John D. Cook

Automation doesn’t always save as much time or effort as we expect. The xkcd cartoon above is looking at automation as an investment. Does the work I put in now eventually save more work than I put into it? Automation may be well worth it even if the answer is “no.” Automation can be like […]

The sha function, also known as the Dirac comb, is denoted with the Cyrillic letter sha (Ш, U+0428). This letter was chosen because it looks like how people visualize the function, a long series of vertical spikes. The function is called the Dirac comb for the same reason. This function is very important in Fourier […]

Suppose project completion time follows a Pareto (power law) distribution with parameter α. That is, for t > 1, the probability that completion time is bigger than t is t-α. (We start out time at t = 1 because that makes the calculations a little simpler.) Now suppose we know that a project has lasted […]

There are a couple common uses of the term distribution in math. The most familiar is probability distribution, such as a beta distribution, a Poisson distribution, etc. Less familiar but still common is distributions in the sense of generalized functions, like the Dirac delta distribution. Anybody with much exposure to math will have heard of a […]

I’m making a couple changes to my Twitter accounts. First, I’m winding down @PerlRegex. I’ll stop tweeting there when my scheduled tweets run out. I suggest that everyone who has been following @PerlRegex start following @RegexTip instead. The latter is more general, but is mostly compatible with Perl. Second, I’m reviving my @DSP_Fact. I stopped […]

I was listening to a classic music station yesterday, and I heard the story of a professional pianist whose hand was injured in an accident. He then started learning trumpet and two years later he was a professional trumpeter. I didn’t catch the musician’s name. I was not surprised that a professional in one instrument could become […]

In the previous post, we looked at an elegant equation involving integrals of the sinc function and computed the corresponding integrals for the jinc function. It turns out the analogous equation holds for sums as well: As before, we’d like to compute these two sums and see whether we can compute the corresponding sums for the […]

The sinc function is defined by sinc(x) = sin(x)/x. Philip Woodward introduced the name of the function in 1952, saying it “occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own.” Here’s an elegant equation involving the integrals of the sinc function: When I ran […]

Almost three years ago I put up a web page to let you go back and forth between Unicode code points and LaTeX commands. Here’s the page and here’s a blog post explaining it. I’ve expanded the data the page uses by merging in data from the STIX Project. More queries should return successfully now. […]

Pi The other day I ran across the following continued fraction for π. Source: L. J. Lange, An Elegant Continued Fraction for π, The American Mathematical Monthly, Vol. 106, No. 5 (May, 1999), pp. 456-458. While the continued fraction itself is interesting, I thought I’d use this an example of how to typeset and compute […]

There are six or eight ways to define a Fourier transform. The differences in the various conventions are minor, but they lead to differences in the basic results. So whenever you look up a result, you have to make sure the reference’s definition matches the one you’re expecting. Or maybe you re-derive the result. This is good […]

This is what the book Social Media Mining calls the Big Data Paradox: Social media data is undoubtedly big. However, when we zoom into individuals for whom, for example, we would like to make relevant recommendations, we often have little data for each specific individual. We have to exploit the characteristics of social media and […]

Richard Guy’s Strong Law of Small Numbers says There aren’t enough small numbers to meet the many demands made of them. In his article by the same name [1] Guy illustrates his law with several examples of patterns that hold for small numbers but eventually fail. One of these examples is 3! – 2! + 1! = […]

Here are some of the non-technical books I’ve mentioned in blog posts this year. I posted the technical list a couple days ago. Maybe I should say “less technical” rather than “non-technical.” For example, Surely You’re Joking, Mr. Feynman is a book about a physicist, but it’s at least as much a human interest book as […]

“Nature does not consist entirely, or even largely, of problems designed by a Grand Examiner to come out neatly in finite terms, and whatever subject we tackle the first need is to overcome timidity about approximating.” H. and B. S. Jeffreys, Methods of Mathematical Physics, 2nd ed., Cambridge University Press, 1950, p. 8. Related post: […]

RSS feeds RSS lets you subscribe to blogs. It also lets you read posts in peace, free from distracting peripheral ads. This explains why Google would kill off the world’s most popular RSS reader. Blogs used to display an icon linking to the site’s RSS feed, and any still do. Blogging software still creates RSS feeds, […]

