This got super long. tl;dr: There is no recovery. If the exponential trend is no longer a good fit for the data, the entire trend should be replaced with another function, which will not show a recovery.

I hope it's not piling on to respond to Michael Hauber. One thing to be careful about is whether we are all using language the same way.

Typically, year over year data is broken into two components: the trend, which is predictable, and the noise, which is random and therefore unpredictable. For monthly or daily data, there is also a seasonal (or periodic) component, which is usually subtracted out before further analysis is performed. Therefore, the PIOMAS volume data is usually presented as an anomaly graph, where the estimated seasonal component has been subtracted out.

There are two assumptions that are typically made about the noise. The first is that the variance is constant, so the vertical spread of the noise is unchanging over time. The second is that the noise is uncorrelated, so knowledge about the past noise does not give any information for predicting the future noise. In practice, one or both of these assumptions are not valid. If this is not accounted for, model forecasts are generally less accurate, but not necessarily fatally. On the other side, attempting to account for these generally makes the model more complicated.

Another required assumption about the noise is that the mean noise is zero. In other words, the actual measured values spend as much time above the trend as below, over sufficiently long time periods. If the mean for the noise is not zero or appears to be changing over time, that should really be captured in the trend.

The trend is a deterministic function describing the predictable component of the data. Which function to use is a difficult question. In the absence of a physical model, less complicated trend functions are almost always better. If a more complicated function is used, it is very easy to incorporate the noise into the trend function. This is called overfitting the data, and it generally makes predictions worse. Simple trend functions, even when you know they are too simple, often provide more reliable predictions than complicated trend functions which overfit the data.

Which leads to "recovery". As Michael is using the term, it does not refer to the noise, so it must refer to the trend. (Denialists do not distinguish between the trend and the noise, so they use "recovery" to refer to the noise.) When the long term trend is toward ice loss, a recovery means an increase in ice in the short term trend.

In Reply #2945, Michael refers to a chart of PIOMAS volume to argue that the previous downward trend has broken and has been replaced by an upward trend. Unfortunately, his chart is not displaying for me, so I'm not sure of what he's looking at. From context, it sounds like he is looking at a chart with an exponential trend with indicated confidence intervals. (Possibly year over year July average volume?) The data from the last two years is now far outside the confidence intervals.

From this, Michael concludes that the exponential trend, which had been accurate through 2012, has broken, and should be replaced by a rising trend. If Michael is able to confirm that this is his argument, I would appreciate it.

In response, if the trend function changes between different time periods, the resulting function can never be considered a simple function. I would agree that the large differences in the past two years between the exponential trend and the observed data mean that the exponential function is not a good fit. However, I would replace this function by another simple function. Possibilities include a straight line. If that is deemed too simple, I would consider adding one (but only one) of x^2, x^1.5, and x ln x terms, and see if that results in an improved fit.

I would be extremely reluctant to conclude that the trend is represented by one function before a certain date and a different function after that date, unless there is a strong physical reason to do so. Tying into another recent discussion in this thread, if someone were to present convincing evidence that Fukushima had effects which permanently changed the behavior of the atmosphere, I might support using different functions to represent ice volume before and after that event. I would be more likely to support this if there were significant evidence that no reasonably simple function could adequately represent the ice trend.

My insistence on simple functions does have a consequence I should acknowledge. No trend represented by a simple function will ever show a recovery. Essentially, any function complicated enough to be able to show a recovery will be too complicated to use as a trend function by my standards. Anyone who is determined to find a recovery is likely to consider this a drawback. I just consider it a reflection of reality.

There is a way out. If simple functions do not represent the trend well enough, the trend can be represented by splines or other "non-parametric" curve fitting techniques. If it became clear that a linear function or other relatively simple parametric function could not capture important behavior of the trend, I would consider a spline or other approach. If the ice turns out to have alternating periods of loss and recovery, this could turn out to be necessary.

The danger here is that it is very easy to overfit the data. If a spline fit shows a recovery, especially one at the end of the data, it may be that the recovery is real, but it could also be a case of overfitting the data. While I'm more likely to accept the use of a spline than a break in the trend resulting in a different trend function, I'd also be extremely wary of announcing a recovery based on the behavior of a spline fit alone.