<![CDATA[Networks in Climate Science]]>

<![CDATA[Networks in Climate Science]]>What follows is draft of a talk I’ll be giving at the Neural Information Processing Seminar on December 10th. The actual talk may contain more stuff—for example, more work that Dara Shayda has done. But I’d love comments now, so I’m posting this now and hoping you can help out.

You can click on any of the pictures to see where it came from or get more information.

Preliminary throat-clearing

I’m very flattered to be invited to speak here. I was probably invited because of my abstract mathematical work on networks and category theory. But when I got the invitation, instead of talking about something I understood, I thought I’d learn about something a bit more practical and talk about that. That was a bad idea. But I’ll try to make the best of it.

I’ve been trying to learn climate science. There’s a subject called ‘complex networks’ where people do statistical analyses of large graphs like the worldwide web or Facebook and draw conclusions from it. People are trying to apply these ideas to climate science. So that’s what I’ll talk about. I’ll be reviewing a lot of other people’s work, but also describing some work by a project I’m involved in, the Azimuth Project.

The Azimuth Project is an all-volunteer project involving scientists and programmers, many outside academia, who are concerned about environmental issues and want to use their skills to help. This talk is based on the work of many people in the Azimuth Project, including Jan Galkowski, Graham Jones, Nadja Kutz, Daniel Mahler, Blake Pollard, Paul Pukite, Dara Shayda, David Tanzer, David Tweed, Steve Wenner and others. Needless to say, I’m to blame for all the mistakes.

Climate variability and El Niño

Okay, let’s get started.

You’ve probably heard about the ‘global warming pause’. Is this a real thing? If so, is it due to ‘natural variability’, heat going into the deep oceans, some combination of both, a massive failure of our understanding of climate processes, or something else?

Here is chart of global average air temperatures at sea level, put together by NASA’s Goddard Institute of Space Science:

You can see a lot of fluctuations, including a big dip after 1940 and a tiny dip after 2000. That tiny dip is the so-called ‘global warming pause’. What causes these fluctuations? That’s a big, complicated question.

One cause of temperature fluctuations is a kind of cycle whose extremes are called El Niño and La Niña.

A lot of things happen during an El Niño. For example, in 1997 and 1998, a big El Niño, we saw all these events:

El Niño is part of an irregular cycle that happens every 3 to 7 years, called the El Niño Southern Oscillation or ENSO. Two strongly correlated signs of an El Niño are:

1) Increased sea surface temperatures in a patch of the Pacific called the Niño 3.4 region. The temperature anomaly in this region—how much warmer it is than usual for that time of year—is called the Niño 3.4 index.

2) A decrease in air pressures in the western side of the Pacific compared to those further east. This is measured by the Southern Oscillation Index or SOI.

You can see the correlation here:

El Niños are important because they can cause billions of dollars of economic damage. They also seem to bring heat stored in the deeper waters of the Pacific into the atmosphere. So, one reason for the ‘global warming pause’ may be that we haven’t had a strong El Niño since 1998. The global warming pause might end with the next El Niño. For a while it seemed we were due for a big one this fall, but that hasn’t happened.

Teleconnections

The ENSO cycle is just one of many cycles involving teleconnections: strong correlations between weather at distant locations, typically thousands of kilometers. People have systematically looked for these teleconnections using principal component analysis of climate data, and also other techniques.

The ENSO cycle shows up automatically when you do this kind of study. It stands out as the biggest source of climate variability on time scales greater than a year and less than a decade. Some others include:

• The Pacific-North America Oscillation.
• The Pacific Decadal Oscillation.
• The North Atlantic Oscillation.
• The Arctic Oscillation.

For example, the Pacific Decadal Oscillation is a longer-period relative of the ENSO, centered in the north Pacific:

Complex network theory

Recently people have begun to study teleconnections using ideas from ‘complex network theory’.

What’s that? In complex network theory, people often start with a weighted graph: that is, a set $N$ of nodes and for any pair of nodes $i, j \in N,$ a weight $A_{i j},$ which can be any nonnegative real number.

Why is this called a weighted graph? It’s really just a matrix of nonnegative real numbers!

The reason is that we can turn any weighted graph into a graph by drawing an edge from node $j$ to node $i$ whenever $A_{i j} >0.$ This is a directed graph, meaning that we should draw an arrow pointing from $j$ to $i.$ We could have an edge from $i$ to $j$ but not vice versa! Note that we can also have an edge from a node to itself.

