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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
Measure of covariance of components of a random vector
a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving
Covariance_matrix
Measure of joint variability in statistics
t } {\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
Cross-covariance
Concept in digital signal processing
uncorrelated if and only if their cross-covariance matrix K X Y {\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }} matrix is zero. In the case of two
Cross-correlation_matrix
Measure of the joint variability
calculating covariance Analysis of covariance Autocovariance Covariance function Covariance matrix Covariance operator Distance covariance, or Brownian
Covariance
}+\operatorname {K} _{\mathbf {X} \mathbf {Y} })\end{aligned}}} The cross-covariance matrix between two complex random vectors Z , W {\displaystyle \mathbf
Complex_random_vector
Type of algorithm
which is a cross-covariance matrix when P and Q are seen as data matrices. It is possible to calculate the optimal rotation R based on the matrix formula
Kabsch_algorithm
Concepts in probability and statistics
variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable
Covariance_and_correlation
Concept in probability and statistics
a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is
Autocovariance
Random variable with multiple component dimensions
refers to the transpose of the indicated vector: By extension, the cross-covariance matrix between two random vectors X {\displaystyle \mathbf {X} } and Y
Multivariate_random_variable
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
Covariance and correlation
Coherence Convolution Correlation Correlation function Cross-correlation matrix Cross-covariance Cross-spectrum Digital image correlation Phase correlation
Cross-correlation
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
Algorithm that estimates unknowns from a series of measurements over time
_{k}} is the covariance matrix of the observation noise, v k {\displaystyle \mathbf {v} _{k}} . Additionally, the cross covariance matrix is also needed
Kalman_filter
Correlation of a signal with a time-shifted copy of itself, as a function of shift
of the autocorrelation matrix are real and non-negative. The auto-covariance matrix is related to the autocorrelation matrix as follows: K X X = E
Autocorrelation
Function in probability theory
multivariate random fields (to refer to the covariance of a variable with itself, as opposed to the cross covariance between two different variables at different
Covariance_function
Method of data analysis
different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between
Principal_component_analysis
Classification algorithm
of random variables with a known covariance matrix into a set of new variables whose covariance is the identity matrix, meaning that they are uncorrelated
Whitening_transformation
Statistical tool
used in statistics and econometrics to provide an estimate of the covariance matrix of the parameters of a regression-type model where the standard assumptions
Newey–West_estimator
Concept in probability theory
\mathbf {W} } are called uncorrelated if their cross-covariance matrix and their pseudo-cross-covariance matrix is zero, i.e. if K Z W = J Z W = 0 {\displaystyle
Uncorrelatedness
Topics referred to by the same term
together, and may refer to: Covariance matrix, a matrix of covariances between a number of variables Covariance or cross-covariance between two random variables
Covariance_(disambiguation)
Estimation method that minimizes the mean square error
C_{XY}} is cross-covariance matrix between x {\displaystyle x} and y {\displaystyle y} , the C Y {\displaystyle C_{Y}} is auto-covariance matrix of y {\displaystyle
Minimum mean square error estimator
Minimum_mean_square_error_estimator
Generalization of the one-dimensional normal distribution to higher dimensions
{\displaystyle 1\leq j\leq k} . The inverse of the covariance matrix is called the precision matrix, denoted by Q = Σ − 1 {\displaystyle {\boldsymbol {Q}}={\boldsymbol
Multivariate normal distribution
Multivariate_normal_distribution
Statistical relationship
used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables X i / σ ( X i )
Correlation
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
Statistical measure of how far values spread from their average
positive semi-definite square matrix, commonly referred to as the variance-covariance matrix (or simply as the covariance matrix). If X {\displaystyle X} is
Variance
Study of high-dimensional data
high-dimensional statistical phenomenon can be found in the problem of covariance matrix estimation. Suppose that we observe X 1 , … , X n ∈ R p {\displaystyle
High-dimensional_statistics
Regularization technique for ill-posed problems
Gaussian) a covariance matrix C M {\displaystyle C_{M}} representing the a priori uncertainties on the model parameters, and a covariance matrix C D {\displaystyle
Ridge_regression
Solution to the spacecraft attitude determination problem
before the advent of Wahba's problem and its several optimal solutions. Covariance analysis for Black's solution was subsequently provided by Markley. Firstly
Triad_method
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
Statistical property
the value of x {\displaystyle x} . More generally, if the variance-covariance matrix of disturbance ε i {\displaystyle \varepsilon _{i}} across i {\displaystyle
Homoscedasticity and heteroscedasticity
Homoscedasticity_and_heteroscedasticity
Statistical estimation technique
It requires knowledge of the covariance matrix for the residuals. If this is unknown, estimating the covariance matrix gives the method of feasible generalized
Generalized_least_squares
Way of inferring information from cross-covariance matrices
canonical variates analysis, is a way of inferring information from cross-covariance matrices. If we have two vectors X = (X1, ..., Xn) and Y = (Y1, ..
