Learning association between multiple effects and causal outcomes

Question

We have some data represented as follows: on each row there are 10 observable effects, and 3 non-observable causes out of 8 that were guessed by scholars. Effects are real numbers, and causes are discrete categories.

We want to train a system that would accept observed effects, and guess three most likely causes.

So how to represent "3 out of 8 causes", and what would be the best approach and how to use such a learning algorithm?

Edit: Observables are characteristics of cracks in a structure. Causes are things like load, wind, temperature. Example sample is like

1.2 2.3 45 6.7 5.5 12 3.4 1.1 5.6 2.3 load weather temperature

It would be better if we could determine the rank of causes. E.g. most important is load, then weather, etc. Because the sample is gathered this way.

Answer 1

There are only 56 ways to choose 3 items out of 8 options. If you have enough data you could fit a multinomial logit regression with a response variable with 56 levels. "Enough" might be quite a bit however... (although this probably applies for any solution).

This model could then be used to predict the combination of 3 variables for any given values of the 10 effects.