Statistics and the Mystery of the Feet
Robert Israel
have been dealing with a mystery lately: five detached feet in sneakers
have been found on beaches along the coast over the last year: four left
feet and one right foot. The right foot matches one of the left feet
(though they were found at different locations). See e.g.
<http://www.ctv.ca/servlet/ArticleNews/story/CTVNews/20080710/severed_feet_080710/20080710?hub=TopStories>
It occurred to me that this might make an interesting problem in statistics,
which I propose to the group:
assume there are n two-footed individuals who could be sources of the feet,
and each of their feet has probability p of being found, all feet being
independent.
From the numbers of unmatched left feet, unmatched right feet, and matched
pairs that are found, estimate n and p.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Mensanator
<isr...@math.MyUniversitysInitials.ca> wrote:
> As you may have seen in the news, police here in British Columbia
> have been dealing with a mystery lately: five detached feet in sneakers
> have been found on beaches along the coast over the last year: four left
> feet and one right foot. The right foot matches one of the left feet
> (though they were found at different locations). See e.g.
>
> It occurred to me that this might make an interesting problem in statistics,
> which I propose to the group:
> assume there are n two-footed individuals who could be sources of the feet,
> and each of their feet has probability p of being found, all feet being
> independent.
> From the numbers of unmatched left feet, unmatched right feet, and matched
> pairs that are found, estimate n and p.
Trying to sell a screenplay to Numb3rs? (Is that show still on?)
Paul Rubin
> As you may have seen in the news, police here in British Columbia
> have been dealing with a mystery lately: five detached feet in sneakers
> have been found on beaches along the coast over the last year: four left
> feet and one right foot. The right foot matches one of the left feet
> (though they were found at different locations). See e.g.
> <http://www.ctv.ca/servlet/ArticleNews/story/CTVNews/20080710/severed_feet_080710/20080710?hub=TopStories>
>
> It occurred to me that this might make an interesting problem in statistics,
> which I propose to the group:
> assume there are n two-footed individuals who could be sources of the feet,
> and each of their feet has probability p of being found, all feet being
> independent.
> From the numbers of unmatched left feet, unmatched right feet, and matched
> pairs that are found, estimate n and p.
I assume that you are assuming that there is a common cause to the
presence of detached feet (not unreasonable -- Occam's Razor?). Are you
also assuming that the n two-footed (or formerly two-footed) individuals
are all presumed to be victims of this cause (but that some feet simply
remain undiscovered, or not yet detached)? Otherwise the independence
assumption is shaky.
I think that the MLE of n and p would be found by maximizing the expression
nCm * (n-m)Ck * p^(k+2m) * (1-p)^(2n-k-2m)
where iCj is the (i,j) binomial coefficient, k is the number of single
feet found, and m is the number of matched pairs of feet found (for a
total of k+2m feet found). The support for n is [m+k, infinity) and for
p is 0 < p < 1 (unless k=0, in which case n=m and p=1 is possible).
The derivation is to first model the likelihood of finding k+2m feet
using the binomial distribution with parameters 2n and p, then
(conditional on finding that many feet) to model the likelihood that
there are m pairs by asserting (based on independence and common
probability p) that all combinations of k+2m feet out of the 2n feet in
the population are equally likely, and counting the number of
combinations that give m pairs and k "odd" feet.
I suspect the MLE would have to be estimated numerically, rather than
obtaining a closed form for it. (?)
Assuming I'm correct about this, it might make an interesting, if
somewhat morbid, problem in an introductory probability class. Here's
another interesting (to me) question: is the MLE unique?
/Paul
Robert Israel
> On Jul 21, 12:01=A0pm, Robert Israel
> <isr...@math.MyUniversitysInitials.ca> wrote:
> > As you may have seen in the news, police here in British Columbia
> > have been dealing with a mystery lately: five detached feet in sneakers
> > have been found on beaches along the coast over the last year: four left
> > feet and one right foot. =A0The right foot matches one of the left feet
> > (though they were found at different locations). =A0See e.g.
