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Archive for the ‘Math’ Category

Something lovely: oscillography

In Delights, Math, Technology on March 3, 2009 at 4:06 am


Eric Archer, electronic media experimentor, has rigged up a vector art synthesizer with an oscilloscope, a digital pattern generator, and a set of identical cards called Quadrature Wavetable Oscillators, which convert digital information into analog voltages. The outputs are summed on a two-channel mix bus, with the two channels representing the X and Y coordinates in Cartesian space. The oscillations can make beautiful fine-line patterns reminiscent of the engravings on paper currency around the world.

Specialized lathes have been in use for hundreds of years to make complex patterns that are unreproduceable without directly copying them (i.e. photography or digital means). This is the historical art of guilloche (ghee-o-shay’) or Engine Turning. Remember the old 1970’s toy called Spirograph? It operates on a similar principle, producing mathematical curves called epitrochoids via revolving circular gears around each other while a stylus traces their motion. Other combinations of motion can be used, such as mounting the stylus to a rotating disc as it traverses a straight line. Watchmakers and jewelers have long used these techniques for ornamentation on their work. The famous Faberge eggs bear designs engraved by a similar technique.

Inside these sophisticated engraving machines, there are numerous settings to be made among the gears that revolve to cut the pattern. One doesn’t need many meshed revolving gears before it becomes possible to produce endless patterns that are practically impossible to replicate. Hence this technique was adopted very early by national governments to mint their paper currency, postage stamps, and other monetary certificates. The U.S. Treasury is rumored to maintain such a machine, known as a Geometric Lathe, containing ten interlocked pattern-generating discs. The settings of the discs would only be known to a select few, as this information must be guarded from the hands of counterfeiters… at least prior to the digital age we are in now.

More here.
Flickr stream here.
Also: Archer has a gadget that lets you listen to the modulations in visible light. The sun apparently sounds incredible — like “pink static.” Listen for yourself here.


Stop Math Now!

In Finance and business, Math on February 25, 2009 at 4:47 am

Or so it would seem…
This article by Felix Salmon, pinpoints the Gaussian Copula formula, an advance in financial mathematics pioneered by David X. Li, as the source of our financial collapse.

With his brilliant spark of mathematical legerdemain, Li made it possible for traders to sell vast quantities of new securities, expanding financial markets to unimaginable levels.

His method was adopted by everybody from bond investors and Wall Street banks to ratings agencies and regulators. And it became so deeply entrenched—and was making people so much money—that warnings about its limitations were largely ignored.

Then the model fell apart. Cracks started appearing early on, when financial markets began behaving in ways that users of Li’s formula hadn’t expected. The cracks became full-fledged canyons in 2008—when ruptures in the financial system’s foundation swallowed up trillions of dollars and put the survival of the global banking system in serious peril.

So what is a copula? It’s a statistical model for determining correlation between different random distributions. Correlation is at the heart of the business of financial engineering — if you’re going to construct a CDO out of mortgage obligations, you’d better know the probability that all of them will default at once. Pricing risky assets — the problem the market got horribly wrong — rests on computing the correlation between rare events. Li’s formula had a deadly simplicity. Correlation was reduced to one parameter — it didn’t take into account the relationships between various loans that make up a pool. The correlation between two stocks may not be constant over time, but the copula formula treated it as if it were.

So is financial math evil? I’m inclined to think we need more of it, not less. A simple model was applied too enthusiastically by bankers who didn’t understand the quants — maybe we’d have been spared the current credit crisis if asset managers had more mathematical sophistication.

But read Salmon’s whole article; it’s very smart and very accessible.

Facebook, evolution, and mathematical modeling

In Biology, Delights, Math on February 12, 2009 at 6:09 pm

Slate has a neat article about Facebook’s new “25 things about me” craze. (For those who have remained blissfully ignorant: thousands of users wrote notes about random personal habits or goals, and tagged their friends in an expanding web of navel-gazing.) Turns out it can be modeled like an epidemic. A user is “contagious” for about one day — the day he tags a bunch of his friends in the note. After being tagged, most users respond within one day. Then response frequency drops off exponentially.

Here’s a nice Nature Review about the mathematics of modeling infectious disease.

biological infectiousness of influenza, HIV, and malaria

biological infectiousness of influenza, HIV, and malaria

The number of individuals that an infected person infects is given by a probability distribution. The probability that an infected person will infect another person within a small interval is

b(t) s dt

b is infectiousness, dt is an arbitrarily small amount of time, and s is the probability that the other person is infected.
If a group of individuals all have the same infectiousness, then the number of secondary infections that are caused by each infectious individual is a random number drawn from the Poisson distribution with mean R, where R is the expected number of new infected victims.

The interesting thing here is that the whole field of mathematical modeling of disease transmission isn’t going to be just a biological subject forever. It’s also going to be a behavioral subject. The idea that cultural ideas propagate and evolve like organisms isn’t new — it’s as old as Dawkins and his notion of “memes.” But back in the sixties he couldn’t have predicted just how concrete the similarities would be — that we could see the exact same differential equations governing Facebook crazes as malaria outbreaks. Watch as epidemiologists get drafted as marketing consultants in the next few years.

Diffusion geometry: data takes shape

In Math on February 10, 2009 at 4:34 am

Graph of a set of 1000 documents

Graph of a set of 1000 documents, arranged by similarity.

How can you organize and extract information from huge data sets of digital text documents? How do you develop automatic recommendations based on preference history? Could Google ever personalize your search results to reflect your previous interests? Essentially, this is an applied math problem, and it uses a relatively new technique called diffusion geometry. Here’s an article by Ronald Coifman and Mauro Maggioni, two leaders in the field.

Basically, we want to understand the geometry of the data. We would like data points to lie on a manifold — a smooth surface in high-dimensional space, representing some kind of relatively simple rule (just as a sphere represents the rule “all points are the same distance from the origin.”) Then, we want to guess functions on the data from a few samples, with the goal of predicting values of the functions at new points.

The first step is to define a “similarity” function on the data. For example, if you have a thousand articles, a data point might be a vector consisting of the frequency of each word in the article (note that these vectors are huge!) and the similarity might be the correlation between word vectors when larger than 0.95, and zero otherwise.

Then, if we normalize the similarity, it’s a Markov chain; it describes a random walk, the probability of “jumping” from one document to a similar one. We can imagine a drunken ant crawling at random from one node to another, hitting similar, nearby nodes most often, but occasionally venturing to farther ones.

The eigenvectors of this Markov chain can be used as new coordinates for the data set; we can make a map of the data in space so that the distances between points equal the “diffusion distances” on the original data, the probabilities of reaching one node from another. This gives the desired representation of the data on a low-dimensional surface.

What this means is that the data can create its own rules and categories. Instead of labeling articles “business” or “sports,” we’ll see a cluster in the data that forms its own label. It’s a kind of spontaneous organization.

Here’s a nifty tutorial from Duke; for more mathematical detail, check out this article.