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Extra info for Regression and the Moore-Penrose Pseudoinverse
1) Let h,, h 2 , ... h,, ... , A,). ,An+ 1 . 2) Stationary Probability for a Markov Chain A matrix P is called a stochastic matrix if its entries are nonnegative and its row sums are unity. Such matrices can be used to represent the one step transition probabilities for homogeneous Markov chains. If the process is ergodic (a condition which can be deduced by visual inspection of the transition matrix, Feller [l, Chap. XV]) then lim (P*)” = (xi x i n+ w i x) where x is the unique probability vector (a vector having nonnegative components which sum to unity) satisfying PTX = x.
1) is unique if and only if ( I - H + H ) y vanishes for all y . This can only happen if the projection of all vectors on N ( H ) is zero, which means that N ( H ) consists only of the zero vector. ( I - H + H ) y = 0 for all y , by the way, if and only if H + H = I. This proves (b). 3) has a solution if and only if z is the afterimage of some x under H . This is, by definition, the same as saying z € % ' ( H ) . 10), the last holds true if and only if z = HH+U for some u. 6), it follows that HH'z = ( H H + ) 2=~ HH'u = Z.
Cj cjT)+ 47 getting a good fit. In many statistical contexts, underspecifying the number of regressors results in biased estimates. Over specification (too many regressors) results in loss of accuracy. ” is partially answered by the theory of the analysis of variance and covariance, under certain assumptions. ” is much more difficult and not easily answered in quantitative terms. 6) Theorem: Let e l ,c 2 , ... 1). 1) S,’+l = . (Amcm+l)(sm+cm+l)T c;f+ 1 A m c m + 1 if c , + ~is not a linear combination of c l r .