Blog - El Niño project (part 3) (Rev #6, changes)

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This is a blog article in progress, written by John Baez. To see discussions of the article as it is being written, visit the Azimuth Forum.

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In February, this paper claimed that there’s a 75% chance the next El Niño will arrive by the end of 2014:

• Josef Ludescher, Avi Gozolchiani, Mikhail I. Bogachev, Armin Bunde, Shlomo Havlin, and Hans Joachim Schellnhuber, Very early warning of next El Niño, *Proceedings of the National Academy of Sciences*, February 2014. (Click title for free version, journal name for official version.)

Since it was published in a reputable journal, it created a big stir! But that’s not the main reason we at the Azimuth Project want to analyze and improve this paper. The main reason is that it uses a *climate network*.

Very roughly, the idea is this. Draw a big network of dots representing different places in the Pacific Ocean. For each pair of dots, compute a number saying how strongly correlated the sea surface temperatures are at those two places. The paper claims that when a El Niño is getting ready to happen, the average of these numbers is big. In other words, temperatures in the Pacific tend to go up and down in synch!

Whether this idea is right or wrong, it’s interesting— and it’s not very hard for programmers to dive in and study it.

Two Azimuth members have done just that: David Tanzer, a software developer who works for financial firms in New York, and Graham Jones, a self-employed programmer who also works on genomics and Bayesian statistics. These guys have really brought new life to the Azimuth Code Project in the last few weeks, and it’s exciting! It’s even gotten me to do some programming myself.

Soon I’ll start talking about the programs they’ve written, and how you can help.

But today I’ll summarize the paper by Ludescher *et al*. Their methodology is also explained here:

• Josef Ludescher, Avi Gozolchiani, Mikhail I. Bogachev, Armin Bunde, Shlomo Havlin, and Hans Joachim Schellnhuber, Improved El Niño forecasting by cooperativity detection, *Proceedings of the National Academy of Sciences*, 30 May 2013.

The basic idea is to use a climate network. There are lots of variants on this idea, but here’s a simple one. Start with a bunch of dots representing different places on the Earth. For any pair of dots $i$ and $j$, compute the cross-correlation of temperature histories at those two places. Call some function of this the ‘link strength’ for that pair of dots. Compute the average link strength… and get excited when this gets bigger than a certain value.

The papers by Ludescher *et al* use this strategy to predict El Niños. They build their climate network using correlations between daily temperature data for 14 grid points in the El Niño basin and 193 grid points outside this region, as shown here:

The red dots are the points in the El Niño basin.

Starting from this temperature data, they compute an ‘average link strength’ in a way I’ll describe later. When this number is bigger than a certain fixed value, they claim an El Niño is coming.

How do they decide if they’re right? How do we tell when an El Niño actually arrives? One way is to use the ‘Niño 3.4 index’. This the area-averaged sea surface temperature anomaly in the yellow region here:

**Anomaly** means the temperature minus its average over time: how much *hotter than usual* it is. When the Niño 3.4 index is over 0.5°C for at least 3 months, Ludescher *et al* say there’s an El Niño.

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 3 months. The red line is their ‘average link strength’. Whenever this exceed a certain threshold $\Theta = 2.82$, they predict an El Niño will start in the following calendar year.

The green arrows show their successes. The dashed arrows show their false alarms. You can see a little letter n whenever an El Niño occurred that they failed to predict.

Actually, chart A here shows the ‘learning phase’ of their calculation. In this phase, they adjusted the $\Theta$ so their procedure would do a good job. Chart B shows the ‘testing phase’. Here they used the value of $\Theta$ chosen in the learning phase, and checked to see how good a job it did.

Now I mainly need to explain how they compute their ‘average link strength’.

Let $i$ stand for any point in this 9 × 27 grid:

For each day $t$ between June 1948 and November 2013, let $\tilde{T}_i(t)$ be the the average surface air temperature at the point $i$ on day $t$. You can get these numbers from here:

Let $T_i(t)$ be $\tilde{T}_i(t)$ minus its **climatological average**. For example, if $t$ is June 1st 1970, we average the temperature at location $i$ over all June 1sts from 1948 to 2013, and subtract that from $\tilde{T}_i(t)$ to get $T_i(t)$. They call $T_i(t)$ the **temperature anomaly .**

(A subtlety here: when we are doing prediction we can’t know the future temperatures, so the climatological average is only the average over *past* days meeting the above criteria.)

