Skip to content

Category: Statistics

The general theory, methods, and philosophy of the Science of Guessing What Is.

December 30, 2007 | 1 Comment

Hurricanes have not increased: misuse of running means

Most statistics purporting to show that there has been an increase in hurricanes do not use the best statistical methods. I want to highlight one particular method that is often misused, and which can lead one to falsely conclude that trends (increasing or decreasing) are present when they actually are not. Read my original post to learn more about this.

That technique is the running mean. As you can see in the rather dramatic graphic from Science Daily, a 9-year running mean has been plotted over the actual hurricane numbers (up to 2005 only) in the North Atlantic. It looks like, in later years, a dramatic upswing is taking place, doesn’t it? This type of plot has shown up in many scientific, peer-reviewed papers.

Science Daily hurricane running mean

Don’t be turned off by the equations! Something very surprising is coming at the end of this article and you will be rewarded if you read to the end.

What is a running mean? A p-year running mean converts a series of actual, observed numbers into a statistical estimate, or model, of what a supposed underlying trend of those numbers might actually be. Because it is a model, its use must first be justified. In math, a 5-year running mean looks like

Running mean equation

where the symbol y indicates the hurricane numbers, and the subscripts t, t-1 and so on indicate the time period: time t is now, time t-1 was last year (now minus one year) and so on. The superscript on the symbol y to the left of the equal sign indicates that this is the modified data value plotted, and is not the actual number. Even if you’re afraid of math, this equation should be fairly easy to understand: the current, modified, number is just the mean of the last 4 observations and the most current one.

Additionally, in the mathematics of time series models, an auto-regressive series of order 5 is written like this

Autoregressive formula

which shows how the current data point is predicted by a weighted sum of past values, and where the weights are the coefficients ?. Just let all the ? = 1/5 and you have a similar running mean structure like that above. The point is this: using a running mean implies an underlying statistical time series model. Which is OK, as long as the data support such a model.

Do they for hurricanes? No.

In order to justify using auto-regressive time series models, you start by looking at something called an auto-correlation plot, which is a plot of how each year’s number of hurricanes is correlated with the previous year’s number, and how this year’s number of hurricanes is correlated with the number of hurricane from two years ago and so on: the number of previous years is called the lag. If any of these correlation lags are significant, then you can use an auto-regressive time series model for this data. If none of these correlations are significant, then you cannot.

Here is a picture of the auto-correlation of hurricane number (number of storms s) for the North Atlantic using data from 1966 to 2006.


None of the correlations reach above the horizontal dashed lines, which means that none are significant, and so a simple running mean should not be used to represent North Atlantic hurricane numbers.

So far, so good, right? Now let’s look at some made up, fictional data. Take a look at the following pictures, which are all simulated hurricane numbers; one of them looks pretty close to what the real data looks like. The running-mean even shows a healthy upward trend, no doubt due to global warming. But what do these pictures really show?

Simulated data with 9-year running mean

To get this data (the R code to make it yourself is pasted bellow), I simulated hurricane numbers (Poisson with mean 10) for pretend years 1966 to 2005, four separate times. Each year’s number is absolutely independent of each other year: to emphasize, these are totally random numbers with no relationship through time. I also over-plotted a 9-year running mean (red line). Because all of these numbers are independent of one another, what we should see is a flat line (with a mean of 10). The reason we do not is because of natural variation.

I only had to run this simulation once, but pay attention to the lower-right hand numbers, I got something that looks like the actual North Atlantic hurricane numbers. The 9-year running mean is over-emphasizing, to the eye, a trend that is not there! Actually, this happens to two of the other simulated series. Only one shows what would expect: a (sort of) straight line.

Like I said, I am including the code I used to make these plots so that, if you are curious, you will see how exceptionally easy this is to do.

Good statistical models are hard to do. See some of these posts for more discussion and to find some papers to download.

R code to make the plots. You must first have installed the package gregmisc.

library(gregmisc)
par(mfrow=c(2,2))
for (i in 1:4){
x=rpois(40,10)
plot(1966:2005,x,type='l',xlab="",ylab="",axes=F)
axis(1)
lines(1966:2005,running(x, width=9, pad=TRUE, fun=mean),lwd=2,col="#CC7711")
}
December 28, 2007 | No comments

Were the cannonballs on or off the road first?

