With the ever-increasing urgency to understand the spread of spatially varying processes such as COVID-19 and air pollution, geostatistical analyses have never been more important than today. When we conduct such analyses, we make the assumption that there exists an underlying smooth process whereby points that are close together exhibit similar behaviour. The observations we have access to are assumed to be a set of noisy realisations from the spatial process. One example of this could be an air quality sensor, whereby observations are generated at the specific location of the sensor, and noise can come from a number of sources, one being the sensor’s calibration.

Figure 1: Side-by-side depiction of the predictive mean and predictive uncertainty of Malaria prevalence in Central Africa

Our goal is often to then make predictions at places where we have no observations using the spatial structure we have learned from the set of observations. Of equal importance to these predictions is the accompanying uncertainty. Through this quantity, we are able to express how confident we are about the prediction we are making. Being able to effectively communicate this quantity is of fundamental importance in socio-economic situations whereby future decisions and reasoning must be made under the knowledge of uncertainty.

The question then becomes, “how can we effectively communicate prediction and the associated uncertainty in a meaningful way?”. For 1-dimensional time series problems, this is a straightforward task that can be achieved through a line and a series of transparent bands around the line that depict the predictive mean and credible intervals, respectively. For spatial problems, we are often forced to use two side-by-side maps. A classic example of this can be seen in Figure 1 whereby the task is to predict Malaria incidence in Central Africa.

Communicating uncertainty in this way poses several problems, namely

1. Decision-makers must try and understand two different value scales
2. A visual projection from one the predictive map to the uncertainty map must be made; a task that will often be inaccurate.
3. Twice as much space is required in order to accommodate two maps

A compelling article by Taylor et. al. (2020) on arꞳiv last week proposed using just a single map to try and rectify the above 3 problems. To represent uncertainty in a predictive map, pixelation is applied to the map depicting the predictive mean, with the pixel size being determined by the underlying uncertainty at each point (Figure 2).

Figure 2: A unified alternative to Figure 1. The colour scale still represents the predictive mean, however, predictive uncertainty is now depicted through the size of pixels used.

Whilst not perfect, the benefits of this approach are two-fold. Firstly, two plots have now become one, meaning that there is no more rapid eye flickering from left to right, as could be the case in Figure 1. Secondly, and possibly more importantly, the enlarged pixels used for areas of higher uncertainty prevents fine-scale inference being made about the spatial surface when the uncertainty is really too high to allow this.

This is just one way to depict spatial uncertainty. An alternative approach pointed out to me by Tim Wollock is to overlay a grid onto the image whereby the opacity of the grid at a certain point is proportional to the point’s respective uncertainty. An example of this can be found in Figure 6 of Wakefield et. al. 2019.

The plots in this post were created by adapting the vignette in the Pixelate package.

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