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I'm using ArcGIS 10.3, and I have 20 shapefiles, each representing 1 year worth of data over the same study area, that contain postal code point data, with each point having some NDVI values (growing season mean, growing season peak etc.).

What I would like to do is a time series analysis to see how vegetation changed over the course of 20 years, and if possible to identify areas where the change was significantly different than what would normally be expected. The reason for this is that in one year there was a large storm which wiped out a good chunk of the vegetation in my study area. So I want to identify those areas that were hardest hit by the storm.

I've looked online and through the GIS stackexchange and while I've found some examples of what I want to do (I think) like here:

http://r-video-tutorial.blogspot.com/2016/07/time-series-analysis-in-arcgis.html

or maybe like this:

https://www.umesc.usgs.gov/management/dss/curve_fit.html

They are always using raster data, which I don't have access to.

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You can turn point data into raster data via Point to Raster (or Feature to Raster in the same toolbox), and then apply the raster-based methods you mention.

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