Abstract :
Estimating the information contained in natural data, such as electroencephalography data, is unusually difficult because the relationship between the physical data and the information that it encodes is unknown. This unknown relationship is often called the encoding problem. The present work provides a solution to this problem by deriving a method to estimate the Tsallis entropy in natural data. The method is based on two findings. The first finding is that the physical instantiation of any information event, that is, the physical occurrence of a symbol of information, must begin and end at a discontinuity or critical point (maximum, minimum, or saddle point) in the data. The second finding is that, in certain data types such as the encephalogram (EEG), the variance within of an EEG waveform event is directly proportional to its probability of occurrence.
These two outcomes yield two results. The first is the easy binning of data into separate information events. The second is the ability to estimate probabilities in two ways: frequency counting and computing the variance within of an EEG waveform. These results are used to derive a linear estimator of the Tsallis entropy functional, allowing it to be estimated without deducing the encoding.
This method for estimating the Tsallis entropy is first used to estimate the information in simple signals. The amount of information estimated is highly accurate. The method is then applied to two problems in electroencephalography. The first is distinguishing normal aging from very early Alzheimerʹs disease (mild cognitive impairment), and the second is medication monitoring of Alzheimerʹs disease treatment. The former is done with an accuracy of 92% and the latter with an accuracy of 91%. This detection accuracy is the highest published accuracy in the literature, which suggests that this method for Tsallis entropy estimation is both accurate and useful.
