## 5. Data Stream Derivatives

### Introduction

This Notebook will discuss the construction of Derivatives of a Data Stream: the Directionals and Deviations. We will also discuss some of their properties. First we will examine the difference between a Continuous and Discontinuous Data Stream. We will discuss forming derivatives from each type of Stream. Second we will discuss the Decaying Average Series, the first Derivative of the Data Stream. We will show how starting from the Decaying Average, that we can go scalar and arrive at Deviations or that we can go vector and end up with Directionals. Third, we find that at the root of the two types of Derivatives are the Vector Series and the Scalar Series. A simplification is derived and applied to both the Directionals and Deviations. Finally we examine the differences between Data Streams and their Derivative Series. Below is a list of the contents.

1. Building Derivatives of a Data Stream

A. Continuous & Discontinuous Data Streams
B. False Beginnings
C. Beginning a Discontinuous data Stream or If N < D

2. Deviations & Scalars; Directionals & Vectors

A. A Review of the Lower Derivatives
B. Terminology of Data Streams
C. The Higher Derivatives and their Sets
D. Reflections upon the Connections between Raw Data, their Derivatives, & the Real World

3. Properties of these 4 Average and Change Series

A. The Original or Zeroth term of the 4 Series
B. Simplification or Losing the Change Series, some Derivations
C. Casual Proof that the First Deviation is > or = to the First Directional
D. Streams vs Series
E. Dynamical Directionals

Summary & Conclusions