A blog post accompanying the publication walks through the idea less formally, and I gave an invited talk on it for Methods in Ecology and Evolution. You can watch the talk, and the workshop materials are all open.

Below is an earlier post from when we were developing the model, kept here for context:

Post from 2024-12-01

Over the last couple of years, Tony Ives, Jack Williams and I have been developing a model for estimating coefficients of driver-species relationships, and taxa-taxa interactions from multinomially distributed data. The model works with any data that are multinomially distributed; however, our objective is to be able to understand the drivers of ecological change using palaeo-data.

The point of this work is to go beyond describing patterns of past changes, and get at why ecosystems change. If we can better understand the drivers of change in ecosystems, we can use this knowledge to understand how ecosystems will change into the future under human pressures and environmental change.

A crash course in palaeo-data

There are many forms of data that describe historical changes in environments and ecosystems (e.g., isotopes from ice cores, or diatoms from lake sediments). To reconstruct changes in vegetation, lake sediment cores are often used, and fossilised pollen is extracted from the sediment core. Consider a lake as a big instrument for recording the surrounding environment. Every year, the air is full of pollen from all the wind-pollinated species around the lake. Some of this pollen lands on the lake surface and eventually sinks to the bottom of the lake and leaves a layer like a page in a book (borrowing an analogy from my colleague Rose Gregerson in New Zealand). Every year, another page is added to the book. Each page saying something about the surrounding environment, maybe pine forest takes over from oak woodland, or periods of more fire occur. The book tells the story of the ecosystem.

Figure 1 is a visualisation of the pollen data from a sediment core collected at Story Lake in Northern Indiana. The data are from Schlenker et al., 2024.

Our model

These palaeo-data (Figure 1) show patterns of change in the taxa over the time-scale of thousands of years, but we have no direct observations, or experimental evidence of the drivers of change. What this means is we cannot determine, with any certainty, the causes of ecological change over such long time periods. However, it turns out that, with some very fancy maths by Tony Ives, we can generate estimates of the relationships between: (i) environmental drivers (e.g., changes in temperature or moisture) and taxa in the system; and (ii) taxa-taxa interactions. While these estimates cannot be called causal links, they allow us to test (and lend statistical support to) multiple hypotheses about the drivers of ecological change from palaeo-data.

To determine whether the model is recovering accurate estimates of driver-taxa relationships, and taxa-taxa interactions, we turn to simulation. We simulated data with different parameters and assess how well the model recovers those parameters. The whole process is described in detail in the paper, and here is an example from the paper on the success of the model at recovering driver-taxa relationships (Figure 2). This figure (Figure 2) will be a bit tricky to interpret out of the context of the full paper, the key points are that: each sub-plot represents different simulated conditions; each violin within the sub-plots is a simulated taxon; and the yellow dot is the target value that we want the model to estimate. Across many replicate simulations the median recovered value from the model is very close to the yellow dot, this is good!

In my previous post, I described the phenomenological model used to generate species patterns that mimic the statistical properties of empirical data. How can we use this approach to quantify the relative influence of sources of uncertainty on the statistical inferences we make from palaeoecological data?

I’ll pick up where I left off in the last post. In short, we simulate a whole bunch of species and begin to systematically degrade the data with physical processes (in this case core mixing), sub-sampling strategies (taking 1 cm thick slices of sediment at given intervals), and proxy counting (counting and identifying species in a sub-sample). Because each of these three sources of uncertainty are introduced to the data individually, and combined, at 10 levels each, we end up with 1210 datasets per replicate simulation! There are 30-35 replicate simulations, multiplied by four different scenarios varying the driving conditions. That is terabytes of data! We want to know the relative influence of the three sources of uncertainty (mixing, sub-sampling, and proxy counting) on the two statistical analyses (Fisher Information and principal curves), from the ‘error-free’ benchmark to the most degraded and least intensively sampled.

To compare the Fisher Information and principal curve analyses of the ecological communities across the 1210 datasets we used Feature Analysis for Time series. This further reduces the time-series into a set of ‘features’ that describe the time-series. We can then use Euclidean distance to measure the distance between the features of each dataset and the original benchmark. These distances are then summarised across the replicate simulations (Figure 1).

Ok, the methods are not very easy to describe in simple terms… They are set out in full in the paper, so let’s jump to the exciting results!

The uncertainties are introduced to the data systematically. By introducing the uncertainties individually, we can see how the Euclidean distance increases from our benchmark data (Figure 2). It is interesting the Euclidean distance increases steadily with mixing, while the Euclidean distance does not increase a great deal as the proxy count is reduced. Decreasing sub-sampling intervals are having the strongest influence on the Euclidean distance.

It is useful to see the different influence of the uncertainties when applied individually. However, this result is expected. We know that data become worse if we ruin them! What we want to find out is which sources of uncertainty have the strongest influence, and which ones interact when applied in combination.

In the following figure (Figure 3), two sources of uncertainty are plotted, one on each axis. What we are interested in here is whether the Euclidean distance increases along the diagonal of the plot, indicating an interaction between the uncertainties. By far the strongest influence is from the combined effect of proxy counting and sub-sampling (Figure 3 C). There appears to be little interaction between mixing and either proxy counting and sub-sampling.

In the paper, all three uncertainties are combined and visualised. I won’t go there in this short summary! What does this all mean for palaeoecologists?

