Tuesday, April 10, 2018

Two Beautiful Mathematical Results - Part 2

In Part 1 of this article (see here) we featured the two results below, as well as a simple way to prove these formulas.
Here, we continue on the same topic, featuring and proving the formulas below, which are just the tip of the iceberg.
However cool these formulas might look, the biggest contribution here is a general framework to solve much more general problems of this type. The mathematical level is still relatively simple, accessible to people in their first year of college education, if they attended a solid course on calculus. These results are still easy to prove for people who have been exposed to the basics of complex number theory.
I haven't seen these results published anywhere, but my guess is that they are not new. I encourage readers to post questions on Quora or Stackexchange to find more references on this topic, as Google search is of no use here.The focus here is to get more people interested in mathematics, by featuring fascinating results that are not that hard to prove, even for high school students participating in mathematical Olympiads. Also, it could be an interesting, fresh, original topic for university professors to discuss in their lectures or for exams. Finally, if you have seen many similar formulas on Wikipedia (or elsewhere), and you are wondering how they are derived, you will find the solution in this article.
The last section introduces a new, tough, unrelated problem, still unsolved today, that will be of interest to people with a background in probability and/or number theory.

Sunday, April 1, 2018

I Analyzed 10 MM digits of SQRT(2) - Look at My Findings

This article is intended for practitioners who might not necessarily be statisticians or statistically-savvy. The mathematical level is kept as simple as possible, yet I present an original, simple approach to test for randomness, with an interesting application to illustrate the methodology. This material is not something usually discussed in textbooks or classrooms (even for statistical students), offering a fresh perspective, and out-of-the-box tools that are useful in many contexts, as an addition or alternative to traditional tests that are widely used. This article is written as a tutorial, but it also features an interesting research result in the last section. The example used in this tutorial shows how intuiting can be wrong, and why you need data science.
The main question that we want to answer is: Are some events occurring randomly, or is there a mechanism making the events not occurring randomly? What is the gap distribution between two successive events of the same type? In a time-continuous setting (Poisson process) the distribution in question is modeled by the exponential distribution. In the discrete case investigated here, the discrete Poisson process turns out to be a Markov chain, and we are dealing with geometric, rather than exponential distributions. Let us illustrate this with an example.
Example
The digits of the square root of two (SQRT(2)), are believed to be distributed as if they were occurring randomly. Each of the 10 digits 0, 1, ... , 9 appears with a frequency of 10% based on observations, and at any position in the decimal expansion of SQRT(2), on average the next digit does not seem to depend on the value of the previous digit (in short, its value is unpredictable.)  An event in this context is defined, for example, as a digit being equal to (say) 3. The next event is the first time when we find a subsequent digit also equal to 3. The gap (or time elapsed) between two occurrences of the same digit is the main metric that we are interested in, and it is denoted as G. If the digits were distributed just like random numbers, the distribution of the gap G between two occurrences of the same digit, would be geometric
Do you see any pattern in the digits below? Read full article here to find the answer, and to learn more about a powerful statistical technique.

Friday, March 16, 2018

A Simple Introduction to Complex Stochastic Processes - Part 2

In my first article on this topic (see here) I introduced some of the complex stochastic processes used by Wall Street data scientists, using a simple approach that can be understood by people with no statistics background other than a first course such as stats 101. I defined and illustrated the continuous Brownian motion (the mother of all these stochastic processes) using approximations by discrete random walks, simply re-scaling the X-axis and the Y-axis appropriately, and making time increments (the X-axis) smaller and smaller, so that the limiting process is a time-continuous one. This was done without using any complicated mathematics such as measure theory or filtrations.
Here I am going one step further, introducing the integral and derivative of such processes, using rudimentary mathematics. All the articles that I've found on this subject are full of complicated equations and formulas. It is not the case here. Not only do I explain this material in simple English, but I also provide pictures to show how an Integrated Brownian motion looks like (I could not find such illustrations in the literature), how to compute its variance, and focus on applications, especially to number theory, Fintech and cryptography problems. Along the way, I discuss moving averages in a theoretical but basic framework (again with pictures), discussing what the optimal window should be for these (time-continuous or discrete) time series.
You can read the full article, here
DSC Resources

Monday, February 5, 2018

Are the Digits of Pi Truly Random? - Must Read for Math and Data Geeks

This article covers far more than the title suggests. It is written in simple English and accessible to quantitative professionals from a variety of backgrounds. Deep mathematical and data science research (including a result about the randomness of Pi, which is just a particular case) are presented here, without using arcane terminology or complicated equations.  
The topic discussed here, under a unified framework, is at the intersection of mathematics, probability theory, chaotic systems, stochastic processes, data and computer science. Many exotic objects are investigated, such as an unusual version of the logistic map, nested square roots, and representation of a number in a fractional or irrational base system. 
The article is also useful to anyone interested in learning these topics, whether they have any interest in the randomness or Pi or not, because of the numerous potential applications. I hope the style is refreshing, and I believe that you will find plenty of material rarely if ever discussed in textbooks or in the classroom. The requirements to understand this material are minimal, as I went to great lengths (over a period of years) to make it accessible to a large audience.
The randomness of the digits of Pi is one of the most fascinating, unsolved mathematical problems of all times, having been investigated by many million of people over several hundred years. The scope of this article encompasses this particular problem as part of a far more general framework. More questions are asked than answered, making this document a stepping stone for future research.
This article is structured as follows:
1. General Framework
  • Questions, Properties and Notations about Chaotic Sequences Investigated Here
  • Potential Applications, Including Random Number Generation
2. Examples of Chaotic Sequences Representing Numbers
  • Data Science Step
  • Mathematical Step
  • Numbers in Base 2, 10, 3/2 or Pi 
  • Nested Square Roots
  • Logistic Map
3. About the Randomness of the Digits of Pi
  • The Digits of Pi are Random in the Logistic Map System
  • Paths to Proving Randomness in the Decimal System
  • Connection with Brownian Motions
4. Curious Facts
  • Randomness and The Bad Seeds Paradox
  • Application to Cryptography, Financial Markets, and HPC
  • Exercises
  • Digits of Pi in Base Pi

