We construct an integrated probabilistic model to capture interactions between atoms of a nanocomponent. We then use this model to assess reliabilities of nanocomponents with different structures. Several properties of our proposed model are also described under a sparseness condition. The model is an extension of our previous model based on Markovian random field theory. The proposed integrated model is flexible in that pairwise relationship information among atoms as well as features of individual atoms can be easily incorporated. An important feature that distinguishes the integrated probabilistic model from our previous model is that the integrated approach uses all available sources of information with different weights for different types of interaction. In this paper we consider the nanocomponent at a fixed moment of time, say the present moment, and we assume that the present state of the nanocomponent depends only on the present states of its atoms.

A nanocomponent is a collection of atoms arranged to a specific design in order to achieve a desired function with an acceptable performance and reliability. The type of atoms, the manner in which they are arranged within the nanocomponent, and their interrelationship have a direct effect on the nanocomponent's reliability (survival) function. In this paper we propose models based on the notion of a copula that are used to describe the relationship between the atoms of a nanocomponent. Having defined these models, we go on to construct a ‘nanocomponent’ model in order to obtain the reliability function of a nanocomponent.

We study percolation and Ising models defined on generalizations of quad-trees used in multiresolution image analysis. These can be viewed as trees for which each mother vertex has 2d daughter vertices, and for which daughter vertices are linked together in d-dimensional Euclidean configurations. Retention probabilities and interaction strengths differ according to whether the relevant bond is between mother and daughter or between neighbours. Bounds are established which locate phase transitions and show the existence of a coexistence phase for the percolation model. Results are extended to the corresponding Ising model using the Fortuin-Kasteleyn random-cluster representation.

Often, the statistical analysis of the shape of a random planar curve is based on a model for a polygonal approximation to the curve. In the present paper, we instead describe the curve as a continuous stochastic deformation of a template curve. The advantage of this continuous approach is that the parameters in the model do not relate to a particular polygonal approximation. A somewhat similar approach has been used in Kent et. al. (1996), who describe the limiting behaviour of a model with a first-order Markov property as the landmarks on the curve become closely spaced; see also Grenander (1993). The model studied in the present paper is an extension of this model. Our model possesses a second-order Markov property. Its geometrical characteristics are studied in some detail and an explicit expression for the covariance function is derived. The model is applied to the boundaries of profiles of cell nuclei from a benign tumour and a malignant tumour. It turns out that the model with the second-order Markov property is the most appropriate, and that it is indeed possible to distinguish between the two samples.

The relocation of several thousand members of the Kamba tribe from the Kyulu Hills to the Thange valley near Masongaleni in Kenya provides an excellent opportunity to study the development of the immune response to schistosomiasis mansoni in a population with little or no previous experience of the infection. An adjacent, well-established Kamba community with similar patterns of water contact provides a suitable endemic control population. The immigrants were, uniquely, examined shortly after their arrival in the endemic area, while the prevalence of infection was still low. At this time faecal egg counts peaked atypically around 30 years of age. Over the next 12–18 months infection increased rapidly, especially among teenagers, producing a pattern of infection more typical of endemic communities. This substantially narrows estimates of the time required to develop the important determinants of the age–intensity profile, supporting the notion that changes related to age per se, rather than duration of infection, dominate. Age-dependent factors might include behaviour or physiology, including immune response. This paper provides the background for continuing longitudinal studies on the development of immunological responses to this parasite.

Consider an array of binary random variables distributed over an m1(n) by m2(n) rectangular lattice and let Y1(n) denote the number of pairs of variables d, units apart and both equal to 1. We show that if the binary variables are independent and identically distributed, then under certain conditions Y(n) = (Y1(n), · ··, Yr(n)) is asymptotically multivariate normal for n large and r finite. This result is extended to versions of a model which provide clustering (repulsion) alternatives to randomness and have clustering (repulsion) parameter values nearly equal to 0. Statistical applications of these results are discussed.

In this article we give limiting results for arrays {Xij (m, n) (i, j) Dmn} of binary random variables distributed as particular types of Markov random fields over m x n rectangular lattices Dmn. Under some sparseness conditions which restrict the number of Xij (m, n)'s which are equal to one we show that the random variables (l = 1, ···, r) converge to independent Poisson random variables for 0 < d1 < d2 < · ·· < dr when m→∞ nd∞. The particular types of Markov random fields considered here provide clustering (or repulsion) alternatives to randomness and involve several parameters. The limiting results are used to consider statistical inference for these parameters. Finally, a simulation study is presented which examines the adequacy of the Poisson approximation and the inference techniques when the lattice dimensions are only moderately large.

Let S be a countable set, Q a strictly positive matrix on S × S. The set 𝒢(Q) of one-dimensional Markov random fields taking values in S with conditional probabilities determined by Q has been investigated by Spitzer [4], Föllmer [1] and Kesten [3]. In this paper a new result of Spitzer's is stated and proved, and used to present a specific example (the only one known) of a matrix Q which exhibits phase transition and admits a complete description of 𝒢 (Q).

This paper is concerned with nearest-neighbour systems on the coloured lattice (unordered state space). It extends the paper of Strauss (1975) on clustering in the two-colour case. Comparison is made with the more general methods of Besag (1974). Some tests are developed, and illustrated with an example.

The probability density of binary variables on a lattice with the nearest-neighbor condition is given by the Gibbs Random Field formula. This paper examines some consequences of the result. Approximate formulae for the maximum likelihood estimator of the persistence parameter are derived and discussed, with an example. A comparison is made between two test statistics for independence.

It is shown that the set of Markov random fields and Gibbs states with nearest neighbour potentials are the same for any finite graph. The set of Markov random fields is also shown to be the same as the equilibrium states of time-reversible birth/death processes with nearest neighbour interactions defined on the graph.