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Topic: Covariance matrix

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Random vector

Bivariate Gaussian distribution

Variance

Covariance

Estimation of covariance matrices

Expected value

Multivariate normal distribution

Preimage

Mahalanobis distance

Arg max

Correlation

Principal components analysis

Probability distribution

Vector space

Normal distribution

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	APT - The APT Approach
		To estimate portfolio risk irrespective of the covariance matrix is folly.
		The relation is embedded in the covariance matrix.
		Their meaning is that they span the covariance matrix best; simply put, they reflect the actual covariance between all stocks, irrespective of what real world variable may have led to such covariance at any one point.
	www.apt.com /en/aboutus/theaptapproach.html (2247 words)

	Positive Definite, Positive Semidefinite Covariance Matrix
		Positive semidefinite matrices are important in probability theory because covariance matrices are always positive semidefinite—so all properties of positive semidefinite matrices are properties of covariance matrices.
		Because a correlation matrix is essentially a normalized covariance matrix, results apply equally to them.
		In this sense, a singular covariance matrix indicates that at least one component of a random vector is extraneous.
	www.riskglossary.com /articles/positive_definite_matrix.htm (996 words)

	Reference.com/Encyclopedia/Covariance matrix
		In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector.
		From it a transformation matrix can be derived that allows one to completely decorrelate the data or, from a different point of view, to find an optimal basis for representing the data in a compact way (see Rayleigh quotient for a formal proof and additional properties of covariance matrices).
		The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle.
	www.reference.com /browse/wiki/Covariance_matrix (688 words)

	NationMaster - Encyclopedia: Covariance
		Category: Covariance and correlation In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector.
		Homology theory is covariant because (as is very clear in singular homology) its basic construction is to take a topological space X and map things into it (in that case, simplices).
		By considering a coordinate transformation on a manifold as a map from the manifold to itself, the transformation of covariant indices of a tensor are given by a pullback, and the transformation properties of the contravariant indices is given by a pushforward.
	www.nationmaster.com /encyclopedia/Covariance (963 words)

	Covariance matrix
		In statistics, the covariance matrix generalizes the concept of variance from one to n dimensions, or in other words from scalar-valued random variables to vector-valued random variables (tuples of scalar random variables).
		Since the covariance between a scalar-valued random variable and itself is its variance, it follows that in particular the entries on the diagonal of this matrix are the variances of the scalar components of X.
		From it a transformation matrix[?] can be derived that allows to completly decorrelate the data or from a different point of view to find an optimal basis for representing the data in a compact way.
	www.ebroadcast.com.au /lookup/encyclopedia/co/Covariance_matrix.html (357 words)

	NationMaster - Encyclopedia: Fisher's linear discriminator
		have different means and shared covariance, and then use the maximum likelihood estimate or the maximum a posteriori estimate of the means and covariance.
		In statistics, the method of maximum likelihood, pioneered by geneticist and statistician Sir Ronald A. Fisher, is a method of point estimation, that uses as an estimate of an unobservable population parameter the member of the parameter space that maximizes the likelihood function.
		These projections can be found by solving a generalized eigenvalue problem, where the numerator is the covariance matrix formed by treating the means as the samples, and the denominator is the shared covariance matrix.
	www.nationmaster.com /encyclopedia/Fisher%27s-linear-discriminator (957 words)

	Not Positive Definite Matrices--Causes and Cures
		In addition, one interpretation of the determinant of a covariance or correlation matrix is as a measure of "generalized variance." Since negative variances are undefined, and since zero variances apply only to constants, it is troubling when a covariance or correlation matrix fails to have a positive determinant.
		In regard to the asymptotic weight matrix, the linear dependency exists not between variables, but between elements of the moments (the means and variances and covariances or the correlations) which are being analyzed.
		After decomposing the correlation matrix into eigenvalues and eigenvectors, Ed Cook replaced the negative eigenvalues with small (.05) positive values, used the new values to compute a covariance matrix, then standardized the resulting matrix (diving by the square root of the diagonal values) so that the result was again was a correlation matrix.
	www.gsu.edu /~mkteer/npdmatri.html (2444 words)

	CLUCalc: CLUCalc - A Visual Calculator
		Since the Clifford algebra of 3d-Euclidean space is 8-dimensional, the covariance matrix of a multivector in this space is an 8x8 matrix.
		and wanted to know what the covariance matrix of the resultant multivector is. Note that the multivectors are regarded as the mean values of a stochastic process and their covariance matrices give the second moments.
		The first contains the resultant multivector with its covariance matrix and the second the list of evaluated cross-covariance matrices, one for each pair of cross-covariance matrices passed to EPOp().
	www.perwass.de /CLU/CLUCalcDoc/page_ErrProp.html (531 words)

