# Ways to understand 2-dimensional time-series data

I'm working on 2D time series data where two attributes are depth and temperature. When I plotted depth-vs-temp curve and saw its variation over time, the fluctuation occurs at few places only.

I'm not saying temperature is dependent on depth. But given these data, what should I look into to establish relationships between depth, temperature, time?

Ideas:
Look for depth regions where fluctuation in temperatures occurs over time. And develop time series prediction for these given regions.

What models/papers should I study to get insights out of the data? I want to understand and study what are the various methods that could be tested and visualizations built over these data. It's a data exploration problem.

Data shape: (2000,2,100) (samples,depth-temperature,time-sample) .i.e 2000 depth,temperature samples for each of 100 time stamps.

Thanks.

## 2 Answers

The general solution to your problem is Vector ARIMA (VARIMA) where the 2 endogenous variables that you specify can be related not only to their past values BUT past errors in both series. VAR is a special case of VARIMA as it doesn't assume a specific structure. In both cases VAR and VARIMA one must be concerned with anomalies such as Pulses/Level Shifts/Seasonal Pulses and Local Time Trends. If one doesn't treat these four kinds of omitted deterministic structure your results might be seriously flawed. Furthermore changes in parameters and/or the variance of the error processes need to be considered as the t/F tests are not robust to these violations. Unfortunately VARIMA solutions which don't deal with the pre-mentioned caveats are not very useful. If you find that primarily Y responds to X rather than X responding to Y , one could use a Dynamic Regression (a.k.a' Transfer Function) approach where all of the pre-mentioned conditions can be considered.

You are looking for vector autoregression or VAR (not to be confused with value at risk). Here is a gentle introduction. This kind of model is common in econometrics, where one commonly looks at multiple macroeconomic variables like GDP, joblessness etc., so you may want to consult an introductory econometrics textbook.