An Alternate Formulation for Computing/Validating the Shannon Entropy of Probability Distributions

One of the most pervasive applications in Computing, is the generation of Random numbers, which belong to a certain probability distribution such as a Gaussian (normal) distribution. These probability distributions possess statistical properties such as expected values (mean), variance (standard deviation), p-value, Entropy etc.; out of which Entropy is significant, for quantifying the amount of (useful) information, that a particular instance of a distribution embodies. This quantification of Entropy is of value as a characterizing metric, which determines the amount of randomness/uncertainty and/or redundancy that can be achieved using a particular distribution instance. This is particularly useful for communication, cryptographic and astronomical applications in this day and age. In the present work the Author introduces an alternate way to calculate the approximate value of the Information Entropy (with a variation to the formulation of Information Entropy by Claude Shannon, as known by the scientific community); by observing that a Takens embedding of the probability distribution yields a simple measure of the Entropy; by taking into consideration only four critical/representative points of the embedding. By comparative experimentation, the Author has been able to empirically verify that this alternate formulation is consistently valid: The baseline experiment chosen relates to Discrete Task Oriented Joint Source Channel Coding (DT-JSCC) which utilizes entropy computation to perform efficient and reliable task oriented communication (transmission and reception) as will be elaborated further. The author performed the comparison by employing the Shannon formulation for Entropy computation in the baseline DT-JSCC experiment and then repeating the experiment by employing the Entropy formulation, introduced in this work. Eventually, the accuracy of results obtained (data models generated) were almost identical (differing in accuracy by only ~ 1% overall). Thus, the alternate formulation introduced in this work, provides a reliable means of validating the random numbers obtained from the Shannon formulation and also potentially serves as a simpler, faster, and more computationally optimal method. This is particularly useful in applications, where there is a constraint on the computational resources available, such as mobile and limited devices. The method is also useful as a way of uniquely identifying and characterizing Random probability sources, such as those from astronomical and/or optical (photonic) phenomenon. The author also investigates the impact of incorporating the above notion of Entropy into the Mars Rover IER software and confirms the conclusions in the original article from Jet Propulsion Laboratories, NASA, which describes the ICER Progressive Wavelet Image Compressor.