The archaeological data for each site provides a sequence of occupational durations interlaced with periods of no occupation
In Figure 1 For each occupation, the degree of dependence on terrestrial resources and marine resources will be quantified based on archaeological assemblages and other cultural contexts. One of the theoretical goal of the project is to derive distributions of site occupation times based on characteristics of the island ecosystems (land area, relative isolation from other islands, faunal resources at the beginning of each occupation), stochastic events (earthquakes and volcanic events), and characteristics of the occupant cultures. Additionally, we will to derive a distribution of the duration of site abandonment based on principles from island biogeography and the technology available for human colonization. These distributions, in turn, will be used with observations derived from the archaeological, geological, and botanical field observations to estimate a limited set of ecological parameters. In what follows, we briefly outline the characteristics of our modeling efforts to date.
Duration of site abandonment. Call fA(t) the probability density function for times between abandonments, corresponding to the alpha
In Figure 1 This distribution can be decomposed into a convolution of two component. The first component is a period of ecosystem re-establishment following the previous occupation. This duration will depend, in turn, on the type of subsistence. For the Jomon period, we expect that the previous abandonment arose by overexploitation of the dominant terrestrial mammal. By contrast, recovery of marine resources during the Okhotsk period should occur much more rapidly, and this distribution is irrelevant for the military period.
If a primary subsistence species is driven to extinction, the distribution of times to de novo recolonization can be taken from standard island biogeography theory, with the critical parameters being island size and isolation. When the population is reestablished from low levels, the distribution for this first component follows directly from the density-dependent population growth model.
The second component, following ecosystem re-establishment, is the "rediscovery" period. This is treated as a constant hazard for a given transportation technology, island size, and degree of isolation. Hence, the distribution of times to resettlement is a convolution of the distribution of times to ecosystem re-establishment and a negative exponential distribution of times to rediscovery.
Duration of site occupation: The probability density function, fO(t), for distribution of occupation duration's is a function of variables such as island size, technology (which determines the source of subsistence), harvest (or hunting) efficiency. One particular form for this model is given in
In Figure 2 , but this model will be refined and developed as a part of the project. A closed form for the distribution of fO(t), based on stochastic birth-death models of interest will not likely be analytically tractable (Renshaw 1991). Nevertheless, distributions for fO(t) of arbitrary complexity can be derived by Monte Carlo simulation and used for parameter estimation and hypothesis testing. For a given set of observed variables like island size, isolation, human technology, prey types, as well as a set of constants, we can use Monte Carlo simulation to find the empirical probability density function fO(t) given a series of parameter estimates. The advantage of this approach is that we can explore a number of models, estimating the unknown parameters, regardless of the mathematical convenience or the ability to derive a closed form solution for the probability density function. Additionally, we can quite easily incorporate distributions of uncertainty for the constants that we assume in the model.
Statistical methods. Maximum likelihood will be used to estimate parameters. For a series of I islands, each with Mi sites, each with a sequence of Nij occupations and abandonments like those shown in
In Figure 1 , we can construct a likelihood out of the distributions of occupation and abandonment times, fo() and fa(), given above. Each observation comes as the pair of times (tijk, tijk) and a series of context specific covariates, xijk, such measures as the cultural technology, and island size and isolation. The general likelihood is
where qo and qa are arrays of parameters to be estimated for the model, dijk is an indicator variable that is 1 if tijk defines an abandonment in the past, and is 0 if the abandonment is ongoing to the present, corresponding to t4+ in
In Figure 1
A number of implicit assumption about independence are made for this likelihood. First, we assume all occupational duration's are independent of each other except for characteristics shared through the array of measured covariates xijk. Additionally, the duration of each abandonment is assumed to be independent of the preceding occupational duration (except through xijk). Finally, we assume that the duration of each occupation is independent of the duration of the preceding abandonment.
Maximum likelihood estimates for qo and qa will be found by numerical maximization of L. Most models will require stochastic simulation methods for approximating the distribution of fo(), so that we will develop our own maximization software. Holman (2000; Holman and Wood 1995) has developed a library of routines for estimating such complex likelihoods. The Center for Studies in Demography and Ecology at the University of Washington has made available a network of Sun UltraSPARC workstations that will be used for the numerically intensive model estimation.
A number of models of different parameterizations and complexity will be estimated during the estimation phase of this project. We will use Akaike's information criterion (AIC, Akaike 1992, Burnham and Anderson 1998) to select among models with different parameterizations and structures that most parsimoniously approximates the true underlying model. For all parameter estimates, standard methods (asymptotic variance-covariace matrix, profile likelihood, or bootstrap confidence intervals) will be used to estimate parameter uncertainty. The particular method will depend on model complexity.