0 Interesting numbers

There's a sort-of theorem in mathematics that all numbers are interesting.*  But I'm thinking first of 1729, which is the subject of a story about how it is an interesting number.  The story is that Ramanujan, a brilliant, self-taught mathematician, was in the hospital (he died at only 32).  G. H. Hardy visited him and commented that his taxi cab was number 1729, which wasn't a very interesting number.  Ramanujan replied that it was indeed interesting -- it was the smallest number that was the sum of two cubes, in two different ways.  That is, 10^3 + 9^3 = 12^3 + 1^3 = 1729.  This number appeared also on a cab in an episode of the Simpsons.

There's a different bit of playing with numbers, one of the longest-unproved theorems in mathematical history.  That is Fermat's Last Theorem.  (Itself misnamed, as he never showed a proof for it, and he worked for years after stating it.)  That is, if we use only integers (1,2,3,...), the equation x^n + y^n = z^n has no solutions for n > 2.  He said this in 1637, and it wasn't proven until 1995.

Let's look specifically at n = 3.  I can rewrite Ramanujan's example as:
x^3 + y^3 = z^3 + 1  (where he had x,y,z = 10, 9, 12)
Fermat's equation is:
x^3 + y^3 = z^3   -- and this has no solutions for integers.

That's interesting -- such a small change, and we go from having no solutions, no matter how large we make x,y,z, to having ... how many?  Well, that's a question.  My version here is a more specialized version of so-called 'taxicab numbers' (named in honor of the above story).  You can see some more about them at Durango Bill's.  But I like mine better because of the connection to Fermat's Last Theorem. 

It seems common that answers in mathematics are either 0, 1, or infinity.  Fermat's equation (for n > 2) has 0 solutions.  We already have 1 for my 'Fermat-Ramanujan' equation.  If there's another, not that this is a proof, probably there are an infinity.  So I set my computer to some brute-force searching, and indeed there are more.  Not many.  It found 92 for z going from 12 (the smallest that has a solution) to 2,000,000 (which was pushing the limit of the computer; z^3 at that point is 8,000,000,000,000,000,000).  That suggests that there are an infinity of solutions to the Fermat-Ramanujan equation (a name I just invented, as far as I know), by that rule of thumb.

Challenges:
Can you find some? 
Can you do it by a more elegant method than having a computer pound away?
Can you prove that there _are_ (or are _not_) an infinity of solutions?

I can say that from my search, the solutions are getting pretty sparse as the numbers get larger.  Maybe the step to the next solution goes to infinity?  i.e., there isn't a 'next' one at some point.  Brute force computing can't answer that.  Need some real thought for the job.


* The 'proof' that all numbers are interesting:
0 is interesting because it is the basis for place value mathematics, or because it's the number you can add to any number and get back the original number, or ...
1 is interesting because you can multiply any number by it and get back the original number.
2 is interesting because it is the first prime number (can only be formed by multiplying itself by 1), is the only even prime number, is the first even number, ...
3 is the first odd prime number, ...
4 is the first perfect square number
and so on.  It's quite a while before it starts to be hard for a numbers person to find a number interesting.

Now suppose we keep going up the list, and finally find a number that isn't interesting (for being prime, a perfect square, a perfect cube, ..., a perfect number, a palindromic number, a taxicab number, ...).  Well, then, _that_ would be interesting!  There are so many ways for a number to be interesting, and this number avoids them all.  That's pretty interesting itself!

Detail: Interesting numbers

0 Saturn's Hurricane

Many articles and blog posts about Saturn's north polar hurricane.  I'll point you to the NASA press release for some viewing and discussion of the Saturn side of things.

No complete answer here, but I'll raise the flag that one of the movies and set of papers I watched and read in graduate school was under the title "The Range and Unity of Planetary Circulations".  Additional flag: my bachelor's degree was in applied mathematics.  We argued that if you understood the mathematics behind things, you could rapidly move from one area where a certain math applied to any other area where it applied.  Geophysical Fluid Dynamics is one such.  Notwithstanding the name, it applies to any area where you have a fluid on a rotating body -- whether it's Saturn, the earth, the Sun, or Venus.

Coincidentally, I've been playing with a fluid dynamic model, nominally of the earth, but it could be Saturn just as well, and have a movie which I'm still trying to figure out how to share.  This movie has some characteristics which look a lot like the press release.

Detail: Saturn's Hurricane

0 Century storms

"That's two straight years we've had a 'storm of the century'; those weather guys are idiots!"  Not long after I moved to the Washington, DC area, this happened, and the quote is real.  I'm not sure that the storms involved really were 'storm of the century' events -- events that if we had a long enough record, we'd see happen about 10 times per 1000 years -- but it's something to think of a little quantitatively, particularly in light of my normally abnormal note.

Let's suppose that we're building a house and would like it to last 30 years.  Well, to be specific, let's say we'd like a 99% chance of it lasting that long.  Obviously it has to be able to survive events that we'd expect to happen once per year.  And we can probably ignore things that we'd expect only once in a million years.  But what about a once in 100 year event?  The name misleads us in to thinking that the next time such an event would happen is 100 years after the last time.  While natural reading, it's wrong mathematics.  We could easily be in the unlucky 30 years that sees a 100 year event.  We could even see it twice.  But is there less than a 1% chance of having one 100 year event in a span of 30 years?  That's our design requirement.  If it can be expected more often than that, our house design is not reliable enough.  We need something better. And we'll need to get quantitative.

To deal with situations like this, where we have an expected number of events in a time span, whether it's number of 100 year events in the 30 years we want our house to last, or number of times a phone will ring in the next hour given how many times it usually does, we use the Poisson distribution.  For the Poisson expectation value (lambda) take the number of years you want and divide it by the N in N-year event.  100 at the moment.  In 30 years, then, we expect to see 0.3 '100 year' events.  Of course you can't have 0.3 of an event, you have 0, 1, 2, ... events.  One of the things the Poisson distribution does is to make the translation from our expectation in to something that could happen.  The other, of course, is that it tells us how likely it is to happen.

Number	Probability	Cumulative
0	74.1%	0
1	22.2%	22.2%
2	3.3%	25.5%
3	0.3%	25.8%

More than a 1 in 4 chance (greater than 25% chance) of at least one 100 year storm hitting our house in 30 years!  If we want that confidence, only a 1% risk of our house being destroyed in 30 years, we need to plan not for 100 year storms (which give us 26 times that risk!) but for 3000 year storms!

I used the online calculator at Graphpad Software.  There are a huge number of online calculators, just search on Poisson distribution calculator.  I encourage you to play around with this some, for events you're interested in, and levels of confidence you consider reasonable.

I was surprised at the results here.  One thing it suggests, as my house is more than 30 years old, is that we are already doing some design and construction to these levels.  It isn't a radical new concept.  The other is, when we are considering planning against catastrophic failure, we have to consider extremely rare events.  Most of us expect to live more than 30 years.  The related thing, just ask insurance companies, is that changes to the probabilities of extremely rare events are extremely important.  Changing the average temperature from 15 to 17 C (60 to 63 F) might be ignorable or even welcome, depending on who you are.  But turning the extremely rare 100 year storm in to a rare 10 year storm gives you only a 5% chance of your 'storm of the century' house surviving 30 years. (And instead of 0.4% chance of 3 or more such storms hitting you, it becomes 58%!)

Detail: Century storms

0 Parts per million

One of the sillier arguments against climate change, especially human-affected climate change, is to claim that since CO2 is a trace gas, it can't have any significant effect.  It's now about 395 parts per million in the atmosphere.

It happens that I'm taking some medicine at the moment.  100 mg of one, and 10 mg of another.  To make the math easy, let's say I weigh 100 kg.  mg is milligram, so there are 1000 of them per gram.  1000 grams is 1 kg.

That first medication represents 1 part per million.  The second is a mere 0.1 parts per million.  The body is a complex system.  Small amounts of things can be very important.  Climate is also a complex system.

Let's continue a little in this vein.  Daily nutrition requirements, which I'll take from http://en.wikipedia.org/wiki/Reference_Daily_Intake , include:

• 1000 mg Calcium --> 10 ppm
• 1000 mg Phosphorous --> 10 ppm
• 60 mg Vitamin C --> 0.6 ppm
• 15 mg Zinc --> 0.15 ppm
• 2 mg Manganese --> 0.02 ppm
• 80 micrograms Vitamin K --> 0.008 ppm
• 6 micrograms Vitamin B-12 --> 0.0006 ppm

If you'd like to avoid scurvy, rickets, pellagra, and a host of other illnesses, you have to treat concentrations far, far lower than 400 ppm as being important.

