7. Stochastic and Deterministic Models (1/2)

Models are usually either stochastic or deterministic models, although composite models do exist. Deterministic models are based on a known or hypothesised law of physics, mathematics or some other discipline, so that given input values will always produce the same result. By contrast, stochastic models accept that there is a probability distribution associated with the inputs, with the processes within the model and hence with the output, so that the same input can yield different output values.

Because stochastic models utilise probability density functions in one form or another, they need to be well based in statistical theory. By contrast, deterministic models do not depend on statistics.

One form of stochastic model is where a set of input data to and output data from a process are collected, and used to create a regression equation relating the input variables to the output. In this case, a different sample set will yield a somewhat different regression relationship. Such a model is also an empirical model since it is based on observed data.

Observed data can also produce deterministic models, but only when the data allows the scientist to develop and test the model.

An example of a deterministic model is the Law of Gravity. Sir Isaac Newton is supposed to have imagined the law when he was hit on the head by an apple falling from the tree under which he was sitting.

In this equation, F_g is the force applied by gravity, G is the universal gravitational constant, m are the mass of the two objects and R is the distance between them. This force, of course, applies to both objects. In the case of the apple and the earth, this force is very small relative to the mass of the earth, and so it will have a very small effect on the earth. But the force is much larger relative to the mass of the apple, and so the force is successful in moving the apple towards the earth.

The law of gravity also applies to celestial bodies. The moon, in travelling through space, will seek to follow a straight line and would thus soon disappear in the night sky. But it is attracted to the Earth by gravity. If the effects of gravity are equal to the centrifugal force, or the force attempting to keep the Moon on a straight path, then the Moon will stay in an orbit of a fixed size around the earth. In fact, the Moon is gradually getting further and further from the Earth. Why is this the case?