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Our caveman is now equipped with some powerful computational and predictive tools. He can predict the most likely next point of many useful Data Streams with his Decaying Average. Further he can even predict the range of likely variance from this average, with his Average Deviation. This allows him to save lots of energy by not wasting his valuable time on improbable possibilities. The next tool that we will examine, the Average Directional, allows our caveman to calculate local tendencies or directions.
The Average Directional is similar to the Average Deviation. The main difference is that, while the Average Deviation is the decaying average of the absolute magnitude of the difference from the mean, that the Average Directional is the decaying average of the directional magnitude of the difference from the mean. This is expressed algebraically as follows. This was the defining equation for the Average Deviation.
The Delta-X was defined as the absolute difference between the new data and the past average, i.e. decaying average because we are speaking about a Data Stream. In the pages that follow any time we are speaking about Data Streams, which is most of the time, and refer to an average, deviation, or directional, we are actually referring to a decaying average, decaying average deviation, or decaying average directional. For simplification we will leave the adjective, 'decaying', off all three terms. We will also leave the term 'average' off the deviation and directional. These adjectives, while omitted, are implied.
Because the delta-X in this equation refers to an absolute difference, we will enclose the X with the absolute value signs to signify this. Algebraically this redefines as follows.
Then we rewrite the Deviation with the new terminology:
Now let us redefine delta-X as the difference between the new data and the past average, with the positive or negative sign.
Using the same reasoning that we did above when speaking about Deviations, we come to the following definitional equation for a directional.
Note that a bar over the ÆX denotes the Average Deviation, while an arrow over the ÆX denotes the Average Directional. Verbally the Directional is the Decaying Average of the differences between the new data and the old average. We will repeat that the only difference between the Deviation and the Directional is that the Deviation is the average of absolute differences while the Directional is average of the directional differences. Notice that they both derive from the Decaying Average, one from the absolute differences between the new Data and the past Average, while the other from the directional differences.
As an aside: the Directional has absolutely no meaning in a static Data Set, because it always equals zero. A static Data Set is going nowhere; it has no direction. Locally there might be direction, but globally any direction is balanced out. To get a Directional algebraically, one calculates the mean average of the entire set and then adds up how much each individual data point differs from the mean. Because this mean average is exactly the middle of all the data, simply speaking half of the difference is greater and half is less. When added together they equal zero. Hence our Directional is contextual and only applicable to a Data Stream, not a Data Set.
Let us examine some advantages to computing local tendencies. Our caveman has his Decaying Average Stream to understand and predict the most likely occurrence. He has his Average Deviation to understand and predict his Realm of Probability so that he can save energy. Now this measure helps to predict the direction of the change. It addresses the question: How fast is our Decaying Average going up or down? Let us examine some of the advantages of this computational tool.
Our caveman knows that it is getting colder outside. The caveman who only calculates Decaying Averages and Realms of Probability freezes to death because he doesn't realize that the temperature is dropping quickly and doesn't get to shelter quickly enough. However our caveman who is sensitive enough to also calculate the direction and speed of the change is saved because he realizes the temperature is going down fast and gets inside before the blizzard occurs. Hence the caveman with the ability to calculate Directionals survives while the caveman without the ability perishes. The ability to calculate Directionals confers a necessary evolutionary advantage.
Our manager has got his Decaying Average to predict the expected number of covers and he has also got Decaying Deviation to get his Realm of Probability so that he can call someone in to work in case it gets busy. This guy is pretty good. But the manager who is better watches the rate and direction that the reservations come in. This is akin to the First Directional. If it is an 'increasing' night he calls someone in when it is at the border, if it is a 'declining' night, lots of cancellations, he might even send someone home, thus saving labor costs and still preserving service, by playing tendencies. The restaurant manager who doesn't play tendencies incurs higher labor costs, which weakens the economic health of the business threatening its survival, while the manager who plays the tendencies saves money for the business, which could be used for promotion, which further strengthens the business. The business, which calculates Directionals, even if 'intuitively', saves money on labor and is consequently more fit.
Any good boxer, basketball player, baseball batter, or football defender, to name just a few, to be successful must 'play the tendencies' or they are going to be a step behind. How successful one will ultimately be in competitive sports is determined by how well the athlete can 'play the tendencies' of the opponents. The Directional Series calculates the contextual tendencies of our Living Data Stream.