Events analysis applied to the Active Pulse
We've derived the general expressions for Info Quanta in both the Living Average Grid and the Directional Grid. Let us apply our mathematical findings to the highly significant Active Pulse (a particular Directional) to see what they reveal.
Review: Dynamics required to understand Relationship and Causality
But first let us review our progress to refresh and reinforce understanding. The Living Average and the Directional are the 1st and 2nd derivatives (rates of change) of any data stream that undergoes the Living Algorithm's digestive process. The Living Average Grid consists solely of quanta of info energy. Info energy does the Work of moving the Living Average from location to location. The Directional Grid, on the other hand, consists solely of quanta of info force. An examination of info force reveals the dynamics behind the Living Algorithm System. An analysis based in dynamics, not static states, is required to understand relationship and hence causality.
The highly significant Active Pulse of ones – the Pulse of Attention
The Active Pulse is the graphic representation of the Directional when the Living Algorithm digests a data stream consisting solely of 1s. Because Attention is required to turn on the Living Algorithm's iterative process, the Active Pulse is also known as the Pulse of Attention. Our initial proofs addressed general data streams, where the value of the data can be anything. Let us now turn our focus to a data stream where all the elements are equal to see what that will reveal. This analysis also applies to the more specific case of the Active Pulse whose elements are equal, just ones.
Review of Algebraic Terminology
Living Algorithm digests Instants to produce Moments
Let's first review our algebraic terminology in context to the Living Algorithm's digestive process. A living data stream consists of an ordered series of raw numbers. At each point in the data stream, N, a new piece of data, XN, enters the Living Algorithm System. Because these numbers are 1 dimensional entities without residual impact, we call them Instants. The Living Algorithm digests this piece of data, this Instant. The Living Algorithm digestion process spreads the info energy of the Instant, XN, over time, rather than dissipating the entire energy all at once. This spreading process creates a 2 dimensional Moment. In other words, the Living Algorithm transforms a 1D Instant (the Data, XN) into a 2D Moment. Simply put, the Living Algorithm creates an interaction between time, N, and data, X, that results in Moments.
Data Stream Moments characterized by Derivatives (Rates of Change)
To increase the dimensionality of a string of numbers, the Living Algorithm process produces measures that characterize the rates of change (derivatives) at that particular Moment in the Data Stream. A set of data stream derivatives characterizes the trajectories of each Moment. Thus far in our investigation, we have only concerned ourselves with the first 2 derivatives, the Living Average and the Directional.
Restriction: All Data Stream elements are equal, as in Active Pulse
4.1 In this particular discussion, we have further limited our focus to a data stream where all the members of the set are equal.
Accordingly, this discussion applies to a data stream consisting solely of 1s – the data stream that generates the highly significant Active Pulse.
Directional's Force Quanta characterize the Dynamics of Living Average's Energy Quanta
The above analysis further broke this interaction between time, N, and data, X, into its horizontal and vertical components. This differentiation generated some colored rectangles - deemed info quanta. The Living Average Grid (the grid of colored rectangles) consists of quanta of info energy. In contrast, the Directional Grid consists of quanta of info force. The energy quanta represent the individual contribution of each data point's info energy at each moment to the Living Average (the state of the System). The Directional's force quanta characterize the dynamic interactions of the Living Average's energy quanta.
Grids = Events (letters) by Moments (numbers)
Algebraically, we refer to the horizontal component of the Grids as Events, which are represented by a letter. The vertical components, the columns, of the Grids are Moments in the data stream, and are represented by a number. Accordingly, each info quantum is associated with a specific number and letter (Moment and Event). Summing up each of the colored rectangles (the info quantum) at each moment in time, N, (the columns) determines the value of the data stream measure. The Living Average's energy quanta are topped with a bar, while the Directional's force quanta are topped with an arrow. The following discussion will focus upon the Directional's force quanta, as they reveal the inner dynamics of the intriguing Active Pulse.
