I am going to model a response variable with multiple outcomes, e.g. clear, cloudy, rainy, snowy, and windy, on a set of explanatory variables. However I cannot simply use multinomial regression (or similar models) because the response variable is actually not one variable, since it can be, for example clear and windy at the same time. There are many other possible combinations of two outcomes. I do not want to make a response variable with a lot of levels (clear and windy, cloudy and windy, rainy and windy, snowy and windy, cloudy and rainy, cloudy and snowy, etc.) since it will not be useful. One option is to estimate binary response models for each outcome, which I am also not convenient with as I feel it will not capture the fact the these outcomes are occurring at the same time. I ran into a topic in categorical data analysis book by Bilder, called "Choose all that apply" data. But the model he introduces does not exactly cover my case, I guess. What kind of model do you recommend that captures this characteristic in my response variable?
I would recode the data to accomodate for the fact that one observation can belong into multiple groups. If for example one day is classified as cloudy and windy I would double the entry (predictive variables) but would label one entry as cloudy and the other as windy. If one day has three different classifications I would triple the entry etc. In this case you will have some points that occupy the same place in variable space (multiplied entries) but will belong to different groups. This is not a problem but a feature of your data but it will result in less clear discrimination.
You can then use discriminant analysis or multinomial logistic regression as usual.