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This Notebook has shown that while the Traditional Mean Averages are good descriptors of Static Sets, they are insufficient for describing Live Data Streams. The Mean Averages aren't sensitive enough for a Live Stream. In their insensitivity, they don't acknowledge Decay and they are not responsive enough to Fresh Data. Also the Mean Average isn't relevant enough. A Live Data Stream wants an average that is both sensitive and relevant. Enter the Decaying Average. We derive it and show that it makes up for all the shortcomings of the Mean Average. It is Sensitive to the Present and the Past, and is Relevant. We discover that neural networks are a way of calculating the Decaying Average. We also find that computing Decaying Averages confers an evolutionary advantage on the organism. This average allows the organism to predict, which enables it to anticipate, which assists in survival. We then looked at the difference between the Range of Possibility and the Realm of Probability, discovering that being able to compute the Realm of Probability also enhances survival because it allowed an organism to conserve precious resources. In the search for the Realm of Probability we came upon the arithmetic Average of Changes and the geometric Standard Deviation. These measures were connected by the Exponent of Change, which organisms established experientially. Finally we looked at our final fundamental tool, the Directional. This computational tool is the Decaying Average of the Directional Changes. This Notebook establishes the foundations of the three computational tools that are essential when dealing with Live Data Streams, i.e. Decaying Averages, Deviations & Directionals.
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