Here are some of the books I’ve mentioned in blog posts this year. One of these books may be just the present for a geek in your life. This post looks at technical books: math, science, engineering, programming. I’ll have a follow-up post with non-technical books I’ve written about. (Update: here’s the non-technical list.) Programming Book […]

I saw an impressive chart once of how numerical linear algebra algorithm efficiency have improved over time. I haven’t been able to find that chart, but here is something similar. Thanks to Naveen Palli for pointing this out. Even more remarkable [than Moore’s law] — and even less widely understood — is that in many areas, performance gains due […]

When I was in college, my view of applied math was something like the following. Applied math is mostly mathematical physics. Mathematical physics is mostly differential equations. Numerical solution of differential equations boils down to linear algebra. Therefore the heart of applied math is linear algebra. I still think there’s a lot of truth in […]

The term white noise is fairly common. People unfamiliar with its technical meaning will describe some sort of background noise, like a fan, as white noise. Less common are terms like pink noise, red noise, etc. The colors of noise are defined various ways, but they’re all based on an analogy between the power spectrum […]

A toy problem is a simplified problem meant to be a warm-up to a more complicated problem. I worked on a project earlier this year that was so complex that the write-up of the toy version grew to over 100 pages. We had to make a toy version of the toy version in order to have […]

Hadley Wickham posted a photo on Twitter back in September illustrating R list indices with pepper: Then a few days ago, Jenny Bryan posted on Twitter her follow up, an analogous photo for XML: Related post: R without Hadley Wickham

There are several ways to associate a matrix with a graph. The properties of these matrices, especially spectral properties (eigenvalues and eigenvectors) tell us things about the structure of the corresponding graph. This post is a brief reference for keeping these various matrices straight. The most basic matrix associated with a graph is its adjacency […]

This weekend I mentioned on Twitter that it’s spooky how well dimensional analysis catches errors. If you’re trying to calculate a number of horses, does your final result have units of horses? If it has units of cats or kilograms, something has gone wrong. This is such a simple idea, it’s remarkable that it’s worth checking. […]

Richard Feynman tells a story in Surely You’re Joking, Mr. Feynman that I’m reminded of periodically when I realize something is smaller and less sophisticated than I imagined. [Update: A couple people pointed out in the comments that I got the roles of the two characters in this story reversed, so I’ve corrected this.] Feynman tells the story of […]

A few years ago, a friend told me he was thinking about learning a certain technology because it was really hard to use. This was not something that had to be complex to solve a complex problem, but something that was unnecessarily complex. Why would anyone do that? His reasoning was that as a consultant, […]

Suppose you have data from a discrete power law with exponent α. That is, the probability of an outcome n is proportional to n-α. How can you recover α? A naive approach would be to gloss over the fact that you have discrete data and use the MLE (maximum likelihood estimator) for continuous data. That […]

Here are wordclouds for some of my most popular Twitter accounts. Thanks to Mike Croucher for creating these images. He explains on his blog how to create your own Twitter wordclouds using R. My most popular account is CompSciFact, tweets about computer science and related topics. AlgebraFact is for algebra, number theory, and miscellaneous pure […]

David Mumford wrote a blog post a few weeks ago in which he identified four tribes of mathematicians. Here’s a summary of his description of the four tribes. Explorers are people who ask — are there objects with such and such properties and if so, how many? … Alchemists … are those whose greatest excitement comes from […]

Today I needed to the derivative of the zeta function. SciPy implements the zeta function, but not its derivative, so I needed to write my own version. The most obvious way to approximate a derivative would be to simply stick a small step size into the definition of derivative: f’(x) ≈ (f(x+h) – f(x)) / […]

“Ever since Euclid, mathematical proofs have served a dual purpose: certifying that a statement is true and explaining why it is true. In the future these two epistemological functions may be divorced. In the future, the computer assistant may take care of the certification and leave the mathematician to look for an explanation that humans […]

Anthony Scopatz did an interview for Podcast.__init__ recently talking about xonsh, a command shell that blends Python and some traditions from bash. One line from the interview jumped out at me: … thinking very critically about what shells get used for and what they’re actually good at and what they’re not good at. I’ve wondered about […]

Hilary Mason made an important observation on Twitter a few days ago: You do not want to be an edge case in this future we are building. Systems run by algorithms can be more efficient on average, but make life harder on the edge cases, people who are exceptions to the system developers’ expectations. Algorithms, whether encoded in software or […]

