Thermodynamics of data

The recently introduced concept of generalized thermodynamics is explored here in the context of 1d, 2d and 3d data analysis, performed on samples drawn from a 3d X-ray soil sample image. Different threshold levels are used to binarize the 3d sample, wherefrom relative frequencies of binary patterns are found and then used to address finite size scaling behavior of the response functions as a function of the disorder parameter (equivalent of temperature in thermodynamics). It is found that for different threshold levels response functions for increasing sample sizes approach the thermodynamic limit from different directions, with a crossover reminiscent of a transition from open to periodic boundaries of the Ising model, implying existence of a characteristic correlation scale. It is argued here that this characteristic scale corresponds to the "natural" properties of the data, where correlations within finite size samples are neither underestimated nor overestimated. In the current context of soil this scale may be related to the so-called Representative elementary volume (REV), while in other situations this characteristic scale should be interpreted in the context of the phenomenon under study.