There are two main laws in geography. The first one was proposed by Waldo Tobler: “Everything is associated with others, and close things are more related compared with distant things” [24]. The first law demonstrates the relationship between distance and association. Michael Goodchild came up with the second law, the 1aw of spatial heterogeneity: “The separation of space accounts for the difference between regions, namely, heterogeneity, including spatial local heterogeneity and spatial stratified heterogeneity” [25]. The second law illustrates that the specific values of units were different from the surrounding regions, which could be regarded as hot or cold spots. Based on the laws of geography, spatial autocorrelation analysis was formulated to reveal the spatial dependence and hierarchical spatial enumeration. Appendix 2 provides a demonstration of different types of spatial cluster situations. Each circle represents the variables in specific units, and the circles are associated with each other. The red circles represent indicators with higher values, while the blue circles denote the lower values. The left graph, demonstrating the positive spatial autocorrelation, shows the pattern of clusters with similar values, namely, the red circles tend to be near to each other, and the blue circles surround each other. There is no spatial autocorrelation in the middle graph due to the random distribution of high and low values. Negative spatial autocorrelation is found in the right graph, which means that the high values are surrounded by the low values.

The Moran’s I is one of the most commonly used indicators considering spatial autocorrelation analysis, which consists of global and local Moran’s I. Global Moran’s I is a reflection of the first law, measuring the spatial dependence of the whole research region, while as a transformation of the second law, the local Moran’s I reflects the regional differences. In our study, the global and local Moran’s I findings reveal the whole-level spatial distribution characteristics of the study region and specific cluster regions in the research area, respectively [26].

In this study, the value of global Moran’s I, ranging from − 1 to 1, reflects the overall spatial distribution of HFMD in Shaanxi province. When the index is near 1, a positive spatial autocorrelation is detected [27, 28]. The counties with high incidence rates of HFMD tend to cluster. A zero means that there is no spatial autocorrelation of HFMD, illustrating high and low values scattered randomly in Shaanxi. When the values are distributed around − 1, a negative spatial autocorrelation is observed, indicating that counties with high and low values border each other. The equation of global Moran’s I is as follows:

Where Xi is the incidence rate of HMFD in county i and j. The X¯ is the mean value of the incidence rate of HFMD in Shaanxi. The difference between the mean and absolute values of incidence rate is crucial in determining the positive or negative effects. n is the number of all the counties in Shaanxi. Wij is an important tool in spatial modelling, as it quantifies the spatial dependence between observations, which is normally expressed as an n × n non-negative matrix W:

Where n is the number of spatial units; Wij represents the spatial dependency relationship between region i and region j. The larger the weight value, the stronger the spatial dependency between regions. The spatial weight matrix was constructed based on a contiguity relationship. Therefore, the value on the main diagonal of the matrix is zero, which means that each area is not adjacent to itself, namely, Wij = 0. At the same time, if areas i and j are adjacent, then Wij = Wji. The spatial weight matrix is symmetrical.

Regarding the local Moran’s I, a positive value of the index represents the similarity of region, which means that the regions with high or low incidence rates of HFMD cluster within the same category, while a negative value indicates the opposite, that is, the counties with high incidence rates tended to be near regions with low incidence rates. Based on the value and the significance level, the clusters could be classified as four types, namely, High-high (HH, the regions with high incidence rates are surrounded with other high incidence rate regions), High-low (HL), Low-high (LH), and Low-low (LL). The equation of local Moran’s I is as follows [26]:

Where m0 is a constant across all county-units; the explanation of other parameters is the same as with the global Moral’s I. To further demonstrate the statistically significant level of the incidence rate of HFMD, a map displaying the counties whose local Moran’s I has significant results is presented. The map is also known as a LISA map.

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