Impacts & Influences
There is one last item of note. The first quantum in an Event is referred to as an Impact. The subsequent members of the Event are referred to as Influences. Influences are always generated the same way no matter which Grid they belong to. Multiply the Prior Influence or Impact in the Stream by K, the Scaling Factor, to determine the value of the Current Influence. In other words, Influences go through logarithmic decay with each repetition (iteration) of Living Algorithm process. Conversely, the Impacts for each Grid are determined in a unique fashion. In other words, Living Average Impacts are computed in a slightly different fashion than Directional Impacts. In brief, Living Average Impacts require only one piece of data for computation, the incoming datum, while computing Directional Impacts requires the incoming data and all prior data. This single difference leads to the uniquely living nature of the Living Algorithm System.
Info Quanta in the Active Pulse at the 1st & 2nd Moments
The value of the 1st Moment of every derivative is identical.
4.2 The 1st moment of every Living Algorithm measure (data stream derivative) is identical. It is equal to the initial data byte, X1 = X, divided by the Decay Factor, D. This is true whether it is the initial Living Average, the initial Directional, or the initial Impact of 1st Event, A (Equations 2.5 & 2.13). In the following derivation, we regularly eliminate the subscript of the data point, X, because all the data points in the stream are equal (Eq. 4.1)
The 2nd Moment
4.3 Equation 2.17 provides us with the value of the Directional's 1st Event at the 2nd Moment, A2 arrow.
4.4 Equation 2.22 provides us with the value of the Directional's 2nd Event at the 2nd Moment, B2 arrow. We first eliminate the subscript from the Xs in the equation.
4.5 Due to the fact that all data points in this stream are equal, we can now factor out the X/D term.
4.6 With a simple algebraic manipulation, we can see that the 2nd term in the expression equals K, the Scaling Factor.
4.7 Substituting Equation 4.6 into Equation 4.5 provides us with the desired result – the value of Event B at the 2nd Moment.
4.8 The contribution of both Event A and Event B to the 2nd Moment are the same (Eq. 4.3 and Eq. 4.7)
4.9 Remember: the sum of these two values provides us with the Directional at the 2nd Moment (Eq. 2.23).
Notice how simple this equation is compared to our general term for the 2nd Directional (Eq. 2,24). This reduction is possible because of the equality of the points in the data stream.
Info Quanta in the Active Pulse at the 3rd Moment
4.10 Equation 3.9 provides us with the value of the 1st event, A, at the 3rd moment – a simple scaling of the prior moment in the event.
4.11 The 3rd moment in the 2nd event, B, is an Influence, not the initial Impact. Consequently we can apply the same scaling process (multiply the prior moment in the Event by K) to obtain the value of this Influence - this quanta of info force.
4.12 The 3rd moment in the 3rd event, C3, is an Impact, as it is the 1st element of Event Cs stream. (An Impact occurs whenever there is an alpha-numeric equivalency: A -> 1, B -> 2, C -> 3. Whenever the number of the Moment is greater than the letter of the Event, we have an Influence. The number of the Moment is never less than the letter of the Event, because every Event begins with an Impact. Nothing exists before that point in the data stream.) Accordingly we must first recall Equation 3.14, the general expression for the impact of the 3rd Event.
4.13 We first perform a simple algebraic reduction - combining denominators.
4.14 We then separate the factors. Note again how each of the prior data bytes plays a part in generating the impact of the 3rd Event.
4.15 As each of the data bytes in the stream is equivalent (the given Eq. 4.1), we eliminate the subscripts. This simplification allows us to factor out the ubiquitous X/D term.
4.16 We then perform some standard algebraic manipulations.
Remember the equivalency (Equation 4.6).
4.17 We substitute the Scaling Factor, K, into the above equation and arrive at the familiar result.
4.18 The contribution of each Event to the Directional at the 3rd moment is identical. Each quantum of info force is equal. (Equations 4.10, 4.11 and 4.17)
4.19 The sum of the Events associated with a particular Moment (the column) equals the value of the data stream derivative at this Moment, in this case the Directional (the 2nd derivative) at the 3rd moment.
4.20 Substituting the appropriate values provides us with this nifty simplification.
4.21 Compare the simplicity of the above expression (when all the data bytes are equal) with the following general expression (when the data bytes can be any value) (Eq. 3.5).
4.22 As a crosscheck for our analysis, let's see if we arrive at the same results from another direction. We first eliminate the subscripts.
4.23 We factor out the common X/D term.
4.24 Then break the division into parts.
We again remember the algebraic equivalency (Equation 4.6).
