A Myth of Gaussian Proportions

When I was a freshman in high school, my home room teacher gave us a very nasty assignment during an after-school detention session - to calculate 35 to the 35th power!

This assignment was particularly cruel and unusual punishment because there were no such things as calculators back in 1967.

What I really needed was something I didn't know about until college: a closed-form solution.

Finding closed form expressions for partial sums is a standard calculus exercise. The ur-example of this type of problem is the sum of the first n integers, which is easily shown to be n(n+1)/2.

This closed form expression collapses (n-1) operations into three. Because it yields an exact answer, it is not really a predictor, but, in a sense, it is a model of a process.

When this example is done in a calculus class, a typical accompanying story is how young Gauss solved this problem in record time, totally showing up the teacher who had given out the onerous task of adding the first 100 integers. (The version I always heard was that this was a punishment because the students had been particularly noisy that day. The sadistic mathematical punishments of my high school teacher certainly lends credence to this tale.)

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Improve Your Home Run Chances - Walk Softly But Swing a Small Stick

From Baseball Physics: Anatomy of a Home Run by Davin Coburn in the June 2007 issue of Popular Mechanics, comes some interesting data/trends on home runs. Some of these are contrary to baseball-lifers' opinions, but I assume that some players will take heed of the predictions. After all, one thing not in the article because it is obvious is the direct relation between salary and home-run prowess.

Check out the article for some interesting graphs that illustrate the following findings:

• The sweet spot is larger than previously thought
• Batted Ball Speed (BBS) is more of a determining factor in home runs. This leads to a prediction that increasing the swing speed is better than increasing the bat weight. A corollary to this is that lighter bats (31-32 oz) are ideal bats for pro players.
• Because of the direction of spin when they reach the plate, curve balls can be hit farther than fast balls, even though fast balls leave the bat traveling faster.

Other interesting facts that can be derived in an introductory physics class is that the average pro swing imparts 4145 pounds of force to the ball, and the farthest a ball can be hit (with no wind to help it, and no rarefied air such as in Colorado) is approx 475 ft.

Although no physicist is quoted by name in the article, I believe that some, if not all, of the topics discussed come from Alan Nathan's Physics of Baseball work, described in my earlier post on Willie Mays and Global Warming.

Fractal Solar Wind

Massive solar flare starting and ending on earth's surface. Click to enlarge.Sunspot cycles are hot right now - literally and figuratively. After just posting about Cycle 24 - the about-to-begin 11-year cycle of sunspot activity, I now see a lot of references to scientists at Warwick University reporting on observed fractal nature of the solar wind, and the ramifications for prediction and understanding of sunspot cycles

The articles announcing the findings have been very exuberant about this latest finding, but so far they are short on some crucial facts. They are also woefully inadequate when it comes to reporting previous work.

Most web sites are just reproducing the press release from the University of Warwick:

The researchers, led by Professor Sandra Chapman, have also been able to directly tie these fractal patterns to the Sun's 'storm season'. The Sun goes through a solar cycle roughly 11 years long. The researchers found the fractal patterns in the solar wind occur when the Sun was at the peak of this cycle when the solar corona was at its most active, stormy and complex - sunspot activity, solar flares etc. When the corona was quieter no fractal patterns were found in the solar wind only general turbulence.

From this description it is not clear what the fractal pattern is. The New Scientist site provides more details, including a possible reason for the fractal pattern:

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Visualizing Web Pages - HTML Graphs

FractaLog web graph. Click to enlarge. As an interesting follow up to my recent post on KartOO and Google Browser, check out the HTML Graphlet applet created by Sala (no last name), and posted on the Aharef blog. To use the applet, just enter the URL of the page to be graphed.

The applet constructs multi-colored graph of nodes and edges, with each color representing a different HTML tag. As the graph is produced, it grows outward, with branches sprouting - all in a very kinetic/organic way.