Wednesday, January 31, 2018

Four Interesting Math Problems

The level in this article is for college students familiar with calculus, This material will be also of interest to college professors looking for new material to teach, or for original exam questions, as well as for business data scientists with some spare time, interested in refreshing their math skills. The problems cover real analysis, mathematical algorithms and numerical precision, correct visualizations, as well as geometry. The third problem is the most interesting one in my opinion, and could become a subject of active mathematical research with one new great, unsolved conjecture being proposed, of a probabilistic nature. The last problem has many applications in engineering science.
This article is structured as follows:
1. The Simplest Function Defined by an Infinite Product
  • Exercise
2. Surprising Series for Powers of Number 2
3. From Continuous Fractions to Nested Square Roots and More
  • Algorithm to compute the coefficients
  • Problems
  • Example: Nested Square Root for the Number Pi
  • Conjecture
4. Geometry: Shape Rearrangements and Coverage Problems

Thursday, January 11, 2018

Beautiful Number Theory Problem and Sandbox for Data Scientists

The Waring conjecture - actually a problem associated with a number of conjectures, many now being solved - is one of the most fascinating mathematical problems. This article covers new aspects of this problem, with a generalization and new conjectures, some with a tentative solution, and a new framework to tackle the problem. Yet it is written in simple English and accessible to the layman.
I also review a number of famous related mathematical conjectures, including one with a $1 million award still waiting for a solution, as well as Goldbach's conjecture, yet unproved as of today.  Many curious properties of the Floor function are also listed, and the emphasis is on machine learning and efficient computer-intensive algorithms to try to find surprising results, which then need to be formally proved or disproved.
Content of this article:
1. General Framework
  • Spectacular Result
  • New Generalization of Golbach's Conjecture
  • New Generalization of Fermat's Conjecture
2. Generalized Waring Problem
  • Definitions
  • Main Results
  • Open Problems
  • Fun Facts (Actually, Conjectures!)
3. Algorithms and Source Code
  • Case n = 2: Sums of Two Terms
  • Case n = 4: Sums of Four Terms
4. Related Conjectures and Solved Problems
  • The One Million Dollar Conjecture

Wednesday, December 27, 2017

A Simple Introduction to Complex Stochastic Processes

Stochastic processes have many applications, including in finance and physics. It is an interesting model to represent many phenomena. Unfortunately the theory behind it is very difficult, making it accessible to a few 'elite' data scientists, and not popular in business contexts.
One of the most simple examples is a random walk, and indeed easy to understand with no mathematical background. However, time-continuous stochastic processes are always defined and studied using advanced and abstract mathematical tools such as measure theory, martingales, and filtration. If you wanted to learn about this topic, get a deep understanding on how they work, but were deterred after reading the first few pages of any textbook on the subject due to jargon and arcane theories, here is your chance to really understand how it works.
Rather than making it a topic of interest to post-graduate scientists only, here I make it accessible to everyone, barely using any maths in my explanations besides the central limit theorem. In short, if you are a biologist, a journalist, a business executive, a student or an economist with no statistical knowledge beyond Stats 101, you will be able to get a deep understanding of the mechanics of complex stochastic processes, after reading this article. The focus is on using applied concepts that everyone is familiar with, rather than mathematical abstraction. 
My general philosophy is that powerful statistical modeling and machine learning can be done with simple techniques, understood by the layman, as illustrated in my article on machine learning without mathematics or advanced machine learning with basic excel
1. Construction of Time-Continuous Stochastic Processes: Brownian Motion 
Probably the most basic stochastic process is a random walk where the time is discrete. The process is defined by X(t+1) equal to X(t) + 1 with probability 0.5, and to X(t) - 1 with probability 0.5. It constitutes an infinite sequence of auto-correlated random variables indexed by time. For instance, it can represent the daily logarithm of stock prices, varying under market-neutral conditions. If we start at t = 0 with X(0) = 0, and if we define U(t) as a random variable taking the value +1 with probability 0.5, and -1 with probability 0.5, then X(n) = U(1) + ... + U(n).  Here we assume that the variables U(t) are independent and with the same distribution. Note that X(n) is a random variable taking integer values between -n and +n.
Five simulations of a Brownian motion  (x-axis is the time t, u-axis is Z(t)
What happens if we change the time scale (x-axis) from daily to hourly, or to every millisecond? We then also need to re-scale the values (y-axis) appropriately, otherwise the process exhibits massive oscillations (from -n to +n) in very short time periods. At the limit, if we consider infinitesimal time increments, the process becomes a continuous one. Much of the complex mathematics needed to define these continuous processes do no more than finding the correct re-scaling of the y-axis, to make the limiting process meaningful. 

Two Beautiful Mathematical Results - Part 2

In Part 1 of this article (see   here ) we featured the two results below, as well as a simple way to prove these formulas. Here, we co...