Time Series with increasing dimension

Question

I am stuck with the following problem from research. I am not sure how to model this situation.

I have a vector time series whose dimension increases with time, $t$.

Specifically, let $\mathcal{X}$ be the vector time series data. Let $\mathcal{X}_t$ be the measurement at time $t$. The problem I am facing is that the $\dim{\mathcal{X}_t} = f(t)$. Let us assume, for the sake of simplicity, that $f(t)$ is known and is monotone with $t$.

What tools are available to model such a situation? Is there a PCA method that can be applied in this situation? I am ultimately interested in doing prediction of the time series.

Answer 1

This is very generic question. Here are some ideas.

Every entry in your time series can be thought of as $X_{t,k}$, where $0 \le k < f(t)$ is some index.

1. You can try computing averages/medians (or other group characteristics) of $X_{t, \cdot}$ for a fixed $t$, and examining an ordinary time series of such group characteristics for the entire observation set.
2. You can also possibly think of $X_{t, k}$ as a function in $k$ and characterize it, looking at a sequence of functions $X_t$. Alternatively, alter the dimension, fixing $x$ and consider $X_{t,k}$ as a function in $t$ for each fixed $k$, again ending up with a sequence of functions.