Multiple Regression, Discriminant Analysis and Unevaluated AGCM Predictions of Mar-Apr-May 1996 Rainfall in Northeast Brazil

Seasonal rainfall in the North Nordeste in northeast Brazil occurs mainly from February to May, with heaviest amounts in March and April. Experimental forecasts of North Nordeste rainfall at 1 and 0 month leads are issued using November-January and January-February predictor data, respectively. Two predictors found to deliver substantial forecast skill are (1) the 30oN-30oS portion of the third covariance-based EOF of Atlantic SST for all seasons, and (2) the first EOF of Pacific SST for Dec-Jan-Feb. Both of these EOF patterns are shown in the March 1993 issue of this Bulletin. The Atlantic EOF pattern reflects the SST anomaly immediately off the North Nordeste east coast and the large scale north-south SST gradient structure, while the Pacific EOF pattern serves mainly as an index of the ENSO situation. The amplitude time series of these predictors are used to predict North Nordeste rainfall both with multiple regression (giving a point forecast) and discriminant analysis (giving probabilities for each of five climatologically equiprobable [for 1951-1980] rainfall amount categories).

Details about the EOF analyses, the physical relevance of the predictors, and the two forecasting methods are given in Ward and Folland (1991). Multiple regression develops optimal weights for each predictor in order that the resulting linear equation minimizes squared errors between forecasts and corresponding observations over the training periods (1913-94, 1946-94). In discriminant analysis, categories of rainfall amount are defined, and, given values of the predictors, probabilities of each of the rainfall categories are determined using Bayes' theorem. Less linear constraint is imposed here than in multiple regression, as the probabilities do not necessarily change smoothly as a function of category.

Forecasts are made for three separate North Nordeste rainfall predictands: Nobre (for Feb-May), Hastenrath (Mar-Apr) and Fortaleza/Quixeramobim (FQ) (Mar-May). These are illustrated in Fig. 1. Each of these forecasts is done using both multiple regression and discriminant analysis. The forecasts presented here are only for the two predictands whose periods begin in March, making for a long-lead forecast: Hastenrath and FQ. The Hastenrath rainfall area occupies a central portion of the north Nordeste, while FQ is the rainfall averaged over the two stations, one of which is in the Hastenrath area.

If the amplitudes of the predictor EOFs are changing rapidly during the Nov-Jan period, values from Dec-Jan or only January may be used as predictors, if the more recent SST anomalies are expected to persist. In early March updated forecasts for the predictand periods are issued, using SST data through February.

To estimate forecast skill, multiple regression and discriminant analysis hindcasts for FQ based on the SST for Nov-Jan were made for the 1971-92 period using data from 1913-70, and for the 1981-92 period using data from 1913-80. The eigenvector patterns, computed from 1901-80 data, cause a slight dependency in the first experiment but complete independence in the second. The discriminant analysis forecast skill was assessed by comparing the observed category with the most likely category according to the hindcasts over the period 1971-92 (Table 1), while the point estimate rainfall amounts predicted by multiple linear regression were correlated with observed values. The resulting correlation for the 1971-92 experiment is 0.715 with a bias of +0.09 standard deviations and a root mean squared error (RMSE) of 0.62 standard deviations. For the totally independent (including the eigenvector pattern) period of 1981-92, the correlation is 0.662, with bias of +0.07 and RMSE of 0.70 standard deviations. While the latter results are not quite as high, it is shown in Ward and Folland (1991) that independence of the eigenvector patterns is not nearly as critical to estimation of independent forecast skill as independence of the periods used for statistical model development and for forecast testing.

                           	Observed
  	                Q1    Q2    Q3    Q4    Q5
	     Q1	4       2       0       2       0
	     Q2	0       0       0       0       0
Hindcast     Q3	0       1       1       0       0
	     Q4	0       0       0       0       0
	     Q5	1       2       0       3       6 

Table 1. Hindcasts (i.e. forecasts for already observed times, but with model derived without target years) of FQ rainfall index for 1971-92 using linear discriminant analysis. THE Q's are quintiles (Q1=very dry, Q5=very wet).

Experimental real-time forecasts for FQ using the methods discussed here have been made for each rainfall season since 1987. The forecasters combine the forecasts from discriminant analysis and multiple regression to determine the official forecast category. The forecast- observation correspondence from 1987 to 1995 is very good for the preliminary forecast (hit rate 6.5 out of 9), and slightly worse for the updated forecasts (4.5 out of 9). (Over a large number of cases the updated forecasts would be expected to have more skill.) Table 2 shows the record of real-time forecasts for 1987-95. It is clear that the FQ rainfall index is fairly skillfully predicted from the two SST EOFs--in fact, as much so as most any variable in the extratropical Pacific/North American region in any season.

