An analysis of biological random processes via optimized statistical models

Abstract

The concept of a sampled probability-density vector is defined. It is shown that a relationship may be established between this new estimation means and the random process, expressed by its moment vector. This is a linear transformation using invariant matrices i.e. matrices which are independent of the random process. Thus, in deriving biological probability-density models, instead of using an analytical model, to estimate their parameters and to check the distributional assumptions, a single probability-density vector is computed, subject to some constraints. An optimized statistical model is, thus, obtained, by minimizing a certain loss function, which expresses the inaccuracy of the model. The invariant matrices permitting to obtain the optimized model, starting from the moment vector, are given and the procedure is illustrated by examples. Then, the concept of a parametric probability-density space is defined and it is shown that each of the vectors belonging to this space may express the stationary, ergodic, random process equally well. Some typical constraints in the probability-density space are investigated. It is shown that the normal (Gaussian) law may be regarded as a very strong constraint in the probability-density space, while the integral law, expressing the cumulative distribution function, is a weak one. Between these extreme cases, the large class of the usual constraints are examined, which are determined by the prior knowledge of the process, as well as by some desired model features. Thus, the concept of a constrained probability-density vector is introduced. By using a linear-programming procedure and by observing some peak constraints as well as some slope-sign ones, an optimized model with desired shape is obtained, where a certain value of the variable has a very high probability. This leads to a procedure which enables synaptic models to be derived. In such a model, the constraints in the probability-density space may be regarded as a new expression of the information transmitted in the nervous system. Moreover, the loss function may express the “aptitude” of the random process to realize a given message. Thus, by using the optimized statistical model concept, probabilistic models with desired features for various biological processes may be obtained in a simple and general manner.