Detail: Parts per million

0 Is climate a random walk?

Let's pick up again the discussion between Tamino and me.  He has objected to my use of cumulative sums on the grounds that cumulative sums of random numbers have bad behavior statistically.  He's correct about that statistical point, naturally, which means caution is needed regarding the statistical part of my post on finding a climate normal.

But how concerned should we be as climate scientists?  Crucial to that concern is that climate be, to a fair degree, experiencing random variation.  It isn't, strictly.  As Tamino mentioned, there is certainly a trend in more recent years -- not purely random variation.  His criticism is more one against the cumulative sums method.  As Jim Bouldin mentioned in the comments recently, we do routinely transform variables in order to study topics of scientific interest.  In his case, plants.

I'll start by showing you an illustration of why Tamino is concerned.  This is a plot of the cumulative sum -- of a purely random variable, with uniformly random numbers in the range plus or minus 1 degree.  This is actually about 10 times too large for climate.  So the late period value of 40 really means 4 degrees.

Each tick mark is 1 month, but assumes that each month's random number is completely independent of each other.  That isn't the case, as Tamino has documented.  This curve manages to accumulate its 4 degrees in about 200 years.

Er, it accumulates that climate wandering in only 200 years!.  That and the smooth curves, suggest why we cannot take climate to be random wandering over long periods.  Volcanoes suggest why we cannot do so for short periods either.  Intermediate periods might be ok physically.

The upper and lower curves are growing proportional to the square root of the number of months.  Random walks are known to do this.  We can't tell whether such an accumulation will head for the warming or the cooling side, but we can be confident that over time, the cumulative sum will move away from zero.  I've loaded my spreadsheet for this in Open Office and Excel formats.  I do encourage you to pull one of these down, or write your own, and look at a few dozen examples of what can happen with cumulative sums of random numbers.  As Tamino said, you can find all kinds of interesting results -- that have nothing to do with anything meaningful.

Let's go back to that 4 degree accumulation in 200 years.  That's gargantuan compared to real climate changes.  The 200 years comes out partly because of the nature of randomness used in my spreadsheet, so don't be too wedded to it.  It could easily be more like 2000 years with a better approximation to climate's random nature.  I'll hope that Tamino or one of my more mathematical readers take this part up.  The other thing is, the accumulation is unbounded -- wait long enough and any limit you put, 4 C, 20 C, 100 C, ..., will be passed.  It will take about 25 times as long to pass 20 C warming or cooling, so maybe as fast as 5000 years!?

Regardless, however, of the exact period, the random walk is quite happy to move away from a reference climate by 4 C in a pretty short period.  Ok, long compared to how long I expect to live.  But quite short compared to the 100,000 years or so of an ice age cycle.  Global mean temperatures change by about 5 C in an ice age cycle.  So this 200 (or 2000?) year accumulation of 4 C is enormous, and fast. 

But we don't see that in the ice age records.  Temperatures cool by about 5 C going in to an ice age.  It takes a while.  The thing is, we don't see global warmings and coolings of several degrees occurring routinely each few hundred or few thousand years.  We don't see 20 C changes at all.  What we see instead is that there's a more or less smooth cooling going in to an ice age, and a more or less smooth warming coming out of one.  There are occasional large shifts in regional temperatures (the Younger Dryas, for instance, coming out of the last ice age).  But there are never such large temperature shifts as the random walk looks to expect.

The reason is, climate variation is not a purely random process.  This is no surprise, of course, and no surprise to Tamino either.  The climate system has an idea of what its reference temperature is.  That is, if temperatures are a bit higher than normal, it's more likely that the next change will be a cooling.  Not necessarily the next month, due to the autocorrelation that Tamino has mentioned.  But sooner than later.  -- Unless there's something pushing the climate to a new reference temperature.  I'll illustrate this, from data, in a post to follow.  It provides, by the way, a different way to look for a period of 'normal climate'.  The random walk has no idea whether it is currently warmer or cooler than reference, because there is no real reference.

This gives a sense that over long periods (even if we're not exactly sure of what 'long' means -- 200 years in this example) climate isn't really a random walk.

Volcanoes let us see this return to reference temperature over short periods.  When a major volcanic eruption throws a lot of dust in to the upper atmosphere, the earth cools.  This, itself, is highly non-random.  Throw junk into the upper atmosphere, cool the earth.  Never do we see a warming from it.  No surprise there.  What follows is the interesting part.  Over time, the earth warms back up to about where it was before the eruption.  That time is a few years.  If climate were just a random walk, after that eruption the earth would be as likely to cool as warm.  And a few years after the eruption, the earth could just as easily be markedly cooler as back to its normal.  Yet what we see is the earth returns to its pre-eruption temperature.  Those old enough to remember Pinatubo (1991) experienced this fact themselves.

There certainly is randomness to global climate.  If there weren't, we would be able to make perfect climate predictions.  On the other hand, the randomness is limited somehow.  The climate system does have an idea of what temperature it would be at if it weren't being disturbed by something (volcanoes, El Niño, greenhouse gas releases, etc.) and tends to return towards that.

For the discussion between Tamino and me, I'll suggest:

1. We do want to be cautious about using cumulative sums.  They have bad statistical behavior against random processes.


2. We do not have to give up on cumulative sums in studying climate.  Real climate is not a purely random process.


3. Since climate does seem to have the ability to tell where its 'normal' is, at least over very short (few year) and long (few hundred? few thousand? years) periods, we still have a chance at locating a 'normal' climate.


4. In respect to that, it is reassuring that the length of time over which the normal climate was found in the original post is 90 years -- not just 20-30.


5. "More research is needed"

#5 is the almost universal conclusion for scientific discussions.  I'll hope that Tamino or others take up the question of what more realistic noise would tell us about climate as a random walk processes.  I'll take up a different novel approach to climate myself shortly.  Instead of sums, it will be differences.  They're subject to problems, just as sums are, but they're different problems.  If we get to the same conclusions by both methods, we can be more confident of our conclusions.

Detail: Is climate a random walk?

0 How not to compute trends

Did you know that scientists are lying about the trend in sea ice extent?  That's the conclusion if you apply the popular trend analysis technique those who claim that the earth has cooled, or 'not warmed', since 1998, or 2005.  The probable reason you don't hear about this is that the 'lie' would be that scientists are grossly underestimating how drastic the trend to less ice is if you believe that method.

The method used to claim that there's a cooling trend, or no warming trend, is to cherry-pick a too-recent start year that is exceptionally high and compute the difference between that and a particular recent year (any one, repeat the 'no cooling trend for the last decade' for years afterwards, even if more recent years are warmer).

So I'll take a recent year that had large sea ice extent -- 1996, and compute the trend between there and a recent year that had a low extent -- 2007.  Here's the straight line computed that way, plotted against the observations between 1996 and present.  These data are September ice extents from the NSIDC:

And from eye-inspection of it, it even looks like the average error is about 0.  Sometimes high, sometimes low.  This trend is for ice pack extent to lose about 330,000 km^2 per year, against the about 78,000 km^2 that is computed for a linear trend by climate scientists.  Clearly climate scientists are trying to hide the decline!

I've already done some things more honest than what the 'no warming since 1998' folks do, not least is, I showed you the trend line and the data.  But this is far from sufficient to have a reasonable trend analysis.
Not least of the flaws is, most of the data were ignored.  Related is, this was too short a period to establish a climate trend.  A third is that business about 'average error'.

If I continue the pseudo-skeptics' cherry-picking method, but show you the 'trend' that comes from such an analysis against 30 years of data (1979-2008, this is also the reason for 2008 being the last year shown), the result is:

Now it's clear that this trend computation has nothing to do with representing what's going on with the sea ice extent.  If you use a long enough period to be analyzing climate trends, you're protected against several varieties of silly results.

But suppose you are insisting, for some reason, that the climate system changed fundamentally in 1996, so that nothing before then matters.  (1998 for those cherry-picking on global temperatures.)  Is this trend meaningful then?  Well, no.

The problem is the story of the statisticians target shooting.  One fired and hit the target a foot to the left of center.  The other hit a foot to the right.  They then promptly congratulated each other on the fact that on average they had made bullseyes.  Here is an example of two lines, both of which have zero average error compared to the observations.  If the line is above the data, that's a positive error.  If it's below, that's a negative error.  We compute the error for every year and add them up.  For both lines (the yellow 'least square' line and the orange 'slope-intercept' line) the average is zero.