> >
> ><http://www.ctv.ca/servlet/ArticleNews/story/CTVNews/20080710/severed_...=
> >
> >
> > It occurred to me that this might make an interesting problem in
> > statisti=
> cs,
> > which I propose to the group:
> > assume there are n two-footed individuals who could be sources of the
> > fee=
> t,
> > and each of their feet has probability p of being found, all feet being
> > independent.
> > From the numbers of unmatched left feet, unmatched right feet, and
> > matche=
> d
> > pairs that are found, estimate n and p.
>
> Trying to sell a screenplay to Numb3rs? (Is that show still on?)
Sounds like a good idea. Charlie Eppes is hardly ever bothered by having too
small a sample to draw significant conclusions.
OwlHoot
>
> I assume that you are assuming that there is a common cause to the
> presence of detached feet (not unreasonable -- Occam's Razor?).
Gotta be. But the most baffling aspect is why all trainers?
I've heard that when someone jumps off a high bridge into
water the impact can knock their feet clean off, and perhaps
this is more likely to happen with trainers, due to their large
"footprint", and deep treads which would offer more resistance to
the water. (Apparently they're all large shoes too, size 12 or
something, which lends further support to this grisly theory.
But *are* there any high bridges or cliffs near Vancouver? And
why have feet only been turning up in the last year? Do they
regularly find feet near the Golden Bridge in San Fransisco
and other nigh bridges?
Failing that, it's either a weird fetish of a serial killer or,
it occured to me, a freak accident in an occupation shared by
the former owners of the feet, one in which they would tend
to wear trainers for good grip on slippery surfaces.
In the latter case, the accident would need to be easy to
conceal or explain away, to avoid compensation claims and
enquiries. The only obvious example that comes to mind is
a marine accident, maybe a line of guys hauling on a rope
in a storm, or trying to batten down a hatch, and some cable
or the hatch coming adrift, whips across the deck and lops
off their feet. Skipper then casts the bodies overboard and
scuttles the ship (or, on reaching port, claims the crew
did a runner or were swept over the side by a freak wave).
A quick Google reveals that British Columbia is on the
Pacific coast, and body parts could drift hundreds of
miles especially feet (with air pockets in the soles of
the trainers).
My impression is that a lot of third-world merchant sailors,
on oil rigs for example, tend to be on the small side, asians
for example. So given the large size of the feet, my guess is
that if a marine accident was the cause then it was most likely
a Russian fishing boat operating round those Russian islands
near Alaska. Depends which way the ocean current run though.
Right, now where did I put that pipe and deer stalker hat? ..
Cheers
John R Ramsden
Robert Israel
> On Jul 21, 8:27=A0pm, Paul Rubin <ru...@msu.edu> wrote:
> >
> > I assume that you are assuming that there is a common cause to the
> > presence of detached feet (not unreasonable -- Occam's Razor?).
>
> Gotta be. But the most baffling aspect is why all trainers?
Because they float, and are not biodegradable. The body comes apart
naturally as it decomposes, most is eaten by fish, but the trainers provide
sufficient protection to the bones of the feet.
BTW, one of the feet has now been DNA-matched to a missing man from the
Vancouver area. There is no evidence of foul play: the man had been
depressed before his disappearance.
Michael Press
Robert Israel <isr...@math.MyUniversitysInitials.ca> wrote:
> As you may have seen in the news, police here in British Columbia
> have been dealing with a mystery lately: five detached feet in sneakers
> have been found on beaches along the coast over the last year: four left
> feet and one right foot. The right foot matches one of the left feet
> (though they were found at different locations). See e.g.
> <http://www.ctv.ca/servlet/ArticleNews/story/CTVNews/20080710/severed_feet_080710/20080710?hub=TopStories>
>
> It occurred to me that this might make an interesting problem in statistics,
> which I propose to the group:
> assume there are n two-footed individuals who could be sources of the feet,
> and each of their feet has probability p of being found, all feet being
> independent.
If we are modeling the events, then we should allow that
finding one shod foot will increase the probability that
its mate will be found. Pr(L_k | R_k) = Pr(R_k | L_k) = q > p.
> From the numbers of unmatched left feet, unmatched right feet, and matched
> pairs that are found, estimate n and p.
