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decades, bringing temperatures nearly 0.5 degrees above the long-term average by midcentury. That
above-average warming is off set by slight cooling during subsequent decades, bringing temperatures
back down roughly to their long-term average by century's end. In the east, we see the converse of
this pattern. Slight cooling occurs during the first half-century, when temperatures spend much of the
time below their long-term average. But that cooling is more than overcome by a substantial warming
during the second half, with temperatures ending up nearly 1 degree above the long-term average by
the end of the century.
The variations in surface temperature over the globe as a whole can be fully described
mathematically in terms of just two spatiotemporal patterns of variation in the data. One of these two
patterns (bottom row) is characterized by a linear increase in temperature over the hundred-year
period of just under 1°C in amplitude. In this characterization, the warming is uniform across both
west (panel e) and east (panel f) and is described by a PC series that is a simple upward ramp (for
(middle row) is a roof-shaped temperature variation of 1°C amplitude that plays out oppositely in the
west (panel c) and east (panel d): The west warms during the first half-century and cools during the
later half-century. The east does just the opposite, yielding what looks like an inverted roof. This
second pattern in the temperature record can be described, for the west hemisphere, by a PC series
that increases toward the middle of the century and decreases thereafter, with precisely the reverse
evolution for the east.
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This pattern could be, say, the imprint on the global temperature record of a
single cycle of a multidecadal climate oscillation like the Atlantic Multidecadal Oscillation (AMO)
described in
chapter 3
,
which acts to redistribute heat from one part of the globe to another (e.g., from
west to east), but doesn't change the average temperature of the globe. Adding together the two
patterns yields the total pattern of temperature variation (i.e., top row; adding c and e, gives a, while
adding d and f gives b).
We can rank the two contributing patterns by what fraction of the total variation in the overall
data (temperature, in this case) they explain. These rankings will depend, however, on what baseline
we choose for the data, that is, how we center the data with respect to the
y
-axis. It is conventional to
center the data about their long-term average, as we have done in the above example. By this
convention, the relative temperature departures average to zero over the full hundred-year dataset.
With the data centered that way, the second of the two patterns (c and d) of warming/cooling (in the
west) and cooling/warming (in the east) explains 60 percent of the variation and thus constitutes
PC#1. The global warming pattern (e and f) in our example turns out to be the second most prominent
pattern, explaining 40 percent of the variation in the data. It is PC#2.
In more realistic examples, there are generally a greater number of significant patterns of
variation in the data. Moreover, rarely can they be extracted as cleanly as in this example, since there
is typically some degree of contamination by noise, be it random disorganized temperature variations
or biases and errors in the data. This example nonetheless demonstrates how PCA can efficiently
describe the few leading patterns of variation in a larger dataset, a step that is essential if one is
interested, as with paleoclimate reconstructions, in establishing robust relationships between patterns
in potentially quite large and noisy datasets. Applying PCA in that case can help sort out the climate
signals (the key, most robust patterns of variation in the datasets) from the noise in which they are
immersed.
Using PCA, we were able to represent the main information in each of the various regional
