IEEE Geoscience and Remote Sensing Magazine - June 2016 - 64

vertical resolution [53]. Focus is placed on the data coming
from the Infrared Atmospheric Sounding Interferometer
(IASI), which provides radiances in 8,461 spectral channels
between 3.62 and 15.5 nm , with a spectral resolution of
0.5 cm-1 after apodization [54]. This huge input data along
the high-output dimensionality (the variable is sampled at
137 points in the atmospheric column) makes the direct
application of the previous methods unbearable. Alternatively, noting the high vertical correlation of the profiles, a
simpler strategy is developing a unique GP model that simultaneously predicts all of the PCA-projected state vectors
onto the top p principal components, and solving
K = (K ff + v 2 I N) -1 Y,
where Y columns contain the p scores (i.e., the projected
variables). This approach will be exploited again for RTM
emulation, as described in the "Emulating Radiative Transfer Models Through Gaussian Processes" section.
Figure 4 shows results of applying this strategy using IASI
data to predict (multioutput, 137 dimensions) dew point
temperature profiles. A linear regression (LR) and a GP model were trained using the first 100 principal components of
an IASI orbit (2008-07-17), both using 5,000 samples and
tested in several unseen data. Essentially, it was observed
that GPs largely improve the LR models, with an average gain
of +1.5 K, which is also statistically significant in all regions.

Global

102

efficiencY in gaUSSian proceSS regreSSion
The naive implementation of GPs in (3) and (4) grows as
O (N 3), where N is the number of training samples, which
makes them unfeasible when a large number of training
samples are available. To reduce the GPs' computation
complexity, they are generally computed using approximations. (Other forms of efficiency that involve parallelization and hardware-specific approaches and focus on pure
GP algorithms are intentionally omitted here.) The approximation methods can be broadly classified as sparse, localized regression, and matrix multiplication. Finally, some
recent developments are highlighted in GP efficiency that
exploit random features and particular kernel structures.
SPARSE METhODS
Sparse methods are also known as low-rank covariance matrix
approximation methods and are based on approximating the
full posterior by expressions using matrices of lower rank
M % N , where the M samples are typically selected to wellrepresent the data set (e.g., via clustering or smart sampling).
Since the selected M samples represent all others, these methods are considered global, as opposed to the local methods
described in the next section. These global methods are well
suited to model smooth-varying functions with high correlations (i.e., long length scales), and they use all the predictions data, such as full GPs. The methods in this family are based on substituting the joint prior with a reduced

NP

102

103

−2

0

2 4 6 8 10 12
Bias/Accuracy (K)
Trop

102

103

−2

0

−2

0

2 4 6 8 10 12
Bias/Accuracy (K)

103

−2

0

LR
GPR
p (hPa)

LR
GPR

103

2 4 6 8 10 12
Bias/Accuracy (K)
SP

102

p (hPa)

p (hPa)

2 4 6 8 10 12
Bias/Accuracy (K)
SH

102
LR
GPR

103

LR
GPR
p (hPa)

LR
GPR
p (hPa)

p (hPa)

LR
GPR

NH

102

−2

0

2 4 6 8 10 12
Bias/Accuracy (K)

103

−2

0

2 4 6 8 10 12
Bias/Accuracy (K)

figUre 4. The ME (thin dashed lines) and RMSE (solid lines) throughout the atmospheric column for a linear regression and a GP model
predicting dew point temperature profiles. The results are averaged for the whole globe and considered orbits, as well as for different
regions (i.e., north/south poles, north/south hemispheres, and tropics).

66

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Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - June 2016
