Search results
Results From The WOW.Com Content Network
A contrast is defined as the sum of each group mean multiplied by a coefficient for each group (i.e., a signed number, c j). [10] In equation form, = ¯ + ¯ + + ¯ ¯, where L is the weighted sum of group means, the c j coefficients represent the assigned weights of the means (these must sum to 0 for orthogonal contrasts), and ¯ j represents the group means. [8]
Example of orthogonal factorial design Orthogonality concerns the forms of comparison (contrasts) that can be legitimately and efficiently carried out. Contrasts can be represented by vectors and sets of orthogonal contrasts are uncorrelated and independently distributed if the data are normal.
[3] [4] In the design of experiments for estimating statistical models, optimal designs allow parameters to be estimated without bias and with minimum variance. A non-optimal design requires a greater number of experimental runs to estimate the parameters with the same precision as an optimal design. In practical terms, optimal experiments can ...
Orthogonal polynomials with matrices have either coefficients that are matrices or the indeterminate is a matrix. There are two popular examples: either the coefficients { a i } {\displaystyle \{a_{i}\}} are matrices or x {\displaystyle x} :
This notation is illustrated here for the 2 × 3 experiment. A contrast in cell means is a linear combination of cell means in which the coefficients sum to 0. Contrasts are of interest in themselves, and are the building blocks by which main effects and interactions are defined. In the 2 × 3 experiment illustrated here, the expression
An important example of a centered real stochastic process on [0, 1] is the Wiener process; the Karhunen–Loève theorem can be used to provide a canonical orthogonal representation for it. In this case the expansion consists of sinusoidal functions.
An example of a Riordan array is given by the pair of power series (,) = (, +).It is not difficult to show that this pair generates the infinite triangular array of binomial coefficients , = (), also called the Pascal matrix:
This can be regarded that as the algorithm progresses, and span the same Krylov subspace, where form the orthogonal basis with respect to the standard inner product, and form the orthogonal basis with respect to the inner product induced by .