Multiple Regression and Discriminant Analysis to Predict

Mar-Apr-May-Jun 1997 Rainfall in Northeast Brazil

contributed by Larry Greischar and Stefan Hastenrath

University of Wisconsin, Madison, Wisconsin

In the approach used at the University of Wisconsin to forecast March-to-June precipitation in the Nordeste, several predictors are used with stepwise multiple linear regression and linear discriminant analysis. The predictand includes 27 selected stations in the Nordeste (Hastenrath and Greischar 1993), shown in Fig. 1. The forecasts shown here are made at one month lead--i.e., using data no later than January 1997. The potential predictors of March-June Nordeste rainfall (not all of which are necessarily used in a given prediction model) are listed below along with their correlations with the predictand over the training period, 1921-57. Correlation coefficients are in hundredths, with one and two asterisks indicating significance at the 5% and 1% levels, respectively.

(1) October-January precipitation at the 27 predictand stations. +55**

(2) An index of January meridional surface wind component over the tropical Atlantic, 30^oN-30^oS. -35*

(3) An index of January SST in the equatorial Pacific. -11

(4) An index of January SST in the tropical Atlantic, 30^oN-30^oS. -57

(5) An index of Nov-Dec-Jan SST in the tropical Atlantic, 30^oN-30^oS. -70**

As shown above and described in Hastenrath and Greischar (1993), March-to-June rainfall in the Nordeste is correlated positively with the first and negatively correlated with all the other predictors. This pre-season, predictor values have been: (1) positive pre-season rain, (2) positive (i.e. southerly wind anomaly), (3) negative Pacific SST, (4) positive tropical Atlantic SST index (i.e. warm/cold SST anomalies to the North/South of the Atlantic equator), and (5) positive tropical Atlantic SST index for Nov-Dec-Jan (as in (4)). The equatorial Pacific SST and the pre-season Nordeste rainfall indicate a wetter than normal March-June 1997, while the Atlantic SST field for January 1997 and Nov-Dec-Jan 1996-97 and meridional surface wind component point to dry conditions.

Table 1 shows skill evaluations for eight prediction models, each using a different combination of the predictors listed above. The eighth model uses a neural network rather than stepwise multiple linear regression. Skill (% Variance Explained) Rainfall

------------------------------------------ Forecast

Training Forecast Forecast Mar-Apr-

Model Predictors Period Period 1 Period 2 May-Jun

#/Type Used 1921-57 1958-89 1968-89 1997

------ ---------------- -------- -------- -------- --------

1 SMR (1) 30 35 49 +.39

2 SMR (1),(2) 38 49 69 +.10

3 SMR (1),(4) 49 52 66 -.08

4 SMR (1),(2),(4) 44 58 74 -.09

5 SMR (1),(2),(3),(4) 50 61 74 -.15

6 SMR (1),(3),(5) 62 61 71 -.26

7 SMR (3),(5) 56 58 62 -.51

8 NN (1),(2),(3),(4) 55 66 81 -.03

Table 1. Skill of eight prediction models for Mar-Apr-May-Jun Nordeste rainfall (expressed as a percentage of predictand variance explained), followed by the forecast standardized Nordeste rainfall anomaly for Mar-Apr-May-Jun 1997. The model type is SMR (stepwise multiple regression) or NN (neural network), and predictors numbers (1)-(5) are as shown above.

The models indicate near average Nordeste precipitation, with a very slight tilt toward the low side of the normal. The 1912-56 historical mean and standard deviation are 500 and 200 mm, respectively.

Using the same sets of predictors, Nordeste rainfall was also predicted using linear discriminant analysis, in which five equiprobable categories of rainfall amount are defined and associated with predictor values using Bayes' theorem. In an individual forecast each of the categories is assigned a probability, given the pre-season values of the predictors. Table 11-2 in the March 1995 issue of this Bulletin shows, for predictor models 5 and 7, the five-by-five verification matrices obtained over the 1958-89 period using the earlier years to develop the models. The hit rates are 0.34 and 0.38, corresponding to Heidke skills of 0.18 and 0.22 and expected correlation skills of about 0.55 and 0.65. The quintile probability forecasts for Mar-Apr-May-Jun 1997 using each of these predictor models are:

Probability of.....

Model Predictors Q1 Q2 Q3 Q4 Q5

5 (1),(2),(3),(4) .28 .16 .10 .30 .16

7 (3),(5) .28 .29 .23 .16 .04

Model 7 shows a maximum likelihood of a somewhat drier than normal 1997. Model 5 gives a slight edge to category 4 (wet), although the wide disparity between the two categories with nearly equally high probabilities (1 and 4) does not inspire confidence in this model. If the probabilities given by the two models are averaged for each category, the resulting forecast tends to lean toward the drier categories.

In summary, both the field of meridional wind component and the SST pattern (and the associated OLR pattern showing a northward displaced near-equatorial convection belt, not discussed above) in the tropical Atlantic indicate slightly below normal March-June 1997 Nordeste precipitation. On the other hand, the positive pre-season Nordeste rainfall anomaly and the cool equatorial Pacific SST favor wetter conditions. Based on the several stepwise multiple regression and linear discriminant analysis models, near to somewhat below average precipitation is forecast for the 1997 rainy season, with standardized anomalies of -0.5 to +0.4. This is comparable to the years 1978, 81, 82, 87, and 91.

Hastenrath, S. and L. Greischar, 1993: Further work on the prediction of northeast Brazil rainfall anomalies. J. Climate, 6, 743-758.

Fig. 1. Locations of the 27 selected stations in the Nordeste, used as the predictand by Grischar and Hastenrath.

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