# Difference between time series and trajectory terminology

1. What is the difference between trajectory and time series? To me both seem the same thing.

2. In the 3D diagram (cube picture on left of Fig.2 from the paper titled “Review and comparative evaluation of symbolic dynamic filtering for detection of anomaly patterns”, which illustrates how to assign unique symbol to a partition of a trajectory), what are the labels for the axes?

Based on my understanding, the range $$[1,-1]$$ denotes the range of values taken by the variable of the dynamical system from which the time series is generated and the range $$[0,5]$$ denotes the first 6 time stamps (time axis). But I am not sure if this is correct. Please correct me if wrong.

• Sm1, I took the liberty of inverting the order of your questions, in order to give more emphasis to the question you have in the post title. I also did that because the one about the paper's figure is actually rather off-topic, since it's too specific. Of course feel free to undo my changes if you don't agree with them. – stafusa Aug 14 '19 at 22:58

What is the difference between trajectory and time series?

• A trajectory is a path in the phase space of the system.

Where phase space, or state space, is the space of the variables used to describe the system, such as $$\theta$$ and $$\dot\theta$$ for a simple pendulum. In other words, a trajectory is a description of the time evolution of the system.

• A time series is a record of the value of a given variable or variables at given points or ranges in time.

Therefore, a time series is the same as a trajectory only when the tracked variables describe the state of the system. In the example of a simple pendulum, the time series $$(\theta(t),\dot\theta(t))$$ gives a trajectory, whereas the time series $$\theta(t)$$ doesn't.

As for the question on the paper's Fig.2, the text says:

Let $$\Omega \in \mathbb{R}^n$$ be a compact (i.e., closed and bounded) region, within which the trajectory of the dynamical system, governed by Eq.(1), is circumscribed as illustrated in Fig.2.

Thus, there is no time axis, as all three axes should denote elements of the state vector $$\mathbf{x}$$. The figure is only schematic, so also the ranges shown aren't meaningful.