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I want to use support vector machines to classify landcover of a raster image and compare the results for different spatial resolutions. I will simulate the coarser spatial grain by aggregating adjacent pixels and taking the mean. When I go up to 100 m and 250 m ground pixel size, I have the problem that useful training areas become very sparse (at 250 m there might be not a single pixel belonging to only one landcover class).

Would it be a viable option to train the SVMs with smaller ground pixel areas and use the same SVMs to classify the coarser raster images?

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It's possible! if I understand your question correctly.

I've trained a SVM model on points collected in the field and then fit the model to the entire state of Alaska in attempt to predict a label at any given pixel using the scikit learn method model.fit(X, y) in python.

Here X is your array(image) and y is the target label you wish to predict.

As long as X is the same (spectral bands need to be the same) but different spatial resolutions and you predict the same variable (y), it shouldn't complain.

https://scikit-learn.org/stable/modules/generated/sklearn.svm.SVC.html#sklearn.svm.SVC

I hope this helps a bit!

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