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Equivalently, the Gram matrix of the rows or the columns of a real rotation matrix is the identity matrix. Likewise, the Gram matrix of the rows or columns of a unitary matrix is the identity matrix. The rank of the Gram matrix of vectors in or equals the dimension of the space spanned by these vectors. [1]
Statistical tests are used to test the fit between a hypothesis and the data. [1] [2] Choosing the right statistical test is not a trivial task. [1] The choice of the test depends on many properties of the research question. The vast majority of studies can be addressed by 30 of the 100 or so statistical tests in use. [3] [4] [5]
The Gram matrix of a sequence of points ,, …, in k-dimensional space ℝ k is the n×n matrix = of their dot products (here a point is thought of as a vector from 0 to that point): g i j = x i ⋅ x j = ‖ x i ‖ ‖ x j ‖ cos θ {\displaystyle g_{ij}=x_{i}\cdot x_{j}=\|x_{i}\|\|x_{j}\|\cos \theta } , where θ {\displaystyle \theta ...
Here ||·|| 2 is the matrix 2-norm, c n is a small constant depending on n, and ε denotes the unit round-off. One concern with the Cholesky decomposition to be aware of is the use of square roots. If the matrix being factorized is positive definite as required, the numbers under the square roots are always positive in exact arithmetic.
There is nothing magical about a sample size of 1 000, it's just a nice round number that is well within the range where an exact test, chi-square test, and G–test will give almost identical p values. Spreadsheets, web-page calculators, and SAS shouldn't have any problem doing an exact test on a sample size of 1 000 . — John H. McDonald [2]
A test statistic shares some of the same qualities of a descriptive statistic, and many statistics can be used as both test statistics and descriptive statistics. However, a test statistic is specifically intended for use in statistical testing, whereas the main quality of a descriptive statistic is that it is easily interpretable. Some ...
An n-gram is a sequence of n adjacent symbols in particular order. [1] The symbols may be n adjacent letters (including punctuation marks and blanks), syllables , or rarely whole words found in a language dataset; or adjacent phonemes extracted from a speech-recording dataset, or adjacent base pairs extracted from a genome.
The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (a i, a j), where the elements a i form a basis for the lattice. An integral lattice is unimodular if its determinant is 1 or −1. A unimodular lattice is even or type II if all norms are even, otherwise odd or type I.