Wednesday, May 29, 2019

Gentle Approach to Linear Algebra, with Machine Learning Applications

This simple introduction to matrix theory offers a refreshing perspective on the subject. Using a basic concept that leads to a simple formula for the power of a matrix, we see how it can solve time series, Markov chains, linear regression, data reduction, principal components analysis (PCA) and other machine learning problems. These problems are usually solved with more advanced matrix calculus, including eigenvalues, diagonalization, generalized inverse matrices, and other types of matrix normalization. Our approach is more intuitive and thus appealing to professionals who do not have a strong mathematical background, or who have forgotten what they learned in math textbooks. It will also appeal to physicists and engineers. Finally, it leads to simple algorithms, for instance for matrix inversion. The classical statistician or data scientist will find our approach somewhat intriguing. 
Content
1. Power of a matrix
2. Examples, Generalization, and Matrix Inversion
  • Example with a non-invertible matrix
  • Fast computations
3. Application to Machine Learning Problems
  • Markov chains
  • Time series
  • Linear regression

Tuesday, May 7, 2019

Confidence Intervals Without Pain

We propose a simple model-free solution to compute any confidence interval and to extrapolate these intervals beyond the observations available in your data set. In addition we propose a mechanism  to sharpen the confidence intervals, to reduce their width by an order of magnitude. The methodology works with any estimator (mean, median, variance, quantile, correlation and so on) even when the data set violates the classical requirements necessary to make traditional statistical techniques work. In particular, our method also applies to observations that are auto-correlated, non identically distributed, non-normal, and even non-stationary. 
No statistical knowledge is required to understand, implement, and test our algorithm, nor to interpret the results. Its robustness makes it suitable for black-box, automated machine learning technology. It will appeal to anyone dealing with data on a regular basis, such as data scientists, statisticians, software engineers, economists, quants, physicists, biologists, psychologists, system and business analysts, and industrial engineers. 
In particular, we provide a confidence interval (CI) for the width of confidence intervals without using Bayesian statistics. The width is modeled as L = A / n^B and we compute, using Excel alone, a 95% CI for B in the classic case where B = 1/2. We also exhibit an artificial data set where L = 1 / (log n)^Pi. Here n is the sample size.

Despite the apparent simplicity of our approach, we are dealing here with martingales. But you don't need to know what a martingale is to understand the concepts and use our methodology. 

Saturday, May 4, 2019

Re-sampling: Amazing Results and Applications

This crash course features a new fundamental statistics theorem -- even more important than the central limit theorem -- and a new set of statistical rules and recipes. We discuss concepts related to determining the optimum sample size, the optimum k in k-fold cross-validation, bootstrapping, new re-sampling techniques, simulations, tests of hypotheses, confidence intervals, and statistical inference using a unified, robust, simple approach with easy formulas, efficient algorithms and illustration on complex data.
Little statistical knowledge is required to understand and apply the methodology described here, yet it is more advanced, more general, and more applied than standard literature on the subject. The intended audience is beginners as well as professionals in any field faced with data challenges on a daily basis. This article presents statistical science in a different light, hopefully in a style more accessible, intuitive, and exciting than standard textbooks, and in a compact format yet covering a large chunk of the traditional statistical curriculum and beyond.
In particular, the concept of p-value is not explicitly included in this tutorial. Instead, following the new trend after the recent p-value debacle (addressed by the president of the American Statistical Association), it is replaced with a range of values computed on multiple sub-samples. 
Our algorithms are suitable for inclusion in black-box systems, batch processing, and automated data science. Our technology is data-driven and model-free. Finally, our approach to this problem shows the contrast between the data science unified, bottom-up, and computationally-driven perspective, and the traditional top-down statistical analysis consisting of a collection of disparate results that emphasizes the theory. 
Contents
1. Re-sampling and Statistical Inference
  • Main Result
  • Sampling with or without Replacement
  • Illustration
  • Optimum Sample Size 
  • Optimum K in K-fold Cross-Validation
  • Confidence Intervals, Tests of Hypotheses
2. Generic, All-purposes Algorithm
  • Re-sampling Algorithm with Source Code
  • Alternative Algorithm
  • Using a Good Random Number Generator
3. Applications
  • A Challenging Data Set
  • Results and Excel Spreadsheet
  • A New Fundamental Statistics Theorem
  • Some Statistical Magic
  • How does this work?
  • Does this contradict entropy principles?
4. Conclusions

Variance, Attractors and Behavior of Chaotic Statistical Systems

We study the properties of a typical chaotic system to derive general insights that apply to a large class of unusual statistical distribut...