I repeat: The average is zero.  This is why we don't compute trends by looking for a line with zero average error.  In fact, there are an infinite number of lines that would have zero average error.  Whatever slope you want, I can find an intercept that would give zero average error.  Conversely, this is a cherry-picker's dream come true.  If they even mention actual numbers, which is rare, they can honestly say that their trend has zero average error.  As long as the listeners are sufficient lazy or uninformed, the cherry-pickers are safe.

What is actually done almost all the time in science is to look at the total of the square of the errors.  In that case, the fit line being 3 million km^2 too low (a value of -3) becomes and error of 9 (-3 * -3).  Later, when the line is 3 million km^2 too high, (a value of +3) the error we add is, again, 9 (3 * 3).  The average error for these two is zero.  The sum of the squared errors is 18.  Big difference!  What is normally done is to find the line that has the least squared errors.  Also mentioned as 'least squares method', 'least squares fit', 'ordinary least squares', and probably a number of other names.  You can get quite a bit more elaborate than this, and at times need to.  For such cases, I'll recommend taking a look over at Tamino's place.  For instance, his recent article on How not to analyze tide gauge data.

Still, this will give you a decent start in reviewing arguments.

• Trend analysis must be done over a long enough period
• All relevant data should be considered.
• If this results in a period shorter than normally considered long enough, a specific and strong explanation must be made
• Trends cannot be computed by picking a single year to start from and a single end year and then ignore all years in between
• Average error is not sufficient.  Squared errors should be used, or some other method which avoids the problems that 'average error' has.

Regarding that last point, I'll suggest some looking at the numbers yourself.  I've made up a spreadsheet available in both Excel and OpenOffice formats for you to play with.  In the spreadsheet, page 1, is something called 'power'.  It is just says what power the error is raised to.  For 'average error', the power is 1.  For squared error, it is 2.  You can experiment some looking at lines for power = 3, 4, 5, ....  For all odd powers, you can create a line where the error totals to zero (just keep juggling the 'slope' and 'intercept' until you get there).  For even powers, you cannot get it to zero.  But at some point, some combination of slope and intercept, you'll arrive at the minimum error.  This won't be quite the same line for power = 2 as for 4, 6, 8, ....  See for yourself how different the lines are.  A more advance method, or at least a little more challenging to enter, is to look for the least absolute deviation.  For this you count 3 million km^2 too low as an error of 3 (rather than -3) and an error of 3 million km^2 too high is also 3.

Since I've been mentioning the temperature trends, let's take a look at those.  I'll use the Hadley - CRU temperatures because those are favored by the cherry-pickers for this purpose.  Others don't show 1998 as the warmest year, which defeats the cherry-picking.  Since one of the favorite lines of the cherry-pickers is 'cooling for the last decade', which they've repeated since long after then end of 2007, I'll take the decade 1998-2007.  The cherry-pickers' method gives a trend of -0.0147 K per year.  If it were meaningful, which a decade isn't for temperature, then that'd be a pretty large figure.  Warming over the last century was only around 0.8 K per century, and this would be a cooling of 1.5 K.  Least squares over that period shows a warming trend of 0.0046 K per year.  Over the century that's about 0.5 K, which is a bit less than the 0.8, but markedly closer.  Here's what the two lines look like compared to the data for the 'last decade' of the cherry-pickers:

And here's what it looks like over the whole period of record:

The cherry-pickers' line illustrates two different thing.  1) Why it is we don't use only 10 years to define climate and 2) Why we don't use the cherry pickers' method.  Even though 10 years is too short to define climate, if we use least squares, the results are at least not insane.

As usual, don't take my word for any of this.  If what I've said is already obvious, I hope you found the writing interesting.  If the content is not obvious, get hold of some data yourself, whether the NSIDC (link to the right) ice extents for September, or the HadCRUT (link above) or some other climate data type and start experimenting with the data yourself.  Look, yourself, at what kinds of lines you can run through the data, and how long a period you need before the lines start to stabilize -- that one year more or less doesn't mean you need a very different line.  Also take a look at methods for finding a line through the data.  Average error, I claim, is a very bad method.  But try it out yourself.  See what happens for least square, least 4th power, least 18th, and so forth.

And, of course, if what you find disagrees with what I've said here, post a comment explaining what you did and how what you saw differs from what I discuss here.

In the mean time, I'll suggest that people applying the cherry-picker method are either not honest -- they're trying to mislead you -- or they just don't know much about how to analyze data.  Either way, these aren't the people to learn about climate from, no matter how much you might like their thoughts and conclusions.  That's one of the drawbacks to doing science -- the people doing the best work aren't always the ones you like the most.

Update (3 Aug 2011): It seems that blogger is having problems with comments.  That's particularly unfortunate here as I think there are likely many good comments getting choked by the system.  If you've had this problem, please email them to me at bobg at radix dot net.

Detail: How not to compute trends

0 Reconsidering forecasts and wagers

Comments in two different threads suggest that there's some good room for more discussion.  Yesterday, I noted that if someone would only take a bet at 50 to 1 odds, he wasn't very confident about his side of the bet.  Certainly not the 'just as likely to warm as cool' that was the statement which prompted the bet.  (Nobody here, by the way.)  As M commented, and I assume that he mean in terms of real money, he'd want 5:1 odds even on an even money situation.  I wouldn't mind that myself (again, I don't bet real money, but in examining the mathematics of expectation, that's how it goes).

For illustration of a concept that was in my mind, but not in the prior post, I'll pick up with Alastair's comments about his predictions and preferences for odds.  The thing is, the odds that you're willing to accept also describe what you think is really the case.  At least it's much closer than in the case that your lunch money is riding on the bet and you're already hungry.  Our situation here is betting something that doesn't exist (quatloos) and presuming that we can make a lot of bets, so that we can average out the wins and losses over time -- converging towards the mathematical situation.

I make use of this in group settings when lunch place selection is being discussed.  In a group of, say, 6, all will claim to have no preference between places A and B.  I pull out a coin and say, fine, heads it's A, tails it's B.  An amazing number of people suddenly develop a preference for one or the other.

So it goes with Alastair's estimate of 3.9 million km^2 against mine of 4.4.  His original estimate for uncertainty was 0.1 million km^2, while mine was 0.5 million.  I'll add a curve for him, where the uncertainty is increased to match mine; that'll be the 'Alastair-2' curve:

Here begins the fun and games of the mathematics.  My expectation, if I thought my statistical guess were seriously good, which I don't (hence the 3 different methods for estimating the September cover) is the blue curve.  Alastair's original is the orangish -- the one with the very narrow peak.  And the curve for less confidence is the yellow -- same peak, but broader spread.

To the extent that we believe any of our estimates, and estimates of uncertainty, the probability that we're predicting for a given range of extents is the area under the curve between those.  You see essentially no area under Alastair's original curve for extents over 4.2 million km^2.  That represents a high confidence in a new record.  My statistical guess shows some area under the blue curve for extents under 4.2, meaning that I wouldn't be shocked if a new record were set this year.  I would, on the other hand, be shocked if the ice extent were over 5.5 million km^2.

In his second comment, Alastair mentioned a good science point -- he had changed his forecast method because it hadn't worked in previous years.  That's key to doing science.  Being right is the goal, and the way you learn how to be right is to change your methods as reality disagrees with your expectation.  Mistakes are fine -- learning from them is where you're doing the science.

Alastair also mentioned that he thinks there's a 50% chance of a new record this year.  The old record was 4.3 million km^2.  His original estimates give essentially 100% chance for that.  My modification shows an 86% chance for his less confident estimate.  To get it down to 50%, we have to increase the uncertainty (the standard deviation of the curve) to 4!  That's ... extraordinary, to say the least.  The difference between highest and lowest years is not even that large.  The other way to get the probability of a new record to 50% is to make the center of his expectation to be 4.3 million km^2 -- practically the same as my prediction of 4.4.

Taking my own numbers, the guess translates to a 37.5% chance of a new record this year.  Let's look at that.  Does my intuition complain when it sees that number?  Not really.  If it were to be 98%, or even 86%, I'd be concerned.  But somewhat less than half ... doesn't seem outrageous.  2% would also be unreasonable.  We set the record in only 30 years, which means a 3% chance in each year.  And there's a downward trend in the extent, so the chances of a new record should be increasing year by year.

