# Variable importance (?) for multivariate time series anomaly detection methods

I'm working on anomaly detection methods for multivariate time series $$[\mathbf{x}^{new}_1,\dots,\mathbf{x}^{new}_T]$$ where $$\mathbf{x}^{new}_{i}$$ is $$p-$$dimensional. I won't go into the details of the architecture because it's not relevant to the question, but let's say something like the LSTM encoder-decoder architecture used here:

https://arxiv.org/abs/1607.00148

Basically, you have a statistical model (in my case it's a neural network, but it could be anything), and you train it on normal operation data $$[\mathbf{x}_1,\dots,\mathbf{x}_T]$$, teaching it to reconstruct its input - the model output is thus $$[\mathbf{x}'_1,\dots,\mathbf{x}'_T]$$. It's kind of an autoencoder, but for time series data.

At test time, you feed it a multivariate sequence $$[\mathbf{x}^{new}_1,\dots,\mathbf{x}^{new}_T]$$ and it predicts a sequence $$[\mathbf{x}^{new'}_1,\dots,\mathbf{x}^{new'}_T]$$. If the sequence comes from the same distribution as the training set, i.e., if it's normal operation data, then $$[\mathbf{x}^{new}_1,\dots,\mathbf{x}^{new}_T]$$ and i$$[\mathbf{x}^{new'}_1,\dots,\mathbf{x}^{new'}_T]$$ will be similar, otherwise for some $$i$$ we'll have $$\Vert\mathbf{x}^{new}_i-\mathbf{x}^{new'}_i\Vert>\alpha$$ (a threshold) and we classify that as an anomaly. This is all pretty standard stuff, and well described in the literature.

Now, suppose I want to estimate the contribution to the residual, of each component of the input vector. In other words, I have

$$\boldsymbol{\epsilon}_i=\mathbf{x}^{new}_i-\mathbf{x}^{new'}_i=\mathbf{x}^{new}_i-f(\mathbf{x}^{new}_{1},\dots,\mathbf{x}^{new}_i)=g((x^{new}_{11},\dots,x^{new}_{p1}),\dots,(x^{new}_{1i},\dots,x^{new}_{pi}))$$

and I would like to know which of the $$p$$ components contributes the most to $$\boldsymbol{\epsilon}_i$$.

1. Does it make any sense? After all, $$\boldsymbol{\epsilon}_i$$ probably depends on the whole sequence up to $$i$$, thus on $$i\times p$$ quantities, not on just $$p$$.
2. Has it been done before? What would you call this analysis? Variable importance? Root cause analysis? Do you think it might have more to do with causal inference rather than with forecasting?