P P X, Next, E(u) = E(y) = that is Recall that rows 3 to of P are orthogonal to rows 1 and 2 of P, and notice that the two n columns of the design matrix X are proportional to rows 1 and 2 of P. Hence the means of {u3, must all be 0 by orthogonality. un} …, Next, looking the just the first two rows of these matrices, and given that , after matrix multiplication we obtain Finally, we make the transformation u = Py and develop the of {u1, logLikelihood u2, …, un}. The Jacobian of the transformation is 1 (since the transformation is orthogonal and the absolute value of det(P)=1). Since the inverse of P is PT, we can replace y = P-1u in the of {y1, by PTu instead: logLikelihood y2, yn} …, so Next PTP = I so this can be added inside the two brackets without changing their values. PT Taking the common from both brackets, preserving the correct order of multiplication and P noting that (PT)T = gives:
Next, E(u) = P E(y) = P X, that is Recall that rows 3 to n of P are orthogonal to rows 1 and 2 of P, and notice that the two columns of the design matrix X are proportional to rows 1 and 2 of P. Hence the means of {u3, …, un} must all be 0 by orthogonality. Next, looking the just the first two rows of these matrices, and given that , after matrix multiplication we obtain Finally, we make the transformation u = Py and develop the logLikelihood of {u1, u2, …, un}. The Jacobian of the transformation is 1 (since the transformation is orthogonal and the absolute value of det(P)=1). Since the inverse of P is PT, we can replace y = P-1u in the logLikelihood of {y1, y2, …, yn} by PTu instead: so Next PTP = I so this can be added inside the two brackets without changing their values. Taking the common PT from both brackets, preserving the correct order of multiplication and noting that (PT)T = P gives:
An%20introduction%20to%20REML.docx