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More accurately, the general orthogonality principle states the following: Given a closed subspace of estimators within a Hilbert space and an element in , an element ^ achieves minimum MSE among all elements in if and only if {(^)} = for all .
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]
Given any non-decreasing function α on the real numbers, we can define the Lebesgue–Stieltjes integral () of a function f. If this integral is finite for all polynomials f , we can define an inner product on pairs of polynomials f and g by f , g = ∫ f ( x ) g ( x ) d α ( x ) . {\displaystyle \langle f,g\rangle =\int f(x)g(x)\,d\alpha (x).}
With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of given below. This definition of the 's is the simplest one. It does not appeal to the theory of differential equations.
Orthogonality The property that allows individual effects of the k-factors to be estimated independently without (or with minimal) confounding. Also orthogonality provides minimum variance estimates of the model coefficient so that they are uncorrelated. Rotatability The property of rotating points of the design about the center of the factor ...
The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle Ω, and utilizing the above orthogonality relationships. This is justified rigorously by basic Hilbert space theory.
In general, uncorrelatedness is not the same as orthogonality, except in the special case where at least one of the two random variables has an expected value of 0. In this case, the covariance is the expectation of the product, and X {\displaystyle X} and Y {\displaystyle Y} are uncorrelated if and only if E [ X Y ] = 0 {\displaystyle ...
Plot of the Jacobi polynomial function (,) with = and = and = in the complex plane from to + with colors created with Mathematica 13.1 function ComplexPlot3D. In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) (,) are a class of classical orthogonal polynomials.