Menu Basket Queue

This is a view of a basket of menus at a restaurant in Chincoteague, Virginia. Notice the pattern of marks left as groups of menus scratch the wall when they are returned to the basket. As customers are seated at the restaurant, they are given menus that are removed from the right hand side of the basket. After ordering, the menus are returned to the basket and placed to the right of the remaining menus. When a single is menu returned, it nicks the wall at a location that depends on how many menus are currently in the hands of customers. If few menus are out with the customers, more remain in the basket and the wall marks of this returning menu will be further to the right. If many menus are out with customers, say just before the lunch rush, this returning menu will make a mark on the wall further to the left.

But it is not often that a single menu is returned alone. It is much more likely that a group of menus will be returned to the basket in a bunch. The size of the bunch that is returned is random depending on the size of the party seated. Each of the menus in the bunch makes a mark on the wall as they are returned to the basket. What we see is a steady state distribution of the number of seated customers, with menus in hand, waiting for their order to be taken.

You Must Be This Old To ...

Check your IDs before going out treat-or-treating tonight. From Five Thirty Eight.

♬ Let's All Go to the Lobby ♬

Everybody to the Lobby or the Basement! Got to go out or get that laundry done! Patterns of discrete wear don't keep our collective behavior discreet. (From an NYC elevator).♬ Let's All Go to the Lobby ♬

And so faintly you came tapping...

Here are images from two films from 2014. First, from the movie "The Forger" starring John Travolta and second, from "Kingsman: The Secret Service" starring Colin Firth. In both films secret entries must be made to advance the plot. In the Kingsman it is ultimately to save the people of the world from destruction. In The Forger, it is to steal a Monet painting and replace it with a forgery. Each director accomplishes this entry by looking at the residue of fingers on the pin pad keys. In the case of The Forger, these marks of wear and use are highlighted with a spray liquid that is then observed under a black light. This narrows down a four-digit pin number to 1 in 10,000. But in The Forger, Christopher Plummer, who plays the father to Travolta's character, guesses the correct 4-digit pin, 4867, on the first try!

To me, the way the 6 key has a trailing pattern of use, I would have thought it would have been the last digit of the combination. But no. Was this a bad edit, or am I missing something that makes 4867 so obvious in the first picture? In the second image, from Kingsman, I seem to recall a more automated way that the main character ran through the 10,000 possibilities to come up with the correct pin of 8539. It's been a while since I saw that movie, and I can't recall the details.

Either way, patterns of wear and use either facilitate saving the world or robbing a museum. Your choice.

Daily Double - Maybe

Coming across the Daily Doubles in the TV game show Jeopardy! can quickly propel contestants into the lead.  They do not have to compete with others to come up with the question and they can wager a large sum. In 2014 contestant Arthur Chu upset Jeopardy! traditionalists by bouncing around the board hoping to hit upon a Daily Double. Using data from the fan site J!Archive, Nathan Yau from Flowing Data has tabulated the locations of the Daily Doubles for 31 seasons totaling 13,633 Daily Doubles and shown them as relative frequencies. The darker the color the more likely a Daily Double for that position. Clearly, the fourth row is most favored by a small amount.

Scrabble Distribution

Here is a display of the frequency distribution of the letters used in Scrabble arranged alphabetically.

And here is the frequency distribution of letters arranged by their point value. Although the left-most stack is labeled with the letter "A", it is a stack of all the letters with a one-point value: A,E,I,L,N,O,R,S,T,U  and similarly for the other stacks. Not shown are the two blank tiles.

Top of the Line

Upscale neighborhood. Greatest frequency of wear is on the premium.
Forwarded by a colleague (thanks Jun). Originally, I think, from Reddit.

Gym Freqs

A submission from teacher Jon Purdy. Jon writes of his picture of exercise weights in a gym:

[I] noticed this while exercising today. More wear on the middle weights indicate greater usage, with usage dropping off as the weights get heavier and slight drop off with the lightest weights.

A great example of a discrete frequency distribution of use. Thanks Jon.
Next week, I'll have some measurements of this use and wear.

Testing Poissonness Petals

Last week’s image was of cherry blossom petals that had fallen on a stone walk, a random realization of a spatial Poisson process. In such a process, the probability of some number of petals falling on any stone is proportional to the area of the stone. The mean number of petals falling on the larger stones is 8.71. The mean number of petals falling on the smaller stones is 5.58. Their ratio is 1.56. The ratio of the stones’ area is 1.5. Such a close agreement is what a Poisson process should produce.

Not so fast, implies commenter Kevin. He suggests that we should test whether our observed ratio of 1.56 is significantly different from the ratio of the stones’ area of 1.5. My intuition tells me that, with the large variability in a Poisson process and such a small sample of fallen petals, we would need a much larger difference between the observed and expected ratios for the difference to be significant. But let’s test it.