Canonical_correlation
Statistical method
determine the inertia (i.e. the sum of the singular values) of the covariance matrix of the sub-groups under consideration. Canonical correlation Data
Partial least squares regression
Partial_least_squares_regression
Matrix-valued random variable
limits of the eigenvalues associated with a random variable covariance matrix. This matrix calculated in this way becomes the null hypothesis that allows
Random_matrix
Generalization of gamma distribution to multiple dimensions
conjugate prior of the inverse covariance-matrix of a multivariate-normal random vector. Suppose G is a p × n matrix, each column of which is independently
Wishart_distribution
Shorthand notation for tensor operations
§ Superscripts and subscripts versus only subscripts below. In terms of covariance and contravariance of vectors, upper indices represent components of contravariant
Einstein_notation
Asymptotic variances under heteroskedasticity
heteroskedasticity). As pointed out by Greene, “simply computing a robust covariance matrix for an otherwise inconsistent estimator does not give it redemption
Heteroskedasticity-consistent standard errors
Heteroskedasticity-consistent_standard_errors
Statistical model to calculate the value of multiple quantities as they change over time
(e_{t}e_{t}')=\Omega \,} . The contemporaneous covariance matrix of error terms is a k × k positive-semidefinite matrix denoted Ω. E ( e t e t − k ′ ) = 0 {\displaystyle
Vector_autoregression
mxMatrix("Diag", nrow = 5, ncol = 5, values = 1.0, free = TRUE, name = "U"), mxAlgebra(A %*% L %*% t(A) + U, name="R"), mxExpectationNormal(covariance=
OpenMx
orthogonal matrix Precision matrix — a symmetric n×n matrix, formed by inverting the covariance matrix. Also called the information matrix. Stochastic matrix —
List_of_named_matrices
Cholesky, SVD, Eigenvalue, ...) Matrix Features (rank, symmetric, definitiveness, ... ) Random Matrices (covariance, orthogonal, symmetric, ... ) Different
Efficient_Java_Matrix_Library
Probability distribution on complex matrices
the distribution of n {\displaystyle n} times the sample Hermitian covariance matrix of n {\displaystyle n} zero-mean independent Gaussian random variables
Complex_Wishart_distribution
Estimation method
form of a 2x2 covariance matrix giving the variance in x {\displaystyle x} , the variance in y {\displaystyle y} , and the cross covariance between the
Unscented_transform
Concept in probability theory and statistics
the covariance matrix Σ {\displaystyle \Sigma } which runs in O ( n 3 ) {\displaystyle {\mathcal {O}}(n^{3})} time (using the sample covariance matrix to
Partial_correlation
Asset pricing models
{\displaystyle F} is the covariance matrix of factor returns, and D {\displaystyle D} is a block diagonal matrix of specific returns. The matrix C {\displaystyle
Multiple_factor_models
Multivariate probability distribution
normal distribution with unknown mean and precision matrix (the inverse of the covariance matrix). Suppose μ | μ 0 , λ , Λ ∼ N ( μ 0 , ( λ Λ ) − 1 ) {\displaystyle
Normal-Wishart_distribution
Use of coordinates for representing vectors
advanced contexts, a row and a column vector have different meaning; see covariance and contravariance of vectors for more. A vector in R 3 {\displaystyle
Vector_notation
Counternull Counting process Covariance Covariance and correlation Covariance intersection Covariance matrix Covariance function Covariate Cover's theorem
List_of_statistics_articles
Statistical method
\mathrm {Cov} } is the covariance matrix, to make sure that the factors are uncorrelated, and I {\displaystyle I} is the identity matrix. Suppose C o v ( X
Factor_analysis
Concept in statistical mathematics
feasible generalized least squares with a specific form of the variance-covariance matrix. Two important cases when SUR is in fact equivalent to OLS are when
Seemingly unrelated regressions
Seemingly_unrelated_regressions
Tool for characterizing the response properties of a neuron
prior with covariance proportional to the identity matrix. The ridge parameter sets the inverse variance of this prior, and is usually fit by cross-validation
Spike-triggered_average
Form of causal modeling that fit networks of constructs to data
function of the discrepancy between the observed covariance matrix and the model-implied covariance matrix. Chi-square increases with sample size only if
Structural_equation_modeling
Method for estimating the unknown parameters in a linear regression model
XT X is a Gram matrix, and its inverse, Q = N−1, is the cofactor matrix of β, closely related to its covariance matrix, Cβ. The matrix (XT X)−1 XT = Q XT
Ordinary_least_squares
traits can be measured and analyzed through a phenotypic variance-covariance matrix (P-matrix) which summarizes the dimensions of phenotypic variability and
Developmental_bias
Adaptive filter algorithm for digital signal processing
sample covariance matrix for x ( n ) {\displaystyle x(n)} , and r d x ( n ) {\displaystyle \mathbf {r} _{dx}(n)} is the equivalent estimate for the cross-covariance