There’s something of a controversy whether photographer Roger Fenton placed cannon balls in a road and then took pictures of them. He also took a picture of the same road cleared of cannon balls. Apparently, there is a question whether the cannon balls were ON the road when he got there, or possibly they were OFF and he placed them there to get a more dramatic photo. This drama unfolds at Errol Morris’s New York Times blog.

Whether they were first ON or OFF (Morris uses the capitals letters, so I will, too), excited considerable interest, with hundreds of people commenting one way or the other, each commenter offering some evidence to support his position.

Some people used the number (Morris uses the ‘#’ symbol) and position of the balls, others argued sun shadows, some had some words about gravity, and so on. Morris compiled the evidence used by both sides, ON (cannon balls on first) and OFF (cannon balls placed there by Fenton), and he presented this summary picture (go to his blog to see the full-sized image):

Morris cannonball evidence pic

This is an awful graph: the order of evidence types is arbitrary, it would have been better to list them in order of importance; the use of color is overwhelming and difficult to follow; and, worst of all, the two graphs are on an absolute scale. 288 people supported ON, and 153 OFF, so counting the absolute numbers and comparing them, as this picture does, is not fair. Of course the ON side, with almost twice as many people, will have higher counts in most of the bins. What’s needed is a percentage comparison.

Continue reading “Were the cannonballs on or off the road first?”

December 24, 2007 | No comments

Two differences in perception between global cooling and global warming

As is well known by now, a passel of climatologists in the 1970s, including such personalities as Stephen “It’s OK to Exaggerate To Get People To Believe” Schneider, tried to get the world excited about the possibility, and the dire consequences, of global cooling.

From the 1940s to near the end of the 1970s, the global mean temperature did indeed trend downwards. Using this data as a start, and from the argument that any change in climate is bad, and anything that is bad must be somebody’s fault, Schneider and others began to warn that an ice age was imminent, and that it was mainly our fault.

The causes of this global cooling were said to be due to two main things: orbital forcing and an increase in particulate matter—aerosols—in the atmosphere. The orbital forcing—a fancy term meaning changes in the earth’s distance and orientation to the sun, and the consequent alterations in the amount of solar energy we get as a result of these changes—was, as I hope is plain, nobody’s fault, and because of that, it excited very little interest.

But the second cause had some meat behind it; because, do you see, aerosols can be made by people. Drive your car, manufacture oil, smelt some iron, even breath and you are adding aerosols to the atmosphere. Some of these particles, if they diffuse to the right part of the atmosphere, will reflect direct sunshine back into space, depriving us of its beneficial warming effects. Other aerosols will gather water around them and form clouds, which both reflect direct radiation and capture outgoing radiation—clouds both cool and warm, and the overall effect was largely unknown. Aerosols don’t hang around in the air forever. Since they are heavy, over time they will fall or wash out. It’s also hard to do too much to reduce the man-made aerosol burden of the atmosphere; except the obvious and easy things, like install cleaner smoke stacks.

Pause during the 1980s when nothing much happened to the climate.

Continue reading “Two differences in perception between global cooling and global warming”

December 20, 2007 | No comments

Can increasing fuel economy standards result in more gas consumed?

Yes.

Congress recently passed an increase in fuel efficiency standards for cars, from 25 MPG to 35 MPG, a 40% jump. So you would expect that, when this law goes into force, gasoline usage will go down. That’s what various congresspersons and “environmentalists” are arguing, anyway.

Unintended consequences

Now, the mandated increase is a very large change, and complying with the law is probably beyond current engineering capabilities. That is, automotive engineers will have a difficult time implementing these standards in the time alloted, unless they do the one easy thing available to them, which is to make cars lighter. Lighter cars get higher gas mileages.

Making cars lighter is not hard. You simply take things out of heavy cars or make smaller cars. Problem solved!

Except smaller and lighter cars, all other things being equal, fare far worse in crashes. People know this, and tend to buy a larger vehicle instead. That is, confronted with a choice of a small, more dangerous, car, they will more likely buy a larger SUV or a truck.

Trucks and SUVs do not have to comply with the higher gas mileage requirements. Mileage for these larger vehicles is about 15 MPG (average of city and highway driving).

So instead of buying a safer car that now gets the required 25 MPG, people will be more likely to buy vehicles that are, on average, 60% less efficient!

Thus, more gas will be used than before the higher standards were in place.

Of course, I cannot prove that my scenario is certain to happen, but it is at least not impossible, and even somewhat likely. If I am right, this will be yet another example of good intentions gone bad.