Carport Oil Drips

This is a stain on carport floor of my mother's home.  It comes from the the persistent drip of oil from her parked car and the variability of her parking location. When the car enters the carport from the bottom of this picture, the front of the car (and its motor) stops over the developing stain. The stain pattern running from the bottom to the top in the picture is from parking less or more of the car, respectively, in the carport. The left to right stain is from the car entering the carport more to the left or right.

Notice the prominent lines of oil drops on the right side of the stain. It could be from repetitive and quite accurate placement of the car more on the right of the carport. But it could also be a re-positioning of the car more forward or backward after it stopped on an established an entry line. This illustrates the conditional distribution of bottom to top oil placement given a fixed left to right car approach. Notice that it's likely that the floor is not level, indicated by the lighter, more liquid stain running towards the entry of the carport.

What results is a joint, two-dimensional scatterplot of automobile placement in the carport. The overall pattern of the oil is consistent with a targeted, ideal car placement. With a few other assumptions, such as independence of bottom to top and left to right placement, it is possible to prove that the resulting joint distribution is that of a bivariate normal distribution. More on the details in a future post.

Stepping Up 2

This is an another image of the checkout counter at a deli in Rockville, Maryland. We saw before the pattern of customers standing and awaiting completion of their purchases, and wearing away the floor tiles with their shuffling feet, revealing contours of their use.

Her we see the wear patten made by their purchases that are placed and slid across the counter to be totaled up on the register. The white of the counter top has been worn away to the brown board underneath. There is greater wear near the customer who perhaps sets down several items, but the wear patten narrows as the checker selects one at a time to ring up their prices on the register. What remains is the pattern of support for the joint distribution of purchase placement.

But there's more (as they say on TV commercials). Look along the front edge of the counter nearest the customer. Along this edge we see the bell-shaped marginal distribution of left/right item wear as the items are either placed on or slide off the counter.

Stepping Up

This is an image of the floor in front of the checkout counter at a deli in Rockville, Maryland. Standing and awaiting completion of their purchases, customers have worn away the floor tiles with their shuffling feet, revealing contours of their use. The central brown area shows the concentrated region of the most wear. So frequently has this central spot served customers that the tile has been worn through to its interior, well deeper than its decorative surface. Around that, lesser use has worn off only the pattern of the floor tile. This leaves a white annulus of wear surrounding the central region. Beyond this there is a dirtier area of the floor tile where customers queue up to pay for their purchases. This all results in an image of the somewhat circular contours of the joint frequency distribution of customer foot placement at the checkout counter.

Bang the Drum Differently?

Here is another snare drum pattern of wear showing the joint, two-dimensional pattern of drumstick or brush hits on the head of the drum. Again we can see the central dark and quite circular contour of the region of most wear. This is surrounded by a more irregular discoloring of the drum head, likely from the metal brushes we see here. This is very different from the nearly circular pattern of snare drum wear we have seen earlier. Is this the results of different drummers and their particular style or of similar style drummers playing different types of music? Or perhaps both?

Waldo, revisited

We've seen the scatterplot and marginal distributions of Waldo's location in a collection of Where's Waldo. These were from Ben Blatt at Slate.com. Now Ronald Olsen has added a plot of the contours of a kernel density estimate of the joint frequency distribution of Waldo's location on the facing pages of the Waldo books. Darker color indicates higher density. Olsen then dynamically computes the optimal search path. Try it out. Can you find Waldo quicker?

The Game Changes

Here is a fun interactive display by Noah Velltman of the height and weight of National Football League players from 1920 to 2014, via Statistical Modeling... Clicking on this link allows you to vary the year and observe the joint distribution of height and weight over the years. You can even see how the game as changed from the early 1990s as roles and players have become more specialized resulting in distinct clusters of body types.

Tornado Marginals

Inspired by this map of Waffle House restaurants by latitude and similar to global land mass marginal plots we've seen here before and corresponding world population plots, Tim Brice at the National Weather Service (NWS) in Texas produced maps showing the marginal distributions of tornado touchdowns by longitude and latitude.

Here's the joint distribution, a map of tornado tracks from 1950-2013 from the NWS

 

(Thanks JCT).

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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.

Normal Snare Distribution

Here is the head of a snare drum (thanks Sean) showing the two dimensional, joint distribution of his drumstick hits. The maker, Evans, produces drum heads with two plys of plastic bonded together. In this image the pattern of wear and use reveal themselves through contour lines as closed curves indicating regions of similar frequency of use. The greatest wear is in the lightest colored central region, having seen the greatest frequency of hits. This central region has worn through the first ply showing the remaining plastic support underneath. Sean seems to have a very stable left hand, consistently hitting this small central region in nearly a circular pattern. In this region, the horizontal location of his hits seems independent of the vertical, producing this near circular pattern of the joint distribution. Surrounding this is the darker region of the top ply of plastic. This layer retains more dirt and grime than the underlying supporting plastic. Again the pattern of these hits is nearly circular. It turns out that with simple assumptions, like radial symmetry and independence, the pattern can be shown to be that of a bivariate normal distribution. A result that was thought first published (p. 398) by John Herschel in 1850, but actually discovered much earlier by the American mathematician Robert Adrain in 1808. More details on that in a future post.

But here, perhaps we see slight deviations from normality. The darker ring seems to show slightly  more variability extending vertically and a greater clustering of hits on the bottom of this image. This indicates a bit of skewness towards the top of the picture. Less use and wear is finally shown in the cream colored outer region that has seem very few hits. Thanks again Sean.

Just Marginal Joints

Here is a joint distribution of wear on the counter top of a Home Depot. The counter top was originally white. This has worn away to reveal various layers of shading in the laminate and further into the underlying wood of the table. The innermost area has been actually gouged out by many tools, cans of paint, and other heavy hardware. We see nested triangular patterns of wear revealing the joint distribution of tool placement on the counter and their movement across the checkout scanner.

But wait. There's more, as they say on infomercials. Not only do we get the joint distribution of wear, we also get one marginal distribution of wear along the front edge of the counter. As customers drag their purchases onto or off of the counter they leave a bell-shaped pattern of use along that front edge.