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Friday, July 14, 2006

Illinois Governor's Race: Misreading the polls.

On a political website, I read the following:

IL-Gov: Is Blago pulling away? Hotlist

Fri Jul 14, 2006 at 09:06:42 AM PDT

Rasmussen. 7/6. Likely voters. MoE 4.5% (4/8 results)

Blagojevich (D) 45 (38)
Topinka (R) 34 (44)

Huge turnaround. While Blago may be unliked and corrupt, even that might not be enough to hand the governor's seat to Republicans, not matter how much voters might flirt with Topinka (the only statewide elected Republican in the state). That's how far in the dumps Illinois Republicans are in.

The Blagojevich campaign has run an advertising campaign raising the question "What is She Thinking?" about Topinka and touching on issues ranging from the state budget to Iraq. The Blagojevich campaign has also raised complaints about lease payments by Topinka to a developer who is also a contributor. A better job performance in the state may also have something to do with the turnabout.

So, is this a huge turn-around? Well, maybe, maybe not.

Let's go to the poll itself:

Survey of 500 Likely Voters
July 6, 2006

Election 2006: Illinois Governor
Rod Blagojevich (D) 45%
Judy Baar Topinka (R) 34%

July 13, 2006 After trailing earlier in the campaign season, Illinois Governor Rod
Blagojevich (D) now leads Republican State Treasurer Judy
Baar Topinka by 11-points in his bid for re-election. The latest Rasmussen
Reports poll shows Blagojevich with 45% of the vote to 34% for Topinka.

The Blagojevich campaign has run an advertising campaign raising the
question "What is She Thinking?" about Topinka and touching on issues
ranging from the state budget to Iraq. The Blagojevich campaign has also
raised complaints about lease payments by Topinka to a developer who is
also a contributor. A better job performance in the state may also have
something to do with the turnabout.

The governor, too, has had to contend with charges of impropriety, but
that's old news and he has more money to spend on campaign ads. Many
Republicans in the state concede that he has managed to put Topinka on the
defensive lately.

What is the key? The key is this: the poll was of 500 likely voters. This type of poll is what is known as a proportion. The theory is something like this: when you poll 500 voters, you are interested in, say, the number that say that they will vote for Blagojevich. That number (out of 500) is denoted by the letter "p" and the appropriate distribution is the binomial distribution. Typically, one then divides by the number polled to get a proportion.

The variance turns out to be p*q/n where n = number of voters and p is the assumed proportion (often set to .5 as that is the "null hypothesis"; that is, the initial assumption).

The standard deviation is the square root of the variance, so here we have "sd" = (.5*.5/500)^.5 = .0224. Since we are 95% confident that the true proportion will be within 2 standard deviations of the one that we actually measured, we have that Blagojevich's support is between .45 +- 2*(.0224) = (.4052, .4948).

In the previous poll, it was .38 +-2*(.0224) = (.3352, .4248). Note that these two confidence intervals overlap.

Now that doesn't mean that we can say that there is no change but it does point out that the change might not be as dramatic as it seems at first.

If we now measure the difference between the two proportions, and use the fact that the new variance is the sum of the absolute values of the two variances:

"variance" = 2*.5*5./500, so the standard deviation is: .0316, and so the difference in Blagojevich's support is (.45-.38) +- 2*.0316 = (.007, .133), which means that this difference in support for Blagojevich (between polls) might be less than 1%.

We can say, with confidence, that Blagojevich did not lose support.

Monday, July 10, 2006

What I am up to

First a couple of notes: I found some sources that help with math symbols in HTML:

and this is interesting too:

I am kind of ashamed that I didn't know this earlier.

Next, while looking for good figures to put in this post (I didn't find them yet), I came across a fun little article which relates elementary knot theory with "impossible diagrams" (Escher type stuff)

It is a quick, enjoyable read.