4.25 We substitute K, the scaling factor, for the more complicated expressions.
4.26 No surprise. The hoped for result is the same as before (Eq. 4.20). This redundancy substantiates all features of the current analysis.
4.27 We now take an inferential leap to obtain the general expression for all the Directionals in a data stream that consists of identical elements. Although we didn't perform the standard algebraic manipulations to arrive at this expression, we have crosschecked the results on the computer for the higher-level moments in the data stream. In fact, computer manipulations led the discovery process. Or should I say: the computer graphic visuals led the way. We first perceived the equivalency of the force quanta in each column of the Directional Grid. We then checked our visual perception by checking the data that generated the graph. Only finally did we perform the necessary algebraic manipulations.
4.28 Because all the data bytes in the stream are equivalent, this equation also reveals the values of the quantum of info force (the colored rectangles in the Directional Grid). For instance, at the Nth moment, there are N Events that contribute to the cause. The contribution of each data byte is equivalent. Consequently, we can divide the general expression by N to find the contribution of each of the force quantum for a particular moment. This leaves us with this expression for the force quanta associated with any Directional at Moment N (the Grid's columns).
4.29 Note this deceptively simple algebraic expression only holds true if all the values in the data stream are equal. Otherwise we are left with a bewildering array of increasing complex expressions. For instance, the Directional at only the 3rd moment, X3 arrow, requires a dauntingly complex expression to compute the values of its quantum (Eq. 3.15). This is the result if the values of only the first 3 terms in the data stream are allowed to vary. The complexity only accumulates as we add more data bytes.
The 1st expression, with the C on top, represents the initial Impact of the incoming data point upon the Directional, X3 arrow. This Impact initiates Event C. The remaining expressions represent the values of the Scaled Influences from the preceding Events A & B.
4.30 If we combine like terms, we can simplify this expression. Performing the standard algebraic manipulations, we arrive at the following result. Instead of breaking it into quanta, the 3rd Directional is expressed in terms of the contributions of the individual data points. The values of the individual quantum are hidden inside.
Living Average Grid: Data-based & Quantum-based Analysis Identical. Not so with Directional Grid
We can see that this complex perspective reveals very little, if anything, about the inner workings of the Active Pulse. The increasing complexity of the data-based perspective is an indication that a quantum-based analysis is a more appropriate method for understanding the Living Algorithm process. Note that the data-based and quantum-based analyses are identical for the Living Average Grid only. The Directional Grid requires a quantum-based analysis for a deeper understanding of underlying processes.
Factors behind Directional Simplification
Reiterating, only when all the data points in the stream are equal are we able to apply Equation 4.28 to the right. This incredible algebraic simplification is the general expression for all force quanta in the Directional Grid, not just the contribution of Events A, B & C upon the 3rd Moment (Eq. 4.29).
Two factors contribute to this simplification.
Every Influence, Living Average or Directional, subject to identical logarithmic decay
1) After the Event's initial Impact every subsequent Influence is treated identically (simply multiplied by K, the Scaling Factor) to arrive at the next Influence. In other words, the Influences and Impacts are subject to regular logarithmic decay with each iteration of the Living Algorithm process. This symmetry holds true whether the data points are equal or not.
When Data Points equal, all Power Quanta at the same Moment are Equal
2) When the data points are equal, an even more amazing algebraic fact emerges (established above). The force quanta that contribute to each moment are identical in size. In other words, the quantum from the contributing Events exerts the exact same influence on the Directional. The Directional, the data stream's acceleration, the Living Algorithm's 2nd derivative, is only made possible by Attention. Remember: this connectivity - this temporal spreading that enables data stream dynamics – requires sustained Attention. Accordingly, Attention provides the 'substance' (the scalar) that transforms acceleration quantum into force quantum.
Yet another Living Algorithm Symmetry
The equality of the Influence of each Event upon the Moment when the data is equal is yet another striking symmetry. We say 'yet another' because symmetries are standard business in the Living Algorithm system, especially when the values of the data stream are equal.
Link
Now that we have established some algebraic certainties, we can clear up some uncertainties. Specifically, why is the efficiency of the Active Pulse so much lower than the efficiency in many physical systems? To answer this question, check out the next article in the series – Info Entropy.