For any function of time, denote its moving average over the last 365 days by:

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

Let $i$ be a point in the El Niño basin, and $j$ be a point outside it.~~ Define~~ For~~ the~~ any time lags$\tau$ between 0 and 200 days, define the **time-delayed cross-correlation by:**

` \displaystyle{ C_{i,j}^{t}(-\tau) = \frac{\langle T_i(t) T_j(t - \tau) \rangle - \langle T_i(t) \rangle \langle T_j(t - \tau)} \rangle}{\sqrt{\langle (T_i(t) - \langle T_i(t))^2 \rangle} } }`

~~ if~~ Note that this is a way of studying the~~ delay~~ linear correlation between the temperature anomaly at node$\mathrm{\tau i}$~~ ~~ i~~ \displaystyle{~~~~ C_{i,j}^{t}(-\tau)~~~~ =~~~~ \frac{\langle~~~~ T_i(t)~~~~ T_j(t~~~~ -~~~~ \tau)~~~~ \rangle~~~~ -~~~~ \langle~~~~ T_i(t)~~~~ \rangle~~~~ \langle~~~~ T_j(t~~~~ -~~~~ \tau)}~~~~ \rangle}{\sqrt{\langle~~~~ (T_i(t)~~~~ -~~~~ \langle~~~~ T_i(t))^2~~~~ \rangle}~~~~ }~~~~ }\tau~~ ~~ is~~ and~~ positive~~ the temperature anomaly a time$\tau$ earlier at node $j$.

$C_{i,j}^{t}(\tau) = \langle T_i(t - \tau) T_j(t) \rangle - \langle T_i(t - \tau) \rangle \langle T_j(t) \rangle$

Ludescher *et al* normalize this in a somewhat funny way, defining the **time-delayed cross-correlation** $C_{i,j}^{t}(\tau)$ to be the time-delayed cross-covariance divided by

They consider time lags $\tau$ between 0 and 200 d, where “a reliable estime of the backround noise level can be guaranteed.”

$\sqrt{(\langle (T_i(t) - \langle T_i(t)\rangle)^2 \rangle} \; \sqrt{(\langle (T_i(t-\tau) - \langle T_i(t-\tau)\rangle)^2 \rangle}$

This is something like the standard deviation of $T_i(t)$ times the standard deviation of $T_i(t - \tau)$. Dividing by standard deviations is what people often do to turn covariances into correlations. However, as Nadja Kutz pointed out, it’s unusual to do this when the angle brackets are defined as a moving average over the last 365 days, because then

$\langle \langle f(t) \rangle \rangle \ne \langle f(t) \rangle$

and the usual formal properties of expressions involving nested brackets don’t hold.

Anyway, $C_{i,j}^{t}(\tau)$ is defined in a similar way when $\tau$, starting from

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

Divide the cross-covariances by the standard deviations of $T_i$ and $T_j$ to obtain the cross-correlations.

Only temperature data from the past are considered when estimating the cross-correlation function at day $t$.

Next, for nodes $i$ and $j$, and for each time point $t$, the maximum, the mean and the standard deviation around the mean are determined for $C_{i,j}^t$, as $\tau$ varies across its range.

Define the **link strength** $S_{i j}(t)$ as the difference between the maximum and the mean value, divided by the standard deviation.

They say:

Accordingly, $S_{i j}(t)$ describes the link strength at day t relative to the underlying background and thus quantifies the dynamical teleconnections between nodes $i$ and $j$.

Niño 3.4 is the area-averaged sea surface temperature anomaly in the region 5°S-5°N and 170°-120°W. You can get Niño3.4 data here:

- Niño 3.4 data, NOAA.

Niño 3.4 is just one of several official regions in the Pacific:

• Niño 1: 80°W-90°W and 5°S-10°S. • Niño 2: 80°W-90°W and 0°S-5°S • Niño 3: 90°W-150°W and 5°S-5°N. • Niño 3.4: 120°W-170°W and 5°S-5°N. • Niño 4: 160°E-150°W and 5°S-5°N.

For more details, read this:

• Kevin E. Trenberth, The definition of El Niño, *Bulletin of the American Meteorological Society* **78** (1997), 2771–2777.