Machine Learning Methods for ENSO Analysis and

Machine Learning Methods for ENSO Analysis and Predictions
Carlos H. R. Lima
- Dept. of Civil and Environmental Engineering, University of Brasilia. Brazil. [email protected]
Upmanu Lall
- Earth & Env. Eng., Columbia Water Center, Columbia University. New York, USA. [email protected]
Tony Jebara
- Computer Science, Columbia University, USA. [email protected]
Anthony G. Barnston - International Research Institute for Climate and Society, Columbia University, USA. [email protected]
Motivation
• ENSO plays a vital role on the global climate variability;
• ENSO forecasts are very limited for lead times beyond 6 months;
Our goal: use machine learning methods for better understanding
and improving ENSO forecasts.
 Nonlinear method of dimensionality reduction (MVU) to obtain
covariates from a large climate data set (Tropical Pacific D20 data);
 LASSO regression to shrink model coefficients, since the number of
predictors is large (> 50).
Step 1: Maximum Variance Unfolding
Maximum variance unfolding (MVU) was originally developed by Weinberger
and Saul (2006) and has its origins on Kernel PCA, which uses a nonlinear
mapping of the original data to a transformed which is expected to be linear.
Using the kernel trick, dual PCA can be applied in this space to obtain a lowerdimensional system of the original data.
MVU is a nonparametric approach, where the nonlinear function is not assumed a
priori and the Kernel matrix is obtained from the original data by semidefinite
progamming. The goal is to maximize the sum of the eigenvalues (trace) of a
Kernel matrix while preserving neighbors in the original and transformed space.
Mathematically, MVU can be expressed as
 :  
x i   (x i ), i  1,..., N .
D
ENSO Correlation and the UNB/CWC ENSO Forecast Model
Cross-correlation function of MVU (ticker lines) and PC (thin lines) and NINO34
Forecast Model
Solution : Kernel trick  do not need to compute
NINO(t   )  a  a  NINO(t ) 
the mapping explicitly , but only thedot producs.
Idea : apply PCA in the space
defined by  (x i ) rather than X.
However,  can be huge.
Temporal correlation of the D20 gridded data and MVU (left) and PC (left) modes: first,
second and third from top to bottom.
E.g. for  (w )  w : K (w, z )   (w )   (z )  (w  z )
2
Hence, K ij   (x i ) (x j )
Cross-correlation function of MVU
(thicker lines) and PC (thin lines) modes
with WWV
2
t
 b
l t  24
,l
 MVU 1 (l ) 
 c ,l  MVU 2 (l )  d ,l  MVU 3 (l )
T
10-fold cross-validation: correlation and MSE skills for MVU
(black) and PC(red) models
1982, 1997 and 2014 ENSO Events
Question : given N high dimensiona l inputs xi   , how can we compute outputsy i   ,
D
d
where d  D, such that nearby points remain nearby and distant ones remain distant?
Basics of MVU
NINO3.4 and WWV lagged by 9 months
NINO3.4 and MVU2 lagged by 9 months
Step 2: LASSO regression
Basic idea: it shrinks the model coefficients by minimizing the sum of the mean
squared error with a constraint on the sum of absolute values of the coefficients.
NINO3.4 and MVU3 lagged by 12 months
Climate Dataset
Here we extend previous work (Lima et al., 2009) and apply MVU to the new
and updated NOAA/NCEP GODAS sub-surface ocean dataset. We focus on the
depth of the 200C isotherm of the tropical Pacific ocean, which is a proxy for the
thermocline depth and one of the main carriers of ENSO information.
Details: We restrict our analysis to the Pacific D20 along the latitudinal and
longitudinal bands bounded by 26N and 28S and 122E and 77W, respectively.
The dataset covers the period from January/1980 through May/2014 and consists
of 21001 data points located in an equally-spaced grid cell.
.
Results: Themocline Modes of Variability
MODE
PC1
PC2
PC3
Var exp
MVU1
0.75
-0.51
-0.26
57%
MVU2
0.46
0.51
-0.04
15%
MVU3
0.06
-0.25
0.50
8%
Var exp
24%
16%
7%
80%
47%
NINO3.4
predictions from
MAR/2014
NINO3.4
predictions from
SEP/2014
Summary and Future Work
• More variance explained by MVU modes, more amplitude and less cycles;
• Monotonic incresing trend in the first MVU: trend in the thermocline tilt?
• Patterns of second and third MVU different from those equivalent PCs and more
correlated with NINO3.4;
• LASSO forecast model shrinks coefficients and shows appreciable skills up to 15
month lead time;
• Future work will explore forecasts for other ENSO indices as well as for the
thermocline/SST fields and other ENSO related variables.
Acknowledgment
We thank IRI for providing the climate datasets and K. Q.Weinberger for making
his MVU code available. The first author acknowledges the financial support from
Colorado State University and ORAU to attend Climate Informatics 2014.
References
MVU (thicker lines) and PC (thin lines)
modes for the thermocline data.
Wavelet analysis of MVU (left) and PC
(right) modes 1 (top) to 3 (bottom).
Correlation and variance explained (top),
Kendall’s tau (middle) for temporal trends
and WWV series (bottom).
• Lima, C. H. R., Lall, U., Jebara, T., Barnston, A. G., 2009. Statistical Prediction of ENSO from
Subsurface Sea Temperature Using a Nonlinear Dimensionality Reduction. J. Climate 22, 4501–4519.
• Weinberger, K. Q., Saul, L., 2006. Unsupervised Learning of Image Manifolds by Semidefinite
Programming. Int. J. Comp. Vision 70 (1), 77–90.

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