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  • View in gallery

    The correlation patterns rl for the four field-significant leads. For clarity, the shown fields include all correlations whose magnitude exceeds 0.15, not the derived local significance threshold (which is ∼0.2). The contour interval is 0.05, and the minima–maxima are indicated by plus (+) symbols. Below the color bar, each panel's lead is indicated by L, and the temporal correlation between the 1925–2002 SLPA projection time series on the pattern and NAOI L months later is given by r

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    Results of recalculating the correlation patterns after sequential 3-yr withholding. Since the full time series comprise 78 yr (1925–2002), there are Nw = 76 possible sequential combinations [in which yr {1, 2, 3} {2, 3, 4} ⋯ {76, 77, 78} are withheld]. Each correlation field comprises Ng = 267 grid points. (a) Ratio of the computed field-significance criterion to the corresponding threshold. For each field-significant lead, the 76-experiment mean is given by the open circle, while the vertical bar shows the minima–maxima range of the 76 experiments. (b) Spatial correlations of the patterns derived from the deficient datasets (after withholding) and those derived from the full datasets (rl)

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    The dependence of the predictive patterns on the particular choice of analysis period. The datasets are split into five different periods, given in the lowermost panel. (a) Correlation patterns are recalculated based on the information of that period only. For each lag–subperiod combination, the spatial correlation of the partial pattern with the full pattern (1925–2002, on which the paper is based) is reported. (b) Lagged temporal correlations between the NAOI time series, and the time series that results from projecting the partial SLPA data on the original (1925–2002) pattern. (c) The t-test significance of these temporal correlations, taking note of lost degrees of freedom due to serial correlations. In (c) the four dashed horizontal lines correspond, from the top downward, to p = 0.05, p = 0.01, p = 0.005, and p = 0.001

  • View in gallery

    The vertical solid lines divide the panels into four subareas, corresponding to the number of predictors used [K in Eqs. (3) and (4) and in (b)]. As stated in (a), the correlated time series comprise 78 yr, with, taking note of serial correlations, ∼37 degrees of freedom. In (a) the h symbols report hindcasting skill [solving Eq. (3)] for each combination. The 3-yr random (sequential) withholding cross-validation skills are given in (a) by the open circles (plus symbols). (b) The F-test significance (times 100) of the cross-validation skills reported in (a), using dfK and K for the error and regression degrees of freedom, respectively. (c) Various leads participating in the combinations of (a) and (b) (omitting the “0.5” for brevity)

  • View in gallery

    Results for two of the most skillful lead combinations (numbers 10 and 13, with lead-time combinations shown in each panel).

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    Autoregressive test of all 15 predictor combinations [1 and 2 in (a) through combination 15 in (h)]. All panels show the cumulative probability density function of 600 trials in which the entire cross-validation procedure is repeated, each time with a different “NAOI-like” time series generated from an AR(2) fit to the actual NAOI time series. The solid (dashed) line corresponds to the first (second) combination shown in the panel. The numbers in the lower left give the number of trials whose resultant cross-validation skills equaled or exceeded that of the same predictor combination with the actual (rather than synthetic) NAOI time series. Those actual skills are given by the vertical solid or dashed bars. For each combination two actual skills are shown (very near each other), for the random and sequential cross-validation calculations.

  • View in gallery

    The results with the NP domain including only Trenberth and Hurrell's (1994) NP region, 30°–65°N, 160°E–140°W. The vertical solid lines divide the panels into three subareas, corresponding to the number of predictors used [K in Eq. (3) and in (b)]. As stated in (a), the correlated time series comprise 78 yr, and, taking note of serial correlations, ∼37 degrees of freedom. In (a) the h symbols report hindcasting skill [solving Eq. (3)] for each combination. The 3-yr random withholding cross-validation skill is given in (a) by the various dot-encompassing open symbols (circles for one predictor, diamonds for two predictors, and a triangle for all three). (b) The F-test significance (times 100) of the cross-validation skills reported in (a), using dfK and K for the error and regression degrees of freedom, respectively. (c) Various leads participating in the combinations of (a) and (b) (omitting the “0.5” for brevity).