Recursive least squares filter
Recursive_least_squares_filter
Family of distributions that generalize the multivariate normal distribution
{\displaystyle \Sigma } is a positive definite matrix which is proportional to the covariance matrix if the latter exists. Examples include the following
Elliptical_distribution
List of works published by American statistician Charles Roy Henderson
doi:10.2527/jas1978.4641125x. Henderson, C. R. (1978). "Variance-Covariance Matrix of Estimators of Variances in Unweighted Means ANOVA". Biometrics
Charles Roy Henderson bibliography
Charles_Roy_Henderson_bibliography
Procedure for comparing multivariate sample means
follows a multivariate normal distribution, multivariate variance-covariance matrix homogeneity, and linear relationship, no multicollinearity, and each
Multivariate analysis of variance
Multivariate_analysis_of_variance
Statistical test
which is supposed to follow asymptotically a normal distribution with covariance matrix V, n ( θ ^ n − θ ) → D N ( 0 , V ) {\displaystyle {\sqrt {n}}({\hat
Wald_test
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
the cross product can be equivalently written as matrix multiplication by combining the first operand and the operator into a skew-symmetric matrix, [
Moment_of_inertia
known as a coregionalization matrix. Therefore, the kernel derived from LMC is a sum of the products of two covariance functions, one that models the
Kernel methods for vector output
Kernel_methods_for_vector_output
Signal processing algorithm
corresponding eigenvalue (in the discrete finite-length case, the covariance matrix is Toeplitz and is asymptotically diagonalized by the discrete Fourier
Wiener_filter
Test of normality in frequentist statistics
standard normal distribution; finally, V {\displaystyle V} is the covariance matrix of those normal order statistics. There is no name for the distribution
Shapiro–Wilk_test
Method used in statistics, pattern recognition, and other fields
inverse covariance matrix. These projections can be found by solving a generalized eigenvalue problem, where the numerator is the covariance matrix formed
Linear_discriminant_analysis
Algebraic operation on coordinate vectors
Cauchy–Schwarz inequality Cross product Dot product representation of a graph Euclidean norm, the square-root of the self dot product Matrix multiplication Metric
Dot_product
Number of values in the final calculation of a statistic that are free to vary
that of covariance structures (i.e. the mean structure is not being modeled), this is the number of non-redundant elements of the covariance matrix being
Degrees of freedom (statistics)
Degrees_of_freedom_(statistics)
Form of radar used to create images of landscapes
the estimate of the covariance matrix, the forward-only Capon uses only the forward data vectors to estimate the covariance matrix. Capon can yield more
Synthetic-aperture_radar
Statistical test
test Park test White, H. (1980). "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity". Econometrica
White_test
Statistical amount
{\displaystyle \sigma ^{2}} by the covariance matrix C {\displaystyle \mathbf {C} } and the arithmetic inverse by the matrix inverse (both denoted in the same
Weighted_arithmetic_mean
Second order tensor in vector algebra
matrices. Also, the dot, cross, and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents. The dot
Dyadics
Area of research in signal processing
decomposition of a covariance matrix to carry out the analysis. A breakthrough came about when the eigen-structure of the covariance matrix was explicitly
Array_processing
Measure of variation in statistics
deviation to multiple dimensions. It is the symmetric square root of the covariance matrix Σ {\displaystyle \mathbf {\Sigma } } . S {\displaystyle \mathbf {S}
Standard_deviation
Class of stochastic process
{\displaystyle \{z_{t}\}} is a white noise in the weak sense (the mean and cross-covariances are zero, and the variances are all the same), however it is not strictly
Stationary_process
Nonparametric spectral estimation method
the spectrum of eigenvalues in a singular value decomposition of a covariance matrix, and not directly to a frequency domain decomposition. The origins
Singular_spectrum_analysis
and their cross-correlations, <ΔRi · ΔRj> can be organized as the diagonal and off-diagonal terms, respectively, of a covariance matrix. Based on statistical
Gaussian_network_model
{\displaystyle \mathbf {R} } is the received signal covariance matrix and p {\displaystyle \mathbf {p} } is the cross-correlation vector between the tap-input vector
Adaptive_equalizer
Scientific subjects
to the probability of finding a particle at a given point in space. The matrix mechanics of Werner Heisenberg (1925) makes no mention of wave functions
Branches_of_physics
Correlation as a function of distance
Refutation of a logical fallacy Correlogram – Chart of correlation statistics Covariance function – Function in probability theory Pearson product-moment correlation
Correlation_function