Conversely, if we have any directed graph, we can turn it into a weighted graph by choosing the weight $A_{i j} = 1$ when there’s an edge from $j$ to $i,$ and $A_{i j} = 0$ otherwise.

For example, we can make a weighted graph where the nodes are web pages and $A_{i j}$ is the number of links from the web page $j$ to the web page $i.$

People in complex network theory like examples of this sort: large weighted graphs that describe connections between web pages, or people, or cities, or neurons, or other things. The goal, so far, is to compute numbers from weighted graphs in ways that describe interesting properties of these complex networks—and then formulate and test hypotheses about the complex networks we see in real life.

The El Niño basin

Here’s a very simple example of what we can do with a weighted graph. For any node $i,$ we can sum up the weights of edges going into $i:$

$\sum_{j \in N} A_{j i}$

This is called the degree of the node $i.$ For example, if lots of people have web pages with lots of links to yours, your webpage will have a high degree. If lots of people like you on Facebook, you will have a high degree.

So, the degree is some measure of how ‘important’ a node is.

People have constructed climate networks where the nodes are locations on the Earth’s surface, and the weight $A_{i j}$ measures how correlated the weather is at the $i$ th and $j$ th location. Then, the degree says how ‘important’ a given location is for the Earth’s climate—in some vague sense.

For example, in Complex networks in climate dynamics, Donges et al take surface air temperature data on a grid and compute the correlation between grid points.

More precisely, let $T_i(t)$ be the temperature at the $i$ th grid point at month $t$ after the average for that month in all years under consideration has been subtracted off, to eliminate some seasonal variations. They compute the Pearson correlation $A_{i j}$ of $T_i(t)$ and $T_j(t)$ for each pair of grid points $i, j.$ The Pearson correlation is the simplest measure of linear correlation, normalized to range between -1 and 1.

We could construct a weighted graph this way, and it would be symmetric, or undirected:

$A_{i j} = A_{j i}$

However, Donges et al prefer to work with a graph rather than a weighted graph. So, they create a graph where there is an edge from $i$ to $j$ (and also from $j$ to $i$ ) when $|A_{i j}|$ exceeds a certain threshold, and no edge otherwise.

They can adjust this threshold so that any desired fraction of pairs $i, j$ actually have an edge between them. After some experimentation they chose this fraction to be 0.5%.

A certain patch dominates the world! This is the El Niño basin. The Indian Ocean comes in second.

(Some details, which I may not say:

The Pearson correlation is the covariance

$\Big\langle \left( T_i - \langle T_i \rangle \right) \left( T_j - \langle T_j \rangle \right) \Big\rangle$

normalized by dividing by the standard deviation of $T_i$ and the standard deviation of $T_j.$

The reddest shade of red in the above picture shows nodes that are connected to 5% or more of the other nodes. These nodes are connected to at least 10 times as many nodes as average.)

The Pearson correlation detects linear correlations. A more flexible measure is mutual information: how many bits of information knowing the temperature at time $t$ at grid point $i$ tells you about the temperature at the same time at grid point $j.$

Donges et al create a climate network this way as well, putting an edge between nodes if their mutual information exceeds a certain cutoff. They choose this cutoff so that 0.5% of node pairs have an edge between them, and get the following map:

The result is almost indistinguishable in the El Niño basin. So, this feature is not just an artifact of focusing on linear correlations.

El Niño breaks climate links

We can also look at how climate networks change with time—and in particular, how they are affected by El Niños. This is the subject of a 2008 paper by Tsonis and Swanson, Topology and predictability of El Niño and La Niña networks.

They create a climate network in a way that’s similar to the one I just described. The main differences are that they:

separately create climate networks for El Niño and La Niña time periods;
create a link between grid points when their Pearson correlation has absolute value greater than $0.5;$
only use temperature data from November to March in each year, claiming that summertime introduces spurious links.

They get this map for La Niña conditions:

and this map for El Niño conditions:

They conclude that “El Niño breaks climate links”.

This may seem to contradict what I just said a minute ago. But it doesn’t! While the El Niño basin is a region where the surface air temperatures are highly correlated to temperatures at many other points, when an El Niño actually occurs it disrupts correlations between temperatures at different locations worldwide—and even in the El Niño basin!

For the rest of the talk I want to focus on a third claim: namely, that El Niños can be predicted by means of an increase in correlations between temperatures within the El Niño basin and temperatures outside this region. This claim was made in a recent paper by Ludescher et al. I want to examine it somewhat critically.