It looks like I misplaced a decimal point in figuring things the other day.  Here's a table of areas (probabilities) of various outcomes with respect to my curve and the two I'm using for Alastair:

Outcome	Bob	Alastair-1	Alastair-2
Extent below 4.15	0.25	0.9999	0.77
Extent below 4.30	0.375	1.0	0.86
Extent between 3.8 and 4.0	0.080	0.838	0.223

So our expectations for the ice cover being below 4.15 are really more like 4:1, Alastair.  Does that sound more attractive?  3:1 for the less confident version of your prediction.  For setting a new record, your 50% chance translates to  4:3 odds between us -- you win 3 quatloos if it is a new record, pay 4 if it isn't.

Detail: Reconsidering forecasts and wagers

0 Whiteboard on the end of global warming

The Whiteboard also has a series (now looks to be finished) on whether global warming has 'stopped'.  That's what prompted the series I linked to last week.  The question is pursued statistically, following the lead of Tamino at Open Mind.  The brief summary of the series is 'no'.  But it is well worth looking in to the series for the how and why we can say this, and how strongly we can say it.

Did Global Warming Stop After 1998?
Did Global Warming Stop After 2000?
Did Global Warming Stop in 1940?
Did Global Warming Stop After 2007?

For looking at the converse, global cooling:
Did Global Cooling Stop in 1970?

These are more mathematical takes than my old What cooling trend?  They come to the same conclusion, so those who'd like more math behind their conclusions can get it.  Since I did that post over a year ago, it's probably time for an update.  One of these days ...

Detail: Whiteboard on the end of global warming

0 Wrestling with data

I'll suggest those who haven't been, join me in keeping an eye on a series of posts that Ron Broberg is doing over at The Whiteboard.  As befits a whiteboard, he's showing a lot of the details that get cleaned out of most final publications, even on blogs.  The topic at hand is looking at the temperature records since 1880 and testing ideas on fitting curves to the data.  The series is now to #9 and it's apparent that there will be several more:


http://rhinohide.wordpress.com/2011/01/07/lines-sines-and-curve-fitting-1-oh-my/ (Starts out more on the issue of testing ideas on what we can conclude about temperature trends)
http://rhinohide.wordpress.com/2011/01/08/lines-sines-and-curve-fitting-2-r/ (try fitting the sine and then a line)
http://rhinohide.wordpress.com/2011/01/09/lines-sines-and-curve-fitting-3-double-down/ (Try fitting 2 sine waves)
http://rhinohide.wordpress.com/2011/01/10/lines-sines-and-curve-fittings-4-walk-and-chew-gum/ (Simultaneous line and sine fit.)
http://rhinohide.wordpress.com/2011/01/12/lines-sines-and-curve-fitting-5-a-growth/ (Trying an exponential curve)

http://rhinohide.wordpress.com/2011/01/14/lines-sines-and-curve-fitting-6-backcast-and-forecast/
http://rhinohide.wordpress.com/2011/01/15/lines-sines-and-curve-fitting-7-normal/  (Testing Normality 1)
http://rhinohide.wordpress.com/2011/01/16/lines-sines-and-curve-fitting-8-dagostino/ (Testing Normality 2)
http://rhinohide.wordpress.com/2011/01/17/lines-sines-and-curve-fitting-9-girma/

A sine is a standard oscillation.  It would be a pure tone (rather flute-like) in music.  For a bit more about oscillations and data series, and the language of time series analysis, take a look at my Introduction to Time Series Analysis.

Detail: Wrestling with data

0 Do I have to be good in math to be good at science?

The short answer: No.  Before the longer answer, here's the full comment/question from the teacher:

Question:  Do I have to be good in math to be good at science?

The reason I ask this is that many scientifically inclined students I know are not going to pursue science as they get older because they perceive themselves to be less than stellar in math. This is a shame. And a waste.  Please stress to the young, hormonally-infused people who read your blog (not the adults-one hopes they've figured it out already) that science is a process, just like running, and eating and IMing on the phone. And math is a tool -- a really great, useful tool -- that's part of the process.

Splitting hairs? I don't think so. I've got a few 5th graders who I've shared your blog with, and they were enjoying themselves until they hit the math and freaked out. Nooooooo, I say, fear not the many zeros and exponents. It all makes sense as you practice it (even I can say that).

So ... address my question. My 5th graders will thank you. And, of course, so will I!

I whole-heartedly second everything the teacher said.  The 'enjoying themselves until they hit the math and freaked out' is also not limited to the 5th graders. I've heard from some distinctly older folks about this too.  It's why I'll be making some changes to my posting practices.

For the students, there are two different questions to think about.  One is, what does it mean to be 'good at math', and the other is 'what am I doing when I'm doing science?'.  I'll share some of my thoughts and invite students and teachers to share theirs as well.  Questions, as always, welcome.

First, 'what am I doing when I'm doing science?'  I can speak first hand, as can your 5th graders.  They may not realize it though.  Science is about trying to understand the world around you in a sharable way.  Do you see the word 'math' there?  When you watch clouds and notice that some of them get taller and puffier over time, and the ones that get tallest, puffiest, and darkest are the ones that give you thunderstorms -- you're doing science.  Or you watch bees and notice how they behave.  One thing you notice is that bees normally fly around, or collect pollen, or build their nest.  But then you notice one particular kind of bee occasionally digs through sand.  This is how a friend discovered a new species of bee.  That's science.  Then, of course, she collected some of those bees and examined them to see how it was they could dig through sand (which really is a bizarre thing for a bee to do).  More science, plus a pretty electron microscope picture we have on our wall at home.


When math does show up, it is as a servant of either understanding the world, or for doing the sharing.  So my latest scary post looked at just how much cooling you might get from clouds.  Without the math, we could tell that there could be some cooling, but not whether it would be enough to erase the greenhouse gas warming.  With the math, we understand that it probably can't erase the warming, and just how big the changes would have to be in order to do the erasing.  It's a tool for our understanding here.  You'll be doing this, probably already are, in daily life as well.  As soon as you want to decide whether you have enough money to buy something that's 10% off, you're doing math this way.  Same for deciding whether you have enough to by 12 of something that costs $9.95 each.

It can also be a tool for doing the sharing.  In saying that a cloud is tall, how do you decide what 'tall' means?  You're doing math here.  Either measurement or geometry.  Again, may not be the sort of thing you're encountering yet in daily life, but you certainly will.  Any time you try to decide whether one thing is bigger than another, you're doing this kind of math.

The second question is what it means to be 'good at math'.  The real answer for science is that no, you don't have to be 'good at math'.  But I expect your 5th graders are also not thinking about the right kind of good at math in the first place.  A friend recently observed that when he was in elementary school, he thought he was bad at math.  He learned later, when he was earning a PhD in computer science, that he was good at math -- he just wasn't good at arithmetic.  Still isn't good with arithmetic, but he can do his mathematics just fine.  There is a huge world of mathematics outside of what you're encountering in 5th grade, and many people discover that the new world contains things that are both interesting and fun for them to work on.

Then there's my baseball career, which is similar to what other folks experienced for mathematics.  In 5th grade, I played baseball.  I was without any question one of the worst players in the league, and possibly the worst.  I liked the game, and understood pretty much what I was supposed to do.  But my body just wouldn't do it.  Time passes and I'm playing softball on a team at work.  And I'm not horrible.  In fact, one year I'm our team's representative to the league All Star game.  (Which got rained out :-)  Many scientists I know have similar experiences with their mathematics history -- not being good at math in elementary or even high school, but later on, high school or college the old math started making sense.  This doesn't help your grade on today's math test.  But when you're doing your science and talking about your interesting clouds, or bees, nobody cares what grade you got on today's math test.

Then there's a final version, which fits some good scientists I know.  Namely, they never did get 'good at math'.  Some find their way into kinds of science that don't involve much math.  Such areas do exist.  And some just grit their teeth and do the math they need to in order to do the way more fun science.  The thing is, you can do it.  It's like how in sports you have practices and practice skills for the game.  The fun thing is playing the game.  But you have to do your skills practices to play the game the best you can.  (Just think about the thousands of hours of practice time the Olympic athletes have put in!)  Musicians play scales, which almost never show up in real music.  But it's a skill that is important, so you practice it and it helps you do the fun part better.  At worst, math is like this for you as you go about doing the much more fun science.  (Plus you can usually find a friend who is good at the math and work together on your fun scientific idea.)

