In many of the papers we have read this
semester (and the current two sessions are no exception) we have seen
many examples of spatial regression analysis. That is, we have
looked at the map of correlations or regression coefficients associated
with an index time series. The purpose of this data exercise is
to show you how to make your own regression or correlation maps.
We will use your results in our discussion of decadal variability in
the Antarctic sector during the week of April 21st.
For this exercise we will use the
International
Research Institute for Climate and Society (IRI)'s superb
Data Library and its
computing and visualization engine Ingrid. Ingrid and the Data Library
have been made available via browser by M.B. Blumenthal and others at
Lamont-Doherty Earth Observatory.
To paraphrase what has been said about Emacs: Ingrid is something more
than a computing language for comparing gridded data; and something
less than a religion. That said, I am biased; so please feel
free to employ any computational environment you prefer. Another
good one out there on the internet is the
KNMI Data Explorer.
An introductory tutorial to using the Data Catalog and Ingrid is
here.
Ingrid Function Documentation is
here.
Having a browser tab open to these pages may be useful as you go
through the exercise.
The basic calculation
we wish to make is fairly straightforward. Suppose we have a time
series A(t) and a spatiotemporal field B(x,y,t), each of which have
time mean
zero, in the case of B, at each location in the spatial domain.
- The dimensionless correlation of A and B is a
spatial
pattern over x and y, and can be calculated as
(eq. 1)
r(x,y) = <A*B>/[s(A)*s(B)]
where <...> indicates the average over time of
the quantity inside the angles, * indicates multiplication, and s(A)
and s(B) are the standard deviations of A and B, respectively:
(eq. 2) s(A) = sqrt{sum(A2)/n}
with n = the number of time points for which we have data, sum(A2)
is the sum of squared anomalies relative to
the mean, sqrt is the square root operator, and
recalling that we defined A and B to have time means equal to
zero. You can see that although s(A) is just a number, s(B)
will also be a spatial pattern over x and y.
- The regression of B on A is also a spatial
pattern over x and y:
(eq. 3)
m(x,y) = <B*A>/<A*A>
where the quantity in the numerator is the covariance of A and B, and
in the denominator is the variance of A, and has units of B/A.
- Given eqs. 1-3, derive an equation for the
regression m of B on A in terms of r, s(A) and s(B). (Hint: see
the fine print in my lecture notes
from 2/11/2008, slide 13; and see lecture notes from 4/14/08). If A is a standardized
index, how does the equation simplify? What are the units of the
regression map in this case?
