Accepting Spatial Memory: Some Alternative Ecological Methods

In this post I present a guideline that summarizes how a memory-based model with an increasing pile of empirical verification covering many species – the Multi-scaled Random Walk model (MRW) – may be applied in ecological research. The methods are in part based on published papers and in part based on some of the novel methods which you find scattered throughout this blog.

In the following, let us assume for a given data set that we have verified MRW compliance (using the standard memory-less models or alternative memory-implementing models as null hypotheses) by performing the various tests that have already been proposed in my papers, blog posts and book. Typically, a standard procedure should be to verify (a) site fidelity; i.e., presence of a home range, (b) scale-free space use by studying the step length distribution from high frequency sampling, and (c) the fractal dimension D≈1 of the spatial scatter of relocations in the resolution range between the dilution effect (very small pixels) and the space fill effect (very large pixels).

The obvious next step is to explore specific ecological hypotheses using the MRW as the model for space use. Here follows a quick tutorial:

1. Study the Home range ghost model, I(N) = cNz, to estimate c, the individual’s characteristic scale of space use (CSSU).
   
   Search my blog for methods how to optimize grid resolution, and in particular also consider the recent breakthrough where I show how to estimate CSSU from auto-correlated data. Variations in CSSU quantifies difference in intensity of space use, which logically illuminates aspects of habitat selection.
    
2. After proper estimation of CSSU study the power exponent, z, of the Home range ghost model. If your data lands on z≈0.5 – the default condition – you have verified that the individual has not only utilized its environment in a scale-free manner but also has put “equal weight” into relating to its habitat across the spatial scale range within its home range.
   
   On the other hand, finding 0.2 < z < 0.3 indicates that a model for an alternative movement class, the Markov-compliant MemRW, may be more suitable for your data. 0 < z < 0.3 indicates that the individual has concentrated its space utilization primarily towards finer resolutions, like you would expect from a Markov-compliant kind of cognitive processing. In this case you would also expect to find a larger power exponent for autocorrelated data as shown in the simulation result to the right (z = 0.7 for MemRW, while MRW shows z = 0.47). More detailed procedures should be applied to select model framework, since MemRW and MRW describes qualitatively disparate classes (search Archive for The scaling cube). Observe that the traditional KDE method is not able to discriminate between these classes; hence, typically showing weak N-dependence on area demarcation, due to its implicit assumptions that successive revisits to a local patch are independent events and that the process should be memory-less. 
3. Is the individual’s space use stationary, or is the home range drifting over time?
   
   Spatial autocorrelation in your series of fixes typically has two causes; high-frequency sampling of fixes from space use relative to a slower return frequency (ρ >> 1; see my previous post) or high- or medium-frequency fix sampling under the condition of a drifting home range. Split the data into several subsets of magnitude N_s where the number of fixes (N) in each set is constant. Then study the overlap pattern of incidence I(N) at spatial resolution of CSSU. Low degree of overlap between successive subsets implies a non-stationary kind of home range. By comparing non-adjacent subsets in time one may even quantify the degree of non-stationarity (the speed by which the space use is drifting). These results can then be interpreted ecologically. 
4. What about the fractal dimension of the total set of fixes, for example by applying the box counting method?
   
   By default one expects D≈1 when z=0.5. Deviations from D=1 over specific spatial resolutions can be interpreted ecologically. For example, 1.5 < D < 2 at the coarsest resolutions may indicate that space use is constrained by some kind of borders. However, it could also appear from missing outlier fixes in the set (Gautestad and Mysterud 2012) or a simple statistical artifact (the space fill effect). On the other end, 1 << D can be hypothesized to emerge where the animal has concentrated its space use among a set of fine-scale patches rather than scattering is optimization more smoothly (in a statistical sense) over a wider range of scales. In Gautestad (2011) I simulated central place foraging, where i found 0.7 < D < 1. More sophisticated but logically simple methods can contribute to various system properties and statistical artifacts that contribute to deviation from D≈1, for example by varying the sample size of fixes as illustrated in the Figure to the right (copied from the link above).

The MRW theory also offers several other methods to study ecological and biological aspects of space use. For example, the data may reveal whether the temporal memory horizon has been constrained or unlimited (infinite memory, or remembering previous visits only over a limited, trailing time window). Temporally constrained memory will be shown by example in my next post. For more theoretical or technical details of the methods above please search this blog for the actual term, or find references in the subject index of my book.

REFERENCES

Gautestad, A. O. 2011. Memory matters: Influence from a cognitive map on animal space use. Journal of Theoretical Biology 287:26-36.

Gautestad, A. O. and I. Mysterud. 2012. The Dilution Effect and the Space Fill Effect: Seeking to Offset Statistical Artifacts When Analyzing Animal Space Use from Telemetry Fixes. Ecological Complexity 9:33-42.