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 embedding 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 covers the same Jan. 1876-present period as before, but periods of missing data have been filled. Information on the new data set can be obtained from Rob Allan (rja@dar.csiro.au).

Figure 1 shows the analogue forecast starting from February 1996 and extending through May 1996. The SOI has continued to hover close to zero for the past 3 months. The selected analogues all show similar behavior over the analogue selection period (June 1995 to February 1996) and exhibit similar spread over the forecast period, compared to the forecast issued in December. The analogue forecast shows a weak downward trend from a small positive SOI value in March to weak negative values in April and May. Forecast values for the next three months (in SD units X 10) are:

March 1996 3.2 April 1996 -1.7 May 1996 -5.4

Verification of the forecasts for the previous three months:


		December 1995  	F= 1.5   V=  -5.5
		January 1996	F= 2.3   V=   8.4
		February 1996	F= 8.4   V=   1.0

Figure 2 shows the analogue forecast starting from January 1996 (one month earlier than for Fig. 1), and Fig. 3 starting from December 1995. While the forecasts from these three start times are not greatly dissimilar (in fact, some of the same years are seen to have been selected for all three starting months), a forecast for a rising SOI has tended to change to one of a falling SOI as the observed starting point has increased from what it was in December. The forecasts beginning from December and February have considerable internal spread. The verification of the two previous months' forecasts (Figs. 2, 3) was good for January and mediocre for December.

References

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 measurement error in time series. Nature, 344, 734-741.

Figures

Figure 1. Selected analogues and forecasts based on the SOI up to February 1996. 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.

Figure 2. As in Figure 1, except based on the SOI up to January 1996. The verifying value for February is indicated.

Figure 3. As in Figure 1, except based on the SOI up to December 1995. The verifying values for January and February are indicated.

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