Lee's L

Lee's L is a bivariate spatial correlation coefficient which measures the association between two sets of observations made at the same spatial sites. Standard measures of association such as the Pearson correlation coefficient do not account for the spatial dimension of data, in particular they are vulnerable to inflation due to spatial autocorrelation. Lee's L is available in numerous spatial analysis software libraries including spdep ^[1] and PySAL ^[2] (where it is called Spatial_Pearson) and has been applied in diverse applications such as studying air pollution, ^[3] viticulture ^[4] and housing rent. ^[5]

For spatial data ${\displaystyle x_{i}}$ and ${\displaystyle y_{i}}$ measured at ${\displaystyle N}$ locations connected with the spatial weight matrix ${\displaystyle w_{ij}}$ first define the spatially lagged vector

${\displaystyle {\tilde {x}}_{i}=\sum _{j}w_{ij}x_{j}}$

with a similar definition for ${\displaystyle {\tilde {y}}_{i}}$. Then Lee's L ^[6] is defined as

${\displaystyle L_{x,y}={\frac {N}{\sum _{i}\left(\sum _{j}w_{ij}\right)^{2}}}{\frac {\sum _{ij}({\tilde {x}}_{i}-{\bar {x}})({\tilde {y}}_{i}-{\bar {y}})}{{\sqrt {\sum _{i}({\tilde {x}}_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i}({\tilde {y}}_{i}-{\bar {y}})^{2}}}}}}$

where ${\displaystyle {\bar {x}},{\bar {y}}}$ are the mean values of ${\displaystyle x_{i},y_{i}}$. When the spatial weight matrix is row normalized, such that ${\displaystyle \sum _{j}w_{ij}=1}$, the first factor is 1.

Lee also defines the spatial smoothing scalar

${\displaystyle SSS_{x}={\frac {\sum _{i}({\tilde {x}}_{i}-{\bar {x}})^{2}}{\sum _{i}(x_{i}-{\bar {x}})^{2}}}}$

to measure the spatial autocorrelation of a variable.

It is shown by Lee ^[6] that the above definition is equivalent to

${\displaystyle L_{x,y}={\sqrt {SSS_{x}}}{\sqrt {SSS_{y}}}r({\tilde {x}},{\tilde {y}})}$

Where ${\displaystyle r}$ is the Pearson correlation coefficient

${\displaystyle r({\tilde {x}},{\tilde {y}})={\frac {\sum _{i=1}^{n}({\tilde {x}}_{i}-{\bar {\tilde {x}}})({\tilde {y}}_{i}-{\bar {\tilde {y}}})}{{\sqrt {\sum _{i=1}^{n}({\tilde {x}}_{i}-{\bar {\tilde {x}}})^{2}}}{\sqrt {\sum _{i=1}^{n}({\tilde {y}}_{i}-{\bar {\tilde {y}}})^{2}}}}}}$

This means Lee's L is equivalent to the Pearson correlation of the spatially lagged data, multiplied by a measure of each data set's spatial autocorrelation.

References

1. "Lee's L test for spatial autocorrelation — lee.test".
2. "API reference — esda v0.1.dev1+ga296c39 Manual".
3. Yang D, Ye C, Wang X, Lu D, Xu J, Yang H (2018). "Global distribution and evolvement of urbanization and PM2. 5 (1998–2015)". Atmospheric Environment. 182: 171–178. doi:10.1016/j.atmosenv.2018.03.053.
4. Lu, Yonglong; Yang, Yifu; Sun, Bin; Yuan, Jingjing; Yu, Minzhao; Stenseth, Nils Chr.; Bullock, James M.; Obersteiner, Michael (2020). "Spatial variation in biodiversity loss across China under multiple environmental stressors". Science Advances. 6 (47). doi:10.1126/sciadv.abd0952. PMC   7679164 . PMID   33219032.
5. Hu, Lirong; He, Shenjing; Han, Zixuan; Xiao, He; Su, Shiliang; Weng, Min; Cai, Zhongliang (2019). "Monitoring housing rental prices based on social media:An integrated approach of machine-learning algorithms and hedonic modeling to inform equitable housing policies". Land Use Policy. 82: 657–673. Bibcode:2019LUPol..82..657H. doi:10.1016/j.landusepol.2018.12.030.
6. Lee, Sang-Il (2001). "Developing a bivariate spatial association measure: an integration of Pearson's r and Moran's I.". Journal of Geographical Systems. 3 (4): 369–385. Bibcode:2001JGS.....3..369L. doi:10.1007/s101090100064.