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I asked a question regarding relationship between a categorical and a continuous variables yesterday, and I have another question with regards to how to deal with a categorical and a continuous variable.

c1 = name of the closest metro station

x1 = distance to that metro station

The outcome is affected by both which station and how far away the station is, so, obviously, I would like to combine the information from both variables. The steps I took are:

1) First perform label encoding for c1 -> c1_le

2) Perform one hot encoding to c1_le -> c1_le_encoded

3) Then perform element-wise (not matrix) multiplications c1_le_encoded -> c1_le_encoded_km (Okay maybe these are not the best variable names....)

So in essence, if

c1 = [station1, station2, station1]

c1_le = [1, 2, 1]

c1_le_encoded = [[1, 0],
                 [0, 1],
                 [1, 0]]

x1 = [2, 2, 3]

c1_le_encoded_km = [[2, 0],
                    [0, 2],
                    [3, 0]]

So each data would represent 2km away from station1, 2 km away from station2, and 3km away from station1.

Q1: I feel like this combined variable should represent the information with much more meaning than feeding them separately to a regression/ML model. Is this okay/legal to do? Or could this introduce some distortion in the data?

Q2: If I wanted to standardize my real value variable, x1 -> x1_std, my intuition tells me that I would also have to multiply c1_le_encoded by x1_std not x1. Is there a difference (Of course, given the first step is legal to begin with)?

Thank You

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1 Answer 1

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If I understand the problem correctly, you can just look at the interaction of the two variables. It's not a problem to multiply a categorical variable by a continuous one, since you have to parameterize the categorical one in any case. Just be careful which parameterization you choose.

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