The color of the nodes refer to specific HTML tags:

blue: for links (the A tag)
red: for tables (TABLE, TR and TD tags)
green: for the DIV tag
violet: for images (the IMG tag)
yellow: for forms (FORM, INPUT, TEXTAREA, SELECT and OPTION tags)
orange: for linebreaks and blockquotes (BR, P, and BLOCKQUOTE tags)
black: the HTML tag, the root node
gray: all other tags 

The image at the top of this post is a map of FractaLog as of the date of this posting.

Sala asks that those with flickr accounts post a screenshot of their site tree, using websitesasgraphs for a tag. Click on this link to see a wide variety of web page graphs.

Some of these graphs are more fractal-like than the others. I can only hope that someday the FractaLog graph will look suitably fractal to deserve its name.

Update on Thursday, May 31, 2007 by R.A. DiDio

Note:  Just as I published this post, the link to the Graphlet is not working - nor is the link to Aharef.  I am getting  a "WEb Site Suspended" message.

If it is working for anyone, please let me know with a comment here. 

Solar Cycle 24 Predictions

Cycle 23-24 sun-spot predictions. Click to enlargeIn a post last year titled Solar Activity Modeling: Great Predictions, Lousy Understanding? I described the current state of solar cycle modeling. Solar storm cycles are approximately 11 years in length.

We are now entering Cycle 24, which will start in March 2008, and reach a peak in 2015. The prediction from the NOAA (National Oceanic And Atmospheric Administration) is that "the Earth will soon experience a period of intense solar storms and the exact number of solar storms expected will become clearer in time."

Read the NOAA release of these findings here.  This is an excellent article that describes how sunspots form, the nature of the 11-year cycles, and how the predictions are made. Sun spot activity can be extremely deleterious to transportation and communication infrastructure, and therefore predictions have to be accurate. The care with which the predictions are made is evident in the statement by solar-physicist Doug Biesecker:

“...there are approximately six techniques used to predict the intensity of a solar cycle,” said Biesecker. “The first three are based on statistics and provide a sound historical baseline upon which to forecast future cycles. The other three are based on physics and the sun’s dynamo conveyer belt theory.”

An overview of these techniques can be found in last year's post.

Web Search Visualization & Fractal Maps

KartOO fractal search. Click to enlarge.There is a growing trend in using connectivity-visualization mapping when web searching. Search engines such as KartOO and TouchGraph Google Browswer, attempt to show the linkage among web sites/pages by web-like 2-D graphics, where colors and icon-sizes indicate the strength of the site linkages. The effect can be described best as a "concept map", with nodes representing sites, and connections between nodes representing links. Of the two engines, KartOO is much more visually interesting. (I have included screen shots produced by each engine in a search for "fractal".)

Whether this enhances the search experience is debatable - at present. While KartOO's pr claims that the non-linear nature of the way information is distributed on the internet should be matched in the non-linear nature of the visualization, so far I haven't seen anything that makes me want to switch from Clusty, which does a great job of organizing themes in web searches, and provides the links/themes in a nice old-fashioned list.

I find the same lack of compelling features for the Touch Graph Google Browser. I should note that TouchGraph is a company that designs custom-tailored visual search features for other companies and projects, with a focus on "creating tools that enable decision makers to display, navigate, and analyze complex data simply and intuitively...Individuals and organizations have more vital information at their fingertips than ever before. Traditional search engines provide a way to sift through this data. However, the greatest insights can be achieved not by sifting, but by looking at the big picture to see how items are connected."

TouchGraph fractal search. Click to enlarge.I'm still underwhelmed by what the visual engines do that's better than Clusty. They do produce some fascinating, kinetic displays, because nodes and connections expand when clicking on different parts of the graphic display, and various pop-ups appear with further detail about the sites. Because clicking often produces new sets of nodes, there is a fractal-like quality to the search map. I can see the efficacy of such a search for different types of data, or business functions, but I doubt that they will become the main search option for an ordinary web search.