This approach allows us to make informed decisions around sampling strategies and lab hours spent quantifying proxies. Where should we focus money and effort in a study depending on the research question of interest? We can use this approach to assess how strong the influence of sources of uncertainty are on different statistical methods. Understanding uncertainties is crucial to understanding how reliable our inferences are from the data we have.

In reality, we have imperfect knowledge of a perfect world. In simulation, we have perfect knowledge of an imperfect representation of the world.

Why is this all important? Palaeoecological data provide insight into past ecological and climatic change. This source of data is invaluable to understanding ecological change and potential future ecosystem states. However, the data are highly uncertain, relying on many layers of processing such as lab processing, radiocarbon dating, and age depth modelling. The data are often inconsistently spaced through time and may have large time-gaps between observations. Such sources of uncertainty can, at best, mask ‘true’ signals of change, or at worst, lead to false inferences. One way we can assess the influence of uncertainties on statistical inference is through simulation.

If we start with our perfect knowledge of a simulated world, we can also simulate observations from that world, in the same way that we would sample from the real world. This gives us a benchmark dataset, the original simulated data, and a reduced dataset, the sampled simulated data.

Here is my simulated world (Figure 1). Each species on the right (Figure 1 d) is simulated by a population growth equation. The growth rate of each species is their combined response to the two environmental drivers (Figure 1 b). Their optima and tolerance to each driver (Figure 1 c) determine whether the population growth rate of a species is positive or negative. If the environment is unfavourable, then a species population growth rate becomes negative and it will go locally extinct. Species have the chance to re-establish if conditions become favourable again.

The number of species in (Figure 1) is only a fraction of those simulated. There are actually around 200 potential species, and about 15-40 in existence at any given point in the simulation (depending on the replicate). I simulated a bunch of different driving environments (e.g., one where the environmental driver undergoes an abrupt shift), in the example above (Figure 1) the ‘slow linear’ driver will eventually drive species that existed at the beginning of the simulation locally extinct (unless they were very generalist species!), and new species would establish and thrive.

We then take those simulated species abundances (Figure 1 d), remember this is our ‘perfect’ benchmark data, and degrade them with mixing (i.e., a physical process that alters the core), sub-sampling (i.e., slicing up the core into 1 cm segments at given intervals for processing), and proxy counting (i.e., the counting of species under a microscope from the 1 cm segments after lab processing). Figure 2 shows how we start with absolute abundances of species, and recreate the observational process, ultimately resulting in data that look a lot more like what we see in reality from proxy-data.

This is what a larger subset of the simulated species looks like before and after the virtual degradation and sub-sampling process (Figure 3).

Finally, we analyse both sets of data. In simulation the three sources of uncertainty (mixing, sub-sampling, and proxy counting) are applied individually, and in combination, at increasing levels of severity. So we end up with 1210 (per simulation replicate!) datasets from the ‘error-free’ benchmark to the most degraded. We analyse them all! But here is an example of the benchmark, and one level of degradation (Figure 4).

This figure shows two different multivariate community analyses (Fisher Information and principal curves), and how the results change between the ‘error-free’ data and the degraded and sub-sampled dataset. At the start, I mentioned that uncertainty can lead to incorrect inference. This example is only of one randomly selected model replicate, but it shows how interpretations can be influenced by uncertainties. The 700 year-long period of reversed directionality in Fisher Information (Figure 4 A) can change our inference from a system increasing in stability, to decreasing in stability (a decrease in Fisher Information is interpreted as a system becoming less predictable less stable). The principal curve changing from a gradual increase (a good representation of change driven by the primary driver system), to a sigmoidal shape (Figure 4 B), can alter our interpretation of the ‘true’ rate of species turnover in the system. This example demonstrates well how a Virtual Ecological approach can be used to assess uncertainty and its influence on statistical inference.

LDA uses the word frequencies from a corpus to assign documents in the corpus to arbitrary ‘topics’. For example, if we took a corpus around the subject ‘nuclear’, the corpus will probably be characterised by words like ‘sustainable’, ‘energy’, ‘green’, and ‘disaster’, ‘fallout’, ‘meltdown’, and ‘weapon’, ‘conflict’… and so on. Words will cluster around ‘topics’, that are described by those words. The LDA model is agnostic and does not know what the ‘topics’ are, but we can identify that there are themes in the words that characterise the topics, such as ‘energy’, ‘conflict/politics’, maybe ‘social movements’, and other topics.

‘Climate Justice’ was first mentioned in a published article in 1997 by Ian H. Rowlands, and between 1997 and 2021 has seen 1683 published works on the Scopus database that had enough text (e.g., an abstract) that I could use for the analysis.

Six overall topics emerged from the literature that we assigned the following descriptions to (Figure 1):

• Education and food security
• Sustainable development and policy
• International relations and carbon emissions
• Health, vulnerability and adaptation
• Policy and activism
• Human rights and indigenous people

We can also see how many documents belong to each topic over time (Figure 2). Each document gets a score per topic, and may not belong only to one topic, but we can still reasonably map out the trends. It is also true that there were not many documents published on climate justice in the early years, so interpret the first decade with that in mind.

Topics 2 and 3 around sustainable development and international relations, respectively, make up a consistent, and substantial proportion of the literature. Smaller topics 1, 4, 5, and 6 have seen a small increase over time.

I won’t comment a great deal in this summary on the interpretation of the results as the subject area is Meg’s expertise. I would not be able to do justice to the extensive work she did as the primary author (and of course co-authors Danielle and Johanna!) and qualitative analyst.