John D. Cook

Most sources that present Taylor’s theorem for functions of several variables stop at second order terms. One reason is that one or two terms are good enough for many applications. But the bigger reason is that things get more complicated when you include higher order terms. The kth order term in Taylor’s theorem is a rank k […]

I’ll be presenting at a webinar on Wednesday, December 13 at 1:00 PM Eastern. The title of the presentation is “Seven questions a statistician and answer for an attorney.” I will discuss, among other things, when common sense applies and when correct analysis can be counter-intuitive. There will be ample time at the end of […]

This post relates moment generating functions to the Laplace transform and to exponential generating functions. It also brings in connections to the z-transform and the Fourier transform. Thanks to Brian Borchers who suggested the subject of this post in a comment on a previous post on transforms and convolutions. Moment generating functions The moment generating function (MGF) of […]

The Shannon wavelet has an interesting plot: Given the complexity of the plot, the function definition is surprisingly simple: The Fourier transform is even simpler: it’s the indicator function of [-2π, -π] ∪ [π, 2π], i.e. the function that is 1 on the intervals [-2π, -π] and [π, 2π] but zero everywhere else. The Shannon […]

There are many theorems of the form where f and g are functions, T is an integral transform, and * is a kind of convolution. In words, the transform of a convolution is the product of transforms. When the transformation changes, the notion of convolution changes. Here are three examples. Fourier transform and convolution With the Fourier transform […]

Last week I wrote about Jentzsch’s theorem. It says that if the power series of function has a finite radius of convergence, the set of zeros of the partial sums of the series will cluster around and fill in the boundary of convergence. This post will look at the power series for the gamma function […]

From Orthogonal Polynomials and Special Functions by Richard Askey: At first the results we needed were in the literature but after a while we ran out of known results and had to learn something about special functions. This was a very unsettling experience for there were very few places to go to really learn about […]

The last digits of Fibonacci numbers repeat with period 60. This is something I’ve written about before. The 61st Fibonacci number is 2504730781961. The 62nd is 4052739537881. Since these end in 1 and 1, the 63rd Fibonacci number must end in 2, etc. and so the pattern starts over. It’s not obvious that the cycle should […]

Take a function that has a power series with a finite radius of convergence. Then the zeros of the partial sums will be dense around the boundary of convergence. That is Jentzsch’s theorem. Here are a couple plots to visualize Jentzsch’s theorem using the plotting scheme described in this post. First, we take the function f(z) […]

This post shows a connection between three families of orthogonal polynomials—Legendre, Chebyshev, and Jacobi—and the beta distribution. Legendre, Chebyshev, and Jacobi polynomials A family of polynomials Pk is orthogonal over the interval [-1, 1] with respect to a weight w(x) if whenever m ≠ n. If w(x) = 1, we get the Legendre polynomials. If w(x) = (1 […]

I’ve mentioned the Runge phenomenon in a couple posts before. Here I’m going to go into a little more detail. First of all, the “Runge” here is Carl David Tolmé Runge, better known for the Runge-Kutta algorithm for numerically solving differential equations. His name rhymes with cowabunga, not with sponge. Runge showed that polynomial interpolation […]

The previous post compared bits of information to answers in a game of Twenty Questions. The optimal strategy for playing Twenty Questions is for each question to split the remaining possibilities in half. There are a couple ways to justify this strategy: mixmax and average. The minmax approach is to minimize the worse thing that […]

I’ve had data privacy on my mind a lot lately because I’ve been doing some consulting projects in that arena. When I saw a tweet from Tim Hopper a little while ago, my first thought was “How many bits of PII is that?”. [1] π Things Only Left Handed Introverts Over 6′ 5″ with O+ […]

The Pareto probability distribution has density for x ≥ 1 where a > 0 is a shape parameter. The Pareto distribution and the Pareto principle (i.e. “80-20” rule) are named after the same person, the Italian economist Vilfredo Pareto. Samples from a Pareto distribution obey Benford’s law in the limit as the parameter a goes to […]

Random number generation is typically a two step process: first generate a uniformly distributed value, then transform that value to have the desired distribution. The former is the hard part, but also the part more likely to have been done for you in a library. The latter is relatively easy in principle, though some distributions […]

The beta-binomial model is the “hello world” example of Bayesian statistics. I would call it a toy model, except it is actually useful. It’s not nearly as complicated as most models used in application, but it illustrates the basics of Bayesian inference. Because it’s a conjugate model, the calculations work out trivially. For more on […]

