How does one interpret an element of the "transfer matrix" used to calculate frequency domain granger causality (via VAR models)?

Question

I am attempting to gain a better mathematical understanding for how autoregressive models can be used to infer frequency-domain granger causality. All freq. domain measures of causality that utilize autoregressive models convert the time-domain parameters to a frequency-based representation (i.e. take the Fourier transform of the original parameters), and then take the inverse of that representation to get the so-called "transfer matrix". This transfer matrix is referenced in many causality-related papers, and is the backbone of all freq. domain measures. Here is the derivation for the transfer matrix:

$a_{i,j}(k)$ - Autoregressive coefficient describing influence of node $j$ on node $i$ at lag=$k$

$p$ - Model-order (i.e. how many past time-points to use to predict the present)

$\textbf{A}(k)$ - $Q$-by-$Q$ autoregressive parameter matrix at lag=$k$. This matrix is estimated via model fitting.

$U(t)$ - Innovation noise (i.e. the difference between the model's predictions and observed data at time $t$.)

\begin{align} {Y(t)} = \sum_{k=1}^{p}\textbf{A}(k)Y(t-k) + U(t) \end{align}

$\bar{\textbf{A}}(f)$ - Frequency representation of $\textbf{A}(k)$

$\bar{{a}}_{i,j}(f)$ - Frequency representation of autoregression describing influence of node $j$ on node $i$ at freqeuncy $f$

$\bar{a}_{i,j}(f) = \begin{cases} 1-\sum_{k=1}^{p} a_{i,j}(k) e^{-\textbf{j}2\pi f k} & \text{if $i=j$} \\ -\sum_{k=1}^{p} a_{i,j}(k) e^{-\textbf{j}2\pi f k)} & \text{otherwise} \end{cases} $

\begin{align} \text{where } \textbf{j}=\sqrt{-1} \end{align}

Note that above piecewise equation stems from taking the FT of model coefs. At least as far as I understand it. I'm not actually positive as to why this equation is piecewise and not identical for all $i$/$j$ combinations. I haven't worked with FTs very much.

$\textbf{H}(f)$ - Transfer matrix

\begin{align} \textbf{H}(f)=\bar{\textbf{A}}^{-1}(f) \end{align}

Now for my question--does anyone have an intuitive definition for what the transfer matrix, or a given element of the transfer matrix ($h_{i,j}(f)$), actually represents? I can easily and intuitively understand what $a_{i,j}(k)$ represents, but I've yet to fully grasp what $h_{i,j}(f)$ represents. How does one interpret such a value in layman's terms?

Answer 1

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Hi: At a quick glance, the Geweke, 1982 and 1984 references on the last page of this link are probably necessary reading for understanding it.

I realize that this doesn't answer your question but I'm trying to see if I'm able to send a link without actually typing the link. and I don't know how to do that in a comment. Hopefully it works here.

Answer 2

If you take the Fourier Transform of your AR equation, you get

$$\begin{align} Y(f) &= \sum_{k=1}^p A(k)e^{-j2\pi f k} Y(f) + U(f)\\ \left(I - \sum_{k=1}^p A(k)e^{-j2\pi f k}\right) Y(f) &= U(f) \end{align} $$ Let $\bar{A}(f) = \left(I - \sum_{k=1}^p A(k)e^{-j2\pi f k}\right)$. This should tell you why

$$\bar{a}_{i,j}(f) = \begin{cases} \displaystyle 1 - \sum_{k=1}^p a_{i,j}(k)e^{-2\pi f k}, &\text{if} \ i = j \\ \displaystyle - \sum_{k=1}^p a_{i,j}(k)e^{-2\pi f k}, &\text{otherwise}. \end{cases}$$

Now, $$Y(f) = \bar{A}^{-1}(f)U(f) \\ Y(f) = H(f) U(f)$$

This simply means that to recover your signal, you should multiply your innovation signal with the matrix $H$. You can see it as a transfer function matrix. The transfer function matrix tells you what the linear system output will be for a given input. Each $i$th elemenf of $Y(f)$ is,

$$y_i(f) = \sum_{j=1}^Q h_{ij}(f)u_j(f)$$

Therefore, the matrix $H(f)$ gives you the relationship between the innovation noise (system input) and the observed output. For example, in speech modeling, the innovation noise is white noise or a periodic pulse train (depending on sibilant or vowel) which gets shaped by the matrix $H$.