In this section, the so-called multi-linear regression (MLR)-method developed for GCM- downscaling in Chapter 3 is adapted for short-term climate prediction in the study region, by incorporating such GCM- predictors. In addition, the teleconnective ocean indices already used in the ARIMAex- models will also be employed in the MLR-model. Such linear regression models describing linear relationships between some ocean state- or atmospheric precursors and local climate predictands were applied successfully for short-term, seasonal climate forecasts (e.g.
Hastenrath 1995, Allan et al. 1996, Francis and Renwick 1998, Goddard et al. 2001).
4.4.1 Development of the MLR- methodology
The multi-linear regression (MLR)- method exhibited in Chapter 3 as a particular downscaling technique delivers mathematical relationships between a GCM’s grid data (predictors) and local climate variables (predictands). This basic MLR-method is extended here to also include oceanic indices (SSTs) as additional predictors so that a mixed predictor-set of monthly atmospheric GCM-predictors and ocean state indices is available for seasonal climate forecasting in the study regions. More specifically, similar to ARIMAex-models, the teleconnective ocean predictors are appropriately time-lagged.
The complete GCM/ocean indices- MLR-model is then written as a dynamic regression model:
where is the dependent climate variable vector, is the independent GCM-predictor vector and are the corresponding unknown regression coefficients. The term [ ]
denotes the regression of the optimally-lagged ocean indices with regression coefficients . When this term is shut off, the MLR-method with GCM-predictors of Chapter 3 results.
In the subsequent MLR-analysis regressions with and without the incorporation of the ocean teleconnectors will be done. The coefficients of the regression model, Eq. (4.11), are determined by classical least squares fitting of the observed climate series in the study area.
The determination of the optimal ocean index S and its corresponding optimal lag-time for a particular climate variable X at one the observing stations in the study region is based on the time-lagged cross-correlation between these two time series, as discussed in Chapter 2, namely, the cross-correlation function , defined in Eq. (2.16). For monthly data, the optimal lag-time is then the number of lags k, in months, for which is maximal.
As discussed in Chapter 2 and also in connection with the ARIMAex – models in the previous section, in order to have a precursory causal relationship between an ocean index and a local climate variable, only negative lag-times are of interest here. From cross-correlation plots, similar to Figure 2.30, the maximum correlations and the corresponding negative optimal lag-times
are retrieved for all combinations of the 13 different Indian- and Pacific Ocean state indices (SSTs) and the 4 temperature- and 24 precipitation climate series in the study area. The SSTs are then appropriately time-lagged and incorporated into the MLR-model, Eq. (4.11).
Table 4.11. List of predicting techniques and predictor-sets used in the various experiments of short-term climate prediction.
group
no predictor-set experiment
single-domain model multi-domain MLR with predictor sets SDSM LARS-WG MLR HiRes GCM GCMs ocean indices
GCMs
1 ECHO-G(SDSM) ●
2 ECHO-G(LARS-WG) ●
3 ECHO-G ●
4 HiRes ●
5 GCMs ●
6 GCMs+HiRes ● ●
+SSTs
7 ECHO-G + SSTs ● ●
8 HiRes+SSTs ● ●
9 GCMs+SSTs ● ●
10 GCMs+HiRes+SSTs ● ● ●
11 SSTs ●
Table 4.11 summarizes the various experiments carried out for the short-term prediction with various combinations of predictors used in the general MLR-model, Eq. (4.11), i.e. with either GCM- or SST- predictors or both (+SSTs). Also for comparison, in addition to the MLR-model, the two single-domain statistical and stochastic downscaling tools SDSM and LARS-WG (see Section 3.4), are applied in conjunction with ECHO-G predictors (no.1 and no.2 in Table 4.11) in these short-term (nos. 1 and 2) climate experiments. The various atmospheric predictor variables from the GCM-database which are similar to the GCM-predictor sets used in the MLR-downscaling in Section 3.5, i.e. ECHO-G, HiRes and multi-GCM database (see Section 3.5.2),
[ ] (4.11)
are likewise used in the single- and multi-domain MLR- models (nos. 3 to 6). These GCM-predictors are available for the 20c3m- simulation experiment for years 1971-1999 and climate projection year 2000, based on SRES A2. The combination of GCM- predictors with lagged ocean indices (+SSTs) in the MLR- model is indicated by nos.7 to 11, where the latter refers to a pure SST- regression model as discussed above. For the +SSTs group the optimal lag time for each ocean index is selected, as mentioned above, and will be discussed further in Table 4.12.
