Forecasts of Surface Temperature and Precipitation Anomalies over the U.S. Using Screening Multiple Linear Regression

Screening multiple linear regression (SMLR) is used to predict seasonal temperature and precipitation amounts for locations over the mainland United States. Predictor data consist of northern hemisphere 700-mb heights, near global SSTs and station values of mean temperature and total precipitation amount from the 3-month period prior to the forecast initial time of March 1, 1996. Forecasts for the mean temperature and total precipitation are made for a series of 13 overlapping 3-month periods, at one month intervals, beginning with Apr-May-Jun (AMJ) 1996 and extending through AMJ 1997. Regression relationships were derived from data for the 1955-95 period. Forecasts were produced from single station equations for 59 stations approximately evenly distributed throughout the mainland U.S.

All predictors and predictands were expressed as standardized anomalies relative to the developmental data. Precipitation amounts were transformed by taking their square roots prior to standardization in order to help normalize their distribution. Twenty-five candidate predictors, selected from gridpoint values in regions of known importance for climate prediction, were offered for screening in the regression development. A few predictor locations were chosen on the basis of data examination of the first 20 years of the sample, referred to here as the base period. Information from the most recent 20 years was never used for selection of candidate predictors (Unger, 1996a).

Initial testing indicated that cross-validation cannot be used for SMLR (Unger 1996b) so a variation of a retroactive real time (RRT) validation technique was used. To estimate skill by RRT, a forecast equation was derived from the base period and applied to the next year's data to obtain independent data results. The case was then added to the developmental sample, a new relationship was derived and applied to the following year's data. Independent data statistics accumulate on a year by year basis in exactly the same way as an operational forecast procedure, except retroactively. Forecasts were then obtained for the base period years by application of RRT in reverse: deriving from the "future" years and applying to the most recent year in the withheld period (now the first half of the sample). Each earlier case was then included in the development sample, the relationships re-derived and applied to the next earlier case. This bi-directional RRT (BRRT) validation technique provides that each available case contribute to a skill estimate as independent data in a way similar to cross-validation except with a great reduction in the distortion of results due to redundant sampling (Unger, 1996b).

A forward selection screening procedure was used for equation development. The top 5 terms were selected for each equation. Separate statistics were accumulated for each equation length, so that results for all the one, two, three, four and five term equations were calculated. The optimum equation length was then estimated by an objective learning procedure that used the past performance at each RRT trial to "predict" which equation would perform the best on the next. Verification statistics from this "best guess" forecast were also kept separately and were used to obtain the final skill estimate of the forecasts.

The verification used was the temporal correlation coefficient between forecast and observation on the 40 independent cases at each of the 59 stations. An average correlation coefficient was computed from the root mean squared correlation coefficient with the signs retained both in the squaring process and the final square root. Field significance was measured by comparison of scores from actual target years against scores determined from 500 randomly shuffled target periods. Field significance expresses the percentage of time that the random forecast series outperformed the actual forecasts.

The final forecasts are post-processed to obtain an estimate of the likelihood of the above, normal, or below class being observed, as defined by the terciles of the distribution for each forecast element and location. A forecast is assigned a class on the basis of the forecast distribution and skill. An estimate of the increased likelihood of a given class is made to place the forecast in a format similar to the operational long lead forecasts issued by the CPC (O'Lenic, 1994). Currently these probability assignments are obtained from the relationship between probability of a given class being observed, the inflated SMLR forecast and the predictive skill. (Inflation sets the forecasts variance equal to observed variance at each station.) This relationship is based on forecast performance on independent data. If the correlation skill of the forecast is under approximately .3, the forecast is not assigned to a class and is regarded as a climatological forecast.

The forecasts for AMJ 1996 are shown in Figs. 1 and 3 with the corresponding skill estimates for each station shown in Figs. 2 and 4. Temperature forecasts for JJA 1996 and the associated skill are shown in Figs. 5 and 6 respectively. Shading indicates areas of sufficient skill to assign a tercile category to the forecast. Contours within the shaded areas on the forecast maps indicate estimates of a 5 and 10 percent probability anomaly for the category. Note that the skill estimates are based on the actual forecasts, and not the post processed category assignments, which are presented only for clarity of presentation.

The numbers plotted in Figs. 1, 3, and 5 indicate station values of the regression forecast for the standardized anomaly of temperature or the square root of precipitation amount. Forecasts are damped according to the forecast-observation correlation on independent data so that the squared error between forecast and observation will be minimized. Non-zero numbers plotted outside of shaded regions generally indicate forecast anomalies of substantial magnitude at stations with some skill, but lower than the skill threshold to choose a forecast category with confidence.

Temperature forecasts (Fig. 1) show below normal temperatures along the North Pacific coast, and above normal temperatures in scattered locations in the southern U.S. and eastern New England. The average correlation for this forecast is .25 with a field significance of .000. The forecasts show some contradiction near the Great Lakes, where areas of below normal, near normal and above normal temperatures neighbor each other in a region where the spatial correlation of temperature anomalies are quite high.

Because the regression forecasts are derived from single station equations, inconsistencies, such as those near the Great Lakes for AMJ 1996 can be considered to be similar to an ensemble forecast for the region, with the variability in the forecasts introduced by the station to station variability in the development sample. Thus, different forecasts where temperature anomalies are spatially correlated might be considered to be different "opinions", or possibilities, for the region. While the contradictory predictions may provide useful guidance to assess uncertainty, they diminish the value of the forecasts by failing to provide a realistic final assessment for the region. Future work will be required to provide an objective method to locate and resolve inconsistent forecasts.

Precipitation forecasts for AMJ 1996 show considerably less skill than for temperature forecasts, with an average correlation of .11 and a field significance of .11. Forecasts show a weak tendency for below normal precipitation over the East coast and in central Texas.

The summertime temperature forecast for the U.S. is shown in Figure 5. The East coast and southwestern U.S. show a tendency to be warm, with some pockets of near normal scattered in some localized regions. The average correlation for this map is .23 and the field significance is .002. Precipitation forecasts for summer (not shown) hint at continued dry conditions extending from Ohio to Virginia.

O'Lenic, E., 1994: A new paradigm for production and dissemination of the NWS's long lead-time seasonal climate outlooks. Proceedings of the Nineteenth Annual Climate Diagnostics Workshop. College Park, Maryland, November 14-18, 1994. 408-411.

Unger, D. A., 1996a: Long lead climate prediction using screening multiple linear regression. Proceedings of the Twentieth Annual Climate Diagnostics Workshop. Seattle, Washington, October 23-27, 1995 (in press).

Unger, D. A., 1996b: Skill assessment strategies for screening regression predictions based on a small sample size. Preprints, Thirteenth Conference on Probability and Statistics in the Atmospheric Sciences. San Francisco, CA., February 21-23, 1996. 260-267.

Figures

Figure 1. (left) A 1-month lead screening regression-based temperature forecast for AMJ 1996. Contours are estimated probability anomalies for the specified tercile. Shaded areas delinate the area of correlation skill greater than 0.3. Plotted numbers are station values of the standardized anomaly.

Figure 2. (right) Distribution of skill for the 1-month lead regression forecast for AMJ 1996 temperatures. The values shown are the correlation between forecast and observation for the 1956-1995 period.

Figure 3. (left) Same as figure 1 except for precipitation forecasts.

Figure 4. (right) Same as figure 2 except for precipitation skill.

Figure 5. (left) Same as figure 1 except for JJA 1996.

Figure 6. (right) Same as figure 2 except for JJA 1996.