Final example of doing science, as Beth (who discovered the bees I mentioned above) called while I was writing this note and passed along this story.  Her son John, who is now an artist, discovered a species of wasp.  He was 13 at the time.  The discovery shows, Beth was pointing out, that young people can be better observers than adults.  13 year old John was in the house on a field expedition to India, along with Beth (who is a professional hymenopterist -- studies bees, wasps, ants and such; bees being her favorite) and a very experienced, very famous hymenopterist who specialized in wasps.  But the new species was discovered by the 13 year old.  The wasps were flying around the house, getting caught in netting, and the like.  The adults figured these were just some very common Indian wasp, rather like flies for being so very common -- not some new species.  The 13 year old watched them carefully, and thought that the color pattern was new and interesting.  Certainly he hadn't seen the pattern before (and, thanks to mom's library, he'd seen more than a few types of wasp).  Eventually he caught one and passed it on to the adults to identify.  It worked out that nobody had ever identified this wasp to science before -- and the professional scientists in the house hadn't realized this.  It was the 13 year old with good eyes and persistence (and using no math :-) who made the discovery.

For 5th graders' math concerns, the future holds three possibilities:
a) even if you're currently not good with math and find it unpleasant, in a few years you might be good at it (think how much you've learned in the last 10 years!)
b) in a few years you encounter new kinds of math, and those could be fun and interesting, or you might turn out to be good at them, or both
c) math never does become fun or interesting, and you never get very good at it, so you grit your teeth some to get through what you need to on the way to doing the way more fun and interesting science.

Regardless of which one turns out to be true for you, at worst it means you have to do some exercises you don't like.  You can do it.  Just might be more work, or might mean that you work in a particular area of science (one that doesn't use as much math), or with somebody else who does the math better or more easily than you.  And this last is true for us all.  I have a project myself where I might be turning to a math person for help -- and I was always very good with math and always enjoyed it.  Still, there are people who know more than me in an area of math that I might need in order to get my science done.  This is normal, too.

For the science, though, there is no reason ever to stop.  The universe is a very interesting place!  That means there are many different kinds of things to study.  I'm not a big fan of going to a beach and staring at bees, though Beth would consider that a vacation.  On the other hand, I do like running a whole bunch of numbers from satellites through programs I make up to try to figure out what sea ice is doing; there might be a few people who wouldn't consider that much fun.  And there are many different ways to study the universe.  So keep learning about the universe, and sharing what you learn.  Do it for as many parts of the universe as you like, and in as many different ways as you like.

Update:
A book that might be a help to students thinking they're not good at math, or they can't do it, etc.. is Sheila Tobias' Overcoming Math Anxiety
More specific to pre-teen and early teen age girls is Math Doesn't Suck.

Detail: Do I have to be good in math to be good at science?

0 CO2 and temperature for 800,000 years

First, the figure gives the answer on CO2 and temperature over the last 800,000 years:

Here we have each value of CO2 plotted against the temperature deviation from reference values -- with the temperature being for 1000 years before the corresponding CO2 value. The correlation for this is about 0.89 (R), meaning that you could explain 79 percent (0.89*0.89) of the variation (R^2) in CO2 by looking at the temperature. As is mentioned in my post Does CO2 correlate with temperatures, where we saw equally high correlation between temperature and CO2 (even higher if you give CO2 a 20-30 year lead on temperature), and quite a few times in the comments, correlation is not causation. Could be that both temperature and CO2 are pushed around by something else. More about that in a moment.

It's common to see the claim that temperature 'leads' CO2. Loosely speaking, this means that temperatures generally change before CO2 does, and that there is a consistent pattern to the connection -- if temperature rises, CO2 does as well. This is true; it leads by about 800 years [Caillon and others, 2003]. The amount of lead also seems to depend on whether you're in a glacial period or (as we are now) an interglacial. The correlation between temperature and CO2 is lower (by a very small amount) if you take their values for the same time (drops to 0.88). And it drops by more if we take the temperatures for 2000 years before the CO2 value, to 0.87, indicating from this very simple approach that the lead is between 0 and 1000 years, probably closer to 1000 -- as is found from the more serious approach in the reference.

So there's some interesting science to do, to understand why that lead exists, why it is many hundred years (rather than a few dozen years, or a few thousand), and why the lead time depends on whether you're in a glacial period or an interglacial period. Oddly, to me, most of the time that people mention this lead relationship, they are not referring to any of this.



Rather, they want to conclude, or for you to conclude, that because in the ice age record temperature changes before CO2, that a) the current rise in CO2 is also caused by temperatures (sometimes citing the medieval warm period as the warm time that is causing the current rise in temperature) and b) that CO2 doesn't have any affect on temperatures.

Have another look at the figure.  See that point sitting way the heck away from all others?  That's the most recent value.

You don't need a lot of scientific background to see that whatever it is that produced that CO2 value, it certainly was not the relationship that held for the other 798,000 years of the temperature - CO2 record.  We can be more precise about it.  The best fit line (the one that gives that nice high correlation) through the data points is CO2 = 266 + 8*T.  So, for temperatures at the reference value, we expect a CO2 level of 266 parts per million.  For temperatures 10 C below reference, we expect CO2 levels of 186 parts per million.  Both of these accord fairly well with what we do see in the record -- except for the most recent CO2 value.  As you can see from the plot as well, there is indeed scatter around the best fit line.  The standard deviation is about 11 ppm.  That means we're not surprised to see values 22 ppm away from the line (2 standard deviations), and, given 799 data points, we expect a few to be 33 ppm away.  On the other hand, a value 9 standard deviations away -- the case for the current CO2 levels --  is ludicrous.  Something must have changed.

We can turn this around, and ask: "If the relationship that held for the previous 798,000 years still did, what would the temperature need to have been to give the observed CO2 level -- maybe CO2 is so sensitive to temperature that we simply have a one-time temperature observing problem?"  Certainly the ice sheet temperatures do have their own observing issues.  As I've mentioned, all data have problems.  Still, same as we know that, we also know something about the size of the problem.  So, turn the equation around and solve for the temperature that would correspond to the observed (modern) CO2 level.  That's a temperature anomaly of about 13 C (20 F) -- equal to the entire range from the very warmest interglacial to the very coldest glacial!  And that had to have occurred 600-1000 years ago, by the lead relationship.  Meaning there had to have been an absolutely enormous warming (many times larger than even the highest medieval warm period values) and nobody noticed it ... or else the previous relationship between temperature and CO2 broke down.

Since we know what did cause the CO2 rise -- human activity -- we're not really surprised to see this answer. The old relationship did get broken.  Rather than CO2 rising because the oceans released CO2 to the atmosphere, it is because humans have been burning fossil fuels and making cement.  But we do like to be able to arrive at our conclusions from different directions.  Here, we need only to look at the temperature and CO2 values themselves to see that the relationship that used to hold has broken down in the modern day.  Don't need to know the first thing about isotope geochemistry (which is explained nicely in that faq by Jan Schloerer) to see this.

Let's return, though, to the fact that correlation is not causation.  There are generally 4 options when a correlation is seen: it's chance, A causes B, B causes A, both A and B are caused by something else.  We use statistical tests to decide whether chance is plausible (in this case, as with Does CO2 correlate with temperature?, the answer is a clear no).  So one of the other 3 is involved.  We can reject CO2 being the (sole) cause of temperature changes (it can certainly feed back on temperatures) because it is the temperatures that changes first.  So it's one of two now -- either temperatures drive CO2 (with feedback from CO2 back to temperatures), or something else drives changes in both temperature and CO2 (and perhaps manages to change temperature sooner than CO2).

Oddly, most of the comments about temperature leading CO2 ignore the last option -- in other words, they assume that correlation is indeed causation.  Now ... is there anything in the universe that could affect temperatures?  Could any of those things also affect CO2 levels?  That will be a separate post.  But start thinking about it yourself. Richard Alley gives away the answer in his talk, to which I referred you Tuesday.

Going farther:

I've labelled this a 'project folder' post in part for the questions I mentioned inside it -- why the lead is what it is (or at least was what it was), and so forth.  But also because the things I did you can do yourself.  The data I used were from a BBC news article on ice and CO2.  You can get it yourself and experiment with the lead relationships, find the CO2 response (slope of CO2 versus temperature in the regressions), and the like.

You can go farther as well.  For instance, are the correlation and slope different in different subsections of the record?  Is it different for different temperature ranges? (compute the slope for only temperatures from -10 to -5, -5 to 0, and so forth)  Does the correlation improve if you use the logarithm of CO2 rather than CO2 levels themselves? (if the only thing involved were the radiative properties of CO2 and its connection to temperatures, we would expect a 'yes' here)

Detail: CO2 and temperature for 800,000 years

0 Assessing predictions

It's a little premature to make a detailed assessment of the predictions for September's average extent as the final numbers aren't in. They will be soon, but my focus is actually over on the question of how to go about doing the comparisons. Earlier, I talked about testing ideas, but there, the concern was more one of how to find something that you could meaningfully test. Here, with the September's average extent, we already have a well-defined, meaningful thing to look at.