Large companies take longer to start projects. How much longer? A plausible guess is that project lead time would be proportional to the logarithm of the company size. If a company with n employees has a hierarchy with every manager having m subordinates, the number of management layers would be around logm(n). If every project has […]

Four years ago I wrote about the wildfires in Bastrop, Texas. Here’s a photo from the time by Kerri West, used by permission. Today I visited Bastrop State Park on the way home from Austin. Some trees, particularly oaks, survived the fires. Pines have come back on their own in parts of the park. A volunteer working […]

A medical device company approached me with the following problem. Scientists had written academic journal articles about their product, but the sales force couldn’t understand what they said. My task was to read the articles, then tell the people in sales what the articles were saying in laymen’s terms. One of the questions that came […]

The article Deming, data and observational studies by S. Stanley Young and Alan Karr opens with Any claim coming from an observational study is most likely to be wrong. They back up this assertion with data about observational studies later contradicted by prospective studies. Much has been said lately about the assertion that most published results are false, particularly […]

It’s hard to transfer intellectual property. When I was managing software projects, it would take months to fully transfer a project from one person to another. This was with full access to and encouragement from the original developer. This was a transfer between peers, both part of the same environment with all its institutional memory. […]

The Insight 2015 conference highlighted some impressive applications of big data: predicting the path of hurricanes more accurately (as we saw with hurricane Patricia), improving the performance of athletes, making cars safer, etc. These applications involve large amounts of data. But more importantly they involve new data, not simply greater quantities of data we’ve had before. […]

A/B testing, or split testing, is commonly used in web marketing to decide which of two design options performs better. If you have so many visitors to a site that the number of visitors used in a test is negligible, conventional randomization schemes are the way to go. They’re simple and effective. But if you […]

A few weeks ago I got a message on Twitter saying that IBM’s Watson had identified me as an “influencer” and invited me to the company’s Insight 2015 conference. So that’s where I am this week. I had a brief interview last night. Someone took this photo as we were setting up.

The distance between the Earth and Mars depends on their relative positions in their orbits and varies quite a bit over time. This post will show how to compute the approximate distance over time. We’re primarily interested in Earth and Mars, though this shows how to calculate the distance between any two planets. The planets […]

You may expect that a burst of input will cause a burst of output. Sometimes that’s the case, but often a burst of input results in a long, smoothly decreasing succession of output. You may not get immediate results, but long-term results. This is true of life in general, but it’s also true in a precise sense of differential equations. […]

Suppose a test asks you to place 10 events in chronological order. Label these events A through J so that chronological order is also alphabetical order. If a student answers BACDEFGHIJ, then did they make two mistakes or just one? Two events are in the wrong position, but they made one transposition error. The simplest way […]

Electric lighting has changed the way we sleep, encouraging us to lose sleep by staying awake much longer after dark than we otherwise would. Or maybe not. A new study of three contemporary hunter-gatherer tribes found that they stay awake long after dark and sleep an average of 6.5 hours a night. They also don’t nap […]

Here’s an unusual formula for pi based on the product and least common multiple of the first m Fibonacci numbers. Unlike the formula I wrote about a few days ago relating Fibonacci numbers and pi, this one is not as simple to prove. The numerator inside the root is easy enough to estimate asymptotically, […]

Energy efficiency improvements can pay for themselves in the long run. Financing can make the improvements immediately cash-flow positive, but only if the loan tenor can match the useful life of the equipment. This enables the payments to be low enough that the projected energy savings exceeds the payments. PACE, which stands for Property Assessed Clean Energy, is […]

I help people make decisions in the face of uncertainty. Sounds interesting. I’m a data scientist. Not sure what that means, but it sounds cool. I study machine learning. Hmm. Maybe interesting, maybe a little ominous. I’m into big data. Exciting or passé, depending on how many times you’ve heard the term. Even though each […]

Here’s an unusual formula for π. Let Fn be the nth Fibonacci number. Then As mysterious as this equation may seem, it’s not hard to prove. The arctangent identity shows that the sum telescopes, leaving only the first term, arctan(1) = π/4. To prove the arctangent identity, take the tangent of both sides, use the addition law for tangents, and […]

When I was growing up, I was told that you could never become fluent in a second language, and I believed it. I had no reason not to. I didn’t know anybody who had become fluent at a second language, and I could think of plenty of people who had learned English as a second […]