A Tutorial in Data Science: Lecture 5 – Generating Distributions by Spectral Analysis

by | Jan 12, 2021 | Math Lecture

When two states, i.e. possibly measured outcomes, of the stochastic sampling process of the underlying statistical object communicate, there is a probability of one occurring after the other, perhaps within the internal time (i.e. indexical ordering) of the measurement process, t \in \pi=(1, \cdots, n) for the sample space (\hat{X_1}, \cdots , \hat{X_n}). Arranging the resulting values as a list, there is some chance of one value occurring after the other. Such is a direction of communication between the states those values represent. When two states inter-communicate, there is a positive probability that each state will occur after the other (within the ordering \pi). For such inter-communicating states, they have the same period, defined as the GCD of distances between occurrences. The complex variable of functional communicativity can be described as the real probability of conditioned occurrence and the imaginary period of its intercommunications.

To describe our model by communicative functionals is to follow the Laplacian method of generating the distribution by finite difference equations. A single state, within state-space rather than time-space, is described as a complex variable s=\sigma + \omega i, where \sigma is the real functional relation between state \& system (or part \& whole), while \omega is its imaginary communicative relationship. If we view the branching evolution of the possible states measured in a system under sampling, then the actual sampled values is a path along this decision-tree. The total system, as its Laplacian, or characteristic, representation is the (tensorial) sum of the underlying sub-systems, of which each state belongs as a possible value. A continuum (real-distributional) system can only result as the infinite branching process, as thus each value a limit to an infinite path-sequence of rationalities in the state-system sub-dividing into state-systems until the limiting (stationary) systems-as-states are reached that are non-dynamic or static in the inner spatio-temporality of self-differentiation, i.e. non-dividing. Any node of this possibility-tree can be represented as a state of the higher-order system by a complex-value or as a system of the lower-order states by a complex-function. The real part of this value is the probability of the lower state occurring given the higher-system occuring (uni-directional communicativity), while the imaginary part is its relative period. Similarly, the real function of a higher system is the probability of lower states occurring given its occurence and the imaginary part is relative periods of the state-values.

Consider a binary 2-level tree. The whole tree, as represented by the root node, is described by the complex variable z_0=\theta_0 + \omega_0 and the 2 sub-nodes are defined by z_{1,0} \ \& \ z_{1,1} within the group of z_0. The whole system repeats itself every \omega_0 as thus the finite external-measurement of its absolute time-limit. The internal functional parts, f_{1,0} \ \& \ f_{1,1},

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