Lévy distribution and long correlation times in supermarket sales

Abstract

Sales data in a commodity market (supermarket sales to consumers) has been analysed by studying the fluctuation spectrum and noise correlations. Three related products (ketchup, mayonnaise and curry sauce) have been analysed. Most noise in sales is caused by promotions, but here we focus on the fluctuations in baseline sales. These characterise the dynamics of the market. Four hitherto unnoticed effects have been found that are difficult to explain from simple econometric models. These effects are: (1) the noise level in baseline sales is much higher than can be expected for uncorrelated sales events; (2) weekly baseline sales differences are distributed according to a broad non-Gaussian function with fat tails; (3) these fluctuations follow a Lévy distribution of exponent α = 1.4 , similar to financial exchange markets and in stock markets; and (4) this noise is correlated over a period of 10–11 weeks, or shows an apparent power law spectrum. The similarity to stock markets suggests that models developed to describe these markets may be applied to describe the collective behaviour of consumers.