When Monty Met Erwin

It may be that only Monty Python's exquisite blend of slapstick/farce/erudition/absurdity can take on the infamous cat of Erwin Schrödinger...

Schrödinger, one of the giants of Quantum Mechanics, truly believed that his cat presented an absurdity that undermined the probabilistic interpretation of Quantum Mechanics. QM was not undermined, and still continues to beguile and bedevil all who try to grasp its implications about the nature of the world.

Cecil Adams' "epic" poem The Story of Schrödinger's Cat, is a great riff on the Krazy Kat. A choice excerpt:

Now, you'd say the cat either lives or it don't
But quantum mechanics is stubborn and won't.
Statistically speaking, the cat (goes the joke),
Is half a cat breathing and half a cat croaked.
To some this may seem a ridiculous split,
But quantum mechanics must answer, "Tough @#&!

Of the many thousands of S-Cat web sites, one of my favorites is The Well-Intended, but not quite interactive Schrödinger's cat: A Rather Silly Experiment in Quantum Mechanics.Click here to try your luck at predicting whether you can indeed let the cat out of the bag...

Cartoon info: By Paul Dlugokencky, concept by Zachary H. Levine, for the American Physical Society

There's Danger in Them Thar Equations

A very interesting piece by Howard Wainer in the latest American Scientist (May-June 2007) concerns dangerous equations, which he describes as falling into two classes:

• equations that are dangerous because we know them - they "may pose danger because the secrets within its bounds open doors behind which lies terrible peril," with E=mc² the most obvious candidate
• equations that are dangerous because we don't know them - mot because there is no theory that has yet yielded these equations, but rather because they are not known by those who need to know them. This is especially true for policy makers that base their decision on mathematical models, and specifically statistical models.

Wainer's top choice for most dangerous statistical equation is due to Abraham de Moivre, who showed in 1730 that the standard error of the mean of a sample is the standard error of the mean of the population divided by the square root of the sample size. A significant prediction of this equation is that small sample sizes lead to large fluctuations in sample means. It is this simple statement:

small samples → large fluctuations in sample means,

that provides the biggest danger when not used, or not understood, by both policy makers and the average citizen.

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The Two Einsteins

A lot has been written about the latest Isaacson bio of Einstein, and it's now on all US best-seller lists.  Remarkably, there is another Einstein bio just released by Jüurgen Neffe (actually released 2 years ago in Germany and then most of  Europe where it has been a best-seller equivalent to Isaacson) .

I was lucky enough to get the Inquirer assignment for both of them. See the full review. (Because the Inquirer piece was restricted to 850 words, I will have much more to say about both of these books shortly.)

They're both amazing books, with Neffe's maybe the more interesting... If you have the time, read them both!

Killing the Mud Softly With...

...concrete balls. Big ones.

The mud volcano in Porong, Indonesia, has now been spewing its thick ooze for almost a year  (since May 29, 2006). The stats on how much mud is flowing is scary.  Reports claim that mud flows of up to 126,000 cubic meters (that's 164,801 cubic yards) a day are being recorded.

This is a lot of mud. To put this number into some perspective, it would fill up a 13-story office building with a footprint of 50x50 yards.

Each day.

And there's no sign that the volcano will be stopping anytime soon, if ever. Given the estimated size of the mud source below the volcano and current flow rates, this mud pie could be emptying out for 10's, if not 100's of years!

With nothing traditional working so far (e.g. walls or berms), a radical plan to stop the mud is now underway - thanks to some interesting modeling. The plan, designed by geophysicists, consists of dropping giant concrete balls (weighing up to 250 lbs), linked together on a chain, with four to a chain, into the mouth of the volcano (the largest balls are 16" in diameter). The idea is to "is to make the channel smaller ...narrowing it enough to slow the mud's rise and so decrease its flow rate by up to three-quarters. Forced to go around the chains and balls, the mud will give up some of its energy to friction, vibration and rotation."

The net result is beautifully described: "It will make the mud tired. We're killing the mud softly."

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