Probability Statistics

Download PDF by Robert Hooke: How to tell the liars from the statisticians

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By Robert Hooke

ISBN-10: 0824718178

ISBN-13: 9780824718176

Publication via Robert Hooke

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21). 8), for the operator norm of A, the following holds: A = sup φ =1 Aφ ≤ K 2 . , φi H ψi . , . denotes the real inner product (covariance) in the corresponding Hilbert space; further, the nonnegative real numbers s1 ≥ s2 ≥ · · · ≥ 0 are the singular values with the zero as the only possible point of accumulation, and the corresponding function pairs ψi , φi can be chosen (even in the case of multiple singular values) so ∞ that {ψi }∞ i=1 ⊂ H and {φi }i=1 ⊂ H form complete orthonormal systems.

Since Gm,n is the Cartesian product of Pm and Pn , the Laplacian eigenvalues of the m × n grid are iπ jπ iπ jπ + 4 sin2 = 2 1 − cos + 2 1 − cos , 2m 2n m n i = 0, 1, . . , m − 1; j = 0, . . , n − 1. λi,j = 4 sin2 More generally, denote by Gridd, the d-dimensional cubic grid (d ≥ 2 is an integer) with n = (2 + 1)d vertices, where the vertices are characterized by d-tuples of integers − , . . , −1, 0, 1, . . , such that two vertices are adjacent if and only if their d-tuples differ in exactly one coordinate.

K. (e) Let Sd denote the star graph on n = d + 1 vertices. In fact, the star graph is a complete 2-partite graph, namely, Sd = K1,d . Applying the result of (d), its Laplacian spectrum consists of a single 0, the number n − d = 1 (with multiplicity d − 1) and the single eigenvalue n. Making a d-dimensional representation, based on the d smallest Laplacian eigenvalues, the representatives of the d endpoints form the vertices of a simplex in the (d − 1)-dimensional eigenspace corresponding to the eigenvalue 1, but they are not connected to each other, they are merely connected to the single vertex of degree d, the representative of which is the origin (the center of gravity of the simplex).

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How to tell the liars from the statisticians by Robert Hooke

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