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Forecasting the North Atlantic Oscillation Using North Pacific Surface Pressure

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  • 1 Department of the Geophysical Sciences, The University of Chicago, Chicago, Illinois
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Abstract

A statistical forecasting scheme for the North Atlantic Oscillation (NAO) is presented. The forecasts are based on Pacific Ocean surface pressure. With 21 months lead time, skills are significant and useful but modest. At 15-month lead, forecasts are robust and skillful under stringent cross validation, and further improve at 12-month lead. Cross-validated 1-yr forecasts correlate with 1925–2002 observations at ∼0.45. Other performance measures indicate similar skills, out performing the best autoregressive models of the NAO index. Importantly, skills of North Pacific–based forecasts easily exceed those of North Atlantic–based ones using identical machinery. The results suggest that NAO variability is not exclusively internal to the North Atlantic, but is also a response to upstream forcing from the North Pacific.

Corresponding author address: Dr. Gidon Eshel, Dept. of the Geophysical Sciences, The University of Chicago, 5734 S. Ellis Ave., Chicago, IL 60637. Email: geshel@uchicago.edu

Abstract

A statistical forecasting scheme for the North Atlantic Oscillation (NAO) is presented. The forecasts are based on Pacific Ocean surface pressure. With 21 months lead time, skills are significant and useful but modest. At 15-month lead, forecasts are robust and skillful under stringent cross validation, and further improve at 12-month lead. Cross-validated 1-yr forecasts correlate with 1925–2002 observations at ∼0.45. Other performance measures indicate similar skills, out performing the best autoregressive models of the NAO index. Importantly, skills of North Pacific–based forecasts easily exceed those of North Atlantic–based ones using identical machinery. The results suggest that NAO variability is not exclusively internal to the North Atlantic, but is also a response to upstream forcing from the North Pacific.

Corresponding author address: Dr. Gidon Eshel, Dept. of the Geophysical Sciences, The University of Chicago, 5734 S. Ellis Ave., Chicago, IL 60637. Email: geshel@uchicago.edu

1. Introduction

The North Atlantic Oscillation (NAO)—the time-varying pressure difference between the North Atlantic (NA) subtropical high and Icelandic low—is a primary mode of winter Northern Hemisphere climate variability (e.g., Wallace and Gutzler 1981; Deser and Blackmon 1993; Kushnir 1994). It dominates winter NA climate variability, and strongly affects temperature and precipitation over Europe, the Mediterranean, and the Middle East (Hurrell 1995; Johansson et al. 1998; Eshel and Farrell 2000, 2001; Eshel et al. 2000).

Our understanding of the NAO mechanisms is rudimentary. Most dynamical ideas about the NAO focus on the NA basin itself, invoking, for example, stochastic atmospheric forcing of the ocean mixed layer (Hasselmann 1976; Frankignoul and Hasselmann 1977; Frankignoul 1985). Rossby wave dynamics (Weng and Neelin 1998; Neelin and Weng 1999), wind-driven and thermohaline advection (Latif and Barnett 1994; Battisti et al. 1995), and fully coupled ocean–atmosphere modes (Latif and Barnett 1994, 1996; Cessi 2000). It has also been suggested that the NAO may be teleconnected to the tropical Atlantic (e.g., Robertson et al. 2000), as well as a local manifestation of a hemisphere-wide oscillation centered over the Arctic (Thompson and Wallace 2000).

One possibility that has not yet been fully explored is that the NAO is not only a self-sustained mode of NA climate variability, but also an NA response to upstream forcing by past North Pacific (NP) atmospheric variability. One reason for considering this possibility is that while correlations between the NAO index (NAOI; see the next section for definitions and data sources) and simultaneous NA sea level pressure anomalies (SLPAs) are very high, by construction, NAOI correlations with earlier SLPAs are much lower in the NA than in the NP, and yield very modest forecasting skills (not shown), consistent with Wunsch (1999).