--
Michael Press
OwlHoot
<isr...@math.MyUniversitysInitials.ca> wrote:
> OwlHoot <ravensd...@googlemail.com> writes:
> > On Jul 21, 8:27=A0pm, Paul Rubin <ru...@msu.edu> wrote:
>
> > > I assume that you are assuming that there is a common cause to the
> > > presence of detached feet (not unreasonable -- Occam's Razor?).
>
> > Gotta be. But the most baffling aspect is why all trainers?
>
> Because they float, and are not biodegradable. The body comes apart
> naturally as it decomposes, most is eaten by fish, but the trainers
> provide sufficient protection to the bones of the feet.
>
> BTW, one of the feet has now been DNA-matched to a missing man from
> the Vancouver area. There is no evidence of foul play: the man had
> been depressed before his disappearance.
Ah, right. (I did mention the air cushions in the soles would help
keep them afloat.) Also a quick Google turned up an article which
pointed out that one of the feet was female and in a size 7 shoe.
To model the probabilities, shouldn't one assume the probability
is a function of time, initially zero then rising to a peak and
finally tailing off to zero again to reflect the fact that if a
shoe is not found within a certain time then the chances are it
is buried in sand, or lodged in a tree root or something, and
probably never will be. Sort of like a wave function (no pun
intended).
The integral of this function between 0 and infinity would need
to be less than 1, this being the probability a shoe will never
be found.
I'm not convinced there would be much correlation between pairs
of shoes, and the discovery of only one pair (at separate times
I gather) seems to confirm that.
Also, if the victims died at different times that complicates
things. (But presumably the discovered pair might give one a
reference point, so to speak, of two shoes which must have
started out moreorless together.)
Incidently, if feet in trainers don't regularly turn up on
other beaches, what's so special about Vancouver? Perhaps
there's some quirk of circulating currents there that washes
things out to sea and later sweeps them back in again.
Cheers
John Ramsden
P.S. By "oil rigs" I obviously meant "oil tankers".
jmor...@idirect.com
> In article <rbisrael.20080721163747$2...@news.acm.uiuc.edu>,
> Robert Israel <isr...@math.MyUniversitysInitials.ca> wrote:
>
> > As you may have seen in the news, police here in British Columbia
> > have been dealing with a mystery lately: five detached feet in sneakers
> > have been found on beaches along the coast over the last year: four left
> > feet and one right foot. The right foot matches one of the left feet
> > (though they were found at different locations). See e.g.
>
> > It occurred to me that this might make an interesting problem in statistics,
> > which I propose to the group:
> > assume there are n two-footed individuals who could be sources of the feet,
> > and each of their feet has probability p of being found, all feet being
> > independent.
>
> If we are modeling the events, then we should allow that
> finding one shod foot will increase the probability that
> its mate will be found. Pr(L_k | R_k) = Pr(R_k | L_k) = q > p.
>
> > From the numbers of unmatched left feet, unmatched right feet, and matched
> > pairs that are found, estimate n and p.
>
> --
> Michael Press
On the topic of statistical bias, I have seen some speculation that
there is a systematic difference in the size of left vs right feet
(based on handedness ?)
Also, there was an allegation that one tightens one's shoes to
different levels for right/left feet...
Ray Vickson
wrote:
The local press (Victoria Times Colonist) interviewed some people (non-
experts, I suppose) who claim that there are differences in drift
patterns of left and right shoes, due to their differing shapes and
different turning tendencies in fast-flowing tides and currents.
Whether this has any merit, I don't know.
R.G. Vickson
Russell
<isr...@math.MyUniversitysInitials.ca> wrote:
> Mensanator <mensana...@aol.com> writes:
> > On Jul 21, 12:01=A0pm, Robert Israel
> > <isr...@math.MyUniversitysInitials.ca> wrote:
> > > As you may have seen in the news, police here in British Columbia
> > > have been dealing with a mystery lately: five detached feet in sneakers
> > > have been found on beaches along the coast over the last year: four left
> > > feet and one right foot. =A0The right foot matches one of the left feet
> > > (though they were found at different locations). =A0See e.g.
>
> > ><http://www.ctv.ca/servlet/ArticleNews/story/CTVNews/20080710/severed_...
