Tag Archives: statistics

Evolution of Language

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We know that statistics (which is different from mathematics) plays an important role in various other sciences (mathematics is not a science, it’s an art). But still I would like to discuss one very interesting application to linguistics. Consider the following two excerpts from an article by Bob Holmes:

1. ….The researchers were able to mathematically predict the likely “mutation rate” for each word, based on its frequency. The most frequently used words, they predict, are likely to remain stable for over 10,000 years, making these cultural artifacts, or “memes”, more stable than some genes…..

2. ….The most frequently used verbs (such as “be”, “have”, “come”, “go” and “take”) remained irregular. The less often a verb is used, the more likely it was to have been regularised. Of the rarest verbs in their list, including “bide”, “delve”, “hew”, “snip” and “wreak”, 91% have regularised over the past 1200 years…….

The first paragraph refers to  the work done by evolutionary biologist Mark Pagel and his colleagues at the University of Reading, UK. Also, “mathematically predicted” refers to the results of the statistical model analysing the frequency of use of words used to express 200 different meanings in 87 different languages. They found the more frequently the meaning is used in speech, the less change in the words used to express it.

The second paragraph refers to the work done by Erez Lieberman, Jean-Baptiste Michel and others at Harvard University, USA.  All people in this group have mathematical training.

I found this article interesting since I never expected biologists and mathematicians spending time on understanding evolution of language and publishing the findings in Nature journal. But this reminds me of the frequency analysis technique used in cryptanalysis:

Mathematical Relations

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In this post I will share my perception of relation of mathematics with other academic disciplines. All this is based on my very limited knowledge of various disciplines.

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Shape doesn’t signify anything.

Mathematics deals with study of properties of numbers (or the symbols representing them) and geometric objects (not in classical sense, it can mean manifolds also). In my opinion, there is no partition of mathematics into “applied” or “pure”, but intersections with other subjects. The term applied Mathematics doesn’t make any sense to me. Mathematics is somehow applicable in various places. For me, mathematics is what people call “pure” mathematics (what about “impure” Mathematics??).  Also now I agree with the vastly established belief that art and mathematics are similar, since both involve abstract ideas motivated but physical situations (at some point).

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Truth Lies Deception and Coverups – Democracy Under Fire (Source: http://goo.gl/yUHi93)

All experimental sciences (physics, chemistry, biology, economics) are based on statistics. Since statistics is a young discipline (only a couple of centuries old) many times we get wrong interpretation of results. As far as real life is concerned, study of statistics gives us a powerful tool for predicting future and Probability Theory acts as the connecting link between statistics and mathematics. Understanding of statistics affects us on daily basis since (effective) government policies are framed keeping statistical analysis in mind. Unfortunately, most of universities don’t have separate department for statistics.

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P vs NP Problem in Relationships (http://ctp200.com/comic/6; CC BY-NC 4.0)

Study of algorithms is one of the most important aspect of computer science (I am not talking about software industry…). What surprises me is that Euclid’s division algorithm is  one of the most efficient division algorithm even for computers! The neglected subject of Logic, which is supposed to be foundations of mathematics, flourishes in computer science. P vs NP is another “millennium open problem“.

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Convincing (http://xkcd.com/833/ ; CC BY-NC 2.5)

For me, Economics like Statistics is full of imperfections due to real life complications (so many dependencies to account for). Game Theory appears to be the connecting link between mathematics and economics.

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We all know that the needs of physicists are responsible for development of calculus and study of differential equations. On the other hand, theoretical physics (quantum mechanics, string theory) depends heavily on the developments in algebra.

Galton Board

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Like previous post, in this post I will discuss another contribution of Jacob (Jacques) Bernoulli. The motivation for this post came from Cédric Villain’s recent TED talk. Though I am not a fan of probability theory, but this “toy”, which I am going to discuss, is really interesting.  Consider following illustration from a journal’s cover:

“Galton Board” was invented by Francis Galton in 1894.  It provided a remarkable way to visualize the distribution obtained by performing several Bernoulli Trials in pre-digital computer era.  Bernoulli trial is the simplest possible random experiment with exactly two possible outcomes, “success” and “failure”, in which the probability of success (say, p) is the same every time the experiment is conducted.  If we perform these Bernoulli trials more than one time (say, n times) we get, what we call, Binomial Distribution. We get a discrete distribution like this:

And  when the number of Bernoulli trials is very large (theoretically what we would call infinite number of trials), this Binomial Distribution can be approximated to Normal Distribution, which is a continuous distribution.

The Normal Distribution is important because of the Central Limit Theorem. This theorem implies that if you have many independent variables that may be generated by all kinds of distributions, assuming that nothing too crazy happens, the aggregate of those variables will tend toward a normal distribution. This universality across different domains of science makes the normal distribution one of the centerpieces of applied mathematics and statistics.

Here is a video in which James Grime demonstrates how Galton Board can be used to visualize Normal Distribution approximation of Binomial Distribution for very large number of Bernoulli trials. The trial outcome are represented graphically as a path in the Galton board: success corresponds to a bounce to the right and failure to a bounce to the left.