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Archive for the ‘maths’ Category

Removing outliers

Monday, October 17th, 2011

If you've got a sequence of data that you want to do some statistical analysis on, but you know that some of it is bad, how do you remove the bad data? You could just remove the top 5% and bottom 5% of values, for example, but maybe you don't have any bad data (or maybe you have lots) and that would adverse affect your measurement of the standard deviation.

Suppose that you know that your dataset is supposed to follow a normal distribution (lots do). Then you could remove outliers by measuring the skew and kurtosis (3rd and 4th moments), and just repeatedly remove the sample furthest from the mean until these measurements look correct. This algorithm is guaranteed to terminate since if you only have 2 samples the skew and kurtosis will be 0. You've still got a parameter or two to tune though (how much skew and kurtosis you'll tolerate).

Posted in maths | No Comments »

Import taxes to level the playing field

Thursday, October 6th, 2011

There's a big hurdle for the US manufacturing sector in that many products are much cheaper to produce in China or Taiwan. Part of the reason for this is that those countries have much less strict standards on things like pay, employee working conditions and pollution controls. Another way of looking at it is that US companies are saving money and skirting important social and environmental rules by outsourcing their manufacturing to countries which don't have these rules - with excellent consequences in terms of increased profits and cheaper products, but disastrous consequences in terms of the environment, the welfare of those who make the products, and US manufacturing jobs.

It seems to me that the US government should use its power to tax imports to level the playing field for US manufacturers by removing the incentive to manufacture elsewhere. In other words, the import tax on something should be the difference between what it costs to make something in the US and what it actually cost to make due to laxer regulations. Then, manufacturing for things sold in the US would (over time) move to the optimal locations based on where they were being sold and the where the raw materials were mined or recycled.

Suddenly making all our electronics more expensive by (maybe) a factor of 10 would be enormously disruptive so I suggest ramping up the tax gradually over a period of (say) 10 years or so. That would lessen the blow and give the US manufacturing companies some time to bootstrap. It also gives a great incentive to China to improve working conditions and emissions since doing so essentially wouldn't cost them anything (it would be covered by the corresponding reductions in import taxes).

In the long term, I would expect that the final cost of the manufactured products in question would stay about the same or even become cheaper than what they would be without these taxes. That's because as it becomes more expensive to hire people to do a menial job, it becomes more cost-effective to automate that job. The machine costs more to begin with (you need clever people to build and program it) but once it's up and running the unit cost per produced item is much lower.

There's a lot more to it than that, of course - China still has a big advantage in the expertise it has developed in building things, China sells to other places besides the US, and there are currency, debt, and trade treaty issues which further complicate matters in ways I don't completely understand. Still, I think it's an interesting idea to consider.

Posted in economics | 10 Comments »

Fractal waveform

Tuesday, September 27th, 2011

It would be really interesting to try to make a waveform that is fractal in nature, and actually sounds interesting and musical at whatever frequency you play it at. As you speed it up, some frequencies will go so high they become inaudible and new frequencies move into the audible range at the low end. In normal music, the pitches, effects, tempos and structures all happen at different timescales but in fractal music they would all interact with each other in interesting ways.

Posted in fractals, music | No Comments »

A weird thing in Haskell

Wednesday, September 21st, 2011

Here's something odd I noticed while playing around with the Haskell programming language. Sometimes, a==b does not imply f(a)==f(b). Look:

> 1/0
Infinity
> 1/(-0)
-Infinity
> 0==(-0)
True
> (1/0)==(1/(-0))
False

Posted in language, maths | 2 Comments »

Interactive IFS

Monday, September 19th, 2011

I want to write a program that allows you to explore Iterated Function System (IFS) fractals interactively by moving points around with the mouse.

There's a few different ways to do this, depending on what set of transformations you use.

IFS fractals usually use affine transformations, which encompass translations, rotations, dilations and shears. This can be done with 3 points per transformation - if one considers the points as complex numbers we have the transformation $f(x)=ax + b\overline{x} + c$ . However, rather than controlling $a$ , $b$ and $c$ directly I think it would work better to move the images of some interesting points like $0$ , $1$ and $i$ (i.e. move $c$ , $a+b+c$ and $(a-b)i+c$ ). Then the geometric interpretation of the control points would be easy to understand - they would just be the corners of a transformed rectangle.

However, there are other possible transformations. We could reduce the number of control points to 2 and disallow non-orthogonal transformations, giving $f(x)=ax + b$ and control points $0$ and $1$ mapping to $a$ and $a+b$ .

