Markov Language

From prooffreader.com  is an interactive color-coded matrix of transition probabilities from any given letter on a row to its following letter in a column. For example, along the row beginning with the letter "h", the darkest hue, represents the highest probability (47.42%), is for the letter that most likely follows, which is "e".

The most likely letter to follow "d" is "-", indicating that the most likely choice is no single letter, but instead "nothing". So that "d" most likely is at the end of a word.

There is a similar graphic of reverse transition probabilities, showing letters that most likely precede a given row choice.

It would be fun to simulate how words would behave when primed with this limited behavior of English. We could use our last post and these graphics of letter transition probabilities to simulate a "Markov language".

Dynamic Visualization of Markov Chains

Here is a visual demonstration of Markov Chains by Victor Powell. We've seen his work before in demonstrating conditional probability. These dynamic views of one, two, and then many state Markov Chains.

The program allows for varying transition probabilities, varying speed of travel between the states, and a realization of the resulting time series of state visits. Another very nice visualization.

The Knotted String

We've seen a paper on the Thrown String. Here is one on the Knotted String.University of California at San Diego physicists Raymer and Smith place various lengths of string in a box and film it tumbling for ten seconds. More specifically from their PNAS paper "Spontaneous knotting of an agitated string,"

Most of the measurements were carried out with a string having a diameter of 3.2 mm, a density of 0.04 g/cm, and a flexural rigidity of 3.1 × 10⁴ dynes·cm², tumbling in a 0.30 × 0.30 × 0.30-m box rotated at one revolution per second for 10 sec.

Results from 200 trials noted the proportion of knots formed for various lengths. These results are plotted above. The dependence of this knotted probability on other physical properties of the string are shown in their table below:

They conjecture that the string confinement and rotation promote braiding at the ends of the string, producing the knots. As one report noted  Apple's iPhone earbuds are 139 cm (55 inches) long and thus right at the 50% tangle-rate-sweet-spot at the top of the curve. Shorter earbuds would be welcome.

Sad Drowned Earthworms

Lots of rain this past week has saturated the ground and brought up earthworms attempting to escape drowning. These poor ones weren't lucky. Their last appearances were these squiggly shapes on the sidewalk. They resemble strings thrown on the ground, which is actually an established problem in probability called "The Thrown String". First formulated by J.L. Synge in a question in the Mathematical Gazette in 1968:

A perfectly flexible inextensible string of length L is thrown down at
random on a horizontal table. It is assumed that the form of the string
is represented by x =x(s), y =y(s), these functions possessing derivatives
of all orders for 0<s<L. The experiment is repeated many times.
What is the average value of the rectilinear distance between the ends
of the string?

Synge later explored the problem in the Mathematical Gazette in 1970. He collected experiments of actually throwing stings and measuring the distance between the endpoints. Let D be the distance between the endpoints of a string of length L.

Suggesting that a ratio around 1/3 might be possible. But his final conclusion leaves the problem open:

 Conclusion
The problem of the thrown string is not solved. Perhaps we should
say that it has not been adequately formulated.

Others have considered the problem. Two in particular: Clarke in the Mathematical Gazette (1971) represents the the string as a sequence of line segments J.F.C. Kingman in the Journal of the Royal Statistical Society B (1982) considers the string as a chain and models its dynamics on the way to the floor. But before this approach he considers the string as the realization of a stochastic process. The squiggly string is the continuous sample path of the process. He further argues that if the string is cut at a point P, relative to axes, one of which is tangent at P, the two segments are independent. This leaves us with a stochastic process with continuous sample paths and independent increments. This implies the process is Gaussian. The distance measured is then between two points from a bivariate normal distribution and as such twice the square of the distance is proportional to a chi-square distribution with 2 degrees of freedom.

Of course, none of this helps the poor worms!