Description of neural network model (version 3.1)
A neural network (NN) model has been used by our group to predict the
SST anomalies in the Nino3.4 region in the equatorial Pacific (Tang et
al, 2000). A new NN model is built here to forecast the SST anomalies
over the whole tropical Pacific.
The data used in this forecast came from two datasets: the monthly
COADS
sea level pressure (SLP) data (Woodruff et al. 1987; downloadable from
ftp.cdc.noaa.gov/Datasets/ncep.marine) and the monthly extended
reconstructed sea surface temperature (ERSST version 2; Smith and
Reynolds 2004; downloadable from ftp.ncdc.noaa.gov/pub/data/ersst-v2).
Both data have 2°x 2° grids over global ocean with time
coverage
from January 1950 to April 2004. Anomalies for SLP and SST were
calculated by subtracting the monthly climatology based on the
1950-2003
period.
After removing the linear trend and applying a 3-month running mean to
the gridded data, principal component analysis (PCA) (also called
empirical orthogonal function, EOF, analysis) was performed on the
anomaly data over the tropical Pacific (SLP: 123°E-71°W,
21°S-21°N; SST: 124°E-70°E, 20°S-20°N) with 10
leading SST principal components (PC) and 6 leading SLP PCs retained.
The SST and SLP PCs were then normalized so they have the same
variance. For a given month, we stacked the SLP PC values of 3, 6 and 9
months before this month together with the SLP and SST PC values of
this month. Altogether, this yielded 34 PC time series. An extended EOF
(EEOF) (i.e. singular spectrum analysis) was performed to reduce the 34
PC time series to 12 retained EEOF PC time series, which were used as
the predictors, i.e. inputs of the NN model. The predictand was one of
the 8 leading PCs of the SST anomalies, i.e. the SST PCs were predicted
individually. The predicted SST PCs were then combined with the
corresponding EOF spatial patterns to yield the SST anomaly (SSTA)
forecast over the whole domain of tropical Pacific.
The NN model had a similar structure to the old NN model (Tang et al,
2000), i.e. a feedforward NN with an input layer of 12 neurons, a
hidden layer of neurons, and a single output neuron. The hidden neuron
has only 1 neuron, as forecast skill dropped when more hidden neurons
were used due to overfitting (i.e. the NN model is fitting to noise in
the training data).
A cross-validation scheme was used to estimate the forecast skill of
the NN models. The data record was divided into 8 equal segments. Data
from the one segment were withheld as validation data, while data from
the other 7 segments were used to train the NN model. Thus, independent
forecast was made for the period of the validation data using the
models based on the training data. This procedure was repeated until
all 8 segments were predicted, and the correlation and the root mean
square error (RMSE) between the predicted SSTA and the corresponding
observed SSTA could be calculated over the whole record.
Actually, only 85% of the training data (denoted by D85, randomly
chosen from the 7 segments of training data) were used to train the NN
model as the remaining 15% (D15) were reserved to test for overfitting.
If the MSE on D15 is 10% larger than the MSE on D85, the solution was
rejected. We made 30 runs from random initial model (weight and bias)
parameters, and the run with the smallest MSE on D15 was selected. The
above procedure was repeated 100 times and the ensemble mean of the 100
selected solutions was used as the desired model forecast.
Only the SST PC1, PC2 and PC3 were predicted using the NN models, while
linear regression model was used to predict higher modes, as the higher
modes have poor signal to noise ratios, hence difficult to extract a
robust nonlinear solution. For linear regression, as there was no need
to reserve 15% of the training data to test for overfitting, 100% of
the training data were used to train the model.
The cross-validated forecast skills for the regions Nino1&2, Nino3,
Nino3.4 and Nino4
(stretching from the eastern equatorial Pacific to the western
equatorial Pacific) are in Table 1 for individual decades and for the
whole record:
Table 1 Forecast Skill of the Neural Network Model.
Correlation and RMSE (given in
parenthesis) are cross-validated over Jan. 1950 - Apr. 2004.