First, consider the larger stones. If we count those laid down horizontally, left to right, from top to bottom, my counts (again likely prone to error) are 12,9,10,9,11,13,3,13,11,7. For the larger stones laid down vertically, left to right, from top to bottom, I count 12,6,7,6,11,6,6,10,10,4,7 petals. For the smaller, square stones, left to right, from top to bottom, I count 3,3,2,6,8,4,8,11,4,5,7,6 petals.  Let’s investigate whether these data can be modeled with a Poisson distribution. For example, the mean and variance for the larger stones are 8.71 and 8.61, respectively, values close to the equal mean and variance expected for a Poisson distribution. But we can do more.

The image above is called a Poissonness plot (Hoaglin 1980). This one is for the sample of petals from the larger stones. It allows us to graphically test whether a Poisson distribution is an appropriate model for the petal counts. Let x[k] be the count of stones collecting k petals, we plot on the vertical axis: log(x[k])+log(k!) against k (on the horizontal axis). In such a plot, Poisson counts fall upon a straight line with a slope equal to the logarithm of the Poisson mean. This is what we see for the larger stones. The plot for the smaller stones is also straight.

Kevin suggests testing our observed ratio to see if it is significantly different from the expected ratio of 1.5. Since we are dealing with discrete distributions we can compute the exact probability distribution for this ratio. The larger stones have a length of 9.375. The shorter, square stones have a length of 6.25, for a ratio of 1.5. We take these as the null parameters of two independent Poisson distributions. Our samples of sizes 21 and 12 have sums that are also Poisson. The joint distribution is simply their product, allowing easy computation of the probabilities for the ratio of the means. Its discrete probability distribution is shown below (ignoring the small probability that the denominator is zero, 2.6 x 10^-33 ).

Although this appears to be a darkly colored continuous probability density, it is actually a discrete distribution of probability mass plotted as densely packed individual vertical lines or spikes of probability. For this null distribution our observed ratio of means, 1.56, has an upper tail p-value of 0.39.  This, of course, is not significantly different from 1.5. In fact, our observed ratio would have to exceed 1.89 to be significant at the 5% level. Thanks for the prompting, Kevin.

Pine Needle Tracks

Guest Post by Dan Drake of the University of Puget Sound:

The attached picture is of the entrance to the library at the University
of Puget Sound. It's autumn here in Tacoma and there are lots of fallen
pine needles outside which the students track in. The door opens on the
right (on the left, to entering students) and you can see the
distribution is shifted that way near the door. The pine needles are
concentrated in the middle but show a few outliers, and the light color
in the middle shows dust and dirt brought in -- so there's a discrete
and continuous distribution.

The two dimensions of the carpet show a sort of time series: near the
door, people (and hence the dirt and pine needles) are shifted towards
the side of the door that opens, and as they walk into the library, they
tend to move to the center of the entryway.

Thanks for maintaining the blog -- I (and hopefully my students...)
enjoy it!

Thank you Dan for your interest and submission.
I'll have more on pine needles next week.

Queue Server Distribution

These are the feet of tourists at the Charleston, South Carolina visitor's center. The tourists queue up to get information about tours around the city and its attractions. There are only three servers helping them book their tours. These are the feet of the tourists being served. Others, waiting to be served, are standing to the left of the brass pole. 

If no servers were busy, an approaching tourist would be served at the first position nearest the pole. This is on the right of this picture, where four people in one party are discussing their vacation options. Further down the line, two people are talking to server number two, and further still (at the top left of this picture) a lone visitor is with server number three.

The first server position gets the most business and other servers are called in to fill positions two and three as the queue requires.

We can see the server frequency of use in the next picture. There is more floor scuffing and wear at position one, less at position two, and the smallest amount of wear at position three - a discrete frequency distribution of server use.

Discrete Text Wear

A student's cell phone (thanks Jason) showing the discrete distribution of texting wear. Not many uses of wxyz (9), much more of ghi (4) and mno (6) most likely from Jason's two thumbs resting on these keys during texting. Are the remaining letter sets (numbers) equally likely?

Texting has its own unique spelling and distribution of letter usage. I have been unable to find any published distribution of letter frequencies for texting. Are there any? If they were the same as standard English, the letter sets would have the following frequencies of use:

• abc    12.439
• def     19.243
• ghi     15.089
• jkl        4.925
• mno   16.773
• pqrs   14.347
• tuv     12.851
• wxyz    4.66

Of course relative to these percentages, resting thumbs have caused over amounts of wear on ghi (4) and mno (6). The wear also shows over amounts on jkl (5) as well. The others are all very similar as are the wear patterns, except for wxyz (9) that shows the small amount of wear suggested by the 4.66% frequency of use.

Contours of Use

Here are scatterplot contours of hand placement in opening a door at a health center stairwell. The door itself is the darkest color shown, but it has been painted over with white, blue, and peach(?) colored paint. As thousands of hands push the door open they slowly rub away some of the paint, leaving bivariate contours of greatest use. Horizontal and vertical frequency of hand placement can be readily seen in the paint wear pattern.

A Sad Tally Indeed

Discrete distribution of Golden Gate Bridge Suicides. Locations plotted to the nearest light pole. The bay side is more popular than the ocean side. Any conjectures as to why? From strangemaps.wordpress.com via sfgate.com