# Mathematical notation for subset of a dataset based on temporal conditions

I have a dataset where every sample is taken at a certain time and belongs to a certain person. Every sample contains several features. I want to write (in mathematical notation) that a subset from this dataset is taken. This subset of the dataset are all samples up until some moment in time and can belong to all users. Also, what is the notation for a single sample (which is taken at a certain moment belonging to a certain user)

Let $\mathbf{x}_{i, t}$ denote a sample (feature vector) taken at time $t$, belonging to person $i$.

Let $\mathcal{D}$ denote the full dataset.

Let $\mathcal{D_{t < \tau}} \subset \mathcal{D} = \{ \mathbf{x}_{i, t} \mid t < \tau \}$ denote the subset of all samples taken before time $\tau$.

A single sample for user $i$ at time $t$ would look like: $\mathbf{x}_{i, t} \in \mathcal{D}$

In LaTeX, the above would look like:

Let $\mathbf{x}_{i, t}$ denote a sample (feature vector) taken at time $t$, belonging to person $i$.

Let $\mathcal{D}$ denote the full dataset.

Let $\mathcal{D_{t < \tau}} \subset \mathcal{D} = \{ \mathbf{x}_{i, t} \mid t < \tau \}$ denote the subset of all samples taken before time $\tau$.

A single sample for user $i$ at time $t$ would look like: $\mathbf{x}_{i, t} \in \mathcal{D}$


It looks like you are looking for something like "$x_{ut}\in\mathbb{R}^n$" to denote a sample taken from user $u$ at time $t$ and containing $n$ features.

To indicate the set of samples taken before a time $t_0$ and coming from a set $U$ of users (possibly a subset of the full set of users, you would write

$$\{x_{ut}\,|\, u\in U, t\leq t_0\}.$$