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Is there a way to take a set of data that consists of discrete values and predict a continuous value? Take for instance data that looks like:

sample matrix of jewel data
 color |  size | shape 
['red' ,'large','square']
['blue','small','circle'] 
['blue','small','square']

sample array of price labels
[9.99, 7.00, 6.37]

Can I do Decision Tree Regression on this to predict the price of a jewel with a given set of features? What if some of the data is continuous? Also is there any way I can/should pre-process the categorical data other than onehot encoding?

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Yes, most software implementations of trees will allow you to predict a continuous target variable with all binary predictors. This is because the predictors are only used as splits, and the prediction comes from the average value at a given terminal node. The predictions will not be truly continuous across all terminal nodes in the same way that linear regression is continuous, but in practice, this is generally not a problem. If your tree is under-fitting (not continuous enough) you can always add more terminal nodes. Also, one-hot encoding should be sufficient.

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