Gaussian Random Walk and Gaussian Process applied to Global Mental Health data

The present repository contains results from spatiotemporal models applied to results from a mental health questionnaire, Global Psychotrauma Screen (GPS), used to asses mental health globally during the covid-19 pandemic (Olff et al., 2022). Models are simply a proof of concept, so they were applied to the south-American region only.

Model 1

The first model implements a multivariate Gaussian random walk (GRW) prior with an LKJ prior for covariances and correlations.

ϛ ~ HalfNormal(1)

L, R, SD ~ LKJ(n=E, η=6, sd=ϛ)

Σ = LL^T

w_d,c ~ Normal(0,1)

σ ~ HalfNormal(1)

t_d = √time_d... √time_D

β_d,c = w_d,ct_dσ

B = Σβ_d,c

α_c ~ Normal(0, 1)

μ_c = α_c + B

ϵ_c ~ HalfNormal(0.05) + 1

y_d,c ~ Normal(μ_d,c, ϵ)

Where c…C , C = 7, is the number of countries, d…D, D = 178, is the number of dates when questionnaires were taken. Countries are only countries which provided data within the south-American region. Questionnaires were administered between April and November 2020 (i.e. dates). Note that the GRW is expressed as the product of a standard Gaussian w, a standard deviation σ, and the square roots of times (see Morokof, 1998).

The model was sampled using Markov chain Monte Carlo (MCMC) No U-turn sampling (NUTS) with 2000 tuning steps, 2000 samples, 4 chains.

Results

Prior predictive checks are a bit wild, but models with narrower predictions sample worse.

Posterior estimates show a good approximation and uncertainty.

Predictions from the posterior also indicate sensible and reasonable uncertainty.

Spatially, estimates indicate higher scores for Brazil, followed by Argentina, and the lowest scores are found (curiously) in Chile. Correlations respect to Chile (bottom right panel), however, indicate not relevant association between countries.

Conclusion

Although the model performs well in terms of inference and predictions, showing a reasonable/sensible measurement of uncertainty, the sampling is not ideal, with some parameters showing rather low ESS (above 200 but below 1000).

Model 2

The second model implements a multivariate Gaussian process (GP) prior with an exponential quadratic covariance function. The model is intended for comparison, so it is rather simple.

ℓ ~ HalfNormal(1)

k(x, x') = exp[-(x,x')²/2ℓ²]

f(x) ~ GP(m(x), k(x, x'))

μ = log(f(x))

y ~ Poisson(μ)

Where k(x, x') is an exponential quadratic kernel covariance function, and m(x) is the mean function, x are the dates (sample points), and observed data (y) are the questionnaire aggregated scores.

The model was sampled using Markov chain Monte Carlo (MCMC) No U-turn sampling (NUTS) with 1000 tuning steps, 1000 samples, 4 chains, with ADVI initialization.

Results

Prior predictive checks are relatively reasonable.

Posterior estimates show a good approximation and uncertainty (both posterior and data were re-scaled to z-scores).

Predictions from the posterior seem to underestimate uncertainty in regions with low samples (i.e. dates/countries with fewer questionnaire data).

Spatially, estimates indicate higher scores for Brazil, followed by Argentina and Peru, and the lowest scores are found (curiously) in Chile.

Conclusion

This model sampled better in terms (all ESS over 1000), and has the advantage of not requiring data transformation; namely questionnaire scores (added scores by country, counts) do not need to be transformed. However, predictions cannot properly account for higher uncertainty in regions with lower density.

References

Morokoff (1998). Generating Quasi-Random Paths for Stochastic Processes . https://www.jstor.org/stable/2653031

Olff et al (2022). Mental health responses to COVID-19 around the world. https://doi.org10.1080/20008198.2021.1929754