Free Book: Applied Stochastic Processes

Free Book: Applied Stochastic Processes


Full title: Applied Stochastic Processes, Chaos Modeling, and Probabilistic Properties of Numeration Systems. Published June 2, 2018. Author: Vincent Granville, PhD. (104 pages, 16 chapters.)
This book is intended to professionals in data science, computer science, operations research, statistics, machine learning, big data, and mathematics. In 100 pages, it covers many new topics, offering a fresh perspective on the subject. It is accessible to practitioners with a two-year college-level exposure to statistics and probability. The compact and tutorial style, featuring many applications (Blockchain, quantum algorithms, HPC, random number generation, cryptography, Fintech, web crawling, statistical testing) with numerous illustrations, is aimed at practitioners, researchers and executives in various quantitative fields.

New ideas, advanced topics, and state-of-the-art research are discussed in simple English, without using jargon or arcane theory. It unifies topics that are usually part of different fields (data science, operations research, dynamical systems, computer science, number theory, probability) broadening the knowledge and interest of the reader in ways that are not found in any other book. This short book contains a large amount of condensed material that would typically be covered in 500 pages in traditional publications. Thanks to cross-references and redundancy, the chapters can be read independently, in random order.
This book is available for Data Science Central members exclusively. The text in blue consists of clickable links to provide the reader with additional references.  Source code and Excel spreadsheets summarizing computations, are also accessible as hyperlinks for easy copy-and-paste or replication purposes. The most recent version of this book is available from this link, accessible to DSC members only. 
About the author
Vincent Granville is a start-up entrepreneur, patent owner, author, investor, pioneering data scientist with 30 years of corporate experience in companies small and large (eBay, Microsoft, NBC, Wells Fargo, Visa, CNET) and a former VC-funded executive, with a strong academic and research background including Cambridge University.
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Content
The book covers the following topics: 
1. Introduction to Stochastic Processes
We introduce these processes, used routinely by Wall Street quants, with a simple approach consisting of re-scaling  random  walks to make them time-continuous, with a finite variance, based on the central limit theorem.

Construction of Time-Continuous Stochastic Processes
From Random Walks to Brownian Motion
Stationarity, Ergodicity, Fractal Behavior
Memory-less or Markov Property
Non-Brownian Process

2. Integration, Differentiation, Moving Averages
We introduce more advanced concepts about stochastic processes. Yet we make these concepts easy to understand even to the non-expert. This is a follow-up to Chapter 1.

Integrated, Moving Average and Differential Process
Proper Re-scaling and Variance Computation
Application to Number Theory Problem

3. Self-Correcting Random Walks
We investigate here a breed of stochastic processes that are different from the Brownian motion, yet are better models in many contexts, including Fintech. 

Controlled or Constrained Random Walks
Link to Mixture Distributions and Clustering
First Glimpse of Stochastic Integral Equations
Link to Wiener Processes, Application to Fintech
Potential Areas for Research
Non-stochastic Case

4. Stochastic Processes and Tests of Randomness
In this transition chapter, we introduce a different type of stochastic process, with number theory and cryptography applications, analyzing statistical properties of numeration systems along the way — a recurrent theme in the next chapters, offering many research opportunities and applications. While we are dealing with deterministic sequences here, they behave very much like stochastic processes, and are treated as such. Statistical testing is central to this chapter, introducing tests that will be also used in the last chapters.

Gap Distribution in Pseudo-Random Digits
Statistical Testing and Geometric Distribution
Algorithm to Compute Gaps
Another Application to Number Theory Problem
Counter-Example: Failing the Gap Test

5. Hierarchical Processes
We start discussing random number generation, and numerical and computational issues in simulations, applied to an original type of stochastic process. This will become a recurring theme in the next chapters, as it applies to many other processes.

Graph Theory and Network Processes
The Six Degrees of Separation Problem
Programming Languages Failing to Produce Randomness in Simulations
How to Identify and Fix  the Previous Issue
Application to Web Crawling

6. Introduction to Chaotic Systems
While typically studied in the context of dynamical systems, the logistic map can be viewed  as a stochastic process, with an equilibrium distribution and probabilistic properties, just like numeration systems (next chapters) and processes introduced in the first four chapters.