Suppose you want to prevent your data science team from being able to find out information on individual customers, but you do want them to be able to get overall statistics. So you implement two policies. Data scientists can only query aggregate statistics, such as counts and averages. These aggregate statistics must be based on […]

It’s easy to visualize function from two real variables to one real variable: Use the function value as the height of a surface over its input value. But what if you have one more dimension in the output? A complex function of a complex variable is equivalent to a function from two real variables to two […]

The Kullback-Leibler divergence between two probability distributions is a measure of how different the two distributions are. It is sometimes called a distance, but it’s not a distance in the usual sense because it’s not symmetric. At first this asymmetry may seem like a bug, but it’s a feature. We’ll explain why it’s useful to measure […]

As companies get into data analysis for the first time, many of them are going to start by making the same mistakes that were common a century ago, then gradually recapitulate the development of modern statistics.

Fitting a polynomial to a function at more points might not produce a better approximation. This is Faber’s theorem, something I wrote about the other day. If the function you’re interpolating is smooth, then interpolating at more points may or may not improve the fit of the interpolation, depending on where you put the points. […]

Fourier-Bessel series are analogous to Fourier series. And like Fourier series, they converge pointwise near a discontinuity with the same kind of overshoot and undershoot known as the Gibbs phenomenon. Fourier-Bessel series Bessel functions come up naturally when working in polar coordinates, just as sines and cosines come up naturally when working in rectangular coordinates. […]

I’m experimenting with making animated versions of the kinds of images I wrote about in my previous post. Here’s an animated version of the exponential sum of the day for 12/4/17. Why that date? I wanted to start with something with a fairly small period, and that one looked interesting. I’ll have to do something […]

How do you create a database for testing that is like your production database? It depends on in what way you want the test database to be “like” the production one. Replacing sensitive data Companies often use an old version of their production database for testing. But what if the production database has sensitive information […]

The exponential sum of the day draws a line between consecutive partial sums of where m, d, and y are the current month, day, and two-digit year. The four most recent images show how different these plots can be. These images are from 10/30/17, 10/31/17, 11/1/17, and 11/2/17. Consecutive dates often produce very different images for a couple […]

I just got an evaluation copy of The Best Writing on Mathematics 2017. My favorite chapter was Inverse Yogiisms by Lloyd N. Trefethen. Trefethen gives several famous Yogi Berra quotes and concludes that Yogiisms are statements that, if taken literally, are meaningless or contradictory or nonsensical or tautological—yet nevertheless convey something true. An inverse yogiism is […]

According to this post from Stack Overflow, Perl is the most disliked programming language. I have fond memories of writing Perl, though it’s been a long time since I used it. I mostly wrote scripts for file munging, the task it does best, and never had to maintain someone else’s Perl code. Under different circumstances […]

Let ω(n) be the number of distinct prime factors of x. A theorem of Landau says that for N large, then for randomly selected positive integers less than N, ω-1 has a Poisson(log log N) distribution. This statement holds in the limit as N goes to infinity. Apparently N has to be extremely large before […]

Yesterday I heard that the county I live in, Harris County, is the 3rd largest is the United States. (In population. It’s nowhere near the largest in area.) Somehow I’ve lived here a couple decades without knowing that. Houston is the 4th largest city in the US, so it’s no shock that Harris County the […]

I was playing around with something in Mathematica and one of the images that came out of it surprised me. It’s a contour plot for the system function of a low pass filter. H[z_] := 0.05634*(1 + 1/z)*(1 - 1.0166/z + 1/z^2) / ((1 - 0.683/z)*(1 - 1.4461/z + 0.7957/z^2)) ContourPlot[ Arg[H[Exp[I (x + I […]

Series solutions to differential equations can be grubby or elegant, depending on your perspective. Power series solutions At one level, there’s nothing profound going on. To find a series solution to a differential equation, assume the solution has a power series, stick the series into the equation, and solve for the coefficients. The mechanics are […]

You can find any integer you want as a substring of the digits in π. (Probably. See footnote for details.) So you could encode a number by reporting where it appears. If you want to encode a single digit, the best you can do is break even: it takes at least one digit to specify […]

Terry Tao’s most recent blog post looks at the Chowla conjecture theoretically. This post looks at the same conjecture empirically using Python. (Which is much easier!) The Liouville function λ(n) is (-1)Ω(n) where Ω(n) is the number of prime factors of n counted with multiplicity. So, for example, Ω(9) = 2 because even though 9 […]

