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I wonder if anyone has put into use the autoregressive metric for ARIMA clustering proposed by Corduas and Piccolo (2008). The authors define the distance autoregressive metric between two processes $X, Y$ as:

$$\text{d}(X,Y)=\sum_{j=1}^m \left| (\pi_j,x-\pi_j,y)^2\right|\qquad (1)$$ for $j=1$ to $m$.

So in standard ARIMA notation, the squared distance of two $X,Y$ processes as:

$$\text{d}(X,Y)^2=(\phi x-\theta x)^2/(1-\theta x^2)+(\phi y-\theta y)^2/(1-\theta y^2)-2(\phi y-\theta y)^2/(1-\theta y\theta x)\qquad (2)$$

So I have a few questions.

  1. Does $j$ in (1) refer to the number $m$ ACF and PACF values taken for $N$ lags? If that's the case why do we have to fit in the time series a particular ARIMA model beforehands?
  2. For an ARIMA(0,0,1) model the $\phi$ parameter in (2) is 1?
  3. For an ARIMA(1,0,0) model the $\theta$ parameter in (2) is 0?
  4. In case of a very large value of $\theta x$ or $\theta y$ (close to 1) we get a "ridiculously" large value for squared distance (as described by (2)). Does this have a qualitative meaning?

I'm writing the necessary routines in R (for distance matrix calculation). I'm pretty confident there is nothing implemented yet but you never know for sure. However if anyone knows something more or is interested in this field I'll gladly share my experience.

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