Logistic Map and Fractals
Simulation: Flaws in Popular Random  Number  Generators
Quantum Algorithms

7. Chaos, Logistic Map and Related Processes
We study processes related to the logistic map, including a special logistic map discussed here for the first time, with a simple equilibrium distribution. This chapter offers a transition between chapter 6, and the next chapters on numeration system (the logistic map being one of them.)

General Framework
Equilibrium Distribution and Stochastic Integral Equation
Examples of Chaotic Sequences
Discrete, Continuous Sequences and Generalizations
Special Logistic Map
Auto-regressive Time Series
Literature
Source Code with Big Number Library
Solving the Stochastic Integral Equation: Example

8. Numerical and Computational Issues
These issues have been mentioned in chapter 7, and also appear in chapters 9, 10 and 11. Here we take a deeper dive and offer solutions, using high precision computing with BigNumber libraries. 

Precision Issues when Simulating, Modeling, and Analyzing Chaotic Processes
When Precision Matters, and when it does not
High Precision Computing (HPC)
Benchmarking HPC Solutions
How to Assess the Accuracy of your Simulation Tool

8. Digits of Pi, Randomness, and Stochastic Processes
Deep mathematical and data science research (including a result about the randomness of  p, which is just a particular case) are presented here, without using arcane terminology or complicated equations.  Numeration systems discussed here are a particular case of deterministic sequences behaving just like the stochastic process investigated earlier, in particular the logistic map, which is a particular case.

Application: Random Number Generation
Chaotic Sequences Representing Numbers
Data Science and Mathematical Engineering
Numbers in Base 2, 10, 3/2 or p
Nested Square Roots and Logistic Map
About the Randomness of the Digits of p
The Digits of p are Randomly Distributed in the Logistic Map System
Paths to Proving Randomness in the Decimal System
Connection with Brownian Motions
Randomness and the Bad Seeds Paradox
Application to Cryptography, Financial Markets, Blockchain, and HPC
Digits of p in Base p

10. Numeration Systems in One Picture
Here you will find a summary of much of the material previously covered on chaotic systems, in the context of numeration systems (in particular, chapters 7 and  9.)

Summary Table: Equilibrium Distribution, Properties
Reverse-engineering Number Representation Systems
Application to Cryptography

11. Numeration Systems: More Statistical Tests and Applications
In addition to featuring new research results and building on the previous chapters, the topics discussed here offer a great sandbox for data scientists and mathematicians. 

Components of Number Representation Systems
General Properties of these Systems
Examples of Number Representation Systems
Examples of Patterns in Digits Distribution
Defects found in the Logistic Map System
Test of Uniformity
New Numeration System with no Bad Seed
Holes, Autocorrelations, and Entropy (Information Theory)
Towards a more General, Better, Hybrid System
Faulty Digits, Ergodicity, and High Precision Computing
Finding the Equilibrium Distribution with the Percentile Test
Central Limit Theorem, Random Walks, Brownian Motions, Stock Market Modeling
Data Set and Excel Computations

12. The Central Limit Theorem Revisited
The central limit theorem explains the convergence of discrete stochastic processes to Brownian motions, and has been cited a few times in this book. Here we also explore a version that applies to deterministic sequences. Such sequences and treated as stochastic processes in this book.

A Special Case of the Central Limit Theorem
Simulations, Testing, and Conclusions
Generalizations
Source Code

13. How to Detect if Numbers are Random or Not
We explore here some deterministic sequences of numbers, behaving like stochastic processes or chaotic systems, together with another interesting application of the central limit theorem.

Central Limit Theorem for Non-Random Variables
Testing Randomness: Max Gap, Auto-Correlations and More
Potential Research Areas
Generalization to Higher Dimensions

14. Arrival Time of Extreme Events in Time Series
Time series, as discussed in the first chapters, are also stochastic processes. Here we discuss a topic rarely investigated in the literature: the arrival times, as opposed to the extreme values (a classic topic), associated with extreme events in time series.

Simulations
Theoretical Distribution of Records over Time

15. Miscellaneous Topics
We investigate topics related to time series as well as other popular stochastic processes such as spatial processes.

How and Why: Decorrelate Time Series
A Weird Stochastic-Like, Chaotic Sequence
Stochastic Geometry, Spatial Processes, Random Circles: Coverage Problem
Additional Reading (Including Twin Points in Point Processes)

16. Exercises

Link: Free Book: Applied Stochastic Processes