4.4.2 MLR-seasonal prediction
The predicting performance of the MLR- short-term prediction model with pure SST-predictors can be enhanced further by applying the model to individual seasons of the year. This means that the regression is not only done for the total of the measured monthly (annual) climate series at a station, but also for split data-sets of different seasons of the year, following the three seasonal schemes discussed earlier in Section 2 (see Table 2.4), namely, a division of the 12 months of a year into periods of 2, 3 and 4 seasons. This seasonal regression approach is similar to that of Chapter 3, with the difference that unlike there, where GCM-predictors were used, here the appropriately lagged teleconnective ocean indices are employed as regressors.
Each seasonal linear regression model between ocean state and local climate variables is then identical to Eq. (2.19), only that the corresponding seasonal data is processed for each seasonal scheme. The slope and intercept of the linear regression are determined again by the method of least-squares, and the goodness of the regression fit is measured by the coefficient of determination, R2. The lag operation in Eq. (2.19) is applied repeatedly for different lag-times of the observed ocean variable to find the optimal lag-time , which is the one providing the best predicting skill at a particular climate site. As such, the methodology is similar to the seasonal-scheme- MLR-downscaling approach with GCM-predictors of Section 3.5.4.2.
For the two-season scheme, i.e. the dry season [Nov-Apr] and the wet season [May-Oct], and using the El Niño 1+2 SSTs as predictors - which were found in Chapter 2 overall as the most suitable of the 13 Indian- and Pacific Ocean state indices analyzed, - the results of this analysis are shown for climate station 48459 in Figure 4.4.
One may note from the various panel of Figure 4.4 that, depending on the lag-time τ used, the (predictor, predictand) pairs ( , ) are more or less scattered around the linear regression lines, whose coefficient of determination R2 - which for a simple linear regression is equal to the square of the Pearson correlation coefficient r2 - are listed on top of each panel.
The quantitative results of Figure 4.4 are again summarized in Table 4.12, which lists the correlation coefficients at different lags for the annual-, the dry- and the wet season climate series. One can recognize from the table that the maximum temperature series at this climate station is optimally related with the El Niño 1+2 SSTs for a lag of -2 months, while the optimal lags for Tmin and the precipitation are situated between -3 and -4 months.
This seasonal correlation analysis has been extended to include all 13 ocean indices (see Figure 2.3) and to find the best average indices with their optimal lags for the MLR- seasonal prediction of the three climate variables across the study region. The results are summarized in Table 4.13, which lists the Pearson-correlation coefficients and the optimal lags of all combinations of the three climate series with the 13 ocean indices. One may notice from the table that, although the El Niño 1+2 SST has the strongest teleconnective relationship with the annual temperature (s0), for the 4-season regression schemes, the EPO-index works best for the dry, pre-monsoon- and
Lag = 0 Lag = -2 months Lag = -3 months Lag = -4 months
a = annual, s1=dry season [Nov-Apr], s2= wet season [May-Oct]
Tmax(°C) PCP(mm/day)Tmin(°C)
Figure 4.4. Annual and seasonal regression of 1971-2005 El Niño 1+2 and climate at station 48459 associated with time lag 0, -2, -3 and -4 months showing coefficient of determination (R2) by annual and seasonal regression separated into 2 seasons (a = annual, s1=dry season [Nov-Apr] in grey dots, s2= wet season [May-Oct] in blue dots).
Table 4.12. Cross-correlations squared (R2 ) between El Niño 1.2 and local climate at station 48459 (extracted from the regressions in Figure 4.4) as a function of lag-times for single season (annual) and 2-season schemes (dry and wet).
variable season R2 best lag
(month) lag = -2 month lag = -3 months lag = -4 months
Tmax annual (a) 0.41 0.2 0.03 -2
dry (s1) 0.55 0.50 0.23 -2
wet (s2) 0.38 0.30 0.09 -2
Tmin
annual (a) 0.43 0.54 0.42 -3
dry (s1) 0.55 0.52 0.26 -2
wet (s2) 0.39 0.40 0.25 -3
PCP
annual (a) 0.02 0.11 0.20 -4
dry (s1) 0.14 0.16 0.12 -3
wet (s2) 0.026 0.032 0.03 -3
monsoon1- seasons. For the precipitation, while the El Niño 1+2 SST is the best regressor in the annual regression, the EPO-index works better for the dry-season regression.
From these results one can conclude that the El Niño 1+2 SST are optimal for the annual regressions, the EPO-index should be considered for the seasonal-, namely, the dry-season regressions.
