Best algorithm for correlation between time series?
You haven't given us much information about your problem, so let me give you some hints about how to approach this.

First, look for a model.  Ask yourself: Do you have a plausible biological/physical model for how the dependent variables might depend on the independent variables?  Why do you think it's plausible that the independent variables might help in predicting the dependent variables?  This may help you select an appropriate machine learning tool.

If you don't have a model and don't have any physical/biological reason to expect a causal relationship, then you're doing data mining (and maybe you should reconsider why you are doing this project, or maybe you need to study the underlying application domain better to try to investigate candidate explanations why there might be a causal connection between the two).

Second, classify the data you have.  The right algorithm will depend on what data is in the time series.  Ask yourself: What is the value at each point in time, for the dependent variables and the independent variables?  For each, decide whether you have ordinal data, categorical data, or interval data -- that will play a huge role in narrowing down which machine learning algorithm might be applicable.  Also, ask yourself whether you know anything about their distributions.

Third, visualize your data.  Before you start diving into complex machine learning algorithms, do some data visualization.  Try to find as many ways as you can think of to look at your data.  Sometimes when you stare at data in the right way, that suggests hypotheses or relationships that might be relevant.

So, do all of that.  That will help you think more clearly about the problem, and may help you narrow down the scope of machine learning problems.  Once you've done all of that and can answer all of those questions, you might be in a position to ask a better-crafted, more narrowly scoped question.



Last, don't forget to consider some variety of linear regression: for interval data where you hypothesize some kind of linear relationship, it's a natural first thing to try.