Following earlier attempts to forecast NAO-related variability either dynamically (Griffies and Bryan 1997; Rodwell et al. 1999) or statistically (Johansson et al. 1998), in this paper I entertain the possibility that some NAO variability is a local response to upstream forcing by attempting to construct a statistical NAO forecasting method based exclusively on NP SLPAs. The choice of SLPA as the predictive variable (as opposed to, say, sea surface temperature anomalies) is based on the following. If an NP–NA lagged teleconnection exists, even if NAO variability is driven originally by the NP ocean, the NP oceanic forcing must be transmitted to the NA through the atmosphere. Thus NP SLPAs, even if forced by the ocean, are more directly linked to the phenomenon of interest, the NAO, than are ocean temperatures.

Enhancing skills and lead times, the current results indicate that some of the NAO variability is indeed related to variations outside the NA region. It is entirely possible, but not explored here, that both the NAO and the Pacific forcing are jointly related to a larger phenomenon, perhaps the Northern Hemisphere annular mode (e.g., Thompson and Wallace 2000).

2. Data

The NAO is characterized by the Hurrell (1995) NAOI, the December–March normalized SLP difference between Lisbon, Portugal, and Reykjavik, Iceland (information available online at www.cgd.ucar.edu/~jhurrell/nao.html). Let n = n(tn) denote the NAOI time series, where tn is the annual-resolution time grid for 1 February 1925, 1 February 1926, … , 1 February 2002, with 1 February being the midpoint of the December–March NAO period.

The January 1923–March 2002 monthly resolution SLPA dataset results from the merger of two observational datasets. Before January 1991, I use the Kaplan et al. (2000) data (information available online at ingrid.ldeo.columbia.edu/SOURCES/.KAPLAN/.RSA.COADS_SLP1.cuf). From January 1991 to the present, I use the National Oceanic and Atmospheric Administration/Climate Diagnostics Center (NOAA/CDC) and National Centers for Environmental Prediction Real-Time Marine dataset (online at ftp.cdc.noaa.gov/Datasets/nmc.marine/slp.mean.nc). During a 2-yr overlap period in which both datasets are available, the correlations between the two datasets are nearly perfect throughout most of the Pacific north of ∼15°N, but deteriorate discernibly equatorward. Most calculations were repeated with the merger point shifted to December 1992, without discernible changes.

The trial predictive domain is chosen as 20°–65°N, 140°E–120°W, based on the following considerations. First, the reader may find the choice of excluding the NA from the trial predictive domain a glaring omission; after all, would the NA itself not be its own best predictor? This question is certainly well-posed and sensible. However, part of this paper's motivation is, as stated in section 1, to entertain the possibility that the NAO is partly a response to upstream forcing from the NP. Distinguishing this possibility from the alternative view of the NAO as an intrinsic mode of NA climate variability motivates the NA's exclusion from the trial predictive domain. Next, the divergence of the Kaplan and NOAA/CDC SLPA estimates in the deeper Tropics motivates the exclusion of this area from the Pacific domain, despite the fact that NP anomalies are partly forced by the Tropics (Trenberth and Hurrell 1994). In addition, the longitudinal extent chosen by Trenberth and Hurrell (1994) for the definition of their NP index is extended both east and west by 20° of longitude, thus including the whole NP. This choice reflects an effort to be least conjectural and most inclusive, by entertaining the possibility that SLPA variability patterns other than those projecting strongly on the PNA (Wallace and Gutzler 1981) also influence the tested NP–NA teleconnection. In retrospect, the domain's east–west extension has proven unimportant; the calculations reported in this paper were repeated in their entirety with the Trenberth and Hurrell domain, with only slight deterioration in skill. These results are reported briefly in section 5.

Let 𝗗(x, t) denote the SLPA vector time series, where t and x ≡ (x, y) vary along 𝗗's rows (columns) and denote time (space), respectively.

