The idea behind spatial dependence was captured by Wado Tobler in his first law of geography. Tobler was a professor of geography at the University of California, Santa Barbara. The first law of geography states that “Everything is related to everything else, but near things are more related than distant things”.
To illustrate this point consider three examples.
1) A remotely sensed image of Tuz Golu in Turkey, 2) Air pollution measurements in Europe, 3) Malaria parasite rate in Africa.
The remotely sensed image was acquired using the Landsat TM sensor. The values in the image are reflectance (percentage). In the remotely sensed image (Figure 1 - top left) we see clearly from the colors of the map that similar values tend to occur close by each other whereas dissimilar values occur at larger separations. More information about this dataset is available in Odongo et al. (2014). Remote sensing can be used to detect many different environmental variables and are used to support studies in Geohealth1.
Figure 1 (top left): Reflectance values at Lake Tuz (Tuz Gölü), Turkey (Landsat TM Band 4, 31 August 2008)4. Figure 1 (top right): Air pollution (PM10 in µg m-3) concentration in Europe for 2 April 2009 2. Figure 1 (bottom): A world malaria risk map.
Figure 1 (top right) and Figure 1 (bottom) show point measurements of air pollution concentration (Figure 1 top right) and malaria parasite rate (Figure 1 bottom). It is clear that high values tend to occur nearby other high values and low values occur nearby other low values. The air pollutant in this example is PM10 (particulate matter less than 10 µm in diameter).
To illustrate the concept of spatial dependence consider a transect of values across the image in Figure 1 (top left). This is illustrated in Figure 2. We see clearly that similar values tend to be clustered together.
Figure 2: A transect of reflectance values across the image shown in Figure 1 (top left)
Readers are most likely familiar with the statistical concept of correlation between two variables. For example, we might talk about the correlation between a person’s height (variable 1) and weight (variable 2) or between altitude (variable 1) and temperature (variable 2). If two variables are perfectly correlated the correlation = 1. In spatial statistics we consider the term spatial auto-correlation and spatial association. A common tool for quantifying spatial association is the variogram. The variogram is discussed in the Article on the variogram.
1Hamm, N. A. S., R. J. Soares Magalhães and A. C. A. Clements (2015). Earth Observation, Spatial Data Quality and Neglected Tropical Disesases. PLoS Neglected Tropical Diseases 9(12): e0004164. DOI: 10.1371/journal.pntd.0004164.
3Hay, S. I., C. A. Guerra, P. W. Gething, A. P. Patil, A. J. Tatem, A. M. Noor, C. W. Kabaria, B. H. Manh, I. R. Elyazar, S. Brooker, D. L. Smith, R. A. Moyeed and R. W. Snow (2009). A world malaria map: Plasmodium falciparum endemicity in 2007. PLoS Medicine 6(3): e1000048. DOI: 10.1371/journal.pmed.1000048.
4Odongo, V. O., N. A. S. Hamm and E. J. Milton (2014). Spatio-Temporal Assessment of Tuz Gölü, Turkey as a Potential Radiometric Vicarious Calibration Site. Remote Sensing 6(3): 2494-2513. DOI: 10.3390/rs6032494.
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