Normal vs. Fat-tailed Distributions(vudlab.com)
vudlab.com
Normal vs. Fat-tailed Distributions
http://vudlab.com/fat-tails.html
fat tailed distributions
13 comments
This is great, but I have a slight complaint.
Sliding the kurtosis indicator changes the left distribution, which makes sense. However, it also changes the appearance of the right, normal distribution, which is misleading. Normal distributions have [EDIT: constant] kurtosis. I realize that the appearance is changing because the scale is changing so that the max is always pegged. However, it might be less confusing if the scale remained static and the height of the left distribution simply changed, since that would be a more accurate representation of what's actually happening. That would obscure some details, but I don't think the precise details of the heights of each bar in the distribution are really the point of this page.
Sliding the kurtosis indicator changes the left distribution, which makes sense. However, it also changes the appearance of the right, normal distribution, which is misleading. Normal distributions have [EDIT: constant] kurtosis. I realize that the appearance is changing because the scale is changing so that the max is always pegged. However, it might be less confusing if the scale remained static and the height of the left distribution simply changed, since that would be a more accurate representation of what's actually happening. That would obscure some details, but I don't think the precise details of the heights of each bar in the distribution are really the point of this page.
Agree: This is great, slight complaint...
The scales on the walks should be synchronized to the extremes across both images. I was trying to compare the two walks visually and realized that the scaled didn't match.
The scales on the walks should be synchronized to the extremes across both images. I was trying to compare the two walks visually and realized that the scaled didn't match.
It's more than the matching scales.
The way the scales adjust on-the-fly means that once each graph (inevitably) either hits the near-top or near-bottom of the graph, the line stays there, and doesn't appear to move as the scale adjusts around it - the scale adjusts to keep the line close to the "adjustment point" near the edge of the graph.
The way the scales adjust on-the-fly means that once each graph (inevitably) either hits the near-top or near-bottom of the graph, the line stays there, and doesn't appear to move as the scale adjusts around it - the scale adjusts to keep the line close to the "adjustment point" near the edge of the graph.
yeah i gave myself some grief about that aspect but ultimately decided different scales was more misleading. could go either way.
also, the kurtosis of a normal distribution is 3, not 0.
Many statistical packages (SAS IIRC) subtract 3 from the kurtosis to set a normal distribution at 0. This is an infuriating tendency.
There are some good reasons for this. For example, if Y is the sum of n independent and identically distributed random variables with the distribution of X, then the kurtosis of Y is 1/n times the kurtosis of X. This doesn't hold without subtracting 3.
Yes, on your scale it appears that is the case. Most definitions I've seen have a -3 constant to zero out the normal, but if you say that's weird I won't argue. Incidentally, why didn't you extend the kurtosis scale to allow for a platykurtic distribution?
EDIT: to the original question, I'm not suggesting the distributions should have different scales, but rather that the scale of both should be static.
EDIT: to the original question, I'm not suggesting the distributions should have different scales, but rather that the scale of both should be static.
It's about fat tails, which is probably why it doesn't include platykurtic distributions.
Normal distributions have excess kurtosis equal to zero.
Yep, and for some reason unknown to me, the Excel function for kurtosis actually calculates excess kurtosis. I guess it's a confusion similar to log and ln. In most maths packages log is the natural log and kurtosis is the by-definition kurtosis. Excel? ln is natural log and kurtosis =KURT() + 3.
I worked with an exponential distribution before at work. At first, it seemed like we could model them like gaussian because the part of the data we were interested in was "close enough" to a gaussian. We already wrote code that worked for our other data that was gaussian. As it turns out, I was wrong.
The thing that can't be easily seen in pictures is that exponential distributions move differently than gaussian distributions. When the variance increases for a gaussian, it flattens out. When the variance increases for an exponential, the whole thing spikes out to the right, and the area under the tail actually increases. It really screwed things up until I started treating it for what it really was.
The thing that can't be easily seen in pictures is that exponential distributions move differently than gaussian distributions. When the variance increases for a gaussian, it flattens out. When the variance increases for an exponential, the whole thing spikes out to the right, and the area under the tail actually increases. It really screwed things up until I started treating it for what it really was.
Also, it is important to note that Gaussian is symmetric about its mean, whereas an exponential distribution takes values between 0 and infinity.