3. Methodology

The forecasting scheme is a simple cross-validated multiple regression (e.g., Barnston 1994; Eshel et al. 2000), employing predictors derived from NP SLPAs. The scheme's various stages are described below.

a. Pattern calculation

First, Pacific SLPA patterns best correlated with the NAOI some time later are computed.

To reduce observational noise, the SLPA dataset is subsampled according to lead and averaged over 3 neighboring months, creating lead-specific, annual-resolution subsets of 3-month means:
i1520-0493-131-5-1018-e1
where l is the lead in months with respect to the middle of the NAOI period, 1 February. Thus, for example, the data subset corresponding to lead = 12.5 month, 𝗣12.5, comprises DJF-mean SLPAs for all years, with its time grid, yr, leading tn, the NAOI time grid, by 12.5 months. Consistent with the hypothesized NP–NA lagged teleconnection, I search for predictive skill at lead times 1.5 ≤ l ≤ 24.5 months; given typical atmospheric timescales, or even those of the upper ocean, useful skills, if found, are unlikely to extend beyond ∼2 yr.

To further reduce noise, each lead-specific SLPA subset 𝗣l is decomposed into EOFs, and only enough leading modes are retained to account for 75% of the total variance. The choice of the 75% truncation point is of course rather arbitrary, but the results are essentially independent of the exact choice. Varying this percentage over 65%–85%, or choosing a specific modal cutoff between 10 and 15, made very little difference. This behavior is a manifestation of the fact that for all leads, the covariance matrices' eigenspectra display a very distinct “kink” separating ranges from effective null spaces, thus making an expensive rule N-based cutoff calculations unnecessary. Let 𝗣l, l = 1.5, 2.5, … , 24.5 denote these filtered subsets.

For a given lead l, 𝗣l at each grid point is correlated with the NAOI time series l months later (after linearly best-fit detrending both). Gridpoint correlations are t tested for significance, allowing for loss of degrees of freedom due to temporal persistence according to Livezy and Chen (1983). Local correlations that fail to achieve p < 0.05 significance are set to zero, and p < 0.05 ones are retained unchanged. The resultant correlation map for each considered lead, normalized to unit norm, is denoted rl, l = 1.5, 2.5, … , 24.5.

b. Field significance

The next issue is to establish the field significance of each such correlation map as a whole (Livezy and Chen 1983). For the map to be field significant, the areal extent of locally significant correlations in it must exceed the areal extent that can be expected by chance. If each of the domain's grid points were independent of all others, the problem would have reduced to a binomially distributed set of local tests, each with 5-in-100 chances for passing spuriously (corresponding to the required local p < 0.05). However, because neighboring grid points are often strongly correlated, this test can be too liberal. To address this difficulty, I use the Monte Carlo approach of Livezy and Chen (1983). For the correlation map of each lead, the field-significance statistic is the area-weighted mean absolute correlation (including only locally significant correlations). The field-significance threshold is derived from a 1000-member Monte Carlo population, using synthetic autoregressive (AR) NAOI “realizations.” Each synthetic NAOI time series is correlated with every grid point in the domain, in the same way the actual NAOI time series was, and the area-weighted mean absolute correlation is computed over all locally significant grid points. For each lead, the 1000 areal means form a population whose cumulative probability density function's 95th percentile is the field-significance threshold value for that lead's correlation map. To obtain the most adequate ARMA(m, n) (autoregressive moving-average model of orders m and n, respectively) representation of the NAOI time series, I explore 1 ≤ m ≤ 10, 0 ≤ n ≤ 10. The various models' cross-validation skills (see below) and Akaike information criterion (Box et al. 1994) both single out ARMA(9, 8) and AR(1 ≤ m ≤ 3) as close “best” choices, well ahead of other orders. Experimenting with those models revealed very weak sensitivity of the derived thresholds to the model choice among those best ones. Consequently, the results reported here are based on AR(2), the most parsimonious of the above orders that still allows for long-term oscillations.

