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Estimation
Published in Julian J. Faraway, Linear Models with Python, 2021
Orthogonality is a useful property because it allows us to more easily interpret the effect of one predictor without regard to another. Suppose we can partition X in two, X=[X1|X2] such that X1TX2=0. So now: Y=Xβ+ϵ=X1β1+X2β2+ϵ
Optimal experimental designs
Published in Adedeji B. Badiru, Ibidapo-Obe Oye, Babatunde J. Ayeni, Manufacturing and Enterprise, 2018
Adedeji B. Badiru, Ibidapo-Obe Oye, Babatunde J. Ayeni
There are several areas to which the CP algorithms can be applied. Some types of problems include supersaturated design, mixed-level design, and many other model-driven designs in which orthogonality cannot be retained. The proposed algorithmic procedure can spread out the non-orthogonal pairs of designs more evenly, and consequently, lead to designs with higher efficiency. The CP algorithms can also be applied to design repair problems, in which a submatrix of the design matrix is optimized.
Linear Vector Spaces
Published in Sohail A. Dianat, Eli S. Saber, ®, 2017
Sohail A. Dianat, Eli S. Saber
The orthogonality concept is very useful in engineering problems related to estimation, prediction, signal modeling, and transform calculus (Fourier transform, Z-transform, etc.). In this section, we discuss orthogonal vectors in a given vector space with applications such as Fourier series and discrete Fourier transform (DFT).
Modification of the Maximin and ϕp (Phi) Criteria to Achieve Statistically Uniform Distribution of Sampling Points
Published in Technometrics, 2020
Miroslav Vořechovský, Jan Eliáš
Designs optimized for model prediction should be noncollapsible: if only a subset of input variables is relevant for predicting the response, then the prediction error is related to the uniformity of the projected designs. Another frequent requirement placed on designs is for them to display orthogonality: this ensures that all specified parameters may be estimated independently of any others (estimation of main effects), and that interactions can also be estimated in a straightforward manner. There are numerous design types focused on orthogonality, such as various factorial designs, “orthogonal arrays,” “mutually orthogonal Latin squares,” etc. Some authors simply tend to decrease statistical correlations among vectors of samples of individual variables in designs (Morris and Mitchell 1995; Vořechovský and Novák 2009). As shown by Owen (1992), the bilinear part of the integrand is more accurately estimated if the sample correlations among input variables are negligible.