# Notation for a time series with discrete time points

I want to mathematically describe a vector containing a series of time points. The time points must lie on an equidistant time grid of 15 minutes of a day. Thus, elements of the vector can contain values that occur at times $$[t_1 =$$ 00:00$$, t_2 =$$ 00:15$$, ..., t_n =$$23:45$$]$$.

Example: vector $$\pmb{t} = [$$00:00, 01:15, 02:00, 04:15, 15:45, 18:00, 20:30$$]^T$$

Now I want to describe the vector with $$\pmb{t} = [t_1, \dots, t_m] \in \square$$. However, in my case $$\mathbb{R}$$ does not fit as a set. How would you describe the set of $$\pmb{t}$$?

• Why do you think $\mathbb{R}$ (or rather $\mathbb{R}^m$) doesn't fit as a set? Jun 23 at 19:03
• You could define a mapping that assigns the times to the first $4 \times 24 -1$ natural numbers. You then get a vector in $\{0,1,\ldots,95\}^m$ (and the space can further be reduced if the vector is ordered). Jun 23 at 19:45
• As pointed out by @FedericoPoloni, your vector $\mathbf{t}$ is...well...a vector in $\mathbb{R}^{m}$. In fact, depending on your application, many researchers refer to MATLAB notations when describing vectors induced by equally spaced points. For instance, $\mathbf{t}\triangleq\operatorname{linspace}(0,23.45,m)$. Jun 24 at 7:56
• @FedericoPoloni I thought the set $\mathbb{R}$ does not fit, for two reasons: First, because only a discrete number of values is possible here. Secondly, it is not necessarily clear that a time such as 00:00 is in set $\mathbb{R}$. My thought was that you have to convert it first for that, e.g. in seconds since 1970.
– Emma
Jun 24 at 13:06
• @SPARSE 23.75 actually, if you are measuring in hours. Jun 24 at 15:17