[R] How to interpret the phase spectrum?

[David Scott]

Peter Dalgaard wrote:
> sdlywjl666 wrote:
>> Dear all, 
>> I would like to know whether positive or negative values of the phase spectrum indicate that the time series leads or lags.
>>     In my work, x and y have peak nearly at the same frequency,(eg:f=1/56);and the coherency is peak where f=1/56,the phase is 0.5 where f=1/56.
>>     Can I get the conclusion that x lead y 0.5*56=28 at the frquency f=1/56?
>> if not,how can I compute the lag/lead by phase and frequency.
> 
> Are you referring to a particular piece of software?
> 
> As far as I know, this is completely dependent on choice of notation, so
> the question really only makes sense in a specified context. In the
> cases I remember seeing (I'm no time series expert, though), the phase
> is an _angle_ between 0 and 2*pi or between -pi and +pi, or sometimes in
> degrees, but I suppose it could be scaled to (-1 , 1) or (0, 1) as well.
> Also lead/lag for cyclic functions is a matter of convention; in
> particular, there's no difference between leading and lagging by half a
> cycle.
> 
> 
Following up on Peter's comment. Different authors define the 
cross-covariance and hence cross-spectrum differently. Time series seems 
to me to be plagued by inconsistencies in definitions.

There is a way out though, and when faced with different software, it is 
a step which should always be undertaken before any interpretation is 
attempted. Generate a series, a simple sinusoid will do, change the 
phase to generate a leading or lagged series, and see how the cospectrum 
looks. That is really the only infallible way of determining what the 
software is doing.

David Scott

-- 
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David Scott	Department of Statistics
		The University of Auckland, PB 92019
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Director of Consulting, Department of Statistics