Movie Quote Graphs

 

Statistician Nathan Yau at Flowing Data offers a poster print of his Famous Movie Quotes Poster from the American Film Institute's list "100 Years...100 Quotes". The image above is just a glimpse of the larger poster. Enjoy the fun.

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!

Sobriety Contours

Pattern from the joint distribution of keyhole use labeled with contours of sobriety (thanks Scott) from this source (among many). Perhaps this owner needs the keyhole for drunk people.

Snow Nose

The Washington, DC area was spared from the worst of the big snow storm this week. At home we got only about 4 inches of snow. The video is of the window that our cat frequents. She walks back and forth along the narrow window sill and touches her nose on the window panes. This is likely cage behavior of her indoor life. She leaves behind a grease spot on the panes with each nose touch. The marks are faint and hard to see, with fewer marks on the left-most pane, many more on the center pane, and fewer on the right-most pane. This is a pattern we have seen often. If this was one complete pane of glass, rather than three individual panes, we could see the continuous frequency distribution of her nose touches, likely following a bell-shaped frequency distribution curve. As it is, we see this continuous distribution divided up into three disjoint regions. The individual touches have been placed into the three bins of a histogram of this continuous distribution of nose touches. Overall, if she touched the panes n times, the counts of the touches that fall on the three panes would have a multinomial distribution, with a higher probability of touching in the middle and lower probabilities on either side.

Happy New Year

Is it better to be right or be happy? A British Medical Journal report investigates this research question in their lighthearted Christmas issue. The research sample consisted of one married couple (n=2). The female was blind to the null hypothesis being tested: it is better to be right than happy. The female was assigned the "right" condition. The male was assigned the condition of agreeing with the female's "every opinion and request without complaint." Happiness was measured on a 10 point Likert scale labeled Quality of Life. Unfortunately this study was terminated early because of "severe adverse outcomes." The males Quality of Life fell 4 points in 12 days, whereas the female's Quality of Life increased slightly from 8 to 8.5. The researcher's findings: "The results of this trial show that the availability of unbridled power adversely affects the quality of life of those on the receiving end."

Be nice in the new year and be happy.

Probability of a White Christmas

The map above is the current snow cover (as of 13 December 2013) in the US according to the weather.com. Compare this to National Oceanic and Atmospheric Administration map from the last few years of Christmas morning snow cover over the last five years.

Combining such maps from 1908 to 2010, NOAA maps out the probability of a white Christmas (that is, 1" of snow on the ground).

They also have a report from 1995 that maps the probability of 1", 5", or 10" of snow on Christmas morning.

Except for the Northeast, a many of the areas of the greatest chances are not densely populated. 

I wonder, what is the expected percentage of the US population that experiences a white Christmas?

Happy Holidays.

Scatterplot Waldo

Slate.com staff writer Ben Blatt has examined the wide range of Where's Waldo picture books,

looking for a useful search strategy to find Martin Handford's elusive cartoon character. Blatt plotted the horizontal and vertical page location of Waldo in 68 pictures in, what he calls, the seven "primary" Where's Waldo books. He claims to have sat for three hours with a tape measure in a Barnes & Noble bookstore and measured Waldo's location on each 20" X 12.5" two-page spread. The image above shows these locations and two horizontal bands of 1.5" each: one three inches from the bottom of the page and the other seven inches from the bottom. Blatt found that Waldo can be found in these bands in 53% (36) of the 68 images.

We can see the higher frequency of occurrence in bands by finding the marginal histograms from digitized locations in Blatt's scatterplot (data shown below). Here is the marginal histogram of the vertical locations of Waldo. The regions found by Blatt stand our prominently in the two modal peaks in the histogram.

Horizontally there is a less prominent patten of the locations across the two-page spread.

 

Perhaps we could improve on the two-vertical-strips strategy by concentrating on the far left and far right of the two-page spread with then a glance just left of center?

Here are the data:

horizontal	vertical	horizontal	vertical	horizontal	vertical	horizontal	vertical
1.02	11.97	9.52	7.21	11.79	6.22	8.8	2.95
7.78	10.22	10.51	7.71	14.78	5.72	12.57	3.71
8.51	9.99	11.52	7.97	14.51	4.96	17.76	3.97
9.26	9.46	11.29	6.95	18.03	5.43	18.03	4.21
10.77	10.48	12.19	7.77	18.75	5.43	19.01	4.47
11.99	11.48	12.51	8.24	1.8	3.97	16.81	2.95
13.27	10.48	12.51	7.48	1.31	3.45	1.04	2.72
16.26	11.21	14.51	8.21	2.26	3.18	1.54	1.93
19.48	12	14.25	7.21	3.77	4.21	1.54	1.43
16.26	9.99	15.62	7.48	3.51	3.45	3.28	1.2
17.5	9.99	17.24	7.48	3.8	3.45	7.29	1.93
17.74	9.46	18.49	7.21	5.31	3.97	8.27	1.2
15.5	8.97	19.25	7.48	4.79	2.98	8.53	0.44
5.78	8.24	2.76	6.75	5.54	3.21	10.54	2.45
6.76	7.97	3.8	6.98	6.76	4.44	17.5	2.69
7.52	8.47	3.28	5.96	8.27	4.21	19.48	2.45
8.51	8.47	5.54	6.72	9.03	3.45	18.98	1.93