Section 13.4: Canonical Correlation Analysis
253
Zwiers (1993) discusses the problem of estimating the correlations. Following Glynn and Muirhad (1978) he proposes to improve the straightforward
estimator h by using the formula
Bk = Zk - ~ [mx + my - 2 + ß~ + 2(1- ßn E ~/J ~2l
npk
j # Pk - Pj
(13.54)
with Zk = tanh- 1 (h) and 0 = tanh- 1 (Pk). Glynn and Muirhead show
that Ok is an unbiased estimator of (}k up to O(n- 2 ) and that VAR(O) = ~
up to order O(n- 2 ). An approximate 95% confidence interval for the k-th
canonical correlation is then given by tanh(Ok)± *. However, Zwiers (1993)
found in Monte Carlo experiments that the correction (13.54) represents an
improvement over the straightforward approach but that substantial bias
remains when the sampie size is small.
13.4.4 Example: Central European Temperature
We return now to the Central European temperature in winter, with which
we have dealt already in Section 13.3.4. Werner and H. von Storch (1993)
analysed simultaneously the large-scale circulation, as given by the seasonal
mean sea-Ievel air-pressure (SLP) anomaly over the North Atlantic and Europe, and the Central European temperature given at mT = 11 locations.
The objective of this exercise was to determine to what extent the regional
temperature is controlledby large-scale circulation anomalies. 9
With first 40 years of the full1901-1980 data set the CCA was done. The
data were first projected on EOFs as given by (13.45). The number of EOFs
retained was determined in such a way that an increase by one EOF would
change the canonical correlations onIy little. The first pair of patterns goes
with a correlation of 70%. The patterns (Figure 13.6), which are plotted here
as a "typical simultaneously appearing pair of pattern" (by normalizing the
canonical correlation coefficients to variance one) indicate a simple physically
plausible mechanism: In winters with a persistent anomalous southwesterly
flow, anomalous warm maritime air is advected into Europe so that the winter mean temperatures are everywhere in central Europe above normal, with
typical anomalies of the 1 0 - 2 0 C. If, however, an anomalous high pressure
system is placed south of Iceland, then the climatological transport of maritime air into Central Europe is less efficient, and temperature drops by one
or two degrees.
9In any kind of correlation-baesd analysis a positive result (i.e., the identification of a
high correlation) is no proof that the two considered time series are connected through
a cause-effect relationship. It can very weIl be that a third (unknown) is controlling
both analysed time series. In the present case the physical concept that the large-scale
circulation affects regional temperatures is invoked to allow for the interpretation "control" .
253
Zwiers (1993) discusses the problem of estimating the correlations. Following Glynn and Muirhad (1978) he proposes to improve the straightforward
estimator h by using the formula
Bk = Zk - ~ [mx + my - 2 + ß~ + 2(1- ßn E ~/J ~2l
npk
j # Pk - Pj
(13.54)
with Zk = tanh- 1 (h) and 0 = tanh- 1 (Pk). Glynn and Muirhead show
that Ok is an unbiased estimator of (}k up to O(n- 2 ) and that VAR(O) = ~
up to order O(n- 2 ). An approximate 95% confidence interval for the k-th
canonical correlation is then given by tanh(Ok)± *. However, Zwiers (1993)
found in Monte Carlo experiments that the correction (13.54) represents an
improvement over the straightforward approach but that substantial bias
remains when the sampie size is small.
13.4.4 Example: Central European Temperature
We return now to the Central European temperature in winter, with which
we have dealt already in Section 13.3.4. Werner and H. von Storch (1993)
analysed simultaneously the large-scale circulation, as given by the seasonal
mean sea-Ievel air-pressure (SLP) anomaly over the North Atlantic and Europe, and the Central European temperature given at mT = 11 locations.
The objective of this exercise was to determine to what extent the regional
temperature is controlledby large-scale circulation anomalies. 9
With first 40 years of the full1901-1980 data set the CCA was done. The
data were first projected on EOFs as given by (13.45). The number of EOFs
retained was determined in such a way that an increase by one EOF would
change the canonical correlations onIy little. The first pair of patterns goes
with a correlation of 70%. The patterns (Figure 13.6), which are plotted here
as a "typical simultaneously appearing pair of pattern" (by normalizing the
canonical correlation coefficients to variance one) indicate a simple physically
plausible mechanism: In winters with a persistent anomalous southwesterly
flow, anomalous warm maritime air is advected into Europe so that the winter mean temperatures are everywhere in central Europe above normal, with
typical anomalies of the 1 0 - 2 0 C. If, however, an anomalous high pressure
system is placed south of Iceland, then the climatological transport of maritime air into Central Europe is less efficient, and temperature drops by one
or two degrees.
9In any kind of correlation-baesd analysis a positive result (i.e., the identification of a
high correlation) is no proof that the two considered time series are connected through
a cause-effect relationship. It can very weIl be that a third (unknown) is controlling
both analysed time series. In the present case the physical concept that the large-scale
circulation affects regional temperatures is invoked to allow for the interpretation "control" .