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Covariance Matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the x and y directions contain all of the necessary information; a 2 \times 2 matrix would be necessary to fully characterize the two-dimensional variation. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). The covariance matrix of a random vector \mathbf is typically denoted by \operatorname_, \Sigma or S. Definition Throughout this article, boldfaced u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonal Matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is \left begin 3 & 0 \\ 0 & 2 \end\right/math>, while an example of a 3×3 diagonal matrix is \left begin 6 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 4 \end\right/math>. An identity matrix of any size, or any multiple of it is a diagonal matrix called a ''scalar matrix'', for example, \left begin 0.5 & 0 \\ 0 & 0.5 \end\right/math>. In geometry, a diagonal matrix may be used as a '' scaling matrix'', since matrix multiplication with it results in changing scale (size) and possibly also shape; only a scalar matrix results in uniform change in scale. Definition As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix with columns and rows is diagonal if \forall i,j \in \, i \ne j \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Complement
The Schur complement is a key tool in the fields of linear algebra, the theory of matrices, numerical analysis, and statistics. It is defined for a block matrix. Suppose ''p'', ''q'' are nonnegative integers such that ''p + q > 0'', and suppose ''A'', ''B'', ''C'', ''D'' are respectively ''p'' × ''p'', ''p'' × ''q'', ''q'' × ''p'', and ''q'' × ''q'' matrices of complex numbers. Let M = \begin A & B \\ C & D \end so that ''M'' is a (''p'' + ''q'') × (''p'' + ''q'') matrix. If ''D'' is invertible, then the Schur complement of the block ''D'' of the matrix ''M'' is the ''p'' × ''p'' matrix defined by M/D := A - BD^C. If ''A'' is invertible, the Schur complement of the block ''A'' of the matrix ''M'' is the ''q'' × ''q'' matrix defined by M/A := D - CA^B. In the case that ''A'' or ''D'' is singular, substituting a generalized inverse for the inverses on ''M/A'' and ''M/D'' yields the generalized Schur complement. The Schur complement is named after Issai Schur who used it to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Variance
In probability theory and statistics, a conditional variance is the variance of a random variable given the value(s) of one or more other variables. Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function. Conditional variances are important parts of autoregressive conditional heteroskedasticity (ARCH) models. Definition The conditional variance of a random variable ''Y'' given another random variable ''X'' is :\operatorname(Y\mid X) = \operatorname\Big(\big(Y - \operatorname(Y\mid X)\big)^\;\Big, \; X\Big). The conditional variance tells us how much variance is left if we use \operatorname(Y\mid X) to "predict" ''Y''. Here, as usual, \operatorname(Y\mid X) stands for the conditional expectation of ''Y'' given ''X'', which we may recall, is a random variable itself (a function of ''X'', determined up to probability one). As a result, \operatorname(Y\mid X) itself is a random variable (and is a function of ''X''). Expl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Mean
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space. Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted E(X\mid Y) analogously to conditional probability. The function form is either denoted E(X\mid Y=y) or a separate function symbol such as f(y) is introduced with the meaning E(X\mid Y) = f(Y). Examples Example 1: Dice rolling Consider the roll of a fair die and let ''A'' = 1 if the number is even (i.e., 2, 4, or 6) and ''A'' = 0 otherwise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Distribution
Conditional (if then) may refer to: * Causal conditional, if X then Y, where X is a cause of Y *Conditional probability, the probability of an event A given that another event B * Conditional proof, in logic: a proof that asserts a conditional, and proves that the antecedent leads to the consequent *Material conditional, in propositional calculus, or logical calculus in mathematics * Relevance conditional, in relevance logic * Conditional (computer programming), a statement or expression in computer programming languages *A conditional expression in computer programming languages such as ?: *Conditions in a contract A contract is an agreement that specifies certain legally enforceable rights and obligations pertaining to two or more parties. A contract typically involves consent to transfer of goods, services, money, or promise to transfer any of thos ... Grammar and linguistics * Conditional mood (or conditional tense), a verb form in many languages * Conditional se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Normal Distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be ''k''-variate normally distributed if every linear combination of its ''k'' components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables, each of which clusters around a mean value. Definitions Notation and parametrization The multivariate normal distribution of a ''k''-dimensional random vector \mathbf = (X_1,\ldots,X_k)^ can be written in the following notation: : \mathbf\ \sim\ \mathcal(\boldsymbol\mu,\, \boldsymbol\Sigma), or to make it explicitly known that \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marginal Distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables. This contrasts with a conditional distribution, which gives the probabilities contingent upon the values of the other variables. Marginal variables are those variables in the subset of variables being retained. These concepts are "marginal" because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table. The distribution of the marginal variables (the marginal distribution) is obtained by marginalizing (that is, focusing on the sums in the margin) over the distribution of the variables being discarded, and the discarded variables are said to have been marginalized out. The context here is that the theoreti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive-semidefinite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf. More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number \mathbf^* M \mathbf is positive for every nonzero complex column vector \mathbf, where \mathbf^* denotes the conjugate transpose of \mathbf. Positive semi-definite matrices are defined similarly, except that the scalars \mathbf^\mathsf M \mathbf and \mathbf^* M \mathbf are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called ''indefinite''. Some authors use more general definitions of definiteness, permitting the matrices to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Correlation
In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. When determining the numerical relationship between two variables of interest, using their correlation coefficient will give misleading results if there is another confounding variable that is numerically related to both variables of interest. This misleading information can be avoided by controlling for the confounding variable, which is done by computing the partial correlation coefficient. This is precisely the motivation for including other right-side variables in a multiple regression; but while multiple regression gives unbiased results for the effect size, it does not give a numerical value of a measure of the strength of the relationship between the two variables of interest. For example, given economic data on the consumption, income, and wealth of various individuals, consider the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Precision Matrix
In statistics, the precision matrix or concentration matrix is the matrix inverse of the covariance matrix or dispersion matrix, P = \Sigma^. For univariate distributions, the precision matrix degenerates into a scalar precision, defined as the reciprocal of the variance, p = \frac. Other summary statistics of statistical dispersion also called ''precision'' (or ''imprecision'') include the reciprocal of the standard deviation, p = \frac; the standard deviation itself and the relative standard deviation; as well as the standard error and the confidence interval (or its half-width, the margin of error). Usage One particular use of the precision matrix is in the context of Bayesian analysis of the multivariate normal distribution: for example, Bernardo & Smith prefer to parameterise the multivariate normal distribution in terms of the precision matrix, rather than the covariance matrix, because of certain simplifications that then arise. For instance, if both the prior and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