I think OP is talking about the two sided exponential i.e. laplace distribution which is symmetric about its mean.
From the article: "Both distributions below have standard deviations of 1"
I thought fat tailed distributions don't have a variance. But apparently I'm using the term in a stricter sense than other people. See http://en.wikipedia.org/wiki/Fat-tailed_distribution#Definit... for details if you were wondering, too.
I thought fat tailed distributions don't have a variance. But apparently I'm using the term in a stricter sense than other people. See http://en.wikipedia.org/wiki/Fat-tailed_distribution#Definit... for details if you were wondering, too.
Most statisticians have their own definition of fat-tailed, I like the definition you linked to (polynomial decay), but it is not universally accepted/known. These fat-tailed distributions will have a variance as long as they decay faster than x^(-3).
A good example of a fat-tailed distribution that does not have finite variance: http://en.wikipedia.org/wiki/Cauchy_distribution
the [edit: outer envelope of the] walk should slowly deviate from zero, shouldn't it (the variance of the sum increases with time / a random walk is a walk)? what i am seeing is returning to zero much more strongly than i would have expected. what is the prng that you are using? i suspect it's not that great.
Symmetric random walks, including those with step sizes drawn from a Gaussian process with mean zero, have expectation 0 at any time t. Since there's no drift term, either walk won't be expected to slowly deviate from zero.
This is false. Think about it this way. A random walk is a martingale - you're expectation of where you'll be in the future is. So you're right that, before you start the walk, they have expectation of being at 0 at any time in the future.
But the variance of your probability distribution of where you'll be at time t is linear in t. So say that your variance is v(t)=t. Then at t=1, there is a 32% chance that you'll be outside of the range (-1,1). As you can see, as t increases, you expected to drift further than further.
So while the expectation of x(t) may be 0 for all time, the expectation of |x(t)| scales like sqrt(t) (the standard deviation of the distribution).
But the variance of your probability distribution of where you'll be at time t is linear in t. So say that your variance is v(t)=t. Then at t=1, there is a 32% chance that you'll be outside of the range (-1,1). As you can see, as t increases, you expected to drift further than further.
So while the expectation of x(t) may be 0 for all time, the expectation of |x(t)| scales like sqrt(t) (the standard deviation of the distribution).
Thanks for the clarification! I haven't studied stochastic processes in a while, so the distinction was lost on me; it definitely makes more sense to think of it as a martingale.
What you say is true. Another, related fact is that no matter how far from zero the process has strayed, eventually, with probability 1, it will return to zero.
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It would be awesome if these could be embedded in Wikipedia somehow.
My laptop's fan says that may not be advisable.
P.S. sometimes people overlook how a well-drawn illustration may be superior to photos or video; think anatomical drawings, for instance. In this case I feel the lack of two simple curves superimposed to illustrate the point about the variance.
P.S. sometimes people overlook how a well-drawn illustration may be superior to photos or video; think anatomical drawings, for instance. In this case I feel the lack of two simple curves superimposed to illustrate the point about the variance.
I did not quite get the example with the bets. It is not clear what kind of problem the fat tails bring compared to normal as long as the distribution is symmetrical.
There is a term for dealing with actual distributions that differ from the one you put as basis in your theory. http://en.wikipedia.org/wiki/Robust_statistics
Regression analysis and particulary ANOVA are sensitive to http://en.wikipedia.org/wiki/Heteroscedasticity but may be less sensitive to fat tails as long as the standard deviation is independent from the mean.
There is a term for dealing with actual distributions that differ from the one you put as basis in your theory. http://en.wikipedia.org/wiki/Robust_statistics
Regression analysis and particulary ANOVA are sensitive to http://en.wikipedia.org/wiki/Heteroscedasticity but may be less sensitive to fat tails as long as the standard deviation is independent from the mean.
I could be completely off here, but my understanding of the bet example was if you make bets assuming normal distribution, the worst-case situation you're prepared to handle is going to be different than the worst-case situation that's actually going to happen.
http://blogs.sas.com/content/iml/2013/06/10/compare-data-dis... is pretty good about box plots and historgrams (static) and mentions Chambers' visualization book (besides Cleveland's classic text)
Nice use of d3 that really adds value. I'll use this as an example next time I do time series training at work.
Could I ask what you've used to make the animation repeat? From a quick search I noticed you aren't using setInterval() or d3.timer().
