Posts tagged math

Recently the senior class voted on prom themes and there was also an additional field requesting people to select their friends. This was going to be used as an attempt to forecast possible winners of prom king and queen. My theory was that by using PageRank, an algorithm that Google uses to rank importance of a website, to determine the “most popular” people and thereby predict who will win prom king and queen. 

TL;DR

• tried to survey people of their friends
• use pagerank on the resulting graph determine most popular and predict prom king and queen winners
• failed: not enough participants, people didn’t understand why, PageRank doesn’t seem to correlate with popularity.
• other ideas?

TL

Of the 209 seniors, 104 people voted on prom themes. There were three themes and each person was allowed to put a number for each theme between 1 and 10. The reason for this, rather than a first choice, second choice, third choice system was because some people feel adamantly about one about and really don’t want the other options, while some could careless. Those who careless generally give the themes about the same score, whereas those that really don’t want one theme can give it a low score relative to others. I had many 1,10,1 kinds of votes, and a number of 6,7,8 kinds of votes. Clearly some people felt stronger about this than others (I should’ve graphed participation between girls and boys, I’ll do that tomorrow). To determine the prom theme, the scores were averaged and the highest average was chosen. 

Now while less than half the class voted (almost half), the results are representative of the class. There was a clear theme which had a higher average than the other two, thus a complete class participation wouldn’t likely change the outcome (also those who didn’t vote didn’t care enough about the theme anyway).

At the bottom of the voting form was something for the voter to select all of their friends. The form required at least 3 friends to be selected. AT LEAST, most people only selected 3 friends and then submitted their vote. On a side note this says something about how attentive people read when they are directed from Facebook, or it may be that people didn’t have much time to list their friends (though there was a quick search as you type box).

I ran PageRank on this (initially the R implementation and then I coded my own) friend graph. While I got results of who the “most popular” people were, the results aren’t very conclusive. While the prom voting is, because those are simple stats, doing PageRank on a graph requires most of the graph, ideally the whole graph. With only half of the graph, the results are inconclusive.

Further, PageRank is useful for determining “good quality” web pages because it uses link data, what pages other people link to, to determine good pages. I don’t know if this analogy can apply to people, or at least I’m not sure of how to interpret this. When the “links” are friendship lines, what is being determined by finding the highest ranking people with PageRank? Most friendly, most popular, most influential? I’m not sure and I’d like to know your thoughts on this. 

Recently I’ve been writing F# code to process and determine if a tweet is interesting based on previous tweets that I give it. It uses what’s called a Bayesian classifier to determine if a tweet goes into the interesting or boring category. I’ve learned a lot by writing this and it still doesn’t work exactly as I’d like. The problem is finding the right feature set to train the model against. I’m going to use the top 50 words and their distributions from the interesting tweets as features now and see if that improves the model’s ability to predict tweet interestingness.

The first “open question” that I asked was why one can’t integrate things of the form cos(x^n) or sin(x^n) when n is larger than 1. This wasn’t a serious question, just something to think about. The way that I came to conclusion that it couldn’t be done is realizing that e^(x^n) can’t be done.

By Euler’s extension of complex exponentials, e^øi = cosø + i sinø, one can see that ∫cosø^2 dø would be the real part of (∫e^(iø^2) dø). And since e^(x^2) can’t be done (again, with elementary functions) nor can cosø^2 be done.

The other not so serious question was to explain the, ahem, “graph”. It was a graph of two cycloids with parabolas in the middle. It was done as a piece-wise function so that only part of the parabola’s above the cycloids could be seen. This “graph” ended up looking like something interesting.

Two cycloid Tuesday?

Why can’t you integrate sin(x^n) dx or cos(x^n) dx if n is larger than 1?

p.s. you can but you basically end up inventing functions, specifically why can’t it be done with elementary functions

Some people seem to have these things confused.

I feel like rambling about math so that’s what I’m doing. I’m referring to 0/0 and 1/0. One divided by zero is undefined, whereas zero divided by zero is indeterminate. What’s the difference? Undefined more or less means there is no number, whereas indeterminate means there can be any number.

For example, suppose we wish to find the value of 0/0, we set 0/0 = x, multiply both sides by 0 -> 0 = 0x -> true for everything, thus indeterminate.

As for undefined, suppose that 1/0 = x -> 1 = 0x -> no number will do this. It is common to think of 1/0 as infinity, but it’s not so simple. 1/0 is a number with infinite magnitude, but the direction is unknown. A look at the graph of y = 1/x will better help to explain this.

Depending on the direction that one approaches 1/0, determines the direction of the infinite number. 1/0 could be negative infinity, as can be seen by the graph. If one considers arriving at 1/0 in the complex plane, then 1/0 could be any complex infinity!

I prefer calling 1/0 undefined but some still call it infinity. The JavaScript language is one example that prefers “infinity” over “undefined” even though it has both an “undefined” value and an “Infinity” value (it also has a “NaN” or “Not a Number” value). The case for JavaScript is probably that it’s just easier to handle numbers but setting them to “Infinity” rather than set them to “undefined” because a division by zero would then set the variable to “undefined” and the programmer might be confused, thinking that “why is this variable unassigned?”