c. Deriving predictor time series from the correlation maps

Projecting each year's detrended 3-month-mean SLPAs of the months corresponding to a given lead on the lead's correlation map yields an annual resolution predictor time series pl:
plpltTlrl
where the superscript T denotes the transpose. The predictor pl is a measure of the similarity of each year's SLPAs, 𝗣l, to rl, the “most predictive” pattern (Eshel et al. 2000). Various combinations of the resultant time series pl for all field-significant leads l (or pl of a single field-significant lead) are then used in a (possibly multiple) regression framework, to hindcast the NAOI:
nyξpl1pl2plKyξ,
where 𝗔 is implicitly defined by (3), y is the least squares parameter vector, ξ is noise (defined as the part of the NAOI signal not accounted for by 𝗔y), l1 is lead i, and K is the total number of predictors used in a given inversion (bounded by 1, when using only a single predictor, and the maximum number of leads that were found to be field significant). Because insignificant correlations are set to zero, the projection is unaffected by NP areas poorly correlated with the NAOI, thereby reducing the risk of spurious predictability arising from chance phase alignments of SLPAs and NAOI.

d. Cross validation

In the cross-validation mode, 3 yr are withheld, both randomly and sequentially, from both the SLPA and NAOI datasets 1000 times. The choice of 3 yr is based on the following considerations. Given that a forecast estimates the NAOI value for a single future year, it seems fair to test the model by attempting to forecast a single, but full, degree of freedom. From the NAOI autocorrelation function, a full degree of freedom appears to be 2.1 yr (the 78-element series has ∼37 degrees of freedom), so that the choice of 3 yr represents a conservative estimate of a single degree of freedom. For each of the withholding experiments, the analysis is repeated in its entirety, starting from the EOF filtering of the deficient SLPA matrix for the appropriate 3-month means (excluding the withheld years), detrending the deficient SLPA and NAOI time series, correlation pattern calculation, field-significance testing (with the local and field thresholds taking note of retained years only), and finally deriving deficient projection time series and using them in Eq. (3) to obtain y. Thus with subscripts f and d denoting full and deficient datasets, the cross-validation equation is
fTdd−1Tdnd
with 𝗔 comprising the chosen leads according to Eq. (3) and blending NAOI hindcasts (for retained years) and forecasts (for withheld years). For each of the withholding experiments, the hindcasts are discarded, and the forecasts are retained. All forecasts of a given year (obtained by any of the withholding experiments that happened to include that year) are then averaged, and this mean is the scheme's NAOI prediction for that year. Cross-validation skill is defined as the temporal correlation over 1925–2002 between the predicted NAOI time series and the observed (actual) one. The predicted NAOI time series is derived by averaging the forecasted values of each year over all withholding experiments, 1000 in the random case, and 76 ({1, 2, 3}, {2, 3, 4} ⋯ {76, 77, 78}) in the sequential case.

4. Results

Of the 24 considered leads (1.5–24.5 months), 6 (25%, well above what can be expected to pass the field-significance test spuriously) prove to be field significant: l = 1.5 (December), l = 5.5 (August), l = 13.5 (December), l = 14.5 (November), l = 15.5 (October), and l = 21.5 months (April). Because my primary interest is long-range predictability, below I focus on the latter four leads (Fig. 1). Since the results reported below rely heavily on the predictive spatial patterns shown in Fig. 1, their robustness must be demonstrated. First, patterns very similar to Fig. 1 (not shown) result from applying the same analysis to the da Silva et al. (1994) 1945–93 SLPA dataset. The patterns' robustness is further corroborated by recalculating the correlation patterns Nw = 76 times, in each case after sequential 3-yr withholding as before. Each correlation field comprises Ng = 267 grid points. Figure 2a shows the ratio of the computed field-significance criterion to the corresponding threshold. For each field-significant lead, the 76-experiment mean is given by the open circle, while the vertical bar shows the minimum–maximum range of the 76 experiments. Figure 2b shows the spatial correlations of the patterns derived from the deficient datasets (after withholding) and those derived from the full datasets (rl). Figure 2b shows that ∼97% of the trials correlated spatially (over 267 grid points, representing 70–90 degrees of freedom) with the full patterns at >0.6. Finally, since teleconnection patterns can vary considerably with time, the dependence of Fig. 1's patterns on the particular choice of analysis period is examined in Fig. 3. While there is clearly variability with the choice of analysis period, all but 1 of the 20 correlations considered (five periods × four lead times) yield p < 0.05. In most cases, the NP–NA teleconnection is most prominent during 1964–2002, well above the 1925–2002 results that this paper emphasizes. In this sense, the results shown below can be thought of as a conservative lower bound estimate of the strength of the NP–NA teleconnection.