	Covariance Matrix
		The COV= option must be specified to compute an approximate covariance matrix for the parameter estimates under asymptotic theory for least-squares, maximum-likelihood, or Bayesian estimation, with or without corrections for degrees of freedom as specified by the VARDEF= option.
		In small samples, estimates of the covariance matrix based on asymptotic theory are often too small and should be used with caution.
		The covariance matrix specified by the COV= option can be displayed (using the PCOV option) and is written to the OUTEST= or OUTVAR= data set.
	www.asu.edu /it/fyi/unix/helpdocs/statistics/sas/sasdoc/sashtml/ormp/chap5/sect37.htm (760 words)

	The Estimated Parameter Covariance Matrix :: Tutorial (System Identification Toolbox)
		The amount of uncertainty is measured and described by the covariance matrix of the estimated parameter vector, (this vector is a random variable, since it depends on the random noise that has affected the output).
		The estimated covariance matrix is contained in the estimated model as the property
		The estimate of the covariance matrix is based on the assumption that the model structure is capable of giving a correct description of the system.
	www.mathworks.com /access/helpdesk_r13/help/toolbox/ident/ch3tut71.html (244 words)

	Citations: A Heteroscedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroscedasticity - White ...
		White, Halbert L., 1980, A heteroscedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity, Econometrica, 48, 817-838.
		Thus the researcher may evaluate which specific problem he or she wishes to evaluate explicitly in the GEE context, then employ robust standard errors to mitigate the effects of heterogeneity due to the remaining source(s) of variability.
		These covariance matrix estimators can directly be used in the GMM objective function but expressions of the resulting limiting distributions are still unknown.
	citeseer.ist.psu.edu /context/102510/0 (2360 words)

	Citations: An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator - Andrews, Monahan ...
		Parzen 1957; Priestley 1982) have contributed to the rapid development of HAC covariance matrix estimation procedures (e.g.
		Following Zivot (1995) and Hansen (1995) the test statistic for the ECM test depends on the asymptotic covariance matrix of the sample means of t h and.
		Furthermore, to provide a comparison for the performance of Andrews estimators, the Bartlett window with four lags is used instead to calculate the variances and covariances, denoted by CCR B. 2.3 Phillips and Loretan, Saikkonen, and Stock and Watson s dynamic ordinary least squares method:....
	citeseer.ist.psu.edu /context/271069/0 (1700 words)

	Math Forum Discussions - Shrinkage with known covariance matrix
		Math Forum Discussions - Shrinkage with known covariance matrix
		has some other covariance matrix, are there simple recipies to coming
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/thread.jspa?forumID=231&threadID=1359835&messageID=4610813 (89 words)

	Singular Spectrum Analysis Theory
		Broomhead and King (1986: BK hereafter) applied the ``method of delays'' of dynamical systems theory to estimate the dimension of and reconstruct the Lorenz attractor using singular-value decomposition (SVD) on the trajectory matrix formed by lagged copies of a single series obtained from the system.
		is estimated directly from the data as a Toeplitz matrix with constant diagonals.
		Step (iii): The covariance matrix calculated from the N sample points, using the BK or VG algorithm, is then diagonalized and the eigenvalues
	www.spectraworks.com /Help/mcssa/ssatheory.html (1392 words)

	S.O.S. Mathematics CyberBoard :: View topic - covariance matrix
		I have the question regarding the calculation of the covariance matrix that is used in multiple linear regression models.
		Suppose that B is the vector of regression coefficients, X is the matrix of the levels of the regressor variables, and σ^2 is the unknown variance of errors.
		Note that in linear regression the X matrix is treated as a constant, so there are no problems.
	www.sosmath.com /CBB/viewtopic.php?t=25783 (434 words)

	OCX DLL ActiveX Control Math Libraries for Regression Least Squares, Kalman Filtering, FFT, Digital Signal Processing, ...
		The Kalman Filter is similar to least squares fitting, but allows for real-time updates to the fit.
		The control allows user entries of the Process Noise (Q) matrix, the Dynamics (Phi) matrix, the Partials (H) matrix, the Measurement (Z) vector, the initial Covariance (P) matrix, and the initial State (X) vector.
		A single call to KalmanFtX propagates the state vector and covariance matrix, adds the Q process noise to the covariance matrix, calculates the gain, updates covariance matrix, and then updates the state vector.
	www.mathfunctions.com /kalmanx.html (312 words)

	Ch NAG Statistics Package
		Computes a correlation matrix from a sum of squares matrix
		Product-moment correlation, unweighted/weighted correlation and covariance matrix, allows variables to be disregarded
		Computes partial correlation/variance-covariance matrix from correlation/variance-covariance matrix computed by g02bxc
	www.softintegration.com /docs/package/chnagstat (1718 words)

