Discrete natural neighbour interpolation with uncertainty using cross-validation error-distance fields

Cross-validation error

Global error estimation is a traditional approach to measure the uncertainty of geographic models (Zhang & Goodchild, 2002). Given a set of n paired observed o and modelled m values, the absolute error e_i for each pair is e_i = ∣m_i − o_i∣, and a global estimate of error using a method such as the mean absolute error (MAE) is calculated as (6)${\displaystyle MAE=\frac{1}{n}\sum _{i=1}^n{e}_i}$ that is simply the mean of all the absolute errors (Willmott & Matsuura, 2005).

However, there is little point in doing this for the data cells of natural neighbour interpolation as given property (i) that it is an exact interpolator the estimated value $\widehat{z_i}$ for the data cells will always be the same as the actual value z_i so the absolute errors will always be zero. Therefore, MAE needs to be applied in conjunction with a cross-validation approach that iteratively withholds each data cell p_i from the set of data cells P to produce the set {P − p_i}, and then uses interpolation to estimate the value $\widehat{z_i}$ at the withheld data cell p_i on the basis of a discrete Voronoi diagram V({P − p_i}) that is developed without the withheld data cell. The absolute error e_i for each data cell p_i is then calculated as $e_i=\mid {\hat{z}}_i-z_i\mid$ and the cross-validation MAE can be calculated using Eq. (6).

Even with cross-validation the MAE like all global error estimates, such as the commonly used root-mean-square error (RMSE), are not ideal measures of uncertainty for a spatial interpolation (Zhang & Goodchild, 2002). As non-spatial methods that average errors across space they cannot indicate if errors are consistent across space or if higher errors in one region are balanced out by lower errors in another region. This is a critical limitation of global error estimation methods, as for application purposes it could be very useful to know where the interpolation uncertainty is higher or lower.

Cross-validation error field

One way to communicate the spatial uncertainty of geographical information is to map estimates of error (Zhang & Goodchild, 2002). This has been attempted before for natural neighbour interpolation (Bater & Coops, 2009; Ghosh, Gelfrand & Mlhave, 2012), but as already noted these statistical modelling approaches are contrary to natural neighbour interpolation’s useful properties (v), (vi), and (vii).

Another way to map estimates of error that is consistent with the properties of natural neighbour interpolation is the cross-validation error field (Willmott & Matsuura, 2006). This process begins in a similar manner to the cross-validation MAE, but once e has been calculated for each data cell, rather than average the errors using Eq. (6) the errors are assumed to be spatially autocorrelated and interpolation is used to interpolate e to estimate an absolute error field $\hat{e}$. This use of localised absolute errors is highly advantageous as it is consistent with property (iii) of natural neighbour interpolation and allows for error estimates to reflect local changes in the spatial-autocorrelation of the phenomenon being interpolated, with lower errors in more autocorrelated areas and higher errors less autocorrelated areas.

However, while the cross-validation error field does indicate where interpolation errors are likely to be higher, it cannot be used directly as a measure of uncertainty for natural neighbour interpolation as ultimately the interpolation is calculated using all n data cells and given property (i) of natural neighbour interpolation is that it is an exact interpolator we know we will have zero error and hence zero uncertainty at the data cells.

On the basis of Tobler’s first law of geography that “everything is related to everything else, but near things are more related than distant things” (Tobler, 1970), Zhang & Goodchild (2002) recognise that distance is an important component of uncertainty as locations nearer to data should have less uncertainty. This relationship of increasing error with increasing distance to data has even been demonstrated for natural neighbour interpolation (Keller et al., 2015). Therefore, I propose to extend the cross-validation error field idea by incorporating distance to produce a cross-validation error-distance field that will better represent the uncertainty associated with natural neighbour interpolation.

Natural neighbour distances

A positive relationship between natural neighbour interpolation absolute errors and the minimum distance to a data cell has been shown (Keller et al., 2015), so this relationship could be used to predict absolute error as a function of distance from the nearest data point. However, minimum distance to a data cell is a simplistic metric that does not account for the number and spatial configuration of the data cells (Keller et al., 2015). In addition, using the minimum distance from data cells D(P) produces a field that has discontinuities along the edges of the discrete Voronoi polygons (Fig. 1B) that are contrary to the property (ii) of the natural neighbour interpolation method that creates surfaces free of any discontinuities. Therefore, the natural neighbour distance δ is presented as a more appropriate measure of distance that incorporates information about the number, spatial distances, and relative positions of the data cells forming the interpolation.

The method to calculate δ follows a very similar approach to that of calculating the interpolation, and therefore recycles various data structures that are used for the interpolation. For each interpolation cell c_i the Euclidean distances to all data cells are calculated d_j = d(c_i → p_j), and then using the Voronoi diagram V(P) these distances are interpolated via nearest neighbour interpolation to produce D(P) that is the distance to the data cells mapped into the discrete Voronoi polygons (Fig. 2A).

The set V(c_i) can be used again to find the set of relevant data cell distances (7)${\displaystyle D\left(c_i\right)=\left\{c\in D\left(P\right)\mid c\in V\left(c_i\right)\right\}}$

that will form the basis of the interpolation to that cell (Fig. 2B). The natural neighbour distance is then calculated as (8)${\displaystyle \delta \left(c_i\right)=\frac{\sum _{}D\left(c_i\right)}{♯D\left(c_i\right)}}$

that is simply the mean value of the distances for the cells in D(c_i). With δ calculated for all raster cells it becomes evident that unlike minimum distance that contains spatial discontinuities (Fig. 1B) the natural neighbour distance creates a smooth surface free of any discontinuities (Fig. 2C). Also, the minimum distance is an optimistic measure of distance as it only accounts for the closest data cell, whereas by comparison the distances for δ are larger as they recognise that the other data cells involved in the interpolation are further away.

Cross-validation error-distance field

To incorporate δ into the estimate of error to produce a cross-validation error-distance field, the first step is still a cross-validation process in which each data cell is iteratively withheld and an estimate of the value of the withheld data cell is made with the remaining n − 1 data cells. However, the absolute error $e=\mid z_i-{\hat{z}}_i\mid$ is now divided by the natural neighbour distance δ to calculate a rate of error r for each data cell (9)${\displaystyle r_i=\frac{\mid z_i-{\hat{z}}_i\mid }{{\delta }_i}}$ with these rates of error stored so that each data cell becomes p_i = {x_i, y_i, z_i, r_i}. Then when conducting the natural neighbour interpolation, while estimating the value $\hat{z}$ an estimate of the rate of error $\hat{r}$ can be simultaneously produced (Fig. 3A) and used to produce an error estimate (10)${\displaystyle \widehat{e_i}=\widehat{r_i}\times {\delta }_i}$ that when estimated for all cells produces a cross-validation error-distance field (Fig. 3B). The cross-validation error-distance field clearly captures information from the rate of error field (Fig. 3A) and the natural neighbour distance field (Fig. 2C) with lower error estimates in areas that have either low rates of error or natural neighbour distances, and higher error estimates in areas that have higher rates of error and/or natural neighbour distances. Therefore, the cross-validation error-distance field captures uncertainty information relating to local variation in both the autocorrelation of the underlying phenomenon field being interpolated and the spatial distribution of the data cells providing data for the interpolation.