Table 4.13. Average cross-correlation coefficient r and average lag-time (lag) between El Niño 1.2 and the local climate following single season (annual) and 4-season schemes.
vari- able indices annual (s0) 4-season scheme (s4)
dry (s1/4) pre-monsoon (s2/4) monsoon1 (s3/4) monsoon1 (s4/4)
r1 lag2 r1 lag2 r1 lag2 r1 lag2 r1 lag2
Tmax
epo -0.38 8.0 0.34 -5.3 -0.11 -6.0 0.21 -3.0 0.35 -4.8 nino12 0.59 -1.8 0.29 -1.8 0.55 -1.8 0.56 -0.8 0.40 -4.0 nino3 0.56 -0.5 0.38 -0.3 0.48 0.0 0.52 0.0 0.35 -3.0 nino34 0.47 0.0 0.41 0.0 0.39 -2.5 0.45 -5.0 0.29 -2.8 nino4 0.36 1.3 0.47 -4.0 0.42 -10.0 0.42 -4.5 0.24 -1.3 noi -0.23 -2.0 -0.34 -1.5 -0.28 -6.0 -0.39 -0.8 -0.30 -1.8 pdo 0.15 2.0 0.31 -5.0 0.25 -9.5 0.38 0.0 0.23 -3.3 pna 0.13 -1.8 -0.04 -5.8 -0.05 -5.5 0.06 -7.5 0.23 -3.0 setio 0.08 0.0 -0.37 -1.8 0.28 -0.3 0.23 -2.3 0.34 -2.5 soi -0.27 -2.0 -0.43 -2.0 -0.28 -6.0 -0.44 -1.3 -0.25 -3.8 swio -0.08 -2.8 -0.40 -2.0 -0.29 -5.5 -0.21 -5.3 0.21 -7.0 wp 0.12 2.0 0.15 -5.5 0.25 -1.8 -0.03 -6.8 0.22 -6.8 wtio 0.30 0.8 0.35 0.0 0.43 -1.0 0.39 -1.5 0.32 -5.5
Tmin
epo 0.51 0.0 0.75 0.0 0.74 -1.0 0.34 -1.0 0.53 -9.0 nino12 0.75 -3.0 -0.02 -8.0 0.73 -1.0 0.46 -1.0 0.21 -3.3 nino3 0.59 -2.0 0.59 -4.8 0.58 0.0 0.50 -1.0 0.45 -1.5 nino34 0.42 0.0 0.47 -4.0 0.40 -2.8 0.49 -1.8 0.41 -1.0 nino4 0.32 1.5 0.29 -4.0 0.41 -10.0 0.50 -3.0 0.34 -1.0 noi -0.16 -2.0 -0.18 -5.3 -0.21 -7.0 -0.45 -2.0 -0.22 -2.8 pdo 0.24 0.0 0.18 -3.0 0.21 -7.3 0.44 0.0 0.12 -2.5 pna 0.16 -3.0 -0.06 -9.0 0.17 -3.0 0.24 -2.5 -0.18 -8.0 setio 0.14 7.0 0.06 -7.0 0.27 0.0 0.23 -1.5 0.13 -3.5 soi -0.16 -3.0 -0.23 -7.0 -0.26 0.0 -0.53 -2.0 -0.23 -2.0 swio 0.15 2.0 -0.18 -4.0 -0.22 -3.0 -0.14 -3.0 0.21 -7.3 wp 0.10 -5.0 -0.07 -3.5 0.20 -1.5 0.12 -4.5 0.24 -4.0 wtio 0.15 1.0 0.22 -5.5 0.34 -1.0 0.45 -5.5 0.04 -5.8
PCP
epo -0.40 2.8 -0.75 -10.0 0.04 -1.7 0.07 -4.3 -0.46 -9.0 nino12 -0.50 1.0 -0.30 -8.8 -0.09 -5.7 -0.26 -0.8 0.47 -7.7 nino3 -0.46 2.6 0.41 -4.5 0.06 -9.5 -0.23 -1.6 0.32 -5.2 nino34 -0.33 3.4 -0.04 -6.2 0.01 -9.2 -0.09 -3.8 0.23 -4.1 nino4 -0.21 3.8 -0.30 -7.0 -0.08 -4.8 0.01 -5.3 0.20 -4.8 noi -0.14 -5.8 -0.12 -5.2 0.15 -5.0 0.03 -4.2 -0.24 -7.0 pdo 0.18 -1.7 0.18 -3.2 0.14 -9.8 0.20 -6.2 0.19 -4.7 pna 0.13 -3.8 -0.01 -7.4 0.21 -7.8 -0.01 -3.8 0.26 -5.6 setio 0.06 -6.6 0.19 -7.2 0.08 -5.5 -0.17 -2.3 -0.09 -7.4 soi -0.05 -4.0 0.06 -4.9 0.09 -3.4 0.07 -3.3 -0.20 -7.3 swio 0.01 0.7 0.18 -3.9 0.08 -5.2 -0.10 -6.2 -0.02 -6.0 wp -0.09 2.8 -0.18 -1.4 0.14 -4.8 -0.06 -8.7 -0.14 -3.7 wtio -0.08 6.2 0.12 -4.0 0.03 -4.8 0.02 -4.4 0.37 -9.1
1the best models in each group are highlighted in bold italics
2lags may be fractional, because of averaging