In this piece I want to address a point made in part thirteen and also discuss another important issue I’ve only touched on to date.
The first such is that one could well question whether or not subfossil trees that only live 30 years provides a realistic or stringent test. That’s entirely fair, because the geometric effect on ring widths is by far the strongest in the first few decades of a tree’s life. [This is because ring width declines as x^-2, where x = tree basal area, and the relative value of x changes fastest in the earliest years (by relative I mean as a function of size in the previous year, i.e. relative growth rate).]
The short answer is that it doesn’t matter. With the third and fourth figures shown in part thirteen as reference (subfossil trees whose births and deaths are both perfectly distributed throughout the analysis period), if I now instead use a longer tree age of say 150 years, comparable to the mean subfossil tree lifespan of say the Tornetrask (Scandinavia) or Yamal (Russia) sites, and an increased number of years in the chronology (1500, also more realistic), giving yearly core sample sizes that range from 2 to 24 with a mean of about 13, I get the following result (click to get a clearer pdf version):
Compare this result to the analogous situation from the previous post. The two results are nearly identical, the lone difference being that the flexible spline now performs somewhat better when climatic trends are decreasing than it did previously.
When ring areas are used, conversely, I still get a perfect result, as before, for the raw series themselves and therefore the linear trends fit thereto:
Now, onto the other important issue that I mentioned above. Until now in these posts I’ve concentrated almost exclusively on long term trend estimates, i.e. “low frequency” variation. But now let’s look specifically at the higher frequency (century scale and below) variations in the four different RCS-based climate estimates shown in the first graph above. I know that this component of the variance is constant over time–because that’s how I designed it–which we can see visually well enough by looking at the second graph above (where the estimates follow the true values exactly, the black lines being completely hidden). It’s easier to see if I remove the true values from the graphs, so here are the climatic estimates (“reconstructions”) obtained when a flexible spline is used to smooth the regional curve (those computed using a stiff spline are essentially identical):
Looking at the decreasing climate trend in particular, there is a noticeable (and measurable) increase in the estimated high frequency variance with time (but not when climate trends are positive). When combined with the mis-estimate of the trend, this leads to a pronounced under-estimate of the frequency of years with high climate values in the past (especially near the chronology beginning, where the trend is most poorly estimated). And this effect is actually even worse than shown here, because I removed the earliest ~ 60 years in the chronology due to small sample sizes (years with n < 6).
So, what was already a problem in accurate comparisons of current and former climatic states, related to trend mis-estimation, has been made even worse by this distortion of higher frequency variation, especially at the high end of the climate state variable. And to reiterate, this is the best case scenario for the RCS method when applied to ring widths. What will I get when I start departing from all those ideal states I instituted?
I’ll relax just one constraint first: the one that says that all trees start (and end) their lives at perfectly spaced intervals; that’s fair because we know for sure that we’ll never see a data set that perfect. Specifically, I include more living trees in the sample (24% instead of the 2% above), and those trees are a little older on average than are the subfossil trees (means of 252 vs 149 years), because they are not dying as early. This would be a reasonable scenario for those locations, such as the far northern taiga/tundra, where fluvial processes in flat landscapes uproot mature trees before they are biologically senescent, but yet some locations still exist that have not been so affected, on slightly higher ground say. And I also initiate each tree’s life in random years instead of at perfectly spaced intervals.
Here’s what that will give us, first when using a flexible spline, and then using a stiff one:
I’ve included flexible (loess) curve fits here, along with the raw series and linear trend estimates–they are the dashed black (true values) and grey (estimated) lines. I include the former as a control to show how much of the estimated shorter term variation is induced by the loess procedure alone. To see this, simply look at how much the loess fit (the dashed gray line) deviates from the linear trend estimate (straight, thicker gray line), and do the same for the analogous lines representing the true climate values. We can see that the RCS procedure not only fails to recover the true trend, but also transforms that “lost variance” into increased variance at shorter time scales, roughly century-scale and below. And when climate trends are decreasing, the very high frequency variance again increases with time.
Lastly, if I now add a biological size effect to the rings, one that is strongly unimodal, where ring areas increase rapidly until mid-age, and then decline gradually afterwards (an entirely realistic tree growth pattern), I get this, for flexible and stiff spline RCS methods respectively:
(a) flexible spline