Our concern now is to decide how to compare the observed September average extent with the climatological extent, and a prediction. While mine wasn't the best guess in the June summary at ARCUS, it was mine, so I know what the author had in mind.

Let's say that the true number will be 5.25 million km^2. My prediction was 4.92. The useless approach is to look at the two figures, see that they're different, and declare that my prediction was worthless. Now it might be, but you don't know that from just the fact that the prediction and the observation were different. Another part of my prediction was to note that the standard deviation to the prediction was 0.47 million km^2. That is a measure of the 'weather' involved in sea ice extents -- the September average extent has that much variation just because weather happens. Consequently, even if I were absolutely correct -- about the mean (most likely value) and the standard deviation, I'd expect my prediction to be 'wrong' most of the time. 'Wrong' in that useless sense that the observation differed by some observable amount from my prediction. The more useful approach is to allow for the fact that the predicted value really represents a distribution of possibilities -- while 4.92 is the most likely value from my prediction, 5.25 is still quite possible.

We also like to have a 'null forecaster' to compare with. The 'null forecaster' is a particularly simple forecaster, one with no brains to speak of, and very little memory. You always want your prediction to do better than the null forecaster. Otherwise, people could do as well or better with far less effort than you're putting in. The first 'null forecaster' we reach to is climatology -- predict that things will be the way they 'usually' are. Lately, for sea ice, we've been seeing figures which are wildly different from any earlier observations, so we have to do more to decide what we mean by 'climatology' for sea ice. I noticed that the 50's, 60's, and 70's up to the start of the satellite era had as much or somewhat more ice than the early part of the satellite era (see Chapman and Walsh's data set at the NSIDC). My 'climatological' value for the purpose of making my prediction was 7.38 million km^2, the average of about the first 15 years of the satellite era. A 30 year average including the last 15 years of the pre-satellite era would be about that or a little higher. Again, that figure is part of a distribution, since even before the recent trend, there were years with more or less (than climatology) ice covers.

It may be a surprise, but we also should consider the natural variability in looking at the observed value for the month. Since we're really looking towards climate, we have in mind that if the weather this summer were warmer, there'd be less September ice. And if it were colder, or different wind patterns, there would have been more ice this September. Again, the spread is the 0.47 (at least that's my estimate for the figure).

I'll make the assumption (because otherwise we don't know what to do) that the ranges form a nice bell curve, also known as 'normal distribution', also known as 'Gaussian distribution'. We can then plot each distribution -- from the observed, the prediction, and what climatology might say. They're in the figure:



This is one that makes a lot of sense immediately from the graphic. The Observed and Prediction curves overlap each other substantially, while the curves for Observed and Climatology are so far from each other that there's only the tiniest overlap (near 6.4). That tiny overlap occurs for an area where the curves are extremely low -- meaning that neither the observation nor the climatology is likely to produce a value near 6.4, and it gets worse if (as happened) what you saw was 5.25.

The comparison of predictions gets harder if the predictions have different standard deviations. I could, for instance, have decided that although the natural variability was .47, I was not confident about my prediction method, so taken twice as large a variability (for my prediction -- the natural variability for the observation and for the climatology is what it is and not subject to change by me). Obviously, that prediction would be worse than the one I made. Or at least it would be given the observed amount. If we'd really seen 4.25 instead of 5.25, I would have been better off with a less narrow prediction -- the curve would be flatter, but lower. I'll leave that more complicated situation for a later note.

For now, though, we can look at the people who said that the sea ice pack had 'recovered' (which would mean 'got back to climatology') and see that they were horribly wrong. Far more so than any of the serious predictions in the sea ice outlook (June report, I confess I haven't read all of the later reports). The 'sea ice has recovered' folks are as wrong as a prediction of 3.1 million km^2 would have been. Lowest June prediction by far was a 3.2, but the authors noted that it was an 'aggressive' prediction -- they'd skewed everything towards making the model come up with a low number. Their 'moderate' prediction was for a little over 4.7. Shift my yellow triangle curve 0.2 to the left and you have what theirs looks like -- still pretty close.

To go back to my prediction, it was far better than the null forecaster (climatology), so not 'worthless'. Or at least not by that measure. If the variability were small, however, then the curves would have narrow spikes. If the variability were 0.047, ten times smaller than it is, the curves would be near zero once you were more than a couple tenths away from the prediction. Then the distribution for my prediction would show almost no overlap with the observation and its distribution. That would be, if not worthless (at least it was closer than climatology), at least hard to consider having done much good.

Detail: Assessing predictions

0 Introductory Time Series Analysis

As often happens, this note is prompted by someone doing things that look wrong. Time series analysis is something which has been an interest of mine since I earned my Master's (looking at tidal signals in current meter data). And there are some elements which I think are eminently understandable without diving in to the gory details. So here is my shot at a very short, only marginally mathematical, introduction to time series analysis.

A time series is just a series of observations (of something, anything) through time. The monthly averaged global mean temperatures that I use for demonstrating principles of climate are time series. Monthly Southern Oscillation Index is another time series. Not coincidentally, those are the ones examined in the paper, and are what I'll consider here. But there are innumerable other time series -- daily close of the stock exchange, daily temperature, your weight day by day, and so on.

Much of the language for talking about time series is fairly ordinary. But there are a few terms I'd like to be sure we're using the same. Most important is 'period'. The period of something in your time series is the length of time from peak to peak (or trough to trough). Consider hourly temperatures in your area. They peak each day, around, say 3 PM. The period is then 24 hours. If we take a longer view, and consider daily high temperature, then the temperatures peak each year -- a period of 1 year. If we think of the brightness of the moon, it has a period (full moon to full moon) of 29.53 days. And so on. We could also ask how often the peak occurred per year (or other time of interest). This is the frequency -- how frequently do you hit the peaks. For hourly temperatures, the frequency is 365 per year (or 365 cycles per year). The frequency of the lunar cycle is 12.369 cycles per year. And the seasonal cycle has a frequency of 1 cycle per year.

In talking about periods and frequencies, and scientists tend to use the two terms interchangeably since the value of one can always be converted to a value of the other, we sometimes also hear about long/short periods, or high/low frequency. In the periods, it means what ordinary english would lead you to think -- long periods are periods that take a long time from peak to peak, and short periods are fast from peak to peak. You still have to know what is 'long' for the system to read any given paper correctly. If it's a geologist, they could mean 400 million years when they say 'long period' (the period for continental drift cycling), where a meteorologist might mean 40 years. High frequencies correspond to short periods (since the period is short, the thing happens often -- at high frequency). Low frequencies have long periods. The similarity of term with music is no accident. Low frequency sound (low pitch) has a long period, while high frequency sound has a short period.

So when you hit anything relating to time series, if you know about music, think in terms of frequencies and pitches. Another term that comes up in time series analysis, with a slightly different meaning than in music, is 'harmonic'. Unlike music, with its fifths, thirds, etc., in time series we work more simply. There is the base period/frequency. And then there are the integral multiples of that base frequency. The annual cycle's harmonics are 1 (the base period), 2 (6 month period, 2 cycles per year), 3 (4 month period, 3 cycles per year), 4 (you get the idea), and higher. In practice much of our weather time series can be captured by the first 4 harmonics of the annual cycle. (That's interesting in its own right -- it means that there is relatively little happening at periods of 5, 7, 8, 9 months, even though there's a lot at 4, 6, 12 months.)

As with music, we also are interested in how loud the frequency is. Our measure there is called amplitude. It is half the distance from peak to trough. Where I live, our peak summer high temperatures are about 90 F (32 C), and in the winter, our lowest highs are about ... call it -10 C (14 F). The range is 42 C, so the amplitude of our seasonal cycle is 21 C.

Again following music, there's usually more than one frequency being played at a time. This is certainly true for weather! Many different things happening all the time. In music, the description of all the notes the band or orchestra are playing at a time is called the score. For time series analysis, it is the spectrum. A little more involved in time series because we can look at different spectra (1 spectrum, 2 or more spectra) -- the amplitude spectrum, and the 'power' spectrum. Most work is actually done with the power spectrum, but the amplitude spectrum is easier to understand so I'll stay with that.