Predicting El Niños

People really want to predict El Niños, because they have huge effects on agriculture, especially around the Pacific ocean. However, it’s generally regarded as very hard to predict El Niños more than 6 months in advance. There is also a spring barrier: it’s harder to predict El Niños through the spring of any year.

It’s controversial how much of the unpredictability in the ENSO cycle is due to chaos intrinsic to the Pacific ocean system, and how much is due to noise from outside the system. Both may be involved.

There are many teams trying to predict El Niños, some using physical models of the Earth’s climate, and others using machine learning techniques. There is a kind of competition going on, which you can see at a National Oceanic and Atmospheric Administration website.

The most recent predictions give a sense of how hard this job is:

When the 3-month running average of the Niño 3.4 index exceeds 0.5°C for 5 months, we officially declare that there is an El Niño.

As you can see, it’s hard to be sure if there will be an El Niño early next year! However, the consensus forecast is yes, a weak El Niño. This is the best we can do, now. Right now multi-model ensembles have better predictive skill than any one model.

The work of Ludescher et al

The Azimuth Project has carefully examined a 2013 paper by Ludescher et al called Very early warning of next El Niño, which uses a climate network for El Niño prediction.

They build their climate network using correlations between daily surface air temperature data between points inside the El Niño basin and certain points outside this region, as shown here:

The red dots are the points in their version of the El Niño basin.

(Next I will describe Ludescher’s procedure. I may omit some details in the actual talk, but let me include them here.)

The main idea of Ludescher et al is to construct a climate network that is a weighted graph, and to say an El Niño will occur if the average weight of edges between points in the El Niño basin and points outside this basin exceeds a certain threshold.

As in the other papers I mentioned, Ludescher et al let $T_i(t)$ be the surface air temperature at the $i$ th grid point at time $t$ minus the average temperature at that location at that time of year in all years under consideration, to eliminate the most obvious seasonal effects.

They consider a time-delayed covariance between temperatures at different grid points:

$\langle T_i(t) T_j(t - \tau) \rangle - \langle T_i(t) \rangle \langle T_j(t - \tau) \rangle$

where $\tau$ is a time delay, and the angle brackets denote a running average over the last year, that is:

$\displaystyle{ \langle f(t) \rangle = \frac{1}{365} \sum_{d = 0}^{364} f(t - d) }$

where $t$ is the time in days.

They normalize this to define a correlation $C_{i,j}^t(\tau)$ that ranges from -1 to 1.

Next, for any pair of nodes $i$ and $j,$ and for each time $t,$ they determine the maximum, the mean and the standard deviation of $|C_{i,j}^t(\tau)|,$ as the delay $\tau$ ranges from -200 to 200 days.

They define the link strength $S_{i,j}(t)$ as the difference between the maximum and the mean value of $|C_{i,j}^t(\tau)|,$ divided by its standard deviation.

Finally, they let $S(t)$ be the average link strength, calculated by averaging $S_{i j}(t)$ over all pairs $i,j$ where $i$ is a grid point inside their El Niño basin and $j$ is a grid point outside this basin, but still in their larger rectangle.

Here is what they get:

The blue peaks are El Niños: episodes where the Niño 3.4 index is over 0.5°C for at least 5 months.

The red line is their ‘average link strength’. Whenever this exceeds a certain threshold $\Theta = 2.82,$ and the Niño 3.4 index is not already over 0.5°C, they predict an El Niño will start in the following calendar year.

Ludescher et al chose their threshold for El Niño prediction by training their algorithm on climate data from 1948 to 1980, and tested it on data from 1981 to 2013. They claim that with this threshold, their El Niño predictions were correct 76% of the time, and their predictions of no El Niño were correct in 86% of all cases.

On this basis they claimed—when their paper was published in February 2014—that the Niño 3.4 index would exceed 0.5 by the end of 2014 with probability 3/4.

The latest data as of 1 December 2014 seems to say: yes, it happened!

Replication and critique

Graham Jones of the Azimuth Project wrote code implementing Ludescher et al’s algorithm, as best as we could understand it, and got results close to theirs, though not identical. The code is open-source; one goal of the Azimuth Project is to do science ‘in the open’.

More interesting than the small discrepancies between our calculation and theirs is the question of whether ‘average link strengths’ between points in the El Niño basin and points outside are really helpful in predicting El Niños.

Steve Wenner, a statistician helping the Azimuth Project, noted some ambiguities in Ludescher et al‘s El Niño prediction rules and disambiguated them in a number of ways. For each way he used Fischer’s exact test to compute the $p$ -value of the null hypothesis that Ludescher et al‘s El Niño prediction does not improve the odds that what they predict will occur.