Parts of a whole which carry only relative information
variance-covariance structure, reducing its true rank to D − 1 {\displaystyle D-1} . As a direct consequence, the resulting covariance matrix Σ {\displaystyle
Compositional_data
Eighteenth letter of the Greek alphabet
sigma-function. In probability theory and statistics, Σ denotes the covariance matrix of a set of random variables, sometimes in the form | Σ {\displaystyle
Sigma
Probability distribution
approaching zero. These logarithmic variances and covariance are the elements of the Fisher information matrix for the beta distribution. They are also a measure
Beta_distribution
Cluster analysis problem
X, consisting of a mixture distribution of G components with common covariance, Γ. If we let c 1 … c K {\displaystyle c_{1}\ldots c_{K}} be a set of
Determining the number of clusters in a data set
Determining_the_number_of_clusters_in_a_data_set
Measure of spatial autocorrelation
{\displaystyle x} ; w i j {\displaystyle w_{ij}} are the elements of a matrix of spatial weights with zeroes on the diagonal (i.e., w i i = 0 {\displaystyle
Moran's_I
Indicator for how well data points fit a line or curve
} where the covariance between two coefficient estimates, as well as their standard deviations, are obtained from the covariance matrix of the coefficient
Coefficient_of_determination
Weighted average/moment of some pixel intensities
derived by first using the second order central moments to construct a covariance matrix. μ 20 ′ = μ 20 / μ 00 = M 20 / M 00 − x ¯ 2 μ 02 ′ = μ 02 / μ 00 =
Image_moment
Process of reducing the number of random variables under consideration
In practice, the covariance (and sometimes the correlation) matrix of the data is constructed and the eigenvectors on this matrix are computed. The eigenvectors
Dimensionality_reduction
Type of statistical model
the tth row of matrix U is denoted by u(t), then the sequence of vectors {u(t)} should be iid, with zero mean and some covariance matrix Σ (which is unknown)
Simultaneous_equations_model
Statistical method in psychology
number of factors (1) by inspecting patterns of eigenvalues of the covariance matrix, or (2) treating it as a model selection problem. Existing approaches
Exploratory_factor_analysis
Algebraic object with geometric applications
of a vector can respond in two distinct ways to a change of basis (see Covariance and contravariance of vectors), where the new basis vectors e ^ i {\displaystyle
Tensor
Fundamental theorem in probability theory and statistics
{\textstyle {\boldsymbol {\mu }}=\operatorname {E} [\mathbf {X} _{i}]} and covariance matrix Σ {\textstyle \mathbf {\Sigma } } (among the components of the vector)
Central_limit_theorem
Data analysis technique
cross-tabulations between the categorical variables, and has an analogy to the covariance matrix of continuous variables. Analyzing the Burt table is a more natural
Multiple correspondence analysis
Multiple_correspondence_analysis
Algebra associated to any vector space
coefficient in this last expression is precisely the determinant of the matrix [v w]. The fact that this may be positive or negative has the intuitive
Exterior_algebra
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
Proportion of variance that two traits share due to genetic causes
genetic covariance matrix, the genetic correlation is computed by standardizing this, i.e., by converting the covariance matrix to a correlation matrix. Generally
Genetic_correlation
Geometric object that has length and direction
of 1 K/m becomes 0.001 K/mm—a covariant change in value (for more, see covariance and contravariance of vectors). Tensors are another type of quantity that
Euclidean_vector
Generalization of the binomial distribution
= n p i . {\displaystyle \operatorname {E} (X_{i})=np_{i}.\,} The covariance matrix is as follows. Each diagonal entry is the variance of a binomially
Multinomial_distribution
Sampling methodology in statistics
V_{c}(\beta )} stands for the covariance matrix adjusted for clustering, V ( β ) {\displaystyle V(\beta )} stands for the covariance matrix not adjusted for clustering
Cluster_sampling
Concept in statistics mathematics
{x} }} , where H plays the role of the covariance matrix. On the other hand, the choice of the bandwidth matrix H is the single most important factor affecting
Multivariate kernel density estimation
Multivariate_kernel_density_estimation
New Zealand mathematician (1928–2013)
1017/S002185960003032X. S2CID 85188693. Searle, S. R. (1956). "Matrix Methods in Components of Variance and Covariance Analysis". The Annals of Mathematical Statistics
Shayle_R._Searle
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CROSS COVARIANCE-MATRIX
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CROSS COVARIANCE-MATRIX
CROSS COVARIANCE-MATRIX
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CROSS COVARIANCE-MATRIX
CROSS COVARIANCE-MATRIX
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CROSS COVARIANCE-MATRIX
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