Any particular reason for $\Delta x_t=x_t-x_{t-1}$ or $\Delta x_t=x_{t+1}-x_{t}$

Question

I never know whether to define $\Delta x_t$ (change of $x$)as $$\Delta x_t=x_t-x_{t-1}$$ or $$\Delta x_t=x_{t+1}-x_{t}$$

Is there any particular reason to choose one over the other?

If it is irrelevant, what is customary to do?

Answer 1

It does not matter at all. It's just a matter of preference. You can switch from one to the other by adjusting the $t$'s as necessary.

Answer 2

It is customary to use $\Delta x_t$ for the "forward difference" $x_{t+1}-x_t$, and when these sorts of questions are asked (MathWorld, Wikipedia), to use $\nabla x_t$ for the "backward difference" $x_t-x_{t-1}$. However, there are exceptions to this usage, such as at Harvey Mudd's Riemann Sums tutorial page.

I believe the forward difference may be more common, but both are about equally useful (e.g. Newton's forward difference formula has a backward difference formulation).

In my personal opinion, one thing in favor of the forward difference is that if your sequence only goes forward from $x_0$, you can write $\Delta x_0$, but $\nabla x_0$ would require some explanation. The alternative is to keep track of the fact that the lowest allowed index for $\nabla x_i$ is one more than that for $x_i$.

Answer 3

Sometimes when discussing stochastic processes it is good to use the backward difference $x_t - x_{t-1}$. The reason would be that, at time $t$, we know the past, but not (yet) the future. So we know the backward difference $x_t - x_{t-1}$; but the forward difference $x_{t+1}-x_t$ is a random variable that we do not (yet) know.