The methods discussed in section 3 group purchases by customer. However, the specific purchases made by the customer are of interest only in so far as they enable prediction of his future behavior and how he might respond to things the store might do, e.g. advertisements, sales, changes in products offered, changes in prices.
In general, we might regard the customer as a stochastic process, i.e. what he will buy (and whether he will come to the store at all), depends probabilistically on the state of his larder, and the actions of the store.
A regular customer may visit the store once per week for 5 years, i.e. make 250 visits. Some may make as many as 1,000 visits. Nevertheless, there often won't be enough information to make a very sophisticated model of a customer. Therefore, simplified models are worth considering.
The simplest model is that customer c has probability p(c,i) of buying item i. The matrix ||p(c,i)|| is likely to be approximable by a matrix of much lower rank, i.e. the customers form a space of lower dimension. If this is true, customers can be approximately characterized by a much smaller number of parameters than are needed for a complete probability distribution. This in turn means that accurate information about the customers can be obtained with smaller samples that would otherwise be required. If the assumption of independence of the members of the signature is valid, it still takes quite a lot of information to characterize the customer.
The next more elaborate model might take into account the state of the customer's larder. He won't buy more of certain items until he has consumed what he previously bought. If we regard the customer's state as given by the contents of his larder, we can regard his purchases as determined by a Markov process.
The model might be further elaborated to take into account his probable response to sales, etc. Economists would be tempted to try to ascribe a demand curve, most likely just two numbers--the demand at a base price and an elasticity.
We will not pursue these elaborations further in this article, but it seems likely that the most useful information to supermarket companies doing data mining will involve the probabilities of response of different kinds of customers to different stimuli.