A Color-Coded One Sentence Explanation of the Fourier Transform

Ingenious use of concepts and color to help explain the formula for the (inverse) Fourier Transform. Devised by graphics/systems programmer Stuart Riffle, he includes a scale factor 1/N and as he has explained deals with the inverse of the Fourier Transform, but this is still clever. Similar to Epps' geometric explanation of the Characteristic Function in probability theory, that we illustrated some time ago with a cut-out from a CD canister.

Via Revolutions Analytics via Stuart Riffle post on i-programmer.

Pine Needles in a Circle

A view of the cover of a 'sanitary sewer,' (and why would you want any other kind?!). And as promised last week, notice the accumulation of pine needles around the edge of the cover. On this day the wind was quite gusty, blowing along this sidewalk from the bottom of the image to the top. Due to this wind direction, many more pine needles have accumulated and piled up around the cover at the bottom than at the top. This forms a histogram of the frequency distribution of wind action distributed around the circumference of the sewer cover.

If we had data situated around the circumference of a circle we could display it as a circular dot plot as shown below from the book "Circular Statistics" by Fisher. These are arrival times, on a 24hr clock, are for 254 patients at an intensive care unit. Few arrive in the morning, many more arrive in late afternoon and early evening.

An estimate of the density of circular sample can be computed using something like the code for a circular density curve in the programming language R, as shown below.

 

Such graphical tools are the beginnings of modeling on spheres and other manifolds studied under the general heading of directional statistics.

A probability density function defined on the real line is sometimes wrapped around a circle. We have earlier seen that the results of such wrapping give rise to the characteristic function, a fundamental tool of probability modeling.

A Model of a Characteristic Function

I was looking at a CD holder and noticed that it could represent the unit circle and provide a model of a characteristic function. It reminded me of a paper by Epps, available here, whose abstract begins: "The value of a characteristic function of a random variable X at some real number t is the center of mass of the distribution of tX wrapped around the unit circle in the complex plane." I have cut the sides of the CD holder to leave a normal distribution wrapped around a circle. The balance point of this model is the center of mass mentioned by Epps. As t gets smaller, approaching zero, the wrapped distribution gets more concentrated around a modal point. Also, the balance point moves toward the edge of the unit circle, directly under the mode of the wrapped distribution. The rate at which this balance point moves, at the edge of the unit circle, is the mean of the the random variable X.