Click here<\/a> to get the book. For Data Science Central members only.<\/span><\/p>\nContent<\/strong><\/span><\/p>\nThe book covers the following topics:<\/span>\u00a0<\/span><\/p>\n1. Introduction to Stochastic Processes<\/strong><\/span><\/p>\nWe introduce these processes, used routinely by Wall Street quants, with a simple approach consisting of re-scaling \u00a0random \u00a0walks to make them time-continuous, with a finite variance, based on the central limit theorem.<\/span><\/p>\n\n- Construction of Time-Continuous Stochastic Processes<\/span><\/li>\n
- From Random Walks to Brownian Motion<\/span><\/li>\n
- Stationarity, Ergodicity, Fractal Behavior<\/span><\/li>\n
- Memory-less or Markov Property<\/span><\/li>\n
- Non-Brownian Process<\/span><\/li>\n<\/ul>\n
2. Integration, Differentiation, Moving Averages<\/strong><\/span><\/p>\nWe 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.<\/span><\/p>\n\n- Integrated, Moving Average and Differential Process<\/span><\/li>\n
- Proper Re-scaling and Variance Computation<\/span><\/li>\n
- Application to Number Theory Problem<\/span><\/li>\n<\/ul>\n
3. Self-Correcting Random Walks<\/strong><\/span><\/p>\nWe investigate here a breed of stochastic processes that are different from the Brownian motion, yet are better models in many contexts, including Fintech.<\/span>\u00a0<\/span><\/p>\n\n- Controlled or Constrained Random Walks<\/span><\/li>\n
- Link to Mixture Distributions and Clustering<\/span><\/li>\n
- First Glimpse of Stochastic Integral Equations<\/span><\/li>\n
- Link to Wiener Processes, Application to Fintech<\/span><\/li>\n
- Potential Areas for Research<\/span><\/li>\n
- Non-stochastic Case<\/span><\/li>\n<\/ul>\n
4. Stochastic Processes and Tests of Randomness<\/strong><\/span><\/p>\nIn 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.<\/span><\/p>\n\n- Gap Distribution in Pseudo-Random Digits<\/span><\/li>\n
- Statistical Testing and Geometric Distribution<\/span><\/li>\n
- Algorithm to Compute Gaps<\/span><\/li>\n
- Another Application to Number Theory Problem<\/span><\/li>\n
- Counter-Example: Failing the Gap Test<\/span><\/li>\n<\/ul>\n
5. Hierarchical Processes<\/strong><\/span><\/p>\nWe 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.<\/span><\/p>\n\n- Graph Theory and Network Processes<\/span><\/li>\n
- The Six Degrees of Separation Problem<\/span><\/li>\n
- Programming Languages Failing to Produce Randomness in Simulations<\/span><\/li>\n
- How to Identify and Fix\u00a0 the Previous Issue<\/span><\/li>\n
- Application to Web Crawling<\/span><\/li>\n<\/ul>\n
6. Introduction to Chaotic Systems<\/strong><\/span><\/p>\nWhile typically studied in the context of dynamical systems, the logistic map can be viewed \u00a0as a stochastic process, with an equilibrium distribution and probabilistic properties, just like numeration systems (next chapters) and processes introduced in the first four chapters.<\/span><\/p>\n\n- Logistic Map and Fractals<\/span><\/li>\n
- Simulation: Flaws in Popular Random \u00a0Number \u00a0Generators<\/span><\/li>\n
- Quantum Algorithms<\/span><\/li>\n<\/ul>\n
7. Chaos, Logistic Map and Related Processes<\/strong><\/span><\/p>\nWe 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.)<\/span><\/p>\n\n- General Framework<\/span><\/li>\n
- Equilibrium Distribution and Stochastic Integral Equation<\/span><\/li>\n
- Examples of Chaotic Sequences<\/span><\/li>\n
- Discrete, Continuous Sequences and Generalizations<\/span><\/li>\n
- Special Logistic Map<\/span><\/li>\n
- Auto-regressive Time Series<\/span><\/li>\n
- Literature<\/span><\/li>\n
- Source Code with Big Number Library<\/span><\/li>\n
