Properties of multivariate data investigated by fractal dimensionality

Elaborated data-mining techniques are widely available today. Nevertheless, many non-linear relations among variables remain undiscovered in multi-dimensional datasets. To address this issue we propose a method based on the concept of fractal dimension that explores the structure of multivariate data and apply the method to simulated data, as well as to local field potentials recorded from cat visual cortex. We find that with changes in the analysis scale, the dimensionality of the data often changes, indicating first that the data are not simple fractals with one unique dimension and second, that, at a certain scale, important changes in the geometric structure of the data may occur. The method can be used as a data-mining tool but also as a method for testing a model's fit to the data. We achieve the latter by comparing the dimensionality of the original data to the dimensionality of the data reconstructed from a model's description of the data (here using the general linear model). The method provides indispensable help in estimating the complexity of non-linear relationships within multivariate datasets.