Question about mathematical model

[willtebbutt]

Pedantry: it's clear to me why one would define a time series to be an element of the Cartesian product of "time" and "X", but it's not clear to me why the set TS is needed. Would you mind explaining why we can just work with the Cartesian product? If it's useful as a definition to have floating around for e.g. use in defining functions of multiple time series, why does it need to correspond to a strict subset of the Cartesian product? i.e. why not just make it equal to the Cartesian product?

[tpgillam]

It’s a while since I wrote this.

But I think this is due to the permitted orderings? Can’t remember how mathcal T is defined

[willtebbutt]

$\mathcal{T}$ is just time, and needs to have a total order.

[tpgillam]

The ordering I meant was that knots have to be ordered such they are strictly increasing in time.

I think the issue is that T x X just represents a single knot, rather than the whole time series? Not sure about the best way to write this... in general I think about it in the 'functional interpretation' a bit further down the page. In fact, maybe we should just focus on that and delete some of the earlier section?