Monday, October 30, 2017

Logistic Map, Chaos, Randomness and Quantum Algorithms

The logistic map is the most basic recurrence formula exhibiting various levels of chaos depending on its parameter. It has been used in population demographics to model chaotic behavior. Here we explore this model in the context of randomness simulation, and revisit a bizarre non-periodic random number generator discovered 70 years ago, based on the logistic map equation. We then discuss flaws and strengths in widely used random number generators, as well as how to reverse-engineer such algorithms. Finally, we discuss quantum algorithms, as they are appropriate in our context.
Highlights
  • Java, Perl and Excel random number generators compared
  • Historical considerations
  • Backdoor planted by the NSA in some of these systems
  • Original material on complex random number generators
  • Image encryption
  • Periodicity detection, disctinctness quantum algorithm (big data)  
  • Practical solutions
  • Need for new programming language for quantum computing
  • Cool animated gif
  • Post-quantum cryptography
  • Generators based on irrational numbers
The article is not too long, as most of the technical details are provided in the numerous references. It covers many topics ranging from computer science, algorithms, big data, to probability theory and mathematics. The level is simple enough to be read by non-experts, yet of great value for the experts as well. Click here to read this new article.

Wednesday, October 25, 2017

Graph Theory: Six Degrees of Separation Problem

This famous statement -- the six degrees of separation -- claims that there is at most 6 degrees of separation between you and anyone else on Earth. Here we feature a simple algorithm that simulates how we are connected, and indeed confirms the claim. We also explain how it applies to web crawlers: Any web page is connected to any other web page by a path of 6 links at most.
The algorithm below is rudimentary and can be used for simulation purposes by any programmer: It does not even use tree or graph structures.  Applied to a population of 2,000,000 people, each having 20 friends, we show that there is a path involving 6 levels or intermediaries between you and anyone else. Note that the shortest path typically involves fewer levels, as some people have far more than 20 connections. 
Starting with you, at level one, you have twenty friends or connections. These connections in turn have 20 friends, so at level two, you are connected to 400 people. At level three, you are connected to 7,985 people, which is a little less than 20 x 400, since some level-3 connections were already level-2 or level-1. And so on. 
To read the full article with simulator, application to web crawling, some maths, and source code, click here
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Tuesday, October 17, 2017

How Mathematical Discoveries are Made

In one of my previous articles, you can learn the process about how discoveries are made by research scientists, from exploratory analysis, testing, simulations, data science guesswork, all the way to the discovery of a new theory and state-of-the-art statistical modeling,including new, fundamental mathematical/statistical equations.
This is a unique occasion to discover the creative path followed by inventors. Traditional research papers do not explain how a new paradigm was found, instead they focus on showing that the author's methodology is sound, original, and leads to useful results. My article fills the gap, and even though much of the material is accessible to non-experts, it also features deep results accessible to graduate students and professional PhD statisticians. If you work on a PhD thesis or in research (Academia or research laboratories from government and companies such as Microsoft, Google, Facebook, Intel or IBM) this article provides insights on how the brain works to come up with a discovery in analytical fields.  Each researcher obviously has her own approach to creativity, so I do not claim that this is the only path to innovation. Here I only share my own thought process.
I invite you to read or re-read the article (with major updates added recently), especially section 3. It is posted here, and as in all research articles, it features a number of open questions and challenges, that you might want to solve. Interestingly, it started as a little mathematical problem.

Wednesday, October 4, 2017

Interesting Problem for Serious Geeks: Self-correcting Random Walks

Section 3 was added on October 11. Section 4 was added on October 19.  An award is offered to solve any of the open questions, click here for details

This is another off-the-beaten-path problem, one that you won't find in textbooks. You can solve it using data science methods (my approach) but the mathematician with some spare time could find an elegant solution. Share it with your colleagues to see how math-savvy they are, or with your students. I was able to make substantial progress in 1-2 hours of work using Excel alone, thought I haven't found a final solution yet (maybe you will.) My Excel spreadsheet with all computations is accessible from this article. You don't need a deep statistical background to quickly discover some fun and interesting results playing with this stuff. Computer scientists, software engineers, quants, BI and analytic professionals from beginners to veterans, will also be able to enjoy it!
The problem
We are dealing with a stochastic process barely more complicated than a random walk. Random walks are also called drunken walks, as they represent the path of a drunken guy moving left and right seemingly randomly, and getting lost over time. Here the process is a self-correcting random walk, also called controlled random walk, in the sense that the walker, less drunk than in a random walk, is able to correct any departure from a straight path, more and more over time, by either slightly over- or under-correcting at each step. One of the model parameter (the positive parameter a) represents how drunk the walker is, with a = 0 being the worst. Unless a = 0, the amplitude of the corrections decreases over time to the point that eventually (after many steps) the walker walks almost straight and arrives at his destination. This model represents many physical processes, for instance the behavior of a stock market somewhat controlled by a government to avoid bubbles and implosions, and it is defined as follows:
Let's start with X(1) = 0, and define X(k) recursively as follows, for k > 1:
and let's define U(k), Z(k), and Z as follows:
where the V(k)'s are deviates from independent uniform variables on [0, 1], obtained for instance using the function RAND in Excel. So there are two positive parameters in this problem, a and b, and U(k) is always between 0 and 1. When b = 1, the U(k)'s are just standard uniform deviates, and if b = 0, then U(k) = 1. The case a = b = 0 is degenerate and should be ignored. The case a > 0 and b = 0 is of special interest, and it is a number theory problem in itself, related to this problem when a = 1. Also, just like in random walks or Markov chains, the X(k)'s are not independent; they are indeed highly auto-correlated.
Figure 1: Mixture-like distribution of Z (estimated) when b = 0.01 and a = 0.8
Prove that if a  > 0, then  X(k) rapidly converges to 0 as k increases. Also prove that the limiting distribution Z
  • always exists,
  • always takes values between -1 and +1, with min(Z) = -1 and max(Z) = +1,
  • is symmetric, with mean and median equal to 0
  • and does not depend on a, but only on b (if b not too close to 0)
For instance, for b =1, even a = 0 yields the same triangular distribution for Z, as any a  > 0.
If a  > 0 and b = 0, (the non-stochastic case) prove that 
  • Z can only take 3 values: -1 with probability 0.25, +1 with probability 0.25, and 0 with probability 0.50
  • If U(k) and U(k+1) have the same sign,then U(k+2) is of opposite sign  
And here is a more challenging question: In general, what is the limiting distribution of Z? Also, what happens if you replace the U(k)'s with (say) Gaussian deviates? Or with U(k) = | sin (k*k) | which has a somewhat random behavior?
Click here to read more, including partial solution.

Monday, October 2, 2017

9 Off-the-beaten-path Statistical Science Topics with Interesting Applications

You will find here nine interesting topics that you won't learn in college classes. Most have interesting applications in business and elsewhere. They are not especially difficult, and I explain them in simple English. Yet they are not part of the traditional statistical curriculum, and even many data scientists with a PhD degree have not heard about some of these concepts.
The topics discussed in this article include:
  • Random walks in one, two and three dimensions - With Video
  • Estimation of the convex hull of a set of points - Application to clustering and oil industry
  • Constrained linear regression on unusual domains - Application to food industry
  • Robust and scale-invariant variances
  • Distribution of arrival times of extreme events - Application to flood predictions
  • The Tweedie distributions - Numerous applications
  • The arithmetic-geometric mean - Fast computations of decimals of Pi
  • Weighted version of the K-NN clustering algorithm
  • Multivariate exponential distribution and storm modeling

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...