One of the things we do in looking at climate time series is average the data -- construct a moving average, for instance. The moving average says to take the first (some number, let's say 12) months of data and average them together. Then step forward (move) 1 month, and average the next 12. Repeat until you're at the last 12 months of data. As I've suggested for understanding global climate, you want quite a bit more than 12 months of data in your averaging. But we can also try to understand weather. A 12 month average will clobber most of what is happening shorter than 12 month periods (but not absolutely all of it, a point even scientists seem to forget -- it only completely clobbers the 12 month period and its harmonics), and let us look at what is happening on periods longer than 12 months (but some of that, too, gets damped). In musical terms, averaging suppresses the high notes, while leaving the bass line relatively unaffected.

Suppose what we really want is to suppress the bass line and enhance the treble -- suppress the climate frequencies and focus on the weather. Rather than averaging, which is a smoothing operation that suppresses the high frequencies, we would take differences. Say take the difference between months 12 months apart. We can think of the temperature as being a certain amount of weather, plus a certain amount of climate. The climate will be nearly the same 12 months apart, so when we do the subtraction, it is cancelled out and we have only the difference in weather those two months. Differencing is a sharpening operation that suppresses the bass line and enhances the treble. It is, however, an extremely biased operation -- not only does it suppress some and enhance others, but the degree of enhancement is proportional to the frequency. Unlike the averaging operation, which leaves low enough frequencies unchanged, the differencing affects all frequencies and does so strongly.

Changing time series data is called a filtering. The averaging and differencing operations are different filters. There are many, many more that we could use. Any time we do use a filter, though, we should be careful that it isn't creating problems for us. This is one reason I try always to work with the most nearly original data possible -- no filtering has been done that could obscure the effects I'm trying to work with.

If you're a more visual person than a music person, the spectrum (guess where we stole that word from!) also has some color traits. If the amplitudes are all about the same, regardless of the frequency, then we call it a 'white' spectrum. (White light is approximately equal contributions of all colors.) Instead of period, for light we think of wavelength. Frequency is still frequency, but now it means how many times per meter you see the peaks. High frequency light is blue. Low frequency light is red. When we do the moving average, we are making the spectrum redder. When we do a differencing, we are making the spectrum bluer. Most climate-related time series have red spectra -- there are higher amplitudes at longer periods (wavelengths). Our year to year variations are on the order of 0.1 C, but the ice age cycles are 5 C, for instance.

Detail: Introductory Time Series Analysis

0 Statiscal significance of 150 years of data over 4.5 billion years

The subject line is close to a recent search query that lead someone here, and echoes a comment that's not unusual in blog comment sections. The thing about it is, it's not a very strong question.

One part of the question's failure is that it isn't really a proper statistical question. Given limits of search strings, that's no surprise. But it does show up in comments (usually in the vein of assertion "150 years is too small a statistical sample of 4.5 billion years of climate.") where such limits don't apply.

The basis of any good question is to try to understand something. If what you are trying to understand is not statistical, then pursuing statistics is not going to help you. "What is your height?" is marginally statistical. If you measure the length of a short metal bar many times, which I did in freshman physics lab, you'll get slightly differing answers. So you may well answer statistically with your mean and a sample standard deviation. This is only marginally statistical, in that it is purely descriptive statistics and no hypothesis testing is involved.

Your question, however, could be non-statistical: Did it rain in my back yard last night? How much rain did my rain gauge capture? If so, you will be wasting your time if you chase after statistical tables. More at hand, someone raising statistics in a blog debate about a non-statistical question is wasting your time.

But suppose that what you're trying to understand truly does require statistical considerations beyond minor description. What would such a question look like? One could be "Given the population of voters in the USA, how many randomly selected voters would I need to ask a yes/no question in order to have a standard error in my sample of 3% or less as compared to asking the everybody?" or, for more climate-related flavor "How many satellite observations with a (known) standard error of measurement would I need to find an average sea surface temperature whose standard error would be less than 0.1 K?"

Required is a description of the 'population' -- all possible observations (the population of voters in the USA, satellite observations) -- of your 'sample' (ask some number, which we want to determine, of voters, or satellite observations -- and what statistic you are trying to estimate (% of voters who like your candidate, mean sea surface temperature).

So, back to the original question. Does it describe the population? No. 'data' certainly doesn't limit us to anything in particular. I'll guess that it is global mean temperature, of the earth (maybe it's Mars -- I've read interesting papers about Martian climatology), that is the point of interest. But we shouldn't have to guess, a good question is clear on what it is asking about. Does it describe a sampling method? Not really. 150 years, well, I'll guess that this means 'take the most recent 150 years'. Most importantly, however, does it describe what statistic one is trying to estimate? No. Again, I'll guess: that it is "Global mean temperature over the entire history of the earth."

Putting it all together, the statistical question more reasonably posed is something like "Does the last 150 years of global mean air temperatures provide a good estimate of the global mean air temperature through the history of the earth?" (better would be to define 'good', say 'standard error within 0.2 K')

That's the statistical side. But since we're not interested solely in statistical questions regarding climate, at least not most of us, we also have to ask, "Is this a physically meaningful question?" The person who did the search, I don't know what they have in mind. They could indeed have in mind a question for which the mean temperature of the planet throughout its history is exactly the right number to answer.

Usually, though, comments about the 150 years vs. 4.5 billion surface in debates about modern climate and modern climate change. I'll have to invite contributions on what relevance the planetary mean temperature from 4.5 billion years ago to ... oh, let's say 30 million years ago ... has to questions of current climate and climate change. Certainly one wants to know how climate has changed through all of time, and why. I'm one such person. But at hand is those who argue that the global mean over that entire period matters.

What's important regarding human responses to climate change is the climate on human time scales. The last 150 years handily covers me back through my great-grandparents. The next 150 years will cover my children, grandchildren, and possibly great-grandchildren. Even a 'mere' million years ago, much less 4.5 billion, there weren't any humans around to think about climate change at all. So the 4.5 billion years is, for 'what do we do now' questions, a spectacularly large red herring.

Human society infrastructure also dictates a much shorter term (than 4.5 billion years) concern. Almost every mile of paved road in the world is less than 150 years old. Almost every mile of railroad. Almost all port structures. Absolutely all air transport terminals are less than 150 years old. Absolutely all electrical distribution structure, all phone lines, and even more so all cell phone towers are less than 150 years old. Coal, oil, natural gas distribution networks, again, are almost entirely or entirely less than 150 years old. Many of the world's major cities are less than 150 years old (I'll count Chicago here, as the great fire in 1871 erased so much of the city).

The climate-related concern here is that all these things were constructed based on what climate was like around the time of their construction. That climate drove what the standards would be for, say, tolerance to flooding, tolerance to drought, high winds, high rain rates, high and low temperatures, and so on. If climate changes outside the range that the infrastructure was built for -- almost all of it globally being much less than 150 years old -- then there is a serious risk that the structure will fail when it encounters brand new climate conditions.

In this vein, then, comments about the 'we did fine in the medieval warm period' are a different flavor of red herring. Chicago, Sao Paulo, Melbourne, Johannesburg, ... didn't exist back then. They did not 'do fine' in the medieval warm period, they never encountered it. Even cities that did, such as London, Rome, Xi'an, did so with far smaller populations than today's. London, ca. 1100, had a population around 18,000, and is about 1000 times larger today (taking the metro area). Rome was about 20,000 around 1100 AD, vs. 200 times as large now. Xi'an was among the largest of cities in the world then, and may have been about 500,000 around 1100 AD. But almost 20 times that today. (mostly Wikipedia figures). While a Rome or London of 10-20,000 survived a medieval warm period ok, Rome and London of many millions have never see it before.

As usual, we can get far by asking two questions: "Is this question statistically meaningful?" "Is the question physically meaningful?" After looking at the former, regarding the original search string, and not infrequent comment, we see that it isn't a meaningful statistical question. After rephrasing it to something that is statistically good, we check to see whether it makes physical sense. Turns out, for the sort of thing I'm concerned about and others use it, that even the rephrased question isn't physically meaningful. Knowing that global mean temperature for the last 4.5 billion years was 20 C (I make up a number) as opposed to 15 C still doesn't tell us whether major modern cities, and their millions of residents, will be able to manage likely climate changes. Nor does it tell us what adaptations to take, much less how expensive it would be to make those adaptations. And it doesn't tell us what the costs, both dollars and lives, would be if there were no adaptation.

Detail: Statiscal significance of 150 years of data over 4.5 billion years

0 Binomial probabilities

This also goes under the name of 'Bernoulli trials'. The idea is that you have a circumstance in which one of two things will happen. And you're going to try out the process many times. This could be tossing coins, or dice, or a batter going to the plate (or wicket) many time, a basketball or hockey player taking a number of shots, and so on. After a bunch of trials, you then ask what the chances are that you got that many heads (or sixes, hits, baskets, ...) or more. Even better is when you try to figure out the chances before you start the trials.

