Relationships between variables


Our current research work involves analysis of relationships between variables based on historical data. In general, the dataset is in the form of time-series consisting of a few variables that describe given system. In other words, our data can be stored in a matrix, say X, where xij corresponds to the value of the ith variable in the particular (jth) time point. Our target goal is to distinguish between dependent/independent variables in a given system based on the data.
I'm not a statistician, unfortunately, so I'm not sure if this is the proper naming convention, i.e. dependent/independent variables with respect to the variables that are linked by causal relationships - and this is actually what we are looking for.

My first question would be if we're going to the right direction - we started with a correlation-based methods. Basic correlation analysis indicates the strength and direction of a linear relationship between given two variables. We would like to use some more advanced analysis that would be able to judge non-linear relationships. I read that this could be done using a correlation ratio. However, I have hard time finding any information that would give some guidelines about how to do it (more specifically, how to interpret the correlation ratio value with respect to a non-linear relationship between given two variables).

I will appreciate if you could recommend me any source of information about the correlation ratio and/or any other method that could be use to achieve our goal.

Thanks in advance,