Breeding data for birds, with Lat, Long, month. I can't use 'time since start' for the monthly data and month is circular (12 and 1 are more similar than 12 and 6, for example). I'd also prefer not to lump by sub-region (yes, residuals from time series correlations might do, but I'd have to lump to get proportions since each breeding record is once a year). Using a cos transform cos(month/6*pi) gives a more sensible value than 'month' but I'm not sure this is valid...
With a similar task (time in an ocean system influenced by tides) I have found transforming to a *two* variate system helpful, i.e. in terms of new variables *x=cos(2*pi*t/tau)* and *y=sin(2*pi*t/tau)* where *tau* is the known repeat period.
This can be handy for plotting. One plotting approach is to colour-code for the variable of interest, yielding a circle that is colourized around the angle. Another is to indicate that variable by increasing the radius point by point, so that when the variable is large you see spikes coming out of the circle. You can also work in univariate space by defining phase as *phi=atan2(y,x)* and then plotting or doing stats analysis on *phi*.
Lots of other things come to mind but your first step may be to plot, which is why the above suggests relate to that.
Giancarlo, Dan, Allen: thanks for your responses. I think you're right that I need both a cos and a sin variate to capture the subtleties of month. The problem that I then have is the spatial component of the analysis, or more explicitly, how to either calculate Moran's I or run a semivariogram/correlogram with two variables or possibly a linear combination of the two. Any thoughts? I wonder if it would make sense to average Moran's I calculated separately for each of the two components?
Hi Graeme. Maybe this is too simplistic, but could you not take the absolute value of each month minus 6? So 4 would become abs(4-6) = 2. This would rescale your values such that the difference between adjacent locations is equal to the shortest number of months between them.
Hi Graeme,
It would be interesting to know what method you ended up using.
Dan
So I'm still not sure I have the best final approach, but what I ended up doing was looking at the difference between pairs of points as chord length across a unit circle (i.e., mapping months onto the arc) and comparing these values to the geographic distance between breeding records. Putting these two together gives me a correlogram-like function that in my case, indicates a lack of autocorrelation/spatial synchrony for my study species. I had to bootstrap though to test for significance, because my data were non-randomly located in space.
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