One of a series, also see editions: #1, #3, #4, #5

While we wait for the prediction parameters to firm up (El Niño is now even money), it’s worth checking what a naive statistical prediction for 2014 looks like. By *naive* I mean one that ignores influences external to the snow depth data (like El Niño), not one with weak statistics.

The obvious approach, used in the plot above, is to treat our 60 years of snow depth data (from Spencers Creek, near Charlotte Pass, Australia) as equal, and do simple stats on them as a whole. Michael Paine has done a fine job of that:

That approach gives a “naive” 2014 prediction as follows:

- Average peak depth : 198 cm
- Median peak depth : 186 cm (the 50/50 depth — half deeper, half thinner)
- Chance of exceeding 2 m : 40%

All Good? Well, not quite. Fundamental to treating all years of data as equal is the assumption of *stationarity*, the idea that the statistical properties don’t change with time. Whoops…

There’s actually a strong downtrend, consistent with long-standing global warming-based predictions.

How to handle that? The standard approach is to separate the effects: separate the trend from the variation around it. The trend gives us the best estimate depth to expect in 2014, and the difference between the data and the trend across the whole record gives us the likely variation around that. Applying that approach to the Spencers Creek peak depths gives us this, for 2014:

Viewed that way, the naive 2014 prediction plunges 20 cm:

- Average peak depth : 177 cm
- Median peak depth : 165 cm
- Chance of exceeding 2 m : 28%

But there’s more. What about those “variations about the trend”, the so-called *residuals*? Are we sure that they’re stationary¹? Here’s the residuals plotted as absolute values (negatives shown as positive) for ease of interpretation:

It looks like there’s a trend there too, but careful … it’s not all that strong statistically (R² = 0.05, p ~ 10%, 2-tails). Experience leads me to expect that there probably is a trend in the residuals, because that’s common in real world physical parameters — the variation tends towards being proportional to the current average². If there was proportionality, we’d expect the two trends to be similar, and they are³:

So what do we get if we assume a downtrend combined with residuals proportional to the trend estimate? The 2014 peak depth distribution then looks like this:

So the naive 2014 prediction becomes:

- Average peak depth : 177 cm
- Median peak depth : 165 cm
- Chance of exceeding 2 m : 26%

Not exactly pretty. The chance of exceeding 3 m is so small that it’s hard to see on that graph. It’s about 1.3 to 3% (depending how you estimate it). It’s now likely that we’ve already seen the last-ever 3 m season; even a Pinatubo would probably not be enough to produce one⁴.

Notes:

1. “*Homoscedastic*” (yes, seriously … these guys are statisticians).

2. That is where the concept of the coefficient of variation (standard deviation divided by mean) comes from. At the most basic level, for a real physical parameter like snow depth, the negative residual cannot exceed the trend estimate (that would imply a negative depth), so negative residuals at least have to depend on the trend estimate to some extent. Where the variance is large relative to the trend estimate, the dependence has to be strong, at least for a continuous distribution. (But there are plenty of hydrological parameters with a large lump at zero — a discontinuous distribution. Think daily rainfalls.)

3. They’re not quite proportional — the residuals trend is too big — but note that the statistical estimates of *both* trends are pretty uncertain here, because the peak depths are so variable. It’s likely that proportionality is incomplete — there is probably a component of the residuals distribution that is invariant and another bit that is proportional. Unfortunately the statistical data is insufficient to determine that. You could perhaps get it from a mechanistic model, like that of Hennessy et al.

4. The last 3 m season way back in 1992 was very likely the result of the short-lived dip in global temperature following the June 1991 Pinatubo volcanic eruption in the Philippines, which was that century’s biggest. Three metre seasons used to be fairly common; there were three in the first 20 years of data. The last 20 years had none.

[…] trend), and extrapolate the resulting uncertainty band, assuming 50% residual proportionality as previously adopted for the Spencers Creek […]