Transcription of Lecture 13: Simple Linear Regression in Matrix Format
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11:55 Wednesday 14thOctober, 2015 See updates and corrections ~cshalizi/mreg/ Lecture 13: Simple Linear Regression in MatrixFormat36-401, Section B, Fall 201513 October 2015 Contents1 Least Squares in Matrix The Basic Matrices .. Mean Squared Error .. Minimizing the MSE ..42 Fitted Values and Residuals .. Expectations and Covariances ..73 Sampling Distribution of Estimators84 Derivatives with Respect to Second Derivatives .. Maxima and Minima .. 115 Expectations and Variances with Vectors and Matrices126 Further Reading1312So far, we have not used any notions, or notation, that goes beyond basicalgebra and calculus (and probability). This has forced us to do a fair amountof book-keeping, as it were by hand.
The matrix inversion and multiplication then handles all the book-keeping to put these pieces together to get the appropriate (sample) variances, ... If we project a vector u on to the line in the direction of the length-one vector v, we get vvTu (39) (Check the dimensions: u and v are both n 1, so vT is 1 n, and vTu is
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