But let's be concrete. I find it easier to think of a particular example first, and then think about more general cases. (Some people prefer the other way around; people are different. But they're not writing this blog :-) Consider tossing a coin 5 times. What are the chances that we get 3 heads?

There's only 1 way to get 5 heads -- get a head on the first throw, and the second, and the third, and the fourth, and the fifth. 'And' is an important word in probability, meaning that we should multiply the probability of the individual events involved. Since we're using a fair coin with a 1/2 chance of turning up heads, this means 1/2 * 1/2 * 1/2 * 1/2 * 1/2. So, 1/32 chance of turning up 5 heads in 5 throws.

But let's think about getting 4 heads in 5 throws. We have 5 different ways of getting that. I'll list them off with 'H' meaning a heads, and 'T' meaning tails. They are:
THHHH or HTHHH or HHTHH or HHHTH or HHHHT.
Each one of these has a 1/32 chance of happening. The other important probability word is present. 'or' means to add the chances. So there is a 5/32 chance of getting 4 heads in 5 throws of a fair coin. (Or for your team to win 4 games out of 5 between evenly matched teams, or for a player to hit 4 shots out of 5 when he has a 50% shooting percentage), and so on.

It gets more complicated with 3 heads, 2 tails. And a lot more complicated when it's 493 tosses of the coin. That's where we want the general formula to do the legwork for us. When we get to cases where one event is more than 50% likely, again, it becomes nice to have a more general formula. I've made up a little spread sheet in Open Document format, if there's interest.

For the case at hand, about 'Grumbine scientists', we're wondering what the chances are that there'd be 5 or more scientists in a group of 493 people. By the way, speaking of family odds, the Bernoullis had an extraordinary run of mathematicians (hence the name for this bit of probability) and mathematical physicists ('Bernoulli effect' in fluid dynamics). In the case of 'scientist', the coin is weighted heavily against coming up that way. I made up a probability of 99.9% that a given person (in the US) was not a scientist who had published in the scientific literature in the last 20 years. Don't know that this is correct, which we'll return to. Nor, as we've already discussed and had comments on, are we confident that the number 493 is right or even very close.

With only 1 person in 1000 qualifying, we wouldn't be surprised to see 0 of 493 turn out to be scientists. The actual calculation gives 61% of the time that we'd expect 0. 30% of the time, we'd expect only 1. Conversely, the chance of something happening is 100% minus the chance of it not happening. Having 2 or more scientists show up, then, is only about a 9% chance. We shouldn't worship at the altar of the 5% level, but it's a good rule of thumb for getting started. With a 9% chance, we're not very impressed to see 2 or more scientists in this group of 493. But chances of getting 2 scientists are 7.4%. Between that, and some rounding in the earlier figures, there's only a 1.4% chance of finding 3 or more scientists in a group of 493 people. And that beats the 5% requirement handily. It's 0.2% for 4 or more, and 0.016% for 5 or more. This gets to a level where, as a matter of the probability, we'd be pretty confident that something real was going on.

But there's a joker in this deck, and I'm it. There is a selection effect problem. Namely, this group contains me -- because I started looking at the subject on the grounds that I was in it. Any group that contains me is guaranteed to have at least 1 scientist, 1 left-handed person, 1 runner, and so on. If I'm the person selecting the group, then we have to not count me. That leaves us with only 4 identified Grumbine scientists for the purpose of our research. As that's still at the 0.2% level, we're still pretty confident that something real is going on. Or at least if not, it's a pretty surprising coincidence.

Suppose that the number I made up for fraction of people who are scientists is too low. Let's say it's 0.3% instead of 0.1%. Then our chances of getting 4 or more scientists in the sample of 493 rises to 6.3%, which would not pass our standard for 'probably not chance'. So it is important to get a good idea of the figure. Now I'm pretty sure that the true figure isn't as high as that. It would mean 1 million people in the US had published in the last 20 years, and that just doesn't seem plausible. Even 300,000 (the 0.1%) strikes me as high, but I was trying to err on the high side in the first place.

Some odds and ends:

• We had some difficulty in finding data to work with.
  
• Once found, we saw that the data had some serious quality control problems
• Our starting point included a selection bias problem
• In trying to evaluate our conclusion, we discovered that the conclusion depended on an assumption we'd made without much evidence (the 0.1%)

All of this is common. To do science, you have to be ready to go back over your whole process to verify that where each step was not extremely strong, it at least doesn't change your conclusions. If it could, you have to mention this. Something that happens, though, in science is that once it's been mentioned, issues don't necessarily get rehashed in every paper for ever after. Inexperienced readers of science sometimes complain, because of this, about scientists 'hiding' problems. It isn't hidden, it's there in the scientific literature. It's just that the scientific literature assumes you're all big kids and have done your homework in reading the previous work. Citations aren't there for decoration.

Partly, this set of notes is to illustrate the Central Skill of a Scientist note. Although it turned out that the original idea, of there being surprisingly many Grumbines in science, is probably acceptable, it could have turned out otherwise. At that point, move on to other ideas. Scientists have many ideas, which is one of the secondary skills. So moving on to others is not a big deal.

Then again, having passed this far, we are in a position to ask -- again -- "So what?". More politely, "What would be shown even if the idea were true?" If there really were some exceptional number of Grumbines (or Bernoullis or Darwins to name some much better known families) in science, what would that mean? Unfortunately, nothing particular. Maybe it is a sign that there's a genetic contribution to entering science. Maybe it's a sign that there are family environment features which, for some reason, are common in this crowd. And maybe it really is just chance. These are more reasons to not get too wedded to ideas.

Detail: Binomial probabilities

0 Feedback

I'm actually referring to an important mathematical and physical concept, rather than the comments you give (which I also value).

Feedback is an extremely common process. Perhaps the most common case that people know is feedback involved public address systems. You have a microphone to pick up sound, which goes to an amplifier, and then out a speaker. If you put the microphone too near the speaker, then the microphone picks up some minor sound (say the sound of someone putting papers on the stand near the microphone), it goes through the amplifier and the louder sound comes out the speaker. The microphone now picks up that sound, it gets amplified even more and comes out the speaker again. Repeat through a number of cycles and the speaker is now as loud as it can be. This is a runaway feedback. Not all feedback is runaway.

A feedback can also be limited. Consider the atmosphere in this case. It has a temperature, carbon dioxide level, and water vapor level. Both carbon dioxide and water vapor are greenhouse gases, so if there is more of them, the temperature goes up. Suppose we put enough carbon dioxide into the atmosphere that its greenhouse effect (alone) raised temperature by 1 degree. If the atmosphere is warmer, we expect it to have more water vapor. That extra water vapor, let's say, produces another 0.5 degrees of warming. But, since temperature has gone up again, there's still more warming -- another half of this, for 0.25 degrees. This feeds back through increasing the water vapor to make another 0.125 degrees warming. Keep cycling through this feedback and it turns out to add another 1 degree of warming (1 = 0.5 + 0.25 + 0.125 + ...) after you've gone through an infinite number of steps. It's pretty close after a limited number of steps (99.9% of the way there after 10 cycles). We can look at these feedbacks in terms of how much of a response (multiplier) there is to a 1 degree kick to the system. I made up the number 0.5 here (every step is 0.5 times the previous). That's actually about the right size, in addition to being simple to work with. If the response multiplier is, say, 0.75 instead, then instead of getting back a total warming of 2 degrees, we would get back 4 degrees. The formula is
total = 1 / (1 - x)
where x is this multiplier, and the multiplier has to be less than 1.

Feedbacks can also produce cycles. A common illustration of this is predators and prey -- sharks and fish, or foxes and hares. Suppose one year, there are more fish than usual. The sharks, then, have a lot to eat and give birth and raise more sharks than usual. But, now that there are more sharks, they eat up even more fish than usual. That decreases the number of fish below the usual level. With fewer than normal fish, the sharks start dying off. That eventually gives us fewer than normal sharks, so the fish population recovers. And so on. The two populations could keep cycling (if the balances are right), or one could crash (depends on whether the extra sharks eat up too many fish, or whether the time of few fish lasts too long for any sharks to survive).

In the climate system, there are definitely some amplifying feedbacks and some cycling feedbacks. There don't appear to be any runaway feedbacks. That's not terribly reassuring, however, because life can get quite unpleasant even if temperatures don't run off to boiling the oceans.

Detail: Feedback