How to analyze multiple variable time series - suggest references

Question

I have multiple environmental time series variables (for example: temperature, dissolved oxygen, conductivity, depth) measured every few minutes for several months. The variables are measured at different intervals but I could create an interpolated data set such that all data are placed at the same times.

I want to be able to understand how these variables are related to each other. For example I would be interested in a question like:

Is an increase in depth correlated with a decrease in temperature?

The variables may change at the same time, but more likely there will be some lag time between a change in depth and a change in temperature. Plus I would like to look at many variables - not just 2.

I am not interested in cycles or trends within one time series - just how one may affect another.

I am not sure where to start - is this just a multiple regression? How do I take lag time into account? Could you suggest topics I could read about?

Answer 1

You should look to Box-Jenkins Transfer Function modeling. It was Chapter 11 in their book. http://books.google.com/books?id=jyrCqMBW_owC&printsec=frontcover&dq=box+jenkins+time+series&hl=en&sa=X&ei=sDO-U4aCC8WZyASIkoLoBg&ved=0CDYQ6AEwAQ#v=onepage&q=box%20jenkins%20time%20series&f=false

Here is a link that explains the differences between regression vs Transfer Function models. http://www.autobox.com/cms/index.php/afs-university/intro-to-forecasting/doc_download/24-regression-vs-box-jenkins

Answer 2

No reference, but I guess the standard time-series way to do it is to use some kind of auto-regressive model (AR-model)

$$X_t = c + \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon_t$$

You can find it in any textbook on time-seies analysis. The $\varphi_i$ are vectors of parameters which must be fitted, and the pre-chosen order $p$ determines the maximum lag time (further, $\varepsilon_t$ is a Gaussian error term).

The AR-model is simply a multivariate linear regression using the previous $p$ results as predictor and the current result as target. Obviously, you can extend this ansatz to any other multivariate statistic method. For instance, you can incorporate $x_i x_j$ to model interactions, use a neural network, etc.

As an alternative and easier method, you can also model only a single predictor given the instances of all predictors at $p$ previous times. This is more convenient since many methods in statistics consider a scalar output. Also, the interpretation is a bit easier. For example, in your mentioned case (depth-temperature) and for linear regression, you would model the temperature and then -- if there is some correlation -- find significant magnitude of the model parameters related to depth.