Flickering diatoms: A comment on Wang et al (2012)

In November 2012, Wang et al (2012) reported evidence of precursor signals of a critical ecosystem transition in a diatom stratigraphy from Erhai, a lake in Yunnan, China. Today, Jacob Carstensen, John Birks and I have published a comment in Nature that suggests the results of Wang et al (2012) are an artefact of their data processing.

Some ecosystems have alternative stable states, for example forest vs. savanna,  macrophyte vs. plankton-dominated shallow lakes, and coral vs. algae-dominated reefs. The transition from one stable state to another can be abrupt, difficult to reverse, and result in the loss of ecosystem services. Because of these characteristics, there is considerable interest in identifying early-warning signals that indicate that a critical threshold is approaching, potentially allowing preventive action to be taken.

Several early-warning signals have been suggested:

Wang et al seek some of these early warning signals in a diatom stratigraphy from Erhai prior to a critical transition in the diatom assemblages in 2001, examining changes in diatom diversity and composition (using axis one of a detrended correspondence analysis). They find an increase in standard deviation (SD) and skewness, and a decrease in autocorrelation (AR1) as the transition approaches (NB the change in AR1 is the opposite of what is expected). However, the data processing, interpolation of uneven-resolution data to annual resolution followed by detrending, is critical to the analysis in Wang et al.

In common with many short-cores, the age-depth relationship in the Erhai core is not linear. A half centimentre (Wang et al’s sample thickness) typically represents more years at the base of the core than near the top. This could be due to changes in the sedimentation rate, or could simply be because the older, deeper sediments have compacted and lost some water. The cause of the uneven age-depth model is not important, the effects are.

Sample resolution against date for the Erhai core. A LOESS (red line) highlights the major trends.

Sample resolution against date for the Erhai core. A LOESS model (red line) highlights the major trends.

The effects are two-fold. First, the samples near the base of the core, span more time and so have greater time averaging. This will tend reduce the SD near the base of the core relative to the top. Second, linear interpolation will change the spectral properties of the data. Where the resolution is coarse, there will be more interpolated values, tending to increase the AR1 of the data, and further decreasing the SD of the data. These effects are easily demonstrated with a simulation, taking numbers from a Normal distribution and subjecting them to aggregation and interpolation, as shown in the figure and table below.

Left plot shows random numbers from a Normal distribution with a mean of zero and a standard deviation of 1. The middle plot shows the same data aggregated  over 2 (blue) or 4 (red) years. The right plot shows the same data interpolated to annual resolution.

Left plot shows random numbers from a Normal distribution with a mean of zero and a standard deviation of 1. The middle plot shows the same data aggregated over 2 (blue) or 4 (red) years. The right plot shows the same data interpolated to annual resolution.

Standard deviation and autocorrelation coefficient for the data from a Normal distribution shown in the above figure, and the same data after aggregation and aggregation and interpolation
Raw data Aggregated Interpolated
2 years 5 years 2 years 5 years
SD 0.97 0.73 0.37 0.62 0.28
AR1 -0.03 0.61 0.92

These changes in SD and AR1 as sample resolution changes are consistent with the results of Wang et al.

The effects of the uneven sample resolution is the main problem with Wang et al. A second problem is their choice of detrending procedure. A linear trend in data can be removed by fitting a linear model to the data and using the residuals. More complex trends can be removed by using the residuals from a smoother, of which there are many. I would have used a LOESS (locally weighted scatterplot smoothing) model to detrend the data. Wang et al write that they use simple exponential smoothing. Actually they use a moving average model (MA).

The DCA from Wang et al (black) with, left, a LOESS fit (blue), middle a simple exponential smooth (red), and right, the smooth used by Wang et al (red).

The DCA from Wang et al (black) with, left, a LOESS fit (blue), middle a simple exponential smooth (red), and right, the smooth used by Wang et al (red). The simple exponential smooth is fitted with ses( ) in the R library forecast with the smoothing parameter estimated by the function as 0.9999.

Neither the simple exponential smooth nor the smooth used by Wang et al appear to be useful. Both closely resemble the data, and the Wang et al smooth has a higher variance than the data — it roughens rather than smooths the data. The model has large residuals where there are abrupt changes in the direction of data. These changes in direction tend to be largest in the upper part of the core where there is least aggregation and interpolation.

In our comment, we included a simulation that showed the observed trends in SD, skewness and AR1 in Wang et al are no larger than expected by chance by applying their methods to a simulated timeseries that resembles their DCA with added white noise.

We also attempted to repeat the Wang et al’s analyses with more appropriate (but still not ideal) methods, using a LOESS to detrend the data and not interpolating it before analysis. We find that all the patterns reported by Wang et al disappear or are greatly weakened and probably not different from chance.

Given that Wang et al’s results cannot be replicated with more appropriate methods, and conform with the null expectation of their data processing, we conclude that their results are an artefact of uneven sampling and methodological choice.

Wang et al were permitted by Nature to publish a reply. It is not persuasive.

They claim that our LOESS model “does not capture explicitly the unequal time increments in the sediment data”. I have no idea what the basis of this claim is — the LOESS takes the age of each sample as the x-variable. Yes, we could have weighted the samples in the LOESS according to their duration, do Wang et al really believe that would make any material difference? If so, they should demonstrate this.

They complain that the LOESS does not “confront the dating error problem” and we don’t replicate their full range of sensitivity tests. Are they arguing that their results would re-emerge from the noise with some combination of dating error and window width? If so, why didn’t they attempt to find this combination. Not a compelling argument.

They even argue that the trend towards low resolution down-core should be ignored because some of the more recent samples also have low resolution. Our simulation demonstrates that the change in resolution is sufficient to drive the patterns they find, regardless of a couple of recent samples with anomalously low resolution.

They complain our simulation “does not fully account for the temporal aggregation in the actual DCA data…”. Actually it does, we write in our methods section “The time series were aggregated to mimic the observed sample spacing.” Despite this, Wang et al claim “The model does not use the real time periods in the sediment data derived from the dating model.” Did they not read our methods section, or are they arguing that because we can only “mimic” the true aggregation, and have to assign whole rather than partial years to each sample, our results are somehow invalid? Again, a demonstration would help their argument here.

They conclude that the main implication of our comment is that a wide range of smoothing functions should be evaluated. I would demur. The choice (and implementation) of smoothing function by Wang et al was poor, but the real problem was in their treatment of unevenly sampled sediment.

Advertisements

About richard telford

Ecologist with interests in quantitative methods and palaeoenvironments
This entry was posted in Peer reviewed literature and tagged , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s