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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…

ContinueAdded by Vincent Granville on May 9, 2019 at 11:30am — No Comments

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…

Added by Vincent Granville on May 4, 2019 at 12:30pm — No Comments

So many fascinating and deep results have been written about the number (1 + SQRT(5)) / 2 and its related sequence - the Fibonacci numbers - that it would take years to read all of them. This number has been studied both for its applications (population growth, architecture) and its mathematical properties, for over 2,000 years. It is still a topic of active research.…

ContinueAdded by Vincent Granville on April 25, 2019 at 7:30am — No Comments

*Summary:** Finally there are tools that let us transcend ‘correlation is not causation’ and identify true causal factors and their relative strengths in our models. This is what prescriptive analytics was meant to be.*

Just when I thought we’d figured it all out,…

ContinueAdded by Vincent Granville on April 24, 2019 at 7:30pm — No Comments

I describe here the ultimate number guessing game, played with real money. It is a new trading and gaming system, based on state-of-the-art mathematical engineering, robust architecture, and patent-pending technology. It offers an alternative to the stock market and traditional gaming. This system is also far more transparent than the stock market, and can not be manipulated, as formulas to win the biggest returns (with real money) are made public. Also, it simulates a neutral,…

ContinueAdded by Vincent Granville on April 15, 2019 at 10:00am — No Comments

*Summary:** A new business model strategy based around intermediary platforms powered by AI/ML is promising the most direct path to fastest growth, profitability, and competitive success. Adopting this new approach requires a deep change in mindset and is quite different from just adopting AI/ML to optimize your current operations.…*

Added by Vincent Granville on April 8, 2019 at 11:00pm — No Comments

We investigate a large class of auto-correlated, stationary time series, proposing a new statistical test to measure departure from the base model, known as Brownian motion. We also discuss a methodology to deconstruct these time series, in order to identify the root mechanism that generates the observations. The time series studied here can be discrete or continuous in time, they can have various degrees of smoothness (typically measured using the Hurst exponent) as well as long-range or…

ContinueAdded by Vincent Granville on April 1, 2019 at 1:00pm — No Comments

I present here some innovative results from my most recent research on stochastic processes. chaos modeling, and dynamical systems, with applications to Fintech, cryptography, number theory, and random number generators. While covering advanced topics, this article is accessible to professionals with limited knowledge in statistical or mathematical theory. It introduces new material not covered in my recent book (available …

ContinueAdded by Vincent Granville on March 21, 2019 at 7:30am — No Comments

Determining the number of clusters when performing unsupervised clustering is a tricky problem. Many data sets don't exhibit well separated clusters, and two human beings asked to visually tell the number of clusters by looking at a chart, are likely to provide two different answers. Sometimes clusters overlap with each other, and large clusters contain sub-clusters, making a decision not easy.

For instance, how many clusters do you see in the picture below? What is the optimum number…

ContinueAdded by Vincent Granville on March 13, 2019 at 6:00pm — No Comments

Many times, complex models are not enough (or too heavy), or not necessary, to get great, robust, sustainable insights out of data. Deep analytical thinking may prove more useful, and can be done by people not necessarily trained in data science, even by people with limited coding experience. Here we explore what we mean by deep analytical thinking, using a case study, and how it works: combining craftsmanship, business acumen, the use and creation of tricks and rules of thumb, to provide…

ContinueAdded by Vincent Granville on March 7, 2019 at 1:46pm — No Comments

In this data science article, emphasis is placed on *science*, not just on data. State-of-the art material is presented in simple English, from multiple perspectives: applications, theoretical research asking more questions than it answers, scientific computing, machine learning, and algorithms. I attempt here to lay the foundations of a new statistical technology, hoping that it will plant the seeds for further research on a topic with a broad range of potential…

Added by Vincent Granville on February 23, 2019 at 11:00am — No Comments

Many of the following statistical tests are rarely discussed in textbooks or in college classes, much less in data camps. Yet they help answer a lot of different and interesting questions. I used most of them without even computing the underlying distribution under the null hypothesis, but instead, using simulations to check whether my assumptions were plausible or not. In short, my approach to statistical testing is is model-free, data-driven. Some are easy to implement even in Excel. Some…

ContinueAdded by Vincent Granville on February 13, 2019 at 7:00pm — No Comments

For background to this post, please see Learn Machine Learning Coding Basics in a weekend. Here,we present the glossary that we use for the coding and the mindmap attached to these classes and upcoming book. About 80 terms are included in the glossary, covering Ensembles, Regression, Classification,…

ContinueAdded by Vincent Granville on February 12, 2019 at 12:31pm — No Comments

**Logistic regression (LR)** models estimate the probability of a binary response, based on one or more predictor variables. Unlike linear regression models, the dependent variables are categorical. LR has become very popular, perhaps because of the wide availability of the procedure in software. Although LR is a good choice for many situations, it doesn't work well for *all* situations. For example:

- In propensity score…

Added by Vincent Granville on February 7, 2019 at 3:23pm — No Comments

This is another interesting problem, off-the-beaten-path. It ends up with a formula to compute the integral of a function, based on its derivatives solely.

For simplicity, I'll start with some notations used in the context of matrix theory, familiar to everyone: T(*f*) = *g*, where *f* and *g* are vectors, and T a square matrix. The notation T(*f*) represents the product between the matrix T, and the vector *f*. Now, imagine that the…

Added by Vincent Granville on February 3, 2019 at 5:30pm — 1 Comment

First days after the celebration of the New Year is the time when looking back we can analyze our actions, promises and draw conclusions whether our predictions and expectations came true. As 2018 came to its end, it is perfect time to analyze it and to set trends for the next year. The amount of data generated every minute is enormous. Therefore new approaches, techniques, and solutions have been developed.…

ContinueAdded by Vincent Granville on January 29, 2019 at 11:43am — No Comments

Extract from the upcoming Monday newsletter published by Data Science Central. Previous editions can be found here. The contribution flagged with a + is our selection for the picture of the week. To subscribe, …

ContinueAdded by Vincent Granville on January 27, 2019 at 3:20pm — No Comments

Extract from the upcoming Monday newsletter published by Data Science Central. Previous editions can be found here. The contribution flagged with a + is our selection for the picture of the week. To subscribe, …

ContinueAdded by Vincent Granville on January 20, 2019 at 12:15pm — No Comments

This article was written by Ajit Joakar.

In this longish post, I have tried to explain Deep Learning starting from familiar ideas like machine learning. This approach forms a part of my forthcoming book. I have used this approach in my teaching. It is based on ‘learning by exception,' i.e. understanding one concept and it’s limitations and then understanding how the subsequent concept…

ContinueAdded by Vincent Granville on January 16, 2019 at 9:48am — No Comments

*Summary:** Here are our 5 predictions for data science, machine learning, and AI for 2019. We also take a look back at last year’s predictions to see how we did.*

It’s that time of year again when we do a look back in order to offer a look forward. What trends will speed up, what things will actually happen,…

ContinueAdded by Vincent Granville on December 20, 2018 at 6:30pm — No Comments

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