each other.
v. t.
To renew; to reestablish; as, to restore harmony among those who are variance.
n.
One who makes peace by reconciling parties that are at variance.