Thank you Aher ! Maybe Your advice is useful for analysing point data in some special circumstances. (1) However sometimes we couldnot to make sure every grid has at least one sample. For example, if we want to analyse the spatial distribution of crime in a city. Then we may create a gridnet to evaluate the crime frequency in each grid cell. As we know, the crime event only occured in some special area. Though many grid cell will not have crime event. In this circumstance , we need not to keep each grid cell to have at least one crime event, beacuse the research purpose is to discovery the spatial pattern of crime distribution. Then the question is how to decide the grid size and grid number when we analysing some data like crime occurence? (1)Another question, point data is noly one type of many data types. There are some other data type, such as polygon data and line data.  When we analyse these non-point data, how to decide the grid size and number?

Thank you again!

If you are looking at crime data where you got a high spatial variability in occurences, you could use kernel density estimation and choose the bandwidth accordingly. I would start with grid size and bandwidth values so small that local peaks become visible.

Concerning non-point data: I would convert them to raster or point format. I would use a grid size that takes into account the spatial variation of underlying variables that might influence crime occurences. Maybe you could use grid sizes that are small enough to resolve important spatial social borders like streets between neighborhoods, etc!?

Great! The key problem is how to quantify the spatial variability! Is there any universal evaluation method or index ? Do you mind introducing some relational paers or literatures?

Thank you again!

Hi,

consider the Nyquist–Shannon–Kotelnikov theorem or simply  the sampling theorem. It will give you the correct answer to your sampling problem.

Regards,

I do not think that there is a method to determine the grid number and grid size. Instead you have to do a range of all sizes between the largest and the smallest; see an example in this paper in which we used resolution 400-700 (not grid but similar idea). From the example, you can see that our result is little to do with resolution.

Jiang B. and Jia T. (2011), Zipf's law for all the natural cities in the United States: a geospatial perspective, International Journal of Geographical Information Science, 25(8), 1269-1281.

Analysis must be based on a range of all sizes, since MAUP is inevitable.

@Xiaofeng Gao

....The key problem is how to quantify the spatial variability! Is there any universal evaluation method or index ? Do you mind introducing some relational paers or literatures?

Spatial variability or spatial heterogeneity can be quantified by ht-index; the higher the ht-index, the more heterogeneous or more variable.

Jiang B. and Yin J. (2014), Ht-index for quantifying the fractal or scaling structure of geographic features, Annals of the Association of American Geographers, 104(3), 530–541.

As Kazimierz said, you need to look to the sampling theorem. Or you may need to look to the scale-space representation, which addresses the level of details in your signal (here in your question 100 x 100 matrix).