Are you just calling redraw as quickly as the CPU runs the code or am I missing something?
Could I ask what you've used to make the animation repeat? From a quick search I noticed you aren't using setInterval() or d3.timer().
Are you just calling redraw as quickly as the CPU runs the code or am I missing something?
I think the biggest fat tail distribution of all is the things that developers are asked to do on a day to day basis. The fat tail is where frameworks break down, where development processes and procedures break down. Most of the stuff I do is stuff that I have never done before and will never do again.
oops - I was thinking of the "long tail" not the fat tail.
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Sure, given a real valued random variable,
it has a cumulative distribution.
If that distribution is differentiable,
then the random variable also has a
density (distribution). Fine. Alas,
that doesn't mean that in practice
we should try to find what the
distribution is!
Actually, in practice, it's generally difficult to know with much accuracy what a distribution is. Then, for the OP, in practice it's much more difficult to know much about the tails or when they are fat or not or if fat how fat.
The OP wants to claim that the normal distribution applies to heights of people. I can believe that this is only roughly true!
Actually, the usual way we come to a normal distribution is from the central limit theorem (CLT); in practice we want something like the mechanism of the CLT to apply.
When do we get the CLT? Sure: If for some positive integer n our random variable Y is the sum, divided by the square root of n, of n independent, identically distributed (i.i.d.) samples of some distribution with, say, a mean and a finite variance.
If n is 12, then can start to entertain normality if don't want accuracy in the tails. If want high accuracy in the tails of the normal distribution from the CLT, then I'd recommend some careful work and otherwise not trust the accuracy.
A place where have a better shot at getting accuracy in a tail is the exponential distribution. The leading case is: Suppose we have a Geiger counter that goes "click" when it detects a radioactive decay. If the click rate is low so that the chances of two or more decays giving only one click are low, and real random variable T is the time until the next click, then under usual situations in practice T will have quite accurately exponential distribution.
More generally, the stochastic process of such clicks is a Poisson process and an example of an arrival process. Well with weak assumptions, for positive integer n, the sum of n independent arrival processes approaches a Poisson process as n approaches infinity. This result is the renewal theorem with a proof in W. Feller's volume II.
An example of a use of the renewal theorem is arrivals at a Web site. So, for each person on the planet there is an arrival process for that person at that site. If assume that the people are independent, then the Web site sees the sum of those arrivals and should see, over intervals of time much shorter than one day, a good approximation to a Poisson process.
So, that's some 'applied probability' where we might work with tails.
Mostly in applied probability, just f'get about accuracy in the tails!
Actually, in practice, it's generally difficult to know with much accuracy what a distribution is. Then, for the OP, in practice it's much more difficult to know much about the tails or when they are fat or not or if fat how fat.
The OP wants to claim that the normal distribution applies to heights of people. I can believe that this is only roughly true!
Actually, the usual way we come to a normal distribution is from the central limit theorem (CLT); in practice we want something like the mechanism of the CLT to apply.
When do we get the CLT? Sure: If for some positive integer n our random variable Y is the sum, divided by the square root of n, of n independent, identically distributed (i.i.d.) samples of some distribution with, say, a mean and a finite variance.
If n is 12, then can start to entertain normality if don't want accuracy in the tails. If want high accuracy in the tails of the normal distribution from the CLT, then I'd recommend some careful work and otherwise not trust the accuracy.
A place where have a better shot at getting accuracy in a tail is the exponential distribution. The leading case is: Suppose we have a Geiger counter that goes "click" when it detects a radioactive decay. If the click rate is low so that the chances of two or more decays giving only one click are low, and real random variable T is the time until the next click, then under usual situations in practice T will have quite accurately exponential distribution.
More generally, the stochastic process of such clicks is a Poisson process and an example of an arrival process. Well with weak assumptions, for positive integer n, the sum of n independent arrival processes approaches a Poisson process as n approaches infinity. This result is the renewal theorem with a proof in W. Feller's volume II.
An example of a use of the renewal theorem is arrivals at a Web site. So, for each person on the planet there is an arrival process for that person at that site. If assume that the people are independent, then the Web site sees the sum of those arrivals and should see, over intervals of time much shorter than one day, a good approximation to a Poisson process.
So, that's some 'applied probability' where we might work with tails.
Mostly in applied probability, just f'get about accuracy in the tails!
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