Download PDF by B. C. Carlson: Special functions of applied mathematics

By B. C. Carlson

ISBN-10: 0121601501

ISBN-13: 9780121601508

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1. 1, and this section will deal with the special problem mentioned in the Preface, of a mosaic pattern where X(r) is either 1 or 0 (presence or absence). 2 ds of a switch to no vegetation if there were vegetation present. l . The stochastic equation for the 0,1 variable Xs is, moreover, (1) implying an exponential correlation function of A. 2 , viz Ps = e-("t +",)s, (s > 0). It was not immediately clear that such a model was appropriate or even admissible in such a two-dimensional (2) 48 THE STATISTICAL ANALYSIS OF SPATIAL PATTERN Table II Observations on a population of Antennaria umbrinella (from Pielou, 1964, Table 1) Event s;: 1 2 3 00 135 133 124 114 129 01 (10) 20 28 43 59 44 11 45 39 33 27 27 Total 4 5 Total 6 I] 1 27 200 200 200 200 200 200 [0;: 00 + %(01)] 878 [1 ;: 11 + %(01)] 322 1200 context (it may, however, be shown that it is consistent with vegetation areas bounded by random straight lines; see Switzer, 1965) but the correlation function (2) does in fact fit the observed correlations reasonably well (Bartlett, 1964a).

Xi-l "Xi+l' . . Xn) and can only depend on Xi and values at 'neighbouring' sites. Consider in particular site 1. We have L(x) - L(x]) = Xl {G I (Xl) + L/XjGI,j(X\, Xj) + .. } where the dashed summations do not involve site 1. Suppose site k (=1=1) is not a neighbour of site 1. The L(x) - L(XI ) must be independent of Xk for all x. k (x 1, Xk) = O. Similarly the higher-uHler G functions involving X 1 and Xk must be null. Similar results ° ° 42 THE STATISTICAL ANALYSIS OF SPATIAL PATTERN also follow for any pair of sites that are not neighbours, and the theorem follows.

10 1 1 1 1 1 1 2 1 I . 1 1 I . I (4) _ (5) 6 1 1 1 . I 2 . (15) 1 . I (16) 100 10 5 11 6 4 2 10 7 4 4 8 8 6 3 6 6 Total 54 THE STATISTICAL ANALYSIS OF SPATIAL PATTERN written iJ ] C [ S_iJ(h-lul)W1(U)dU 2 (7) =CJ2, say, where A = h 2 . In particular, for the clustering model used for these simulations, for which w(r) = Amoexp{ -(xi + YI )/4a 2 } /(4rra 2 ), (8) we have C = Amo and Wi (u) = (4rra 2 )_'/2 exp( -Y2U 2 /a 2 ). The variance-mean ratio becomes 1 + moI2 /A where J=2 h SO (9) = 1 + moP, O--u/h)(4rra2)-Y2exp{-u2/(4a2)}du.

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Special functions of applied mathematics by B. C. Carlson


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