We could move to quadratics $f(x) = ax^2 + bx + c$ and move $c$ , $a+b+c$ and $c+ib-a$ , and with 4 points we can do cubics $f(x) = ax^3 + bx^2+cx+d$ (in which case we would probably use control points $f(0)$ , $f(1)$ , $f(i)$ and $f(1+i)$ ).

We can even go all the way to $f(x) = ax^2 + b\overline{x}^2 + c|x|^2 + dx + e\overline{x} + g$ (6 control points) if we wanted to go really crazy - that might be a bit unwieldy though.

I'd also like to be able to associate a colour with each transformation so that the colour of a point in the final image depends on the sequence of transformations that led to that point. Perhaps $C = \alpha C_{prev} + (1-\alpha)C_{transform}$ for some $\alpha$ .

Posted in fractals | No Comments »

Energy saving idea

Thursday, September 8th, 2011

One of the most inefficient things we do with energy is heat up water with it only to let most of that water (and most of the energy) go right down the drain when we take showers (baths are a little better because more of the energy goes into heating the house in the winter, but they do use more water).

So I think that if we were serious about wanting to save energy one thing we could do is try to encourage shorter showers by making people aware of how much energy they use. What I imagine is a little display that shows you how much energy you've used since the start of your shower (if you like you could factor in the cost of the water and sewage as well as the energy used to heat the water). For maximum effectiveness, I think the value displayed by the meter should be in units of local currency, and correspond to the amount you'd spend taking a shower of that length every day for a year - that way it feels like you're spending money really fast (of course, you'd really only be spending it at 1/365th of that rate so it's a bit of a psychological hack/cheat).

I think many people would appreciate such a device for the money it would save them, though I suppose some people might believe that long, guilt-free showers are worth that extra money.

Posted in economics | No Comments »

Issues as political proxies

Tuesday, September 6th, 2011

Suppose you own a large successful business which makes money by telling customers things they want to hear - reassuring stories, comforting platitudes and advice and guidance about how to live their lives. Suppose also that, for tax reasons, you are not allowed to use your influence over your customers to push them towards voting for one particular candidate over another, and you're also not allowed to donate any of the company's profits to political parties or candidates.

However, you'd still prefer to have one candidate elected over another because your preferred candidate might lower your taxes or give you more freedom to run your business the way you want to run it, or maybe just because he's a good customer. How could you covertly support that candidate?

One thing you could do is a pick a couple of social issues which aren't fundamentally a big deal to you one way or another but which differentiate your preferred candidate from their opposition and which the opposition is unlikely to change their minds on (perhaps because they are objectively correct in their position). Then you can use your platform to tell your audience that your preferred position on said social issues is vitally important, and deciding the wrong way on them will lead the country to ruin. You don't even need to mention the names of the political candidates or the upcoming election to your audience at all - they can figure out themselves what they need to do.

For this reason I think we need to avoid making "tax deductions for political neutrality" deals - it's too easy for the organizations in question to be covertly politically non-neutral and the tricks they use cause pressure to move candidates away from objectively correct positions in this kind of issue.

Posted in economics, philosophy | 6 Comments »

How do we get there from here?

Sunday, September 4th, 2011

So we have some ideas about how we want the world to look - the next question is "How do we get there from here?" It seems to be very difficult to get anything changed at least in US politics because there are so many entrenched interests, but here's the best idea I've had about it so far.

We use this fantastic information transfer medium of the internet to get as many people interested, involved and well informed as possible. We get these people to vote in on-line elections (that are at least to begin with unofficial, non-binding and informal but are as secure as possible and only open to registered, authenticated voters). We then try to persuade politicians to take these polls into account (as well as what they suppose the opinions of the rest of the electorate to be) when making their decisions. Participating in this system costs the politician nothing at first (since when they disagree with what the poll says they can say "oh that's just the opinion of a small minority of people, most people have the opposite opinion"), but as more and more people participate in these polls they eventually become impossible to ignore ("it's the will of the people"). When politicians vote against the will of the people, we call them out on it and hopefully get them voted out of office in the next election. Once the system has sufficient momentum, we start to field candidates who run on a platform of voting according to the results of these polls rather than their own opinions. Then eventually we can transition away from having elected politicians at all and just have a system of direct delegated democracy so that the people can vote (directly or by proxy) on every piece of proposed legislation. This is much less susceptible to corruption by corporations, because decisions are not made by wealthy minority.