             Year:   87   88   89   90   91    92  93   94   95
  Prelim forecast     1    4    5    2    4   1.5   2    5    4
  updated forecast    1    5    5    3    4    2    2    4   4.5
  observed            1    4    5    2    4   1.5   1   4.5   5

Table 2. Verification of experimental real time forecasts of NE Brazil rainfall (predictions of March-May rainfall at FQ). 1=very dry, ..., 5=very wet. The number of correct categorical forecasts out of 9 (hit rate) is 6.5 for the preliminary (1 month lead) forecast, and 4.5 for the updated (zero lead) forecast.

1996 Forecast

Figures 2 and 3 show the monthly time series of the Atlantic and Pacific SST anomaly predictors used in the regression and discriminant analysis prediction models. Both predictor values are near average at the end of 1995, with little change between December and January.

Atlantic: SST has been mostly above average between the equator and 30oN since November 1995. SST was also above average off the southeast coast of Brazil in January and in the Gulf of Guinea in November. Elsewhere SST is near average. These SST anomalies (apart from the Gulf of Guinea) favor drier conditions in NE Brazil.

Pacific: SST is below average in the equatorial central and east Pacific, in a pattern that is normally associated with above average rainfall in NE Brazil. However, there are negative SST anomalies north of 25oN that have the opposite rainfall influence, and the net effect of this predictor is weak.

Example multiple regression equations for the 1-month lead forecast for the Hastenrath (for Mar-Apr) and FQ (for Mar-May) rainfall indices (standardized rainfall anomaly units), based on 1913-1995 data, are:

where the EOF time coefficients (A=Atlantic EOF, P=Pacific EOF) are not standardized. (The Atlantic series varies between about -2.5 and 1.4 between 1981 and 1995, while the Pacific series varies between about -4.0 and 9.7).

For the 1-month lead linear regression predictions, we calculate the average of predictions made using training periods 1913-95 and 1946-95, and SST anomalies for November, December and January. The result is:

  predictand         forecast   quint  quint range  stand. error 
  Hastenrath          -0.12       3   -.16 to .27          0.57
     FQ               -0.19       2   -.70 to -.16         0.65

The Hastenrath forecast is just above the Q2/Q3 (dry/average) quintile boundary, and the FQ forecast is just below the Q2/Q3 boundary. The multiple regression forecast for the Nobre (Feb-May) predictand is also very close to the Q2/Q3 boundary. The standard errors (in standard deviation units) associated with these forecasts express the inherent uncertainty based on the training period statistics.

The discriminant analysis produces the following probabilities that the Hastenrath and FQ indices will be in each of the quintiles for the 1-month lead time:

                very dry     dry     average    wet   very wet					        

  Hastenrath     .14         .33       .29      .17      .07
        FQ       .25         .25       .17      .26      .07

The discriminant analysis predictions are less consistent than the linear regression predictions, with some bimodal behavior, but all favor below average rainfall.

Our best estimate forecast is for DRY/ AVERAGE conditions (quint 2/3 boundary) for each of the rainfall indices (Hastenrath, FQ, Nobre).

As in 1994 and 1995, a dynamical prediction of Northeast Brazil rainfall was made using a version of the UKMO climate atmospheric general circulation model (AGCM). A similar version of the AGCM showed very high skill in simulating interannual variability of Northeast Brazil rainfall when forced with observed SST. The 1994 and 1995 AGCM forecasts were in good agreement with the statistical predictions. However, skill with persisted SST anomalies has not yet been fully assessed. The AGCM was run from Jan 23, Jan 31 and Feb 1 start dates, using persisted January SST anomalies, to the end of May. All three samples give below average March-May rainfall for NE Brazil, with a net value about 10% below average for the season compared to the model climatology.

Although there is agreement between the statistical and dynamical predictions, confidence in predictions of near-average conditions is generally lower than that for extremes, so overall confidence in this forecast is moderate.

UPDATED (ZERO-LEAD) FORECASTS, WITH INCLUSION OF FEBRUARY DATA

While zero-lead forecasts are not encouraged for this Bulletin, they appear here as auxiliary information accompanying long-lead forecasts for the same targets. In February the Atlantic predictor decreased sharply while the Pacific predictor increased slightly (see Figs. 2, 3). Consequently the zero-lead statistical forecasts indicate wetter conditions (average at the time of writing) than the 1-month lead forecast.

Ward, M.N. and C.K. Folland, 1991: Prediction of seasonal rainfall in the North Nordeste of Brazil using eigenvectors of sea surface temperature. Int. J. Climatol., 11, 711-743.

Figures

Figure 1. Locations of the stations used in the Hastenrath rainfall time series, and the Fortaleza and Quixeramobim stations. The Nobre rainfall time series is based on stations throughout the bounded region indicated.

Figure 2. Amplitude time series for the Atlantic eigenvector for Jan 1990 to Feb 1996. Positive values (e.g. SST anomalies warm in north tropical Atlantic, cool in south tropical Atlantic) are associated with drier conditions.

Figure 3. Amplitude time series for the Pacific eigenvector for Jan 1990 to Feb 1996. Positive values (e.g. SST anomalies warm in the central-east equatorial Pacific, cool in the northwest and southwest Pacific) are associated with drier conditions.