Win-Vector Blog

We describe ergodic theory in modern notation accessible to interested computer scientists.

The ergodic theorem (http://en.wikipedia.org/wiki/Ergodic theory (link)) is an important principle of recurrence and averaging in dynamical systems. However, there are some inconsistent uses of the term, much of the machinery is intended to work with deterministic dynamical systems (not probabilistic systems, as is often implied) and often the conclusion of the theory is mis-described as its premises.

By “interested computer scientists” we mean people who know math and work with probabilistic systems1, but know not to accept mathematical definitions without some justification (actually a good attitude for mathematicians also). Read more…

What is R²? In the context of predictive models (usually linear regression), where y is the true outcome, and f is the model’s prediction, the definition that I see most often is:

In words, R² is a measure of how much of the variance in y is explained by the model, f.

Under “general conditions”, as Wikipedia says,
R² is also the square of the correlation between the actual and predicted outcomes:

I prefer the “squared correlation” definition, as it gets more directly at what is usually my primary concern: prediction. If R² is close to one, then the model’s predictions mirror true outcome, tightly. If R² is low, then either the model does not mirror true outcome, or it only mirrors it loosely: a “cloud” that — hopefully — is oriented in the right direction. Of course, looking at the graph always helps:

The question we will address here is : how do you get from R² to correlation?

Read more…

We share our admiration for a set of results called “locality sensitive hashing” by demonstrating a greatly simplified example that exhibits the spirit of the techniques. Read more…

We give a simple explanation of the interrelated machine learning techniques called kernel methods and support vector machines. We hope to characterize and de-mystify some of the properties of these methods. To do this we work some examples and draw a few analogies. The familiar no matter how wonderful is not perceived as mystical. Read more…

Nina Zumel recently gave a very clear explanation of logistic regression ( The Simpler Derivation of Logistic Regression ). In particular she called out the central role of log-odds ratios and demonstrated how the “deviance” (that mysterious
quantity reported by fitting packages) is both a term in “the pseudo-R^2″ (so directly measures goodness of fit) and is the quantity that is actually optimized during the fitting procedure. One great point of the writeup was how simple everything is once you start thinking in terms of derivatives (and that it isn’t so much the functional form of the sigmoid that is special but its relation to its own derivative that is special).

We adapt these presentation ideas to make explicit the well known equivalence of logistic regression and maximum entropy models. Read more…

Logistic regression is one of the most popular ways to fit models for categorical data, especially for binary response data. It is the most important (and probably most used) member of a class of models called generalized linear models. Unlike linear regression, logistic regression can directly predict probabilities (values that are restricted to the (0,1) interval); furthermore, those probabilities are well-calibrated when compared to the probabilities predicted by some other classifiers, such as Naive Bayes. Logistic regression preserves the marginal probabilities of the training data. The coefficients of the model also provide some hint of the relative importance of each input variable.

While you don’t have to know how to derive logistic regression or how to implement it in order to use it, the details of its derivation give important insights into interpreting and troubleshooting the resulting models. Unfortunately, most derivations (like the ones in [Agresti, 1990] or [Hastie, et.al, 2009]) are too terse for easy comprehension. Here, we give a derivation that is less terse (and less general than Agresti’s), and we’ll take the time to point out some details and useful facts that sometimes get lost in the discussion. Read more…

Having a bit of history as both a user of machine learning and a researcher in the field I feel I have developed a useful perspective on the various trends, flavors and nuances in machine learning and artificial intelligence. I thought I would take a moment to outline a bit of it here and demonstrate how what we call artificial intelligence is becoming more statistical in nature. Read more…

We have been living in the age of “big data” for some time now. This is an age where incredible things can be accomplished through the effective application of statistics and machine learning at large scale (for example see: “The Unreasonable Effectiveness of Data” Alon Halevy, Peter Norvig, Fernando Pereira, IEEE Intelligent Systems (2009)). But I have gotten to thinking about the period before this. The period before we had easy access to so much data, before most computation was aggregation and before we accepted numerical analysis style convergence as “efficient.” A small problem I needed to solve (as part of a bigger project) reminded me what theoretical computer scientists did then: we worried about provable worst case efficiency.

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Recently, we had a client come to us with (among other things) the following question:
Who is more valuable, Customer Type A, or Customer Type B?

This client already tracked the net profit and loss generated by every customer who used his services, and had begun to analyze his customers by group. He was especially interested in Customer Type A; his gut instinct told him that Type A customers were quite profitable compared to the others (Type B) and he wanted to back up this feeling with numbers.

He found that, on average, Type A customers generate about $92 profit per month, and Type B customers average about $115 per month (The data and figures that we are using in this discussion aren’t actual client data, of course, but a notional example). He also found that while Type A customers make up about 4% of the customer base, they generate less than 4% of the net profit per month. So Type A customers actually seem to be less profitable than Type B customers. Apparently, our client was mistaken.

Or was he? Read more…

In the previous installment of the Statistics to English Translation, we discussed the technical meaning of the term ”significant”. In this installment, we look at how significance is calculated. This article will be a little more technically detailed than the last one, but our primary goal is still to help you decipher statements about significance in research papers: statements like “
”.

As in the last article, we will concentrate on situations where we want to test the difference of means. You should read that previous article first, so you are familiar with the terminology that we use in this one.

A pdf version of this current article can be found here.
Read more…