-----------------------------------------------------------------------------
NINO1&2 CORR. SKILL (RMSE in Deg.C)
period
3-month 6-month
9-month
12-month 15-month
50-59 0.782(0.631) 0.653(0.779) 0.592(0.827)
0.543(0.862) 0.515(0.871)
60-69 0.697(0.553) 0.545(0.646) 0.351(0.733)
0.318(0.761) 0.398(0.722)
70-79 0.739(0.653) 0.545(0.802) 0.308(0.893)
0.093(0.993) 0.085(0.997)
80-89 0.864(0.591) 0.648(0.893) 0.504(1.014)
0.487(1.024) 0.554(0.987)
90-03 0.750(0.737) 0.509(0.948) 0.258(1.072)
0.254(1.074) 0.258(1.069)
50-03 0.765(0.653) 0.565(0.842) 0.391(0.942)
0.320(0.983) 0.316(0.979)
-----------------------------------------------------------------------------
NINO3 CORR. SKILL (RMSE in Deg.C)
period
3-month 6-month
9-month
12-month 15-month
50-59 0.765(0.484) 0.618(0.614) 0.486(0.746)
0.430(0.774) 0.349(0.785)
60-69 0.778(0.420) 0.610(0.539) 0.537(0.571)
0.556(0.568) 0.504(0.596)
70-79 0.890(0.411) 0.742(0.596) 0.563(0.726)
0.453(0.781) 0.359(0.829)
80-89 0.828(0.557) 0.628(0.772) 0.611(0.788)
0.677(0.734) 0.701(0.721)
90-03 0.788(0.545) 0.614(0.700) 0.483(0.778)
0.451(0.791) 0.455(0.790)
50-03 0.812(0.500) 0.642(0.660) 0.524(0.735)
0.492(0.750) 0.450(0.770)
-----------------------------------------------------------------------------
NINO3.4 CORR. SKILL (RMSE in Deg.C)
period
3-month 6-month
9-month
12-month 15-month
50-59 0.743(0.492) 0.617(0.591) 0.459(0.748)
0.375(0.783) 0.290(0.798)
60-69 0.824(0.352) 0.639(0.492) 0.585(0.516)
0.614(0.507) 0.478(0.581)
70-79 0.922(0.342) 0.790(0.541) 0.641(0.667)
0.564(0.714) 0.468(0.766)
80-89 0.816(0.529) 0.664(0.678) 0.664(0.674)
0.751(0.596) 0.770(0.584)
90-03 0.871(0.423) 0.740(0.579) 0.636(0.662)
0.605(0.687) 0.554(0.723)
50-03 0.842(0.444) 0.704(0.588) 0.594(0.666)
0.558(0.686) 0.494(0.719)
-----------------------------------------------------------------------------
NINO4 CORR. SKILL (RMSE in Deg.C)
period
3-month 6-month
9-month
12-month 15-month
50-59 0.689(0.372) 0.587(0.419) 0.461(0.500)
0.327(0.531) 0.238(0.550)
60-69 0.813(0.259) 0.642(0.342) 0.573(0.367)
0.610(0.358) 0.475(0.409)
70-79 0.856(0.324) 0.707(0.449) 0.630(0.484)
0.547(0.521) 0.469(0.549)
80-89 0.794(0.354) 0.701(0.412) 0.721(0.389)
0.797(0.339) 0.796(0.340)
90-03 0.873(0.305) 0.761(0.408) 0.644(0.478)
0.643(0.481) 0.542(0.528)
50-03 0.812(0.339) 0.692(0.422) 0.602(0.464)
0.553(0.484) 0.471(0.514)
-----------------------------------------------------------------------------
For real time forecast, the linear trend is added back to the predicted
SST anomalies, i.e. the value of the linear trend at the most recent
month is added to the future NN forecasted values, which increases the
forecast skills slightly (Table 2).