- Solving the Stochastic Integral Equation: Example<\/span><\/li>\n<\/ul>\n
8. Numerical and Computational Issues<\/strong><\/span><\/p>\nThese 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.\u00a0<\/span><\/p>\n\n- Precision Issues when Simulating, Modeling, and Analyzing Chaotic Processes<\/span><\/li>\n
- When Precision Matters, and when it does not<\/span><\/li>\n
- High Precision Computing (HPC)<\/span><\/li>\n
- Benchmarking HPC Solutions<\/span><\/li>\n
- How to Assess the Accuracy of your Simulation Tool<\/span><\/li>\n<\/ul>\n
8. Digits of Pi, Randomness, and Stochastic Processes<\/strong><\/span><\/p>\nDeep mathematical and data science research (including a result about the randomness of\u00a0 p, which is just a particular case) are presented here, without using arcane terminology or complicated equations.\u00a0 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.<\/span><\/p>\n\n- Application: Random Number Generation<\/span><\/li>\n
- Chaotic Sequences Representing Numbers<\/span><\/li>\n
- Data Science and Mathematical Engineering<\/span><\/li>\n
- Numbers in Base 2, 10, 3\/2 or p<\/span><\/li>\n
- Nested Square Roots and Logistic Map<\/span><\/li>\n
- About the Randomness of the Digits of p<\/span><\/li>\n
- The Digits of p are Randomly Distributed in the Logistic Map System<\/span><\/li>\n
- Paths to Proving Randomness in the Decimal System<\/span><\/li>\n
- Connection with Brownian Motions<\/span><\/li>\n
- Randomness and the Bad Seeds Paradox<\/span><\/li>\n
- Application to Cryptography, Financial Markets, Blockchain, and HPC<\/span><\/li>\n
- Digits of p in Base p<\/span><\/li>\n<\/ul>\n
10. Numeration Systems in One Picture<\/strong><\/span><\/p>\nHere 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 \u00a09.)<\/span><\/p>\n\n- Summary Table: Equilibrium Distribution, Properties<\/span><\/li>\n
- Reverse-engineering Number Representation Systems<\/span><\/li>\n
- Application to Cryptography<\/span><\/li>\n<\/ul>\n
11. Numeration Systems: More Statistical Tests and Applications<\/strong><\/span><\/p>\nIn 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.<\/span>\u00a0<\/span><\/p>\n\n- Components of Number Representation Systems<\/span><\/li>\n
- General Properties of these Systems<\/span><\/li>\n
- Examples of Number Representation Systems<\/span><\/li>\n
- Examples of Patterns in Digits Distribution<\/span><\/li>\n
- Defects found in the Logistic Map System<\/span><\/li>\n
- Test of Uniformity<\/span><\/li>\n
- New Numeration System with no Bad Seed<\/span><\/li>\n
- Holes, Autocorrelations, and Entropy (Information Theory)<\/span><\/li>\n
- Towards a more General, Better, Hybrid System<\/span><\/li>\n
- Faulty Digits, Ergodicity, and High Precision Computing<\/span><\/li>\n
- Finding the Equilibrium Distribution with the Percentile Test<\/span><\/li>\n
- Central Limit Theorem, Random Walks, Brownian Motions, Stock Market Modeling<\/span><\/li>\n
- Data Set and Excel Computations<\/span><\/li>\n<\/ul>\n
12. The Central Limit Theorem Revisited<\/strong><\/span><\/p>\nThe 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.<\/span><\/p>\n\n- A Special Case of the Central Limit Theorem<\/span><\/li>\n
- Simulations, Testing, and Conclusions<\/span><\/li>\n
- Generalizations<\/span><\/li>\n
- Source Code<\/span><\/li>\n<\/ul>\n
13. How to Detect if Numbers are Random or Not<\/strong><\/span><\/p>\nWe explore here some deterministic sequences of numbers, behaving like stochastic processes or chaotic systems, together with another interesting application of the central limit theorem.<\/span><\/p>\n
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