Let's say I have price history for a set of currency pairs { AUD/CAD, EUR/USD, NOK/SEK, NZD/USD, ... } . I seek to introduce a new currency XXX priced as a linear combination of these pairs such that new pairs { AUD/XXX, CAD/XXX, EUR/XXX, ... } have minimum volatility.
In other words, I am trying to 'separate' currencies out of their pairs in a most likely fashion.
Any thoughts on how to do this?