>
> > > statisti=
> > cs,
> > > which I propose to the group:
> > > assume there are n two-footed individuals who could be sources of the
> > > fee=
> > t,
> > > and each of their feet has probability p of being found, all feet being
> > > independent.
> > > From the numbers of unmatched left feet, unmatched right feet, and
> > > matche=
> > d
> > > pairs that are found, estimate n and p.
>
> > Trying to sell a screenplay to Numb3rs? (Is that show still on?)
>
> Sounds like a good idea. Charlie Eppes is hardly ever bothered by having too
> small a sample to draw significant conclusions.
> --
> Robert Israel isr...@math.MyUniversitysInitials.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
>
> - Show quoted text -
Yeah, I love the show but it is annoying that
however small the sample size, it always fits
his model perfectly.
Cheers,
Russell
Rod
"Robert Israel" <isr...@math.MyUniversitysInitials.ca> wrote in message
news:rbisrael.20080721163747$25...@news.acm.uiuc.edu...
Peter
For the missing feet problem, with k=5, the results are quite boring. If m=0, the "best" population size of victims is infinite - logically, if you think of it. If m=1, MLE for n is 5. If m=2, it would be 3.
Now, I might be wrong about this (as I might be about all the rest) but it seems to me that, once you have an estimate for n, a reasonable estimate of the probability of a foot being found should simply be k/(2*est_n).
Robert Israel
> What's your answer?
Please don't top-post. My answer is at the bottom.
> "Robert Israel" <isr...@math.MyUniversitysInitials.ca> wrote in message
> news:rbisrael.20080721163747$25...@news.acm.uiuc.edu...
> > As you may have seen in the news, police here in British Columbia
> > have been dealing with a mystery lately: five detached feet in sneakers
> > have been found on beaches along the coast over the last year: four left
> > feet and one right foot. The right foot matches one of the left feet
> > (though they were found at different locations). See e.g.
> >
>
>><http://www.ctv.ca/servlet/ArticleNews/story/CTVNews/20080710/severed_feet_0
>><80710/20080710?hub=TopStories>
> >
> > It occurred to me that this might make an interesting problem in
> > statistics,
> > which I propose to the group:
> > assume there are n two-footed individuals who could be sources of the
> > feet,
> > and each of their feet has probability p of being found, all feet being
> > independent.
> > From the numbers of unmatched left feet, unmatched right feet, and
> > matched
> > pairs that are found, estimate n and p.
> > --
I'll use a maximum likelihood estimator. The likelihood function
L(n,p) is the probability of finding s unmatched left feet, r unmatched right
feet, and b matched pairs, given n and p.
L(n,p) = n!/(s! r! b!(n-s-r-b)!) p^(s+r+2b) (1-p)^(2n-s-r-2b)
We have L(n+1,p)/L(n,p) = (1-p)^2 (n+1)/(n+1-s-r-b)
which would be 1 for n = (s+r+b + p^2 - 2 p)/(2 p - p^2)
and is less than 1 if n is greater than that. So for maximum likelihood
we want (s+r+b)/(2 p - p^2) >= n >= (s+r+b + p^2 - 2 p)/(2 p - p^2).
On the other hand, setting \partial L/\partial p to 0 we get
p = (s + r + 2 b)/(2 n). Substituting that in to the inequalities,
I get
(s+r+2b)^2 - 4 b n >= 0
4 b n^2 - (s+r+2b)(s+r+2b-4) n - (s+r+2b)^2 >= 0
So if a = s+r+2b (and assuming b > 0), we must have
a^2/(4 b) >= n >= (a-4 + sqrt((a-4)^2 + 16 b)) a/(8*b)
with p = a/(2 n).
In the case at hand, s=3, r=0, b=1, a=5, this says
6.25 >= n >= 3.20 approximately. Looking at the integer
possibilities 4, 5, 6, we find that the maximum likelihood
occurs at n=4. This would indicate that there are no additional sources
of feet out there. On the other hand, the difference in likelihood
between n=4 and n=5 is quite small, so the statistical evidence is very
weak.
