Given the common problem of missing data in real-world applications from various fields , such as remote sensing, ecology and meteorology, the interpolation of missing spatial and spatio-temporal datacan be of tremendous value . Existing methods for spatial interpolation, most notably Gaussian processes and spatial autoregressive models, tend to suffer from  (a) a trade-off between modelling local or global spatial interaction , (b) the assumption there is only one possible path between two points , and (c) the assumption of homogeneity of intermediate locations between points . Addressing these issues  , we propose a value propagation method, inspired by Markov reward processes (MRPs) , as a spatial interpolation method, and introduce two variants thereof: (i) a static discount (SD-MRP) and (ii) a data-driven weight prediction (WP-MRP) variant. Both these interpolation variants operate locally, while  implicitly accounting for global spatial relationships in the entire system through recursion.  We evaluated our proposed methods by comparing the mean absolute errors and running times of interpolated grid cells to those of 7 common baselines . Our analysis involved detailed experiments on two synthetic and two real-world datasets over 44 total experimental conditions. Experimental results show the competitive advantage of MRP interpolation on real-world data, as the average performance of SD-MRP on real-world data under all experimental conditionswas ranked significantly higher than that of all other methods, followed by WP-MRP. On synthetic data, we show that WP-MRP can perform better than SD-MRP given sufficiently informative features. We further found that, even in cases where our methods had no significant advantage over baselines numerically, our methods preserved the spatial structure of the target grid better than the baselines .
The interpolation of spatial data can be of tremendous value in various applications , such as forecasting weather from only a few measurements of meteorological or remote sensing data . Existing methods for spatial interpolation, such as variants of kriging and spatial autoregressive models, tend to suffer from at least one of the following limitations: (a) the assumption of stationarity , (b) the assumption of isotropy , and (c) the trade-off between modelling local or global spatial interaction . Addressing these issues in this work , we propose the use of Markov reward processes (MRPs) as a spatial interpolation method, and we introduce three variants thereof: (i) a basic static discount MRP (SD-MRP) , (ii) an accurate but mostly theoretical optimised MRP (O-MRP), and (iii) a transferable weight prediction MRP (WP-MRP) . All variants of MRP interpolation operate locally, while also implicitly accounting for global spatial relationships in the entire system through recursion. Additionally, O-MRP and WP-MRP no longer assume stationarity and are robust to anisotropy. We evaluated our proposed methods by comparing the mean absolute errors of their interpolated grid cells to those of 7 common baselines , selected from models based on spatial autocorrelation, (spatial) regression, and deep learning. We performed detailed evaluations on two publicly available datasets (local GDP values, and COVID-19 patient trajectory data). The results from these experiments clearly show the competitive advantage of MRP interpolation , which achieved significantly lower errors than the existing methods in 23 out of 40 experimental conditions, or 35 out of 40 when including O-MRP .
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Given the common problem of missing data in real-world applications from various fields
The interpolation of spatial data can be of tremendous value in various applications
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remote sensing, ecology and meteorology, the interpolation of missing spatial and spatio-temporal datacan be of tremendous value
forecasting weather from only a few measurements of meteorological or remote sensing data
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most notably Gaussian processes
such as variants of kriging
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at least one of the following limitations:
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a trade-off between modelling local or global spatial interaction
the assumption of stationarity
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there is only one possible path between two points
of isotropy
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assumption of homogeneity of intermediate locations between points
trade-off between modelling local or global spatial interaction
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in this work
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a value propagation method, inspired by
the use of
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,
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introduce two
we introduce three
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static discount
basic static discount MRP
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and
,
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a data-driven weight prediction
an accurate but mostly theoretical optimised MRP (O-MRP), and (iii) a transferable weight prediction MRP
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variant. Both these interpolation variants
. All variants of MRP interpolation
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also
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Additionally, O-MRP and WP-MRP no longer assume stationarity and are robust to anisotropy.
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and running times of
of their
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. Our analysis involved detailed experiments on two synthetic and two real-world datasets over 44 total experimental conditions. Experimental results
, selected from models based on spatial autocorrelation, (spatial) regression, and deep learning. We performed detailed evaluations on two publicly available datasets (local GDP values, and COVID-19 patient trajectory data). The results from these experiments clearly
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on real-world data, as the average performance of SD-MRP on real-world data under all experimental conditionswas ranked significantly higher than that of all other methods, followed by WP-MRP. On synthetic data, we show that WP-MRP can perform better than SD-MRP given sufficiently informative features. We further found that, even in cases where our methods had no significant advantage over baselines numerically, our methods preserved the spatial structure of the target grid better than the baselines
, which achieved significantly lower errors than the existing methods in 23 out of 40 experimental conditions, or 35 out of 40 when including O-MRP