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 x_i is "embedded" in an E dimensional space defined by a sequence of lagged coordinates (x_t, x_t-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 x_t, 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 deter-mines 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 November 1996 and extending through February 1997. The SOI has weakened from moderate positive to near zero values over the past three months. There is still little spread of the analogs over the selection period (March to November) Two interesting features of the analogs are (I) the first appearance (as best analog) of the 1974-75 aborted El Niño event, and (ii) the high proportion of analogs from the 19th century (6 of the 10) and the 1960s (remaining 3 of 10). The behavior of the analogs during the forecast period (December to February) is fairly consistent, maintaining slightly positive SOI values. Forecast values for the next three months (in SD units X 10) are:

December 1996 0.2

January 1997 4.0

February 1997 0.9

Verification of the previous 3 months' forecasts:

September 1996 F= -1.0 V= 6.9

October 1996 F= -1.2 V= 4.2

November 1996 F= 0.1 V= -0.1

The forecast issued early September dropped to zero too rapidly, due to the influence of the first analog selected in that forecast as discussed in the September Bulletin.

Figure 2 shows the analogue forecast starting from October 1996 (one month earlier than for Fig. 1), and Fig. 3 starting from September 1996. Some of the same years are seen to have been selected for two or all three of the starting months. The forecasts have been hovering around slightly positive to near normal SOI for each of the three start times.

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 November 1996. Points corresponding to the October, November or December 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 October 1996. The verifying value for November is indicated.

Fig. 3. As in Fig. 1, except based on the SOI up to September 1996. The verifying values for October and November are indicated.

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