Peter
I did not treat left and right feet differently and only distinguished between unmatched and matching feet. It doesn't matter for the solution. Multiplying the combinatorial stuff with p^a*(1-p)^(2n-a), the non-constant terms in the log-likelihood for the sample at hand are
lfactorial(n)-lfactorial(n-4)+lfactorial(2*n-5)-lfactorial(2*n)
+5*log(p)+(2*n-5)*log(1-p)
Optimizing this, and treating the problem as a two-parameter rather than a one-parameter issue, as I should have I guess, the solution is as the one by Robert Israel: n=4 and p=5/8. The fact that the number of feet that are cut off is likely to be small bothers me a bit for the interpretation of p as a bernoulli parameter - if the number of feet out there is getting less and less, in a distinctly finite population, does it make sense to assume that the probability of finding one is constant?
George Kahrimanis
of left and right, we do not *need* to make that distinction in the L
function. My L function differs from his by the constant factor
left_only! * right_only! / single!
so that the likelihood ratios are the same.
My solution agrees exactly with Paul Rubin's. Here is how I justify it.
For each victim: the probability to find both feet is p^2 (we actually
find m pairs), the Pr to find exactly one foot is p(1-p) (we find k
unmatched feet), and the probability of no foot is (1-p)^2 (the
corresponding count is n-m-k). It is easy from here.
(NOTE. It feels intuitively more rigorous when we use more details of
the data in the likelihood function, but it aint necessarily so.
If a model involves imprecise probabilities it is *possible* that,
by including more details, the likelihood function become less
precise! For a trivial example, consider including the location
of each discovered foot in the likelihood function; our simple model
can only say "I do not know". We could discuss, in another thread,
what criterion to use for resolving this question.)
Though the asymmetry of the data is immaterial in the L function, it
is important in assessing the goodness of fit. I leave this issue for
another message. (Even if the "p value" is very small we do not *have*
to abandon the simple model, but at least we ought to consider other
models -- as some posters have done already.)
I do not think that the abstract likelihood function is the last word
of this problem. I am loking forward to a comparison of concrete
hypotheses, like a Russian ship or a serial killer, each with a
range for n and p.
Question: how would you compare two hypotheses, one coming with
narrow ranges for n and p containing the M-L estimates, and the
other coming with much wider ranges, also containing the M-L
estimates? The first hypothesis seems to score higher in predictive
power, I dare say.
~ George Kahrimanis
Frederick Williams
>
> As you may have seen in the news, ...
It was reported by the beeb:
http://news.bbc.co.uk/1/hi/world/americas/7463305.stm. Has the mystery
been solved?
--
He is not here; but far away
The noise of life begins again
And ghastly thro' the drizzling rain
On the bald street breaks the blank day.
porky_...@my-deja.com
wrote:
> Robert Israel wrote:
>
> > As you may have seen in the news, ...
>
> It was reported by the beeb:http://news.bbc.co.uk/1/hi/world/americas/7463305.stm. Has the mystery
> been solved?
>
or may be it's somehow related to Global Warming, just like the
Redtide in Florida or jellyfish in Mediterranean?
Ray Vickson
<isr...@math.MyUniversitysInitials.ca> wrote:
> As you may have seen in the news, police here in British Columbia
> have been dealing with a mystery lately: five detached feet in sneakers
> have been found on beaches along the coast over the last year: four left
> feet and one right foot. The right foot matches one of the left feet
> (though they were found at different locations). See e.g.
>
> It occurred to me that this might make an interesting problem in statistics,
> which I propose to the group:
> assume there are n two-footed individuals who could be sources of the feet,
> and each of their feet has probability p of being found, all feet being
> independent.
> From the numbers of unmatched left feet, unmatched right feet, and matched
> pairs that are found, estimate n and p.
> --
> Robert Israel isr...@math.MyUniversitysInitials.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada
One problem with all this is that feet in sneakers float, but feet in
other shoe types sink. So, we are seeing feet only from the sub-
population of footloose sneaker-wearers, which gives no indication of
the real prevalence of water-borne detached feet of in all types of
footware. The situation reminds of Operations Research war stories
(literally) in which armor plating on WWII fighter planes was being
assessed, but by examining the bullet marks on returning planes. More
important were the holes on the planes that did not return and were
thus never examined.
R.G. Vickson