Time series analysis and digital signal processing are closely related. Unfortunately, the two fields use different terms to refer to the same things. Suppose you have a sequence of inputs x[n] and a sequence of outputs y[n] for integers n. Moving average / FIR If each output depends on a linear combination of a finite number of previous […]

Authors will often say that “without loss of generality” they will assume that a differential equation has no first order derivative term. They’ll explain that there’s no need to consider because a change of variables can turn the above equation into one of the form While this is true, the change of variables is seldom […]

Mathematical writing is the opposite of business writing in at least one respect. Math uses common words as technical terms, whereas business coins technical terms to refer to common ideas. There are a few math terms I use fairly often and implicitly assume readers understand. Perhaps the most surprising is almost as in “almost everywhere.” […]

If you take the fractional parts of the set of numbers {n cos nx : integer n > 0} the result is uniformly distributed for almost all x. That is, in the limit, the number of times the sequence visits a subinterval of [0, 1] is proportional to the length of the interval. (Clearly it’s not true […]

Probability has surprising uses, including applications to things that absolutely are not random. I’ve touched on this a few times. For example, I’ve commented on how arguments about whether something is really random are often moot: Random is as random does. This post will take non-random uses for probability in a different direction. We’ll start […]

There are many conventions for describing points on a sphere. For example, does latitude zero refer to the North Pole or the equator? Mathematicians tend to prefer the former and geoscientists the latter. There are also varying conventions for longitude. Volker Michel describes this clash of conventions colorfully in his book on constructive approximation. Many […]

The latest episode of Star Trek Discovery (S1E4) uses the word “supercomputer” a few times. This sounds jarring. The word has become less common in contemporary usage, and seems even more out of place in a work of fiction set more than two centuries in the future. According to Google’s Ngram Viewer, the term “supercomputer” […]

I’ve written a page that will show a different exponential sum each day, images along the line of the post Exponential sums make pretty pictures. Here’s page: https://www.johndcook.com/expsum/ Here are a few sample images. Small changes in the coefficients can make a big change in the appearance of the graphs.

Overfitting happens when a model does too good a job of matching a particular data set and so does a poor job on new data. The way traditional statistical models address the danger of overfitting is to limit the number of parameters. For example, you might fit a straight line (two parameters) to 100 data […]

Exponential sums are a specialized area of math that studies series with terms that are complex exponentials. Estimating such sums is delicate work. General estimation techniques are ham-fisted compared to what is possible with techniques specialized for these particular sums. Exponential sums are closely related to Fourier analysis and number theory. Exponential sums also make […]

If a function of one variable has two local maxima, it must have a local minimum in between. What about a function of two variables? If it has two local maxima, does it need to have a local minimum? No, it could have a saddle point in between, a point that is a local minimum […]

One way to obfuscate code is clever use of arcane programming language syntax. Hackers are able to write completely unrecognizable code by exploiting dark corners of programming techniques and languages. Some of these attempts are quite impressive. But it’s also possible to write clean source code that is nevertheless obfuscated. For example, it’s not at […]

How many musical scales are there? That’s not a simple question. It depends on how you define “scale.” For this post, I’ll only consider scales starting on C. That is, I’ll only consider changing the intervals between notes, not changing the starting note. Also, I’ll only consider subsets of the common chromatic scale; this post […]

Database fields can combine in subtle ways. For example, nationality is not usually enough to identify anyone. Neither is religion. But the combination of nationality and religion can be surprisingly informative. Information content of nationality How much information is contained in nationality? That depends on exactly how you define nations versus territories etc., but for […]

Iteration of the quadratic function f(x) = 4x(1-x) is a famous example in chaos theory. Here’s what the first few iterations look like, starting with 1/√3. (There’s nothing special about that starting point; any point that doesn’t iterate to exactly zero will do.) The values appear to bounce all over the place. Let’s look at a […]

The previous post looked at a particular cellular automaton, the so-called Rule 90. When started with a single pixel turned on, it draws a Sierpinski triangle. With random starting pixels, it draws a semi-random pattern that retains features like the Sierpinski triangle. There are only 256 possible elementary cellular automata, so it’s practical to plot […]

A couple months ago I wrote about how a simple random process gives rise to the Sierpinski triangle. Draw an equilateral triangle and pick a random point in the plane. Repeatedly pick a triangle vertex at random and move half way from the current position to that vertex. The result converges to a Sierpinksi triangle. […]