For the evaluation of results from remote sensing and high-resolution spatial models it is often necessary to assess the similarity of sets of maps. This paper describes a method to compare raster maps of categorical data. The method applies fuzzy set theory and involves both fuzziness of location and fuzziness of category. The fuzzy comparison yields a map, which specifies for each cell the degree of similarity on a scale of 0 to 1. Besides this spatial assessment of similarity also an overall value for similarity is derived. This statistic corrects the cell-average similarity value for the expected similarity. It can be considered the fuzzy equivalent of the Kappa statistic and is therefore called K_Fuzzy. A hypothetical case demonstrates how the comparison method distinguishes minor changes and fluctuations within patterns from major changes. Finally, a practical case illustrates how the method can be useful in a validation process.

People with low income often experience higher exposures to air pollutants. We compared the exposure to particulate matter (PM1, PM2.5 and PM10), Black Carbon (BC) and ultrafine particles (PNCs; 0.02-1μm) for typical commutes by car, bus and underground from 4 London areas with different levels of income deprivation (G1 to G4, from most to least deprived). The highest BC and PM concentrations were found in G1 while the highest PNC in G3. Lowest concentrations for all pollutants were observed in G2. We found no systematic relationship between income deprivation and pollutant concentrations, suggesting that differences between transport modes are a stronger influence. The underground showed the highest PM concentrations, followed by buses and a much lower concentrations in cars. BC concentrations in the underground were overestimated due to Fe interference. BC concentrations were also higher in buses than cars because of a lower infiltration of outside pollutants into the car cabin. PNCs were highest in buses, closely followed by cars, but lowest in underground due to the absence of combustion sources. Concentration in the road modes (car and bus) were governed by the traffic conditions (such as traffic flow interruptions) at the specific road section. Exposures were reduced in trains with non-openable windows compared to those with openable windows. People from less income-deprived areas have a predominant use of car, receiving the lowest doses (RDD<1μgh-1) during commute but generating the largest emissions per commuter. Conversely, commuters from high income-deprived areas have a major reliance on the bus, receiving higher exposures (RDD between 1.52 and 3.49μgh-1) while generating less emission per person. These findings suggest an aspect of environmental injustice and a need to incorporate the socioeconomic dimension in life-course exposure assessments.

Fuzzy set map comparison offers a novel approach to map comparison.The approach is specifically aimed at categorical raster maps and applies fuzzy set techniques, accounting for fuzziness of location and fuzziness of category, to create a similarity map as well as an overall similarity statistic: the Fuzzy Kappa. To date, the calculation of the Fuzzy Kappa (or K-fuzzy) has not been formally derived, and the documented procedure was only valid for cases without fuzziness of category. Furthermore, it required an infinitely large, edgeless map. This paper presents the full derivation of the Fuzzy Kappa; the method is now valid for comparisons considering fuzziness of both location and category and does not require further assumptions. This theoretical completion opens opportunities for use of the technique that surpass the original intentions. In particular, the categorical similarity matrix can be applied to highlight or disregard differences pertaining to selected categories or groups of categories and to distinguish between differences due to omission and commission.

The Fuzzy Kappa statistic expresses the agreement between two categorical raster maps. The statistic goes beyond cell‐by‐cell comparison and gives partial credit to cells based on the categories found in the neighborhood. When matching categories are found at shorter distances the agreement is higher. Like the well‐established Kappa statistic the Fuzzy Kappa statistic expresses the mean agreement relative to the expected agreement. The model underlying the expected agreement assumes absence of spatial autocorrelation in both compared maps. In reality however, spatial autocorrelation does lower the expected agreement as matching categories become less likely to be found close‐by. Since most maps have some degree of spatial autocorrelation, the calculated expected agreement is generally higher than the true expected agreement. This leads to counterintuitive results when maps that appear to have considerable agreement obtain negative Fuzzy Kappa values. Furthermore, the Fuzzy Kappa may be biased, as it systematically attributes lower agreement to maps with stronger spatial autocorrelation. This paper proposes an improved Fuzzy Kappa statistic that is based on the same local agreement and has the same attractive properties as the original Fuzzy Kappa. The novelty is that the new statistic accounts for spatial autocorrelation, such that the expected Fuzzy Kappa for maps that are not cross‐correlated is equal to zero. The improved statistic is applied on two cases to demonstrate its properties.