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.
Data and information visualizationData and information visualization (data viz or info viz) is the practice of designing and creating easy-to-communicate and easy-to-understand graphic or visual representations of a large amount of complex quantitative and qualitative data and information with the help of static, dynamic or interactive visual items.
DimensionIn physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it - for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it - for example, both a latitude and longitude are required to locate a point on the surface of a sphere.
Cholesky decompositionIn linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ʃəˈlɛski ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices, and posthumously published in 1924. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.
Kernel principal component analysisIn the field of multivariate statistics, kernel principal component analysis (kernel PCA) is an extension of principal component analysis (PCA) using techniques of kernel methods. Using a kernel, the originally linear operations of PCA are performed in a reproducing kernel Hilbert space. Recall that conventional PCA operates on zero-centered data; that is, where is one of the multivariate observations.
Whole genome sequencingWhole genome sequencing (WGS), also known as full genome sequencing, complete genome sequencing, or entire genome sequencing, is the process of determining the entirety, or nearly the entirety, of the DNA sequence of an organism's genome at a single time. This entails sequencing all of an organism's chromosomal DNA as well as DNA contained in the mitochondria and, for plants, in the chloroplast. Whole genome sequencing has largely been used as a research tool, but was being introduced to clinics in 2014.
Independent component analysisIn signal processing, independent component analysis (ICA) is a computational method for separating a multivariate signal into additive subcomponents. This is done by assuming that at most one subcomponent is Gaussian and that the subcomponents are statistically independent from each other. ICA is a special case of blind source separation. A common example application is the "cocktail party problem" of listening in on one person's speech in a noisy room.
Scientific visualizationScientific visualization (also spelled scientific visualisation) is an interdisciplinary branch of science concerned with the visualization of scientific phenomena. It is also considered a subset of computer graphics, a branch of computer science. The purpose of scientific visualization is to graphically illustrate scientific data to enable scientists to understand, illustrate, and glean insight from their data.
Matrix normIn mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Given a field of either real or complex numbers, let be the K-vector space of matrices with rows and columns and entries in the field . A matrix norm is a norm on . This article will always write such norms with double vertical bars (like so: ).
Diagonal matrixIn 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 , while an example of a 3×3 diagonal matrix is. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size).
One-dimensional spaceIn physics and mathematics, a sequence of n numbers can specify a location in n-dimensional space. When n = 1, the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the position of each point on it can be described by a single number. In algebraic geometry there are several structures that are technically one-dimensional spaces but referred to in other terms. A field k is a one-dimensional vector space over itself.
Visualization (graphics)Visualization or visualisation (see spelling differences) is any technique for creating s, diagrams, or animations to communicate a message. Visualization through visual imagery has been an effective way to communicate both abstract and concrete ideas since the dawn of humanity. from history include cave paintings, Egyptian hieroglyphs, Greek geometry, and Leonardo da Vinci's revolutionary methods of technical drawing for engineering and scientific purposes. Visualization today has ever-expanding applications in science, education, engineering (e.