Now back to the subject at hand: Imagine a knot inside of a torus:


The yellow string of beads is the knot (trefoil). Now suppose that torus (the red solid object) is itself knotted up:


The knot inside the solid torus is called a satellite knot of the knot formed by the solid torus, and the knot formed by the solid torus is called a companion knot of the knot inside the torus. Of course, there is more to the definition than I am telling; for example, the knot inside of the torus (first diagram) must "really" lie in the torus and not in some ball inside of the torus (think of it this way: at the beginning of this process, the knot inside of the torus to be knotted must hit every meridional disk in the beginning torus (the red torus of the first figure). And the example I gave is of a cable knot.

Anyway, at the moment, I am interested in the following kind of object. Consider a knot inside of a companion. Now thicken that knot so that it is a solid torus. Now put another knot inside that. Thicken that knot so that it is a solid torus as well (inside of the two solid tori). Put a knot inside that one, thicken it again, and repeat the process infinitely.

Now take the intersection of this infinite collection of nested tori; kind of like these dolls:
except that the tori are tied up in more and more complicated fashion.

Now this infinite intersection is an example of what is known as a solenoid.

If we are careful about how we nest these tori; that is, if a longitude of each inner torus just goes once around the torus that contains it, we can get a simple closed curve in the limit.

Arnold Shilepsky came up with criteria that guarantees that we get a simple closed curve in the limit; see the references in his selected publication list. (These types of knots were first studied by a real giant of geometric topology: R. H. Bing), and Shilepsky was one of Bing's students).

These knots are very pathological. They are examples of "wild" knots; we'll say a knot is wild if either it is not the image of any differentiable embedding of the circle into 3-space (as in the sense of ordinary calculus) or if the knot cannot be realized by a finite number of straight line segments placed end to end.

But these knots are not just wild; they are so wild that they fail to "pierce a disk" at any point. Think of it this way: if you think of these knots as pieces of thread, there is no way to run this particular thread through an eye of a needle without running an infinite number of "strands" of this thread though the eye at the same time. That is, if one strand of this type of knot goes through the eye, then an infinite number of strands must go through at the same time as well.

Now the question I am interested in is this: how can one tell if two of these types of wild knots are really different; that is, if space can't be deformed to take one of these wild knots onto the other one?

What symmetries do they have? What about their groups?

Saturday, July 01, 2006

A couple of mathematically flavored diaries from the Daily Kos

A couple of mathematically related diaries taken from the Daily Kos: the first is about ecology and discusses some of the mathematical models. The second is about Bayes' law.
I will warn the reader that there are some politics in these diaries; I thought about trying to edit it out but then decided to leave them "as is."

First, a diary from Jmknapp from the Daily Kos.

Mathematical Challenges of Global Ecology Hotlist

Thu Jun 29, 2006 at 07:00:11 PM PDT

All this week a group of climate scientists have been meeting for a Global Ecology workshop at Ohio State University. Abstracts of some of the talks can be found at OSU Mathematical Biosciences Institute

* Marine Carbon Cycle Biogeochemistry from the Bottom Up
* Quantifying the Predictability of Noisy Nonlinear Biogeochemical Systems
* Ecosystem-climate dynamics: What are the phenomena that pose mathematical challenges?
* Stoichiometry of wild plants and crops in the high CO2 world

I attended a lot of the talks, to get a feel for current thinking on the global warming crisis and to watch these scientists in action, on the global warming front line.

The conference generally steered clear of political issues, being concerned with more heady topics like transcritical bifurcations in systems of ordinary differential equations, oscillating relationships in predator-prey populations, hysteresis and catastrophic jumps in system behavior, maximum entropy production models, Navier-Stokes equations, ecological stoichiometry, Redfield ratios of C:N, eigenvalues of Jacobian matrices--fun stuff like that.

Still, it was interesting that Bush would come up from time to time and it was evident that the Bush administration is regarded with total disdain by these scientists. Not surprising, given the all-out assault on science that Bush has led, from evolution to the Big Bang to global warming.

One presenter included a Bush photo in one of his slides:

Underneath the photo was the relationship:

plant -> herbivore + waste

That got quite a few chuckles. Also below the photo was the statement: organisms are just abstract molecules

That thought, the speaker said, "is the only thing getting me through the last administration and a half." Another quipped, "It looks like he's saying 'you know where you can stick it!'"

