The weighted arithmetic mean is similar to an ordinary arithmetic mean, except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.
The principal components of a collection of points in a real coordinate space are a sequence of unit vectors, where the vector is the direction of a line that best fits the data while being orthogonal to the first vectors. Here, a best-fitting line is defined as one that minimizes the average squared distance from the points to the line. These directions constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Principal component analysis (PCA) is the process of computing the principal components and using them to perform a change of basis on the data, sometimes using only the first few principal components and ignoring the rest.
In bioinformatics, neighbor joining is a bottom-up (agglomerative) clustering method for the creation of phylogenetic trees, created by Naruya Saitou and Masatoshi Nei in 1987. Usually used for trees based on DNA or protein sequence data, the algorithm requires knowledge of the distance between each pair of taxa to form the tree.
In statistics, propagation of uncertainty is the effect of variables' uncertainties on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations which propagate due to the combination of variables in the function.
In statistics, a vector of random variables is heteroscedastic if the variability of the random disturbance is different across elements of the vector. Here, variability could be quantified by the variance or any other measure of statistical dispersion. Thus heteroscedasticity is the absence of homoscedasticity. A typical example is the set of observations of income in different cities.
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters or estimate the conditional expectation across a broader collection of non-linear models.
In mathematics, computer science and especially graph theory, a distance matrix is a square matrix containing the distances, taken pairwise, between the elements of a set. Depending upon the application involved, the distance being used to define this matrix may or may not be a metric. If there are N elements, this matrix will have size N×N. In graph-theoretic applications the elements are more often referred to as points, nodes or vertices.
Tikhonov regularization, named for Andrey Tikhonov, is a method of regularization of ill-posed problems. Ridge regression is a special case of Tikhonov regularization in which all parameters are regularized equally. Ridge regression is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters. In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias.
In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function of the independent variable.
In statistics and signal processing, a minimum mean square error (MMSE) estimator is an estimation method which minimizes the mean square error (MSE), which is a common measure of estimator quality, of the fitted values of a dependent variable. In the Bayesian setting, the term MMSE more specifically refers to estimation with quadratic loss function. In such case, the MMSE estimator is given by the posterior mean of the parameter to be estimated. Since the posterior mean is cumbersome to calculate, the form of the MMSE estimator is usually constrained to be within a certain class of functions. Linear MMSE estimators are a popular choice since they are easy to use, easy to calculate, and very versatile. It has given rise to many popular estimators such as the Wiener–Kolmogorov filter and Kalman filter.
Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the variance of observations is incorporated into the regression. WLS is also a specialization of generalized least squares.
In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordinary least squares and weighted least squares can be statistically inefficient, or even give misleading inferences. GLS was first described by Alexander Aitken in 1936.
The sample mean and the sample covariance are statistics computed from a sample of data on one or more random variables.
Equilibrium constants are determined in order to quantify chemical equilibria. When an equilibrium constant K is expressed as a concentration quotient,
The topic of heteroscedasticity-consistent (HC) standard errors arises in statistics and econometrics in the context of linear regression and time series analysis. These are also known as Eicker–Huber–White standard errors, to recognize the contributions of Friedhelm Eicker, Peter J. Huber, and Halbert White.
Distance matrices are used in phylogeny as non-parametric distance methods and were originally applied to phenetic data using a matrix of pairwise distances. These distances are then reconciled to produce a tree. The distance matrix can come from a number of different sources, including measured distance or morphometric analysis, various pairwise distance formulae applied to discrete morphological characters, or genetic distance from sequence, restriction fragment, or allozyme data. For phylogenetic character data, raw distance values can be calculated by simply counting the number of pairwise differences in character states.
In statistics, the reduced chi-square statistic is used extensively in goodness of fit testing. It is also known as mean squared weighted deviation (MSWD) in isotopic dating and variance of unit weight in the context of weighted least squares.
Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods.
T-REX(website) is a freely available webserver, developed at the department of Computer Science of the Université du Québec à Montréal, dedicated to the inference, validation and visualization of phylogenetic trees and phylogenetic networks. The T-REX web server allows the users to perform several popular methods of phylogenetic analysis as well as some new phylogenetic applications for inferring, drawing and validating phylogenetic trees and networks.
In statistics, confirmatory composite analysis (CCA) is a sub-type of structural equation modeling (SEM). Although, historically, CCA emerged from a re-orientation and re-start of partial least squares path modeling (PLS-PM), it has become an independent approach and the two should not be confused. In many ways it is similar to, but also quite distinct from confirmatory factor analysis (CFA). It shares with CFA the process of model specification, model identification, model estimation, and model assessment. However, in contrast to CFA which always assumes the existence of latent variables, in CCA all variables can be observable, with their interrelationships expressed in terms of composites, i.e., linear compounds of subsets of the variables. The composites are treated as the fundamental objects and path diagrams can be used to illustrate their relationships. This makes CCA particularly useful for disciplines examining theoretical concepts that are designed to attain certain goals, so-called artifacts, and their interplay with theoretical concepts of behavioral sciences.