In the meantime, we have to do something about the media. It's no good having a democracy if people are voting against their own interests and blindly following the instructions of corporate mouthpieces. I think this is more of a US problem than a UK one the BBC is much more impartial than private media can be. Here in the US there are massive numbers of people who get all their information from Fox News and conservative talk radio which are really just fronts for organizations like Koch Industries. This is how we get public support for absurd wars and other policies that are disastrous for almost all of the people who are voting for them. The usual method we use as a society for determining which side of an argument is true is the judicial system, so I'm wondering if we can somehow make news organizations liable for things that are not true that they present as news. Don't make the penalty too big because sometimes mistakes happen but make it large enough so that the likes of Fox can't continue their current scam. And if that puts too much power in the hands of judges, then we'll need some entirely new system of checks and balances to prevent abuse there. I guess to avoid stepping on the first amendment there would have to be some kind of voluntary labeling scheme for news organizations, and we would have to learn to take with rather more salt news from sources which don't stand by what they say by participating in this scheme.

We still need to keep the economy growing as fast as possible. Unlike the conservatives, I don't think the way of doing this is reducing taxes on the rich and reducing services on the poor. I think we need more small businesses, and that there are a lot of impediments preventing people from setting up or taking over small businesses. These impediments need to be identified and removed. More small businesses means more competition for large corporations. In the US, creating a functional public healthcare system would be a great benefit for small businesses (companies in the US are can't attract the best employees without providing health insurance plans, which is much more expensive for small companies than for big ones).

Posted in economics, philosophy | No Comments »

Fractals on the hyperbolic plane

Tuesday, August 30th, 2011

Some amazing images have been made of fractal sets on the complex plane, but I don't think I've ever seen one which uses hyperbolic space in a clever way. I'm not counting hyperbolic tessellations here because the Euclidean analogue is not a fractal at all - it's just a repeated tiling.

The hyperbolic plane is particularly interesting because it is in some sense "bigger" than the Euclidean plane - you can tile the hyperbolic plane with regular heptagons for example. Now, you could just take a fractal defined in the complex plane and map it to the hyperbolic plane somehow, but that doesn't take advantage of any of the interesting structure that the hyperbolic plane has. It's also locally flat, so doesn't add anything new. If you use some orbit function that is more natural in the hyperbolic plane, I think something much more interesting could result. I may have to play about with this a bit.

Similarly, one could also do fractals on the surface of a sphere (a positively curved space - the hyperbolic plane is negatively curved and the Euclidean plane has zero curvature).

Posted in fractals | No Comments »

The logic of the minimum wage

Saturday, August 27th, 2011

I've said before that I would prefer to replace the minimum wage with a guaranteed minimum income, but I've since thought of a justification for keeping the minimum wage.

In any economy, there are commodities - things which are easily available and the prices of which are easy to discover such as gold, oil or milk. One very important commodity is the hourly labour of a human who is healthy and competent at following simple instructions but otherwise unskilled. Even without a minimum wage law, there will be a natural price for such labour. It will fluctuate with time and space but will generally stay within a certain range. In some times and places it might actually go higher than the legislated minimum wage (which corresponds to a state of zero unemployment).

Having a legislated minimum wage has the effect of setting a scale factor (or a "gauge", for the physicists) of the economy as a whole - it's sort of like having an electrical circuit that isn't connected to anything and then connecting one part of it to ground. If the minimum wage is set too high then it will cause an inflationary pressure which will dissipate once everyone has a job again. If it's set too low then it will have a negligible effect anyway, since there would be very few people who would be unable to get a job for more than minimum wage. According to this theory, the minimum wage has nothing to do with making sure the poorest members of society are paid a respectable wage (which makes sense since a minimum wage is actually a regressive policy) - it's just an economic control factor.

Now, as an economic control factor, a minimum wage has a number of problems. One is that it takes a while for the inflation to eliminate unemployment after the market rate for labour goes down, so there's always some residual unemployment. Another is that people are not all equal - the labour of some people is just naturally below the basic labour rate (because they are unskilled and also either unhealthy or incompetent). While this is unfortunate, essentially forbidding them from working at all (by preventing employers from hiring them at the market rate) seems to add insult to injury (not to mention creating yet another dreaded poverty trap). A third problem is that there are many other ways that governments "ground" economies - in the electrical circuit analogy they're connecting various bits of a circuit to voltage sources that correspond to "what we think this ought to be" rather than what it actually is, which seems like a good way to get short circuits.