# multivariate time series output analysis

I have several sets of timeseries, representing continuous measurements (10Hz) for the same experiment, that is, a dynamic driving cycle for a car. Each set of timeseries consists on three subsets of different time series:

1. environmental variables (humidity, ambient pressure, ambient temperature; measured continuously but could be summarized by a single value, since they are constant during the experiments)

2. inputs, such as air flow, fuel flow, speed, momentum, exhaust temperature and pressure,...

3. outputs, mainly data about emission (CO2/NOX)

This experiment has been performed several times in the same environment by the same person, and several other in another environment, by another person. When summarizing the outputs of the whole cycle by aggregated values (e.g. NOX as g/km) and not continuous measurements, I can clearly see two clusters differing both in mean and deviation, corresponding to the two different sets of repetitions of the experiment.

The experiments have been performed on the same mechanical object, so I heuristically expect the differences to be explainable by the cumulative effect of small variations in my inputs (lower pressure, higher humidity, small variation in driver response times in braking/accelerating, etc..). Is there any way I can quantify this, and relate the differences in my output to differences in the inputs, by taking the time series as a whole and not using aggregated results?

I would like a way to be able to either say that the difference in my outputs is explainable by variations in my inputs or that it is not and hence something went wrong. I am not sure how mathematically state the problem, but I would like something like:

1. Find a function for $F$, such that $Y_1(t) = F(t, X_1(t)) +\epsilon_1$, for any $(Y_1,X_1)$ belonging to the first group of repetitions, with $\epsilon_1$ error term having some "nice properties" (small, normal distributed, more?). I expect $F$ to be strongly non-linear. [can NARX/NARMAX help in any way?]

2. Test if $Y_2(t) = F(t, X_2(t)) +\epsilon_2$ for any $(Y_2,X_2)$ belonging to the second set of observations. I would like to do it in the most possible quantifiable way

3. If the model fits both sets of repetitions, being able to claim that "the difference between $Y_1$and $Y_2$ is "mostly explained" by the difference between this and that time series belonging to $X_1$ and $X_2$

Thanks in advance for any hint/reference/tutorial you could provide.

• Nice question and I'm sorry it didn't received more attention. I'm not sure what you mean with "When summarizing the outputs of the whole cycle by aggregated values (e.g. NOX as g/km) and not continuous measurements". Do you mean you have discretized the NOX values in groups, e.g., $A=[100,200], B=[200,300]\dots$? This is a bad idea. Also, note that apart from the environmental conditions, another important variable changes between the tests: the experimenter. In an industrial environment, the variation due to changing operator often matters much more than you'd hope it would. – DeltaIV Aug 2 '17 at 10:07