[Previous Article] [Next Article]
Analogue (Non-Linear) Forecasts of the Southern Oscillation Index
Time Series
contributed by Wasyl Drosdowsky
Bureau of Meteorology Research Center, Melbourne, Australia
An analogue selection procedure, based on the non-linear time series forecasting technique of
Sugihara and May (1990), is applied to the Southern Oscillation Index (Drosdowsky 1994).
The time series to be forecast xi is "embedded" in an E dimensional space defined by a sequence
of lagged coordinates (xt, xt-g, x t-2g,..., x t-(e-1)g), where g is the lag interval, usually taken as one
time step. The E+1 closest neighbors (analogues) to the current state, defined by the vector xt, x
t-g, x t-2g,..., x t-(e-1)g, are found and used to construct the smallest simplex containing the current
state. Future states of the system are found by projecting each analogue forward nT, where
n=1,2,..., time steps and taking a suitably weighted average of the analogues. The optimal
embedding dimension E is determined by a trial and error procedure, using the library of patterns
formed by the first half of the time series to predict the evolution at each point of the last half of
the time series. This effectively determines the window over which the analogue is selected.
The forecast system has been tested on time series with known properties. For the SOI, the
optimal em-bedding dimension is found to be of order 9 to 12. The operational scheme has been
used in the monthly Seasonal Climate Outlook issued by the National Climate Centre of the
Australian Bureau of Meteorology since mid-1991. Analogues are selected from the entire
available SOI time series from 1876 to the present time. An element of persistence is included in
the forecast by adjusting the weighted analogue so that the t=0 value agrees with the current
observed base value.
The skill of the analogue system has been examined in hindcast experiments (Drosdowsky 1994),
and is shown in Fig. 9-1 in the September 1994 issue of this Bulletin. For RMSE the one time step
forecasts are approximately equal to persistence while the two or more time step forecasts are
more skillful than persistence within the appropriate range of embedding dimension. The spread of
the analogues during the forecast period can provide a measure of the confidence level of the
forecast.
Beginning with the forecast that appeared in the December 1994 issue, an improved SOI data set
has been used. It covered the same Jan. 1876-present period as before, but periods of missing data
were filled. Information on the data set can be obtained from Rob Allan (rja@dar.csiro.au).
Beginning with the September 1996 issue, SOI values have been calculated using means and
standard deviations from 1933-92 (Information: G.Beard@bom.gov.au).
Figure 1 shows the analogue forecast starting from February 1997 and extending through May.
The SOI has returned to moderate positive over the past three months, slightly higher than those
predicted in the December issue of this Bulletin. There is still little spread of the analogues over
the selection period (June to February), but much more spread over the forecast period, as
expected at this time of year. The selected analog years are now more evenly spread compared
with the December 1996 selection. Forecast values for the next three months (in SD units X 10)
are:
March 1997 8.6
April 1997 8.0
May 1997 8.4
Verification of the previous 3 months' forecasts:
December 1996 F= 0.2 V= 7.2
January 1997 F= 4.0 V= 4.1
February 1997 F= 0.9 V= 13.3
Figure 2 shows the analogue forecast starting from January 1997 (one month earlier than for Fig.
1), and Fig. 3 starting from December 1996. Some of the same years are seen to have been
selected for two or all three of the starting months. The January forecast hovers around near
normal SOI, while the December and February forecasts show a tilt toward positive SOI.
Drosdowsky, W., 1994: Analogue (non-linear) forecasts of the Southern Oscillation Index time
series. Wea. Forecasting, 9, 78-84.
Sugihara, G. and R.M. May, 1990: Nonlinear forecasting as a way of distinguishing chaos from
mea-surement error in time series. Nature, 344, 734-741.
Fig. 1. Selected analogues and forecasts based on the SOI up to February 1997. Points
corresponding to the January, February or March initial condition have been used for selecting
possible analogues. For clarity, only the best five analogues are plotted (light dashed or dotted
lines), labeled with the year and month corresponding to the current month. (The remaining five
analogues are listed to the right.) Heavy solid and dashed curves show the current and forecast
values.
Fig. 2. As in Fig. 1, except based on the SOI up to January 1997. The verifying value for February
is indicated.
Fig. 3. As in Fig. 1, except based on the SOI up to December 1996. The verifying values for
January and February are indicated.
