#### 4.2 'How' to use models

There have been several excellent accounts of the modelling process, concentrating on computer simulation models (e.g. Hamond 1978; Aldenderfer 1981; Freeman 1988; Lake 2001). The following summary draws on these, but attempts to broaden them to cover the field of mathematical models in general.

Step 1: define the problem, and if appropriate the hypothesis.
What is the archaeological question that we are asking? What data are available to us that might be relevant to the question? Do we need more? If so, how do we acquire them?

Step 2: construct a conceptual model.
In this stage we simplify the problem to make it amenable to a mathematical approach. This involves deciding which variables are, and are not, relevant to the problem in hand, how they behave (e.g. what statistical distributions may be used to describe them) and what their limits are. The relationships between them (correlations or associations) also must be built into the model. There is always a trade-off between usefulness and verisimilitude: 'In general the most useful models are the simplest that include all relevant aspects of the real world system' (Lake 2001, 725).

Step 3: choose the appropriate type of model in which to implement the conceptual model.
Is it to be a deterministic model, a stochastic model or a simulation model, for example? Here Freeman's fundamental question '0. Must I simulate?' (Freeman 1988, 140) is of prime importance. Simulation is a major theoretical and practical undertaking, and should be seen as a last resort rather than the option of choice. If an algebraic or statistical solution can be found, it is usually preferable.

Step 4: implementation.
This step may be anything from writing down a simple equation, to several months' work constructing a computer program. Clearly, the simplest route must be sought. Even when a direct mathematical solution is possible, it may be useful to implement it in a general computer package such as Excel, so that the effects of varying the values of the independent variables can be studied rapidly. Indeed, it has been suggested that Excel is preferable to some more specialised simulation packages for implementing simulation models (Sermier, pers comm). Freeman (1988, 142) has issued a warning about the use of random number generators provided with some commercial packages.

Step 5: validation.
Different types of model require different types of validation (Lake 2001, 725). In many cases, what will matter is whether the outcome of the model matches sufficiently closely the relevant data. There is a potential pit-fall here, in that if the details of the model are based on data, a good fit to the data is only to be expected. It may be necessary to use only some of the data initially, while retaining some to test the model (a procedure known as split-sample validation). This is especially important if parameters (e.g. regression coefficients) are to be estimated from data (e.g. in a predictive model), since it raises the possibility of over-fitting, that is, of obtaining estimates of the parameters which fit a particular dataset better than the 'true' values of the parameters would. This in turn can give rise to the observed phenomenon that a model never fits data quite as well as it does the first time around.

Step 6: interpretation.
At the end of the cycle we have to return to the archaeology of the problem. What does it mean that a particular model fits, or does not fit, a particular dataset? In a sense, 'failure to fit' is the more creative outcome, because it is open-ended and points towards refinement or even complete re-casting of the model, but 'fit', while undoubtedly satisfying, is a 'closed' conclusion. The statisticians' saying 'always examine the residuals' is very relevant here — it is the difference between the data and the model that may really shed light on what is happening. One might even say that the fitted model represents a 'processual' conclusion to an archaeological problem, while failure to fit, particularly to a model which has 'succeeded' in other circumstances, may represent the more individualist outcome favoured by a post-processual approach, and we might distinguish between the model (the 'processual' part of the data) and the residuals (the 'extra-processual' part of the data).