Controlling for spatial locations

Introduction

If we have data on the longitudes and latitudes of taxa, sometimes it is useful to control for this spatial location to ensure that our model is capturing deep ancestral relationships rather than shared environmental effects or more recent diffusion among neighbours.

The lon_lat argument in the coev_fit() function allows us to easily control for spatial proximity. This argument takes a data frame of longitude and latitude coordinates (in decimal degrees) for taxa on the phylogeny. If longitude and latitude values are inputted by the user, the function calculates the Euclidean distances between these points on a unit sphere and includes a Gaussian process over these distances for each variable in the model.

A working example

When studying the coevolution of political and religious authority in Austronesian societies, we would like to ensure that our results are due to coevolution over deep cultural time rather than more recent borrowing among societies with close geographic proximity.

Here are the longitude and latitude values for the first five Austronesian societies:

authority$coordinates[1:5, ]

# A tibble: 5 × 3
  id             latitude longitude
  <chr>             <dbl>     <dbl>
1 Aiwoo             -10.3      166.
2 Alune              -3.1      128.
3 AnejomAneityum    -20.2      170.
4 Anuta             -11.6      170.
5 Atoni              -9.7      124.

We can include these spatial coordinates in the model. The distance matrix is scaled to vary between 0 and 1 under the hood to improve model sampling.

fit <-
  coev_fit(
    data = authority$data,
    variables = list(
      political_authority = "ordered_logistic",
      religious_authority = "ordered_logistic"
    ),
    id = "language",
    tree = authority$phylogeny,
    # declare longitude and latitude values
    lon_lat = authority$coordinates,
    # set manual prior
    prior = list(A_offdiag = "normal(0, 2)"),
    # additional arguments for cmdstanr
    parallel_chains = 4,
    iter_sampling = 2000,
    iter_warmup = 2000,
    refresh = 0,
    seed = 1
  )

Running MCMC with 4 parallel chains...

Chain 4 finished in 663.9 seconds.
Chain 3 finished in 666.8 seconds.
Chain 1 finished in 937.3 seconds.
Chain 2 finished in 948.6 seconds.

All 4 chains finished successfully.
Mean chain execution time: 804.1 seconds.
Total execution time: 948.8 seconds.

summary(fit)

Variables: political_authority = ordered_logistic 
           religious_authority = ordered_logistic 
     Data: authority$data (Number of observations: 97)
Phylogeny: authority$phylogeny (Number of trees: 1)
    Draws: 4 chains, each with iter = 2000; warmup = 2000; thin = 1
           total post-warmup draws = 8000

Autoregressive selection effects:
                    Estimate Est.Error  2.5% 97.5% Rhat Bulk_ESS Tail_ESS
political_authority    -0.72      0.55 -2.05 -0.02 1.00     5213     3202
religious_authority    -0.74      0.57 -2.11 -0.03 1.00     5142     3170

Cross selection effects:
                                          Estimate Est.Error  2.5% 97.5% Rhat Bulk_ESS Tail_ESS
political_authority ⟶ religious_authority     1.85      1.25 -0.75  4.25 1.00     2799     2793
religious_authority ⟶ political_authority     1.58      1.32 -1.17  4.14 1.00     2792     3400

Drift parameters:
                                             Estimate Est.Error  2.5% 97.5% Rhat Bulk_ESS Tail_ESS
sd(political_authority)                          1.25      0.74  0.09  2.79 1.00     1677     3694
sd(religious_authority)                          0.96      0.66  0.04  2.43 1.00     2048     3443
cor(political_authority,religious_authority)     0.12      0.33 -0.54  0.71 1.00     6632     5947

Continuous time intercept parameters:
                    Estimate Est.Error  2.5% 97.5% Rhat Bulk_ESS Tail_ESS
political_authority     0.38      0.96 -1.50  2.28 1.00     9242     5774
religious_authority     0.38      0.97 -1.50  2.26 1.00     9177     5522

Ordinal cutpoint parameters:
                       Estimate Est.Error  2.5% 97.5% Rhat Bulk_ESS Tail_ESS
political_authority[1]    -1.56      1.03 -3.58  0.50 1.00     6188     5898
political_authority[2]    -0.83      1.02 -2.81  1.20 1.00     5682     6007
political_authority[3]     1.31      1.06 -0.72  3.44 1.00     4159     5082
religious_authority[1]    -1.90      1.05 -3.96  0.18 1.00     6116     4866
religious_authority[2]    -1.19      1.04 -3.23  0.88 1.00     6325     5471
religious_authority[3]     1.51      1.08 -0.60  3.70 1.00     5441     5222

Gaussian Process parameters for distances:
                          Estimate Est.Error 2.5% 97.5% Rhat Bulk_ESS Tail_ESS
rho(political_authority)      0.10      0.07 0.04  0.22 1.00      860     1169
rho(religious_authority)      0.07      0.05 0.03  0.18 1.00      782      456
sdgp(political_authority)     1.86      0.76 0.59  3.61 1.00     2300     1541
sdgp(religious_authority)     2.37      0.79 1.04  4.19 1.00     2136     2029

Warning: There were 13 divergent transitions after warmup.
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup

The summary output shows that the model has estimated the parameters for two Gaussian process functions over geographic locations, one for each variable.

In addition, it is possible to change the underlying covariance kernel for the Gaussian processes using the dist_cov argument and estimate approximate Hilbert-space Gaussian processes using the dist_k argument. The latter may be particularly useful for large datasets where exact Gaussian processes are intractable. See help(coev_fit) for more details.

- Introduction
- A working example