A more numerically stable two-pass algorithm first computes the sample means, and then the covariance:

This is necessary so that mathematical statements can be made about the sample and statistics computed from it, such as the sample mean and covariance.

and the ML estimator of the covariance matrix from a sample of n observations is

which is simply the sample covariance matrix.

An unbiased sample covariance is

# REDIRECT Sample mean and sample covariance

This method emphasizes the effect of individual column / row pairs on the result, which is a useful point of view with e. g. covariance matrices, where each such pair corresponds to the effect of a single sample point.

The Cholesky decomposition is commonly used in the Monte Carlo method for simulating systems with multiple correlated variables: The correlation matrix is decomposed, to give the lower-triangular L. Applying this to a vector of uncorrelated samples, u, produces a sample vector Lu with the covariance properties of the system being modeled.

If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables X < sub > i </ sub > / σ ( X < sub > i </ sub >) for i = 1, ..., n. This applies to both the matrix of population correlations ( in which case " σ " is the population standard deviation ), and to the matrix of sample correlations ( in which case " σ " denotes the sample standard deviation ).

A distinction must be made between ( 1 ) the covariance of two random variables, which is a population parameter that can be seen as a property of the joint probability distribution, and ( 2 ) the sample covariance, which serves as an estimated value of the parameter.

Given any set of points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian.

The Wishart distribution arises as the distribution of the sample covariance matrix for a sample from a multivariate normal distribution.

* Sample mean and sample covariance

* The discrepancy function is a continuous function of the elements of S, the sample covariance matrix, and Σ ( θ ), the " reproduced " estimate of S obtained by using the parameter estimates and the structural model.

( 9 ) is to make the noise term as much as possible, or the signal model as close to the sample covariance matrix as much as possible.

Subspace ML beamforming is obtained by eigen-decomposition of the sample covariance matrix.

to be the sample covariance.

Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution.

Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix.

The sample data matrix must have full rank or OLS cannot be estimated.

Define a data matrix, X < sup > T </ sup >, with zero empirical mean ( the empirical ( sample ) mean of the distribution has been subtracted from the data set ), where each of the n rows represents a different repetition of the experiment, and each of the m columns gives a particular kind of datum ( say, the results from a particular probe ).

Postulated migration of proteins in a Laemmli gel system A: Stacking gel, B: Resolving gel, o: sample application c: discontinuities in the buffer and electrophoretic matrix

The more extensive the conversational sample, the more accurate and reliable is the " translation matrix ," enabling instantaneous conversion of verbal utterances or written text between the alien language and American English / Federation Standard.

Data of this type is very common in chemistry, for example a liquid-chromatography / mass spectrometry ( LC-MS ) system generates a large matrix of data ( elution time versus m / z ) for each sample analyzed.

If we take the matrix V above to be, not a variance of a random vector, but a sample variance, then it may have a Wishart distribution.

In such applications typically the statistics of matrices is known in advance and one can take as approximate eigenvalue the average eigenvalue for some large matrix sample,

* S / SL ( Split / Splitless ) injector ; a sample is introduced into a heated small chamber via a syringe through a septum – the heat facilitates volatilization of the sample and sample matrix.

The general theory of resonance shifts is used to derive a general expression for the second moment Af of a polycrystalline paramagnetic sample and is specialized to Af.

The first is a wide-ranging sample of successful tonal analyses.

The differentiation, however, is not very much greater, as shown by the fact that Athabascan shows 3.46 stems per meaning slot as against 2.75 for Yokuts, with a slightly greater number of languages represented in our sample: 24 as against 21.

Upon completion of the purging step, additional initiator appropriate for the molecular weight of the sample desired is added, and the system is then warmed to the polymerization temperature, at which the reaction is allowed to go to completion.

Therefore, it is recommended that in such cases the sample be replaced, or if used, the results of dimensional change or dimensional restorability tests be considered as indicative only.

stereo BGS 5031 ) is a good sample of the special, elegant art of English madrigal singing.

The collection is often the sample space of an experiment.

If the list is a statistical sample, we call the resulting statistic a sample mean.

The arithmetic mean of a variable is often denoted by a bar, for example ( read " x bar ") would be the mean of some sample space.

A test result ( calculated from the null hypothesis and the sample ) is called statistically significant if it is deemed unlikely to have occurred, assuming the truth of the null hypothesis.

The expected value of F is ( where n is the treatment sample size )

Reporting sample size analysis is generally required in psychology.

Power analysis is often applied in the context of ANOVA in order to assess the probability of successfully rejecting the null hypothesis if we assume a certain ANOVA design, effect size in the population, sample size and significance level.

Power analysis can assist in study design by determining what sample size would be required in order to have a reasonable chance of rejecting the null hypothesis when the alternative hypothesis is true.

Eta-squared is a biased estimator of the variance explained by the model in the population ( it estimates only the effect size in the sample ).

To do so, a sample of the soil is mixed into a clear container with some water, creating an almost completely saturated liquid.

A formula for calculating an unbiased estimate of the population variance from a finite sample of n observations is:

note that the above online algorithm III is a special case of an algorithm that works for any partition of the sample into sets,:

Based on this sample, the estimated population mean is 10, and the unbiased estimate of population variance is 30.