Express the modelY₁ = Bo + Bixi + B₁x² + €ii=1,2,..., 5where the &; have mean zero, variance o² and are uncorrelated, as a general linear modelin matrix form by specifying Y, X, 3 and e.

Given Yi=β0+β1xi+β11xi2+εi ,i=1,2,......5

Answered: Express the model Y₁ = Bo + Bixi + B₁r?…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter1: Functions

Section1.2: The Least Square Line

Problem 5E

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Find the second-order contact matrix PQ mentioned in the text.
Suppose the x-coordinates of the data (x1, V1), .., (x,, Vn) are in mean deviation form, so that E x; = 0. Show that if X is the design matrix for the least-squares line in this case, then XTX is a diagonal matrix.
Consider the statistical linear model Y = A0 + e with m = vations, a parameter vector 0 = (61,02, 03)" with n = 3 components and e ~ N(0; 200² I6) having mean 0 and component variance 200². A realisation of this model for data Y and design matrix A is given by 8 7 1 6 obser- €1 580.8 -140.2 2 3 1 €2 219.0 7 3 4 €3 02 Өз -64.5 6 3 5 €4 73.4 2 1 5 E5 403.8 5 7 3 €6 (a) Find the Maximum Likelihood (ML) estimate 0 of 0 and the co- variance matrix Cov(@) of the ML estimate ở (b) If 0 has prior distribution 0 ~ N(0, 10²I3), then find the distribu- tion of posterior (0|Y) using ridge regression. (c) Show that the component correlations Corr(0;|Y), (0;|Y)) of (0|Y) are smaller than those of Corr(0;, 0;) of the ML estimate 0 for i + j.
a. Rewrite the data points (2,1), (4,2), (8,6) and (9,6) with new x-coordinates in mean-deviation form. Let X be the associated design matrix. Why are the columns of X orthogonal? b. Write the normal equations for the data in part (a), and solve them to find the least-squares line, y =B₁ + B₁x², * where x = x-5.75.
a. Rewrite the data points (2,1), (4,2), (8,6) and (9,6) with new x-coordinates in mean-deviation form. Let X be the associated design matrix. Why are the columns of X orthogonal? b. Write the normal equations for the data in part (a), and solve them to find the least-squares line, y =B₁ + B₁x², where x = x-5.75. The associated design matrix, X, is 1 - 1.75 2.25 3.25 1 the entries in one column are all 1 and the entries in the other column sum to 0. (Simplify your answer. Use ascending order.) B. Therefore, the columns of X are orthogonal because b. Write the normal equations for the data from part (a). Select the correct choice below and fill in the answer boxes within your choice. (Simplify your answer.) O A. [Bo B₁] = Bo 421-0
Consider the following two data vectors: A = [3,7,11]B=[3,2,1] Calculate the covariance matrix of these two vectors.

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Express the model Y₁ = Bo + Bixi + B₁r? + &i where the ɛ; have mean zero, variance o² and are uncorrelated, as a general linear model in matrix form by specifying Y, X, 3 and e. i=1,2,...,5