During a talk on stability (or instability) of the global climate over Earth's history, the speaker noted that elements of the climatic system develop "without foresight, much like Republicans."

So this ire is definitely directed at Republicans and not all politicians. This is cutting the Democrats some slack, as I recall it was Congressional Democrats who scuttled Gore's carbon tax early on in the Clinton Administration. But still, Republicans are just so egregious in these areas that even an adminstration that "offered a friendly ear" is infinitely preferable. Several pointed out that however that a carbon tax, which could be designed to be revenue neutral, would be a great step towards mitigation of CO2 emissions. So Democrats need to pick up the ball there.

Gore's movie was seen by most. One esteemed professor from the University of Minnesota said, "I learned a few things. Gore was very careful with the science." Also, "The idea that someone could make a movie of a lecture and people would go to see it!" As for Gore's personal stories in the film, "If you want to present an interesting lecture you should include some personal anecdotes." An Oak Ridge scientist organized a group of his colleagues from the Carbon Dioxide Information Center to see the movie & it was very well received.

Dr. John Pastor gave a talk illustrating how there are several time constants involved in CO2 dynamics, and how perturbations in the climate can persist long after the stimulus is removed, so-called reactive transient responses. It is not known whether increased CO2 currently will produce such a response, even if emissions are curtailed.

Dr. John Harte (UC-Berkeley) points out that Milankovich (orbital) cycles explain the timing of the shifts in the CO2/temperature record over the last 400,000 years or more, but those orbital cycles don't explain the magnitude of the temperature changes, which involve various positive feedback factors making the swings larger. He wonders if perhaps the current trend of making models more and more complex, requiring supercomputers to crunch Navier-Stokes equations and the like, involving hundreds of thousands of variables (the GCM model) might need to be rethought, in favor of simpler mathematical techniques, e.g., current explorations of maximum entropy models.

Dr. Rob Armstrong (SUNY/Stonybrook) says that "the patient is sicker than we thought." He points out that acidification of the oceans (as they absorb CO2 and form carbonic acid) may ultimately dwarf global warming effects in terms of ecological damage (coral reef destruction, dead zones). This acidification is a positive-feedback mechanism for atmospheric CO2 levels, as acidified water is less able to absorb CO2, meaning the oceans become less effective as a buffer. Current pH levels are lower (more acid) than any time in the Vostok ice core record (400,000 years) and perhaps several million years.

Dr. Irakli Loladze (Ohio State) explored the effect of increased CO2 on plant growth and nutrient concentrations. Increased CO2 leads to non-uniform changes in N, P, K, S, Ca, Fe, Zn, Cu and many other trace nutrients. Studies have shown that doubling CO2 can lead to 30-40% boost in biomass, and reduce transpiration of water by 23%. However, much of the world suffers from "hidden hunger," that is, bellies are full but nutrient deficiencies exist, leading to anemia, cretinism, infant and maternal mortality, low birth weight, goiter and other maladies. In the US we have supplements to take up the slack, but in other countries like Bangladesh, 97% of calories come from plants (rice primarily) and natural nutrient levels are critical. Most of the trace elements in plants are lower today than heretofore. As a rule of thumb, doubling CO2 causes trace elements (other than C, O and N) to drop by 10%.

Loladze says that cities like Phoenix currently have up to 600 ppm CO2 levels due to the abundance of cars and the formation of CO2 domes over the city.

Corn productivity is up ten-fold, leading to a glut of nutritionally deficient corn on the market. Harte notes that agricultural interests are pushing for ethanol production to reduce this corn surplus, even though it not a long-term solution.

