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Vectors are defined in spherical coordinates by (r, θ, φ), where r is the length of the vector, θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and; φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π).
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal. For such matrices, the half-vectorization is sometimes more useful than the ...
R is a programming language for statistical computing and data visualization. It has been adopted in the fields of data mining, bioinformatics and data analysis. [9] The core R language is augmented by a large number of extension packages, containing reusable code, documentation, and sample data. R software is open-source and free software.
Programming with Big Data in R (pbdR) [1] is a series of R packages and an environment for statistical computing with big data by using high-performance statistical computation. [ 2 ] [ 3 ] The pbdR uses the same programming language as R with S3/S4 classes and methods which is used among statisticians and data miners for developing statistical ...
If the 4th component of the vector is 0 instead of 1, then only the vector's direction is reflected and its magnitude remains unchanged, as if it were mirrored through a parallel plane that passes through the origin. This is a useful property as it allows the transformation of both positional vectors and normal vectors with the same matrix.
The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: […] = [] and [] = […]. The set of all row vectors with n entries in a given field (such as the real numbers ) forms an n -dimensional vector space ; similarly, the set of all column vectors with m entries forms an m ...
Given a subset S of R n, a vector field is represented by a vector-valued function V: S → R n in standard Cartesian coordinates (x 1, …, x n). If each component of V is continuous, then V is a continuous vector field. It is common to focus on smooth vector fields, meaning that each component is a smooth function (differentiable any number ...
If data generated by a random vector X are observed as vectors X i of observations with covariance matrix Σ, a linear transformation can be used to decorrelate the data. To do this, the Cholesky decomposition is used to express Σ = A A'. Then the transformed vector Y i = A −1 X i has the identity matrix as its covariance matrix.