The use of ideas of Information Theory for studying "language" and intelligence in ants

In this review we integrate results of long term experimental study on ant "language" and intelligence which were fully based on fundamental ideas of Information Theory, such as the Shannon entropy, the Kolmogorov complexity, and the Shannon's equation connecting the length of a message ($l$) and its frequency $(p)$, i.e. $l = - \log p$ for rational communication systems. This approach, new for studying biological communication systems, enabled us to obtain the following important results on ants' communication and intelligence: i) to reveal "distant homing" in ants, that is, their ability to transfer information about remote events; ii) to estimate the rate of information transmission; iii) to reveal that ants are able to grasp regularities and to use them for "compression" of information; iv) to reveal that ants are able to transfer to each other the information about the number of objects; v) to discover that ants can add and subtract small numbers. The obtained results show that Information Theory is not only wonderful mathematical theory, but many its results may be considered as Nature laws.

Applications of Universal Source Coding to Statistical Analysis of Time Series

We show how universal codes can be used for solving some of the most important statistical problems for time series. By definition, a universal code (or a universal lossless data compressor) can compress any sequence generated by a stationary and ergodic source asymptotically to the Shannon entropy, which, in turn, is the best achievable ratio for lossless data compressors. We consider finite-alphabet and real-valued time series and the following problems: estimation of the limiting probabilities for finite-alphabet time series and estimation of the density for real-valued time series, the on-line prediction, regression, classification (or problems with side information) for both types of the time series and the following problems of hypothesis testing: goodness-of-fit testing, or identity testing, and testing of serial independence. It is important to note that all problems are considered in the framework of classical mathematical statistics and, on the other hand, everyday methods of data compression (or archivers) can be used as a tool for the estimation and testing. It turns out, that quite often the suggested methods and tests are more powerful than known ones when they are applied in practice.