Dr. Raymond Pierrehumbert (University of Chicago) gave an interesting talk on the idea of "Snowball Earth"--that is, based on calculations that the earth through much, if not all of its history, should have been frozen over, since the sun in the past was dimmer than it is now. Once the earth freezes over, the white surface would reflect solar energy away, tending to lock in the frozen regime. It could take many millions of years for the sitation to reverse, if ever (so-called hysteresis). This paradox was popularized by Carl Sagan. Somehow methane and CO2 concentrations have hit just the right levels over time to keep this from happening. One theory is that CO2 is stabilized by a feedback mechanism where CO2 is removed from the atmosphere by sedimentation into carbonate rock, but gets added again when that rock travels into subduction zones and gets vaporized in volcanic plumes. If CO2 levels rise, rainfall and thus sedimentation rates increase due to the increased temperature, removing CO2--completing a stable negative feedback loop. Of course, lately CO2 levels have been artifically raised dramatically.

There was quite a buzz about a current New York Review of Books article by Jim Hansen, titled "The Threat to the Planet." The author blurb states:

Director of the NASA Goddard Institute for Space Studies and Adjunct Professor of Earth and Environmental Sciences at Columbia University's Earth Institute. His opinions are expressed here, he writes, "as personal views under the protection of the First Amendment of the United States Constitution."
John Pastor said that the article represents "Prometheus unbound." Scientists are seething from an earlier scandal where Hansen was muzzled by the Bush administration, being assigned a 25-year-old political handler censoring his talks. Now the handler is fired (he falsified his resume) the anti-science Bush forces beat a retreat and Hansen is coming out with both barrels blazing.

Much homage was paid to a oceanographic pioneer named Alfred Redfield:

Will have to look him up...


Now, the Bayes' Law Diary:

Statistics 101 Part 11: Why spying is stupid Hotlist

Sat Jul 01, 2006 at 05:26:21 AM PDT

Spying on people.

We folks oppose it. The Bushites say it is necessary. They say that they only spy on terrorists. They say the spying helps them catch terrorists. They are wrong. We can prove it.

See how, below the fold

Thanks to a Kossack who sent me this offline; since he didn't say to cite him, I won't. If he wants to chime in, that's fine too.

He pointed me to an article by Rudmin which is here
Rudmin article. I've adapted their argument, changing things round a bit, and adding some detail (after all, this is a statistics diary).

OK. On to meat of the diary.

Bayes' theorem is a part of elementary probability. A formal statement, with formula, is on the wikipedia page.

prob of A, given B, equals prob. of B, given A, times prob of A, divided by prob of B.

Here's the gist, as applied to the spying situation:

We want to know the probability that a person Bushco spies on is a terrorist. In the formula, then,

A is "person is a terrrorist"
B is "Bushco spies on them"

We need to know
Pr(B|A) - that is, the probability that a person is spied on, given that he is a terrorist
Pr(A) Proabbility that a person is a terrorist
Pr(B) Probability that Bushco spies on a person

Let's make some guesses.

There are about 300 million people in the USA. (That's not a guess). Let's leave out little kids, and make it 250 million

How many terrorists? 1,000? 10,000? Let's be generous to the Bushite and say 10,000 terrorists.

So, Pr(A) = 10,000/250,000,000
= 1/25,000

Now, how many people does Bush spy on?
well, he's looking at all phone records, and now banking records, so that let's out very few people. Say he's spying on 90%. But let's define spying more narrowly. Suppose he is spying on a million people. Then Pr(B) = 1/250.
Next, probability that a person is spied on, given that he is a terrorist. Hmmmm. Terrorists are doing their best to avoid attention. But the NSA is doing their best to find them. Hard to know, so I will plug in various numbers

If NSA is right 10% of the time, then the odds that a person being spied on is a terrorist are 1 in 1,000. Ouch

let's say NSA is right 50% of the time, then odds that a person being spied on is a terrorist is 1 in 200.

Could they be right 90% of the time? I doubt it. But, even if they could, it would mean that the odds are 1 in 111.

Finding terrorists by broad based spying is a violation of our rights. It's also stupid. It can't work well.

Bayes theorem is not esoteric; anyone who's had statistics knows it. NSA is not staffed by idiots. They know Bayes' theorem. They know the spying system can't work.

So, why is Bushco doing it?

Could it be another power grab by a paranoid megalomaniac?

I leave that for you to decide.