Table 2 Forecast Skill of Neural
Network Model (with linear trend included)
-----------------------------------------------------------------------------
NINO1&2 CORR. SKILL (RMSE in Deg.C)
period
3-month 6-month
9-month
12-month 15-month
50-59 0.787(0.631) 0.664(0.779) 0.606(0.827)
0.557(0.862) 0.530(0.871)
60-69 0.708(0.553) 0.564(0.646) 0.381(0.733)
0.354(0.761) 0.427(0.722)
70-79 0.743(0.653) 0.552(0.802) 0.322(0.893)
0.107(0.993) 0.099(0.997)
80-89 0.862(0.591) 0.642(0.893) 0.493(1.014)
0.477(1.024) 0.540(0.987)
90-03 0.752(0.737) 0.507(0.948) 0.243(1.072)
0.236(1.074) 0.242(1.069)
50-03 0.789(0.653) 0.617(0.842) 0.477(0.942)
0.413(0.983) 0.409(0.979)
-----------------------------------------------------------------------------
NINO3 CORR. SKILL (RMSE in Deg.C)
period
3-month 6-month
9-month
12-month 15-month
50-59 0.768(0.484) 0.627(0.614) 0.500(0.746)
0.444(0.774) 0.363(0.785)
60-69 0.784(0.420) 0.623(0.539) 0.552(0.571)
0.570(0.568) 0.519(0.596)
70-79 0.891(0.411) 0.746(0.596) 0.572(0.726)
0.466(0.781) 0.374(0.829)
80-89 0.825(0.557) 0.621(0.772) 0.601(0.788)
0.669(0.734) 0.691(0.721)
90-03 0.785(0.545) 0.606(0.700) 0.470(0.778)
0.437(0.791) 0.443(0.790)
50-03 0.822(0.500) 0.664(0.660) 0.557(0.735)
0.527(0.750) 0.487(0.770)
-----------------------------------------------------------------------------
NINO3.4 CORR. SKILL (RMSE in Deg.C)
period
3-month 6-month
9-month
12-month 15-month
50-59 0.746(0.492) 0.624(0.591) 0.471(0.748)
0.388(0.783) 0.302(0.798)
60-69 0.829(0.352) 0.650(0.492) 0.597(0.516)
0.624(0.507) 0.492(0.581)
70-79 0.923(0.342) 0.792(0.541) 0.646(0.667)
0.571(0.714) 0.477(0.766)
80-89 0.813(0.529) 0.658(0.678) 0.658(0.674)
0.747(0.596) 0.765(0.584)
90-03 0.868(0.423) 0.733(0.579) 0.626(0.662)
0.593(0.687) 0.538(0.723)
50-03 0.847(0.444) 0.714(0.588) 0.608(0.666)
0.573(0.686) 0.511(0.719)
-----------------------------------------------------------------------------
NINO4 CORR. SKILL (RMSE in Deg.C)
period
3-month 6-month
9-month
12-month 15-month
50-59 0.697(0.372) 0.600(0.419) 0.481(0.500)
0.352(0.531) 0.264(0.550)
60-69 0.827(0.259) 0.669(0.342) 0.606(0.367)
0.637(0.358) 0.508(0.409)
70-79 0.858(0.324) 0.712(0.449) 0.638(0.484)
0.557(0.521) 0.482(0.549)
80-89 0.786(0.354) 0.690(0.412) 0.710(0.389)
0.789(0.339) 0.788(0.340)
90-03 0.868(0.305) 0.751(0.408) 0.627(0.478)
0.624(0.481) 0.513(0.528)
50-03 0.825(0.339) 0.715(0.422) 0.634(0.464)
0.591(0.484) 0.517(0.514)
-----------------------------------------------------------------------------
Fig.1 shows the Nino3.4 SSTA predicted by the NN ensemble against the
observations (with linear trend added back).
Fig.2 shows the correlation skill (left column) of the NN model for
3-15 month forecast lead times. The difference of the correlation skill
between the NN model and the linear regression (LR) model (NN minus LR)
is shown in the right column. Areas with positive value (NN better than
LR) are shaded.
Fig.3 shows the RMSE (left column) of the NN model for 3-15 month
forecast lead times. The difference of the RSME between the NN model
and the LR model (NN minus LR) is shown in the right column. Areas with
negative value (NN better than LR) are shaded.
References
Smith, T.M., and R.W. Reynolds, 2004: Improved Extended Reconstruction
of SST [1854-1997]. Journal of Climate, 17, [in press]
Tang, B., W.W. Hsieh, A.H. Monahan and F.T. Tangang, 2000. Skill
comparisons between neural networks and canonical correlation analysis
in predicting the equatorial Pacific sea surface temperatures.
J.Climate, 13: 287-293.
Woodruff, S.D., Slutz, R.J., Jenne, R.L. and Steurer, P.M., 1987: A
comprehensive ocean-atmosphere data set. Bull. Amer. Meteorol. Soc.
6:1239-1250.
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