The best he got (that is, the lowest $p$ -value) was 0.03. This is just a bit more significant than the conventional 0.05 threshold for rejecting a null hypothesis.

Do high average link strengths between points in the El Niño basin and points elsewhere in the Pacific really increase the chance that an El Niño is coming? It is hard to tell from the work of Ludescher et al.

One reason is that they treat El Niño as a binary condition, either on or off depending on whether the Niño 3.4 index for a given month exceeds 0.5 or not. This is not the usual definition of El Niño, but the real problem is that they are only making a single yes-or-no prediction each year for 65 years: does an El Niño occur during this year, or not? 31 of these years (1950-1980) are used for training their algorithm, leaving just 34 retrodictions and one actual prediction (1981-2013, and 2014).

So, there is a serious problem with small sample size.

We can learn a bit by taking a different approach, and simply running some linear regressions between the average link strength and the Niño 3.4 index for each month. There are 766 months from 1950 to 2013, so this gives us more data to look at. Of course, it’s possible that the relation between average link strength and Niño is highly nonlinear, so a linear regression may not be appropriate. But it is at least worth looking at!

Daniel Mahler and Dara Shayda of the Azimuth Project did this and found the following interesting results.

Simple linear models

Here is a scatter plot showing the Niño 3.4 index as a function of the average link strength on the same month:

(Click on these scatter plots for more information.)

The coefficient of determination, $R^2,$ is 0.0175. In simple terms, this means that the average link strength in a given month explains just 1.75% of the variance of the Niño 3.4 index. That’s quite low!

Here is a scatter plot showing the Niño 3.4 index as a function of the average link strength six months earlier:

Now $R^2$ is 0.088. So, the link strength explains 8.8% of the variance in the Niño 3.4 index 6 months later. This is still not much—but interestingly, it’s much more than when we try to relate them at the same moment in time! And the $p$ -value is less than $2.2 \cdot 10^{-16},$ so the effect is statistically significant.

Of course, we could also try to use Niño 3.4 to predict itself. Here is the Niño 3.4 index plotted against the Niño 3.4 index six months earlier:

Now $R^2 = 0.162.$ So, this is better than using the average link strength!

That doesn’t sound good for average link strength. But now let’s could try to predict Niño 3.4 using both itself and the average link strength 6 months earlier. Here is a scatter plot showing that:

Here the $x$ axis is an optimally chosen linear combination of average and link strength and Niño 3.4: one that maximizes $R^2$ .

In this case we get $R^2 = 0.22.$

Conclusions

What can we conclude from this?

Using a linear model, the average link strength on a given month accounts for only 8% of the variance of Niño 3.4 index 6 months in the future. That sounds bad, and indeed it is.

However, there are more interesting things to say than this!

Both the Niño 3.4 index and the average link strength can be computed from the surface air temperature of the Pacific during some window in time. The Niño 3.4 index explains 16% of its own variance 6 months into the future; the average link strength explains 8%, and taken together they explain 22%. So, these two variables contain a fair amount of independent information about the Niño 3.4 index 6 months in the future.

Furthermore, they explain a surprisingly large amount of its variance for just 2 variables.

For comparison, Mahler used a random forest variant called ExtraTreesRegressor to predict the Niño 3.4 index 6 months into the future from much larger collections of data. Out of the 778 months available he trained the algorithm on the first 400 and tested it on the remaining 378.

The result: using a full world-wide grid of surface air temperature values at a given moment in time explains only 23% of the Niño 3.4 index 6 months into the future. A full grid of surface air pressure values does considerably better, but still explains only 34% of the variance. Using twelve months of the full grid of pressure values only gets around 37%.

From this viewpoint, explaining 22% of the variance with just two variables doesn’t look so bad!

Moreover, while the Niño 3.4 index is maximally correlated with itself at the same moment in time, for obvious reasons, the average link strength is maximally correlated with the Niño 3.4 index 10 months into the future:

(The lines here occur at monthly intervals.)

However, we have not tried to determine if the average link strength as Ludescher et al define it is optimal in this respect. Graham Jones has shown that simplifying their definition of this quantity doesn’t change it much. Maybe modifying their definition could improve it. There seems to be a real phenomenon at work here, but I don’t think we know exactly what it is!

My talk has avoided discussing physical models of the ENSO, because I wanted to focus on very simple, general ideas from complex network theory. However, it seems obvious that really understanding the ENSO requires a lot of ideas from meteorology, oceanography, physics, and the like. I am not advocating a ‘purely network-based approach’.

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