Typically Moran Index value should be within a rang of – 1 to +1. But using ArcGIS spatial statistics tool, the outcomes of local moran's Index value even differ in a range of -1 to +4. What should be a reason for the extrem value?
Did You standardize data?
If data were/was standardized based on an assumption of a normal distribution, the results may be affected by the definition of weight function.
Data with inverse distance conceptualizations using the spatial pattern and weights for distance, less than 1 become unsteady.
Existence of extreme values, it is not recommended to perform Moran I analysis for non-neighborhood objects (objects described in the weight matrix only with value 0). Such as objects can be excluded in analysis, or You can select another way of defining the neighborhood (another matrix of weights). In order to eliminate the effect of extreme values You might transform data into the spatial clusters analysis.
A regression line base on The Moran I allows to the interpretate the Moran coefficient: I determined by spatial lagg of variable X.
Below some examples of negative correlation - the target value is not similar to its neighborhood:
1. Spatial outliers implies high–low (a high value in a low value neighborhood).
2. Negative correlation also indicate low–high (a low value in a high value neighborhood).
The locations (spatial outliers) are not spatial clusters.
Dear Wiktor, Thanks a lot for your detail explanation. It will help me a lot for way forward. Will you kind to drop any reference (if any) that explain this issues as well? Looking forward.
really it is very interesting research, I would like to thank both of you
My regards
Please see the attaches and links below:
http://personal.colby.edu/personal/m/mgimond/Spatial/RobustI_writeup.html
https://www.youtube.com/watch?v=vvzZjCKrfb0
Some thoughts about " monotone transformation of Moran’s
I" by Maruyama (2015) https://arxiv.org/pdf/1501.06260.pdf
Does anyone has experience on transformation of original Moran I?
Or what is your opinion about this alternative proposal?
Looking forward. @ Wiktor Halecki @ Hayder Dibs
Did you solve your problems? I alse met it when i use ENVI softeware to calculate Local spatial statistics. I can't figure it out ... could you help me ?
Yinxia Cao I did not find a straight forward answer to the topic but you may find it useful to have a look on this article: https://arxiv.org/pdf/1501.06260.pdf
Thank you~ you can also refer to this website https://www.cnblogs.com/cynchanpin/p/7290963.html. The blogger said the reasons may be from three aspects: 1) the weight matrix should be row-standarization 2)your data should be standarized.(that is, with the mean subtracted and divided by the standard deviation) this opinion is from author Luc Anselin who proposed the local moran's i 3) The distribution of your data is clearly non-normal, and heavily skewed to the right. (this opinion is also from Anselin
Some good answers have been given already in this thread. I would like to add that the feasible ranges of both global and local Moran's I solely depdend on the eigenvalues of the weights matrix. This stems from the design of Moran's I as a ratio of quadratic forms, and the measure behaves similar to other ratios of quadratic forms such as Pearson r. The latter can be thought of as a spatially-weighted measure too, but with all weights being set to 1/n (that is, without space taking effect). In the corresponding matrix, each row would sum to unity, which implies a maximum eigenvalue of 1. Hence, Pearson r is element of the familiar range [-1, 1]. You can observe the same behaviour with Moran's I when you choose row-standardised weights. For other weight configurations (which may be more appropriate in some application scenarios), the range will slightly differ from the interval [-1, 1]. You may want to check out the following papers on this:
de Jong, P., Sprenger, C., & Van Veen, F. (1984). On extreme values of Moran's I and Geary's c. Geographical Analysis, 16(1), 17-24.
Tiefelsdorf, M., & Boots, B. (1997). A note on the extremities of local Moran's Iis and their impact on global Moran's I. Geographical Analysis, 29(3), 248-257.
Tiefelsdorf, M., Griffith, D. A., & Boots, B. (1999). A variance-stabilizing coding scheme for spatial link matrices. Environment and Planning A, 31(1), 165-180.
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