	Consistent Covariance Matrix Estimation in Probit Models with Autocorrelated Errors - Federal Reserve Bank of New York
		Some recent time-series applications use probit models to measure the forecasting power of a set of variables.
		Correct inferences about the significance of the variables requires a consistent estimator of the covariance matrix of the estimated model coefficients.
		A potential source of inconsistency in maximum likelihood standard errors is serial correlation in the underlying disturbances, which may arise, for example, from overlapping forecasts.
	www.newyorkfed.org /research/staff_reports/sr39.html (135 words)

	Expectation and Covariance Matrix of
		In Section 5.2 of Chapter 5 we introduced the gambling test for pseudorandom number generators.
		Recall, that the first 52 coin tosses are needed to initialize the memory of the gambling strategy.
		A Mathematica implementation of the covariance matrix, which exploits symmetries in the problem and employs combinatorical formulas to compute the number of occurrences of certain cases, needs about half an hour of CPU time on a DEC 3000 Alpha workstation to compute the covariance matrix
	random.mat.sbg.ac.at /~ste/diss/node38.html (266 words)

	ED231A: Matrix Arithematic 3
		Σ is the covariance matrix, and C is an arbitrary positive constant.
		of the eigenvlaues of a matrix A is equal to the trace of A, tr(A).
		of the eigenvlaues of a matrix A is equal to the determinant of A, Det(A).
	www.gseis.ucla.edu /courses/ed231a1/notes4/mat31.html (857 words)

	COVARIANCE MATRIX
		A matrix containing the expected values derived from the products of the deviations of pairs of random variables from their means.
		Covariance measures the extent to which two random numbers vary together (i.e., varying at the same rate in the same direction).
		From it a transformation matrix can be derived that allows one to completely decorrelate the data or, from a different point of view, to find an optimal basis for representing the data in a compact way.
	www.websters-online-dictionary.org /definition/english/CO/COVARIANCE+MATRIX.html (611 words)

	PODS: Satellite - Covariance Matrix (Site not responding. Last check: )
		A large covariance usually results when the geometry of the observations is insufficient for the estimator to determine how residuals in observations relate to changes in state parameters.
		The units of the matrix elements are the units of the corresponding row and column state vector element multiplied together.
		The default matrix consists of large values along the diagonal and zeros along the off-diagonal, indicating a lack of confidence in the a priori estimate (determined on the Orbit page) and no correlation among elements.
	www.stk.com /resources/help/stk613/helpSystem/extfile/pods/satellite-03.htm (560 words)

	Capabilities: Matrix language (Site not responding. Last check: )
		Thus, the row and column names of the result are the row names of the first matrix and the column names of the second.
		What they may not have realized is that corresponding to each mathematical definition of a matrix operator is a definition of the operator's effect on the names that can be used to carry the names forward through long and complex matrix calculations.
		Standard errors are obtained from the covariance matrix, and significance tests and confidence intervals are automatically constructed.
	www.stata.com /info/capabilities/matrix/matrix.html (806 words)

	KalmanFilter (JMSL)
		matrix relating the observations to the state vector in the observation equation.
		is the variance-covariance matrix of errors in the observation equation.
		is the variance-covariance matrix of the error vector in the state equation.
	www.vni.com /products/imsl/jmsl/v20/api/com/imsl/stat/KalmanFilter.html (879 words)

	How to Use Tracing Rules to Express Variance/Covariance Matrices: Unstandardized Coefficients
		Similarly, the sling between the predictors is the covariance.
		Covariance of Predictor A with Criterion C. The covariance of A with C follows the same rules as before.
		The covariance of variables B and C in the model is a sum of two components:
	web.missouri.edu /~woodph/psych420/rawpath/rawpath.html (1307 words)

	6.5.4.1. Mean Vector and Covariance Matrix
		The mean vector consists of the means of each variable and the variance-covariance matrix consists of the variances of the variables along the main diagonal and the covariances between each pair of variables in the other matrix positions.
		The mean vector is often referred to as the centroid and the variance-covariance matrix as the dispersion or dispersion matrix.
		Also, the terms variance-covariance matrix and covariance matrix are used interchangeably.
	www.itl.nist.gov /div898/handbook/pmc/section5/pmc541.htm (172 words)

	Program on Random Matrix Theory and its Applications to Statistics and Wireless Communications - IMS
		In this case, even for the independent Gaussian samples, the eigenvalues of the sample covariance matrix are not close to 1, but they spread out in an interval that can be determined by the ratio p/n.
		Especially, we derived the explicit form of the limiting spectral distribution when the Hermitian matrix is a Wigner matrix which becomes the 8th product beyond 7 with known explicit forms of densities of their limiting spectral distributions.
		Consider a N x n random matrix Q = Y + A where Y is a random matrix with centered independent elements having a variance profile and A is a deterministic matrix.
	www.ims.nus.edu.sg /Programs/randommatrix/abstracts.htm (1651 words)

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