Nearest-Neighbour Systems and the Auto-Logistic Model for Binary DataJ. E. BesagJournal of the Royal Statistical Society. Series B (Methodological), Vol. 34, No. 1. (1972), pp.75-83.Stable URL:http://links.jstor.org/sici?sici=0035-9246%281972%2934%3A1%3C75%3ANSATAM%3E2.0.CO%3B2-OJournal of the Royal Statistical Society. Series B (Methodological) is currently published by Royal Statistical Society.Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/journals/rss.html.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academicjournals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers,and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community takeadvantage of advances in technology. For more information regarding JSTOR, please contact [email protected]://www.jstor.orgWed Feb 6 14:49:17 2008Nearest-neighbour Systems and the Auto-logistic Model for Binary Data University of Liverpool [Received February 1971. Revised September 19711 SUMMARY Bartlett (1966) and Whittle (1963), respectively, have proposed alternative, non-equivalent definitions of nearest-neighbour systems. The former, conditional probability definition, whilst the more intuitively attractive, presents several basic problems, not least in the identification of available models. In this paper, conditional probability nearest-neighbour systems for interacting random variables on a two-dimensional rectangular lattice are examined. It is shown that, in the case of 0,1 variables and a homo- geneous system, the only possibility is a logistic-type model but in which the explanatory variables at a point are the surrounding array variables themselves. A spatial-temporal approach leading to the same model is also suggested. The final section deals with linear nearest-neighbour systems, especially for continuous variables. The results of the paper may easily be extended to three or more dimensions. Keywords: NEAREST-NEIGHBOUR SYSTEMS; AUTO-LOGISTIC MODEL; MULTIDIMEN-SIONAL BINARY DATA 1. INTRODUCTION WE shall, principally, be considering a two-dimensional rectangular lattice, each node (or site), (i,j), of which has a random variable, Xi?j, associated with it. There appear in the literature two main definitions of nearest-neighbour models which might be applied to describe the interaction between the variables X4,$in this situation. Whittle (1963) suggested defining nearest-neighbour models in terms of the joint probability distribution of the variables and required that this should be of the product form where is a value of the random variable Xi,j. On the other hand, Bartlett (1966, 1967, 1968) discussed an approach through conditional probabilities, in which he supposed that p{xi,j] all other values}= P{xi,jI xi-l,j~ Xi+l,j, Xi,j-l, ~~,j+~}, (2) that is, dependent only upon the nearest neighbours. Whilst this approach has considerably greater intuitive appeal, it unfortunately presents a number of difficulties. Notably, there is no direct method of evaluating the joint probability distribution on the lattice and, further, the functional form of the conditional probability on the right-hand side of (2) is subject to severe consistency conditions. The latter point was first noted by Levy (1948) concerning normally distributed random variables and subsequently, and more generally, by Brook (1964) who also demonstrated that76 BESAG-Nearest-neighbour Systems [No. 1, any valid model satisfying (2) must also satisfy (1). This situation is similar (and equivalent in one dimension) to that of considering a first-order Markov chain against one of second order: the former is degenerate with respect to the latter but is, of course, of considerable interest in its own right. The object of the present paper is to give some further insight into the nature of conditional probability nearest- neighbour models. 2. CONDITIONAL NEAREST-NEIGHBOURPROBABILITY SCHEMES We consider a conditional probability model for variables situated at each site of a two-dimensional rectangular lattice with given boundary values. We shall be particularly concerned with binary variables; for example, in an ecological context, this might correspond to an array of plants, each of which is either infected (1) or healthy (0), or to the presence (1) or absence (0) of a plant at a site. The perimeter sites of the array may be used to provide the boundary values, although in many practical situations it would seem reasonable to append a boundary of zeros to the array, corresponding, for example, to absence of plants there. In any event, denote the set of boundary sites of the array by B, these surrounding the set of internal sites denoted by I. Then a conditional probability nearest-neighbour model is defined by P{xiajIall other values) =p(xt,j I ~{+1,j, xi,j+J (3)~i-~,j, X~~j-1, for all (i, j) EI, where xi,,. denotes the value of the random variable Xiti at the (i,j) site. We assume here that the model is spatially homogeneous, that is, the function p is independent of the internal position (i,j) on the array. For simplicity of notation, we shall suppose that the perimeter of the array is rectangular, numbering the rows i = O(1) m +1 and the columns j = O(1) n+ 1 but the conclusions will hold for any shape of closed boundary. The explicit evaluation of the joint probability distribution of the inner array variables, conditional upon given boundary values, is not entirely straightforward. Let x denote a realization of the entire array, including boundary values, and