Factor analysisFactor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. For example, it is possible that variations in six observed variables mainly reflect the variations in two unobserved (underlying) variables. Factor analysis searches for such joint variations in response to unobserved latent variables.
Principal component analysisPrincipal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and enabling the visualization of multidimensional data. Formally, PCA is a statistical technique for reducing the dimensionality of a dataset. This is accomplished by linearly transforming the data into a new coordinate system where (most of) the variation in the data can be described with fewer dimensions than the initial data.
Gait (human)A gait is a manner of limb movements made during locomotion. Human gaits are the various ways in which humans can move, either naturally or as a result of specialized training. Human gait is defined as bipedal forward propulsion of the center of gravity of the human body, in which there are sinuous movements of different segments of the body with little energy spent. Varied gaits are characterized by differences such as limb movement patterns, overall velocity, forces, kinetic and potential energy cycles, and changes in contact with the ground.
GaitGait is the pattern of movement of the limbs of animals, including humans, during locomotion over a solid substrate. Most animals use a variety of gaits, selecting gait based on speed, terrain, the need to maneuver, and energetic efficiency. Different animal species may use different gaits due to differences in anatomy that prevent use of certain gaits, or simply due to evolved innate preferences as a result of habitat differences.
Eigendecomposition of a matrixIn linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Eigenvalue, eigenvector and eigenspace A (nonzero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of the form for some scalar λ.
Singular value decompositionIn linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an complex matrix M is a factorization of the form where U is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an complex unitary matrix, and is the conjugate transpose of V.
Eigenvalues and eigenvectorsIn linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.
Matrix decompositionIn the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations , the matrix A can be decomposed via the LU decomposition.
QR decompositionIn linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm. Any real square matrix A may be decomposed as where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning ) and R is an upper triangular matrix (also called right triangular matrix).
Schur decompositionIn the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily equivalent to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix. The Schur decomposition reads as follows: if A is an n × n square matrix with complex entries, then A can be expressed as where Q is a unitary matrix (so that its inverse Q−1 is also the conjugate transpose Q* of Q), and U is an upper triangular matrix, which is called a Schur form of A.
Public healthPublic health is "the science and art of preventing disease, prolonging life and promoting health through the organized efforts and informed choices of society, organizations, public and private, communities and individuals". Analyzing the determinants of health of a population and the threats it faces is the basis for public health. The public can be as small as a handful of people or as large as a village or an entire city; in the case of a pandemic it may encompass several continents.
VarianceIn probability theory and statistics, variance is the squared deviation from the mean of a random variable. The variance is also often defined as the square of the standard deviation. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , , , or .
Law of total varianceIn probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if and are random variables on the same probability space, and the variance of is finite, then In language perhaps better known to statisticians than to probability theorists, the two terms are the "unexplained" and the "explained" components of the variance respectively (cf. fraction of variance unexplained, explained variation).
Polar decompositionIn mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form , where is a unitary matrix and is a positive semi-definite Hermitian matrix ( is an orthogonal matrix and is a positive semi-definite symmetric matrix in the real case), both square and of the same size. Intuitively, if a real matrix is interpreted as a linear transformation of -dimensional space , the polar decomposition separates it into a rotation or reflection of , and a scaling of the space along a set of orthogonal axes.
Cognitive scienceCognitive science is the interdisciplinary, scientific study of the mind and its processes with input from linguistics, psychology, neuroscience, philosophy, computer science/artificial intelligence, and anthropology. It examines the nature, the tasks, and the functions of cognition (in a broad sense). Cognitive scientists study intelligence and behavior, with a focus on how nervous systems represent, process, and transform information.
RiskIn simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environment), often focusing on negative, undesirable consequences. Many different definitions have been proposed. The international standard definition of risk for common understanding in different applications is "effect of uncertainty on objectives".
Risk factorIn epidemiology, a risk factor or determinant is a variable associated with an increased risk of disease or infection. Due to a lack of harmonization across disciplines, determinant, in its more widely accepted scientific meaning, is often used as a synonym. The main difference lies in the realm of practice: medicine (clinical practice) versus public health. As an example from clinical practice, low ingestion of dietary sources of vitamin C is a known risk factor for developing scurvy.
Predictive modellingPredictive modelling uses statistics to predict outcomes. Most often the event one wants to predict is in the future, but predictive modelling can be applied to any type of unknown event, regardless of when it occurred. For example, predictive models are often used to detect crimes and identify suspects, after the crime has taken place. In many cases, the model is chosen on the basis of detection theory to try to guess the probability of an outcome given a set amount of input data, for example given an email determining how likely that it is spam.
Old ageOld age is the range of ages for persons nearing and surpassing life expectancy. People of old age are also referred to as: old people, elderly, elders, seniors, senior citizens, or older adults. Old age is not a definite biological stage: the chronological age denoted as "old age" varies culturally and historically. Some disciplines and domains focus on the aging and the aged, such as the organic processes of aging (senescence), medical studies of the aging process (gerontology), diseases that afflict older adults (geriatrics), technology to support the aging society (gerontechnology), and leisure and sport activities adapted to older people (such as senior sport).
Covariance matrixIn 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. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions.