The four focal field-significant leads (Fig. 1) afford four single-predictor inversions [with K = 1 in Eq. (3)] six double- and four triple-predictor inversions (with K = 2 and 3, respectively), and a single four-predictor inversion (with K = 4), including all four field-significant leads. All of these possibilities are explored, and the results are reported in Fig. 4. In addition, two of the most skillful combinations are reported as time series in Fig. 5.

Since even the best predictor combinations yield a cross-validated skill in the range 0.4–0.5, their utility appears modest. One yardstick against which they may be judged is climatology. Assuming climatological conditions at all times means, by definition, that identically zero SLP anomalies are forecasted. Since the NAOI is derived from anomalies, climatology forecasts will yield zero correlation with observed NAOI. In one-way analysis of variance (ANOVA) parlance, this means that the fit yields no reduction in rms error. Against this backdrop, the modest ∼20% uncertainty reduction that results from a correlation skill of r = 0.45 is nontrivial.

Three of the four focal leads yield single-predictor cross-validation skills with <25 chances out of 1000 to have arisen spuriously. This suggests that the relationship between the NAO and past NP SLPA patterns presented in Fig. 1 are unlikely to be spurious, as are the corresponding predictive skills of the SLPA projections on them. All 15 lead combinations have cross-validations skills with p < 0.04, and three (20%) have p < 0.01, apparently well beyond what can be reasonably expected by chance. Below I quantify the likelihood of such cross-validation skills arising spuriously.

While unlikely, it is possible that the reported cross-validation skills reflect chance alignment of NP SLPA and NAOI noise, and that the NAO is in fact not physically related to past NP atmospheric fluctuations. One way to assess the likelihood of such chance alignments giving rise to the reported cross-validation skills is to replace the NAOI with noise time series, apply the same analysis, and compare the derived skills, which are known to be spurious, to those obtained with the actual NAOI time series. The results of such tests, applied to all lead combinations, are reported in Fig. 6. Because the NAOI time series is not white, to make the test as easy to fail as possible, the noise time series are derived from the AR(2) representation of the NAOI so as to have approximately the same spectral properties and variance as the NAOI time series itself. The entire procedure, starting from EOF filtering, followed by detrending and correlation map calculations, through the cross-validation test, is repeated 600 times, each time with a different realization derived from the AR(2) fit to the NAOI time series. The p < 0.05 and p < 0.01 thresholds for this test are that the actual (NAOI derived) skill will be exceeded by no more than 600 × 0.05 = 30 and 600 × 0.01 = 6 AR realizations, respectively (since we are testing for positive skills, the one-sided test is the appropriate one). All but 2 of the combinations meet the p < 0.05 challenge, and 8 out of 15 (53%) also exceed the p < 0.01 criterion. Thus the actual forecasts outperform all, or very nearly all, trials with synthetic NAOI time series whose skills are spurious by construction. While these results do not eliminate the possibility that the computed predictive skills are spurious, they suggest that the likelihood for that is, on average, 200/(600 × 15) ≈ 0.02, corresponding to p ≈ 0.02 (200 is the sum of all lower-left numbers in Fig. 6, and 600 × 15 is the total number of trials).

5. Result with the Trenberth and Hurrell domain

When the NP domain is defined to include only Trenberth and Hurrell's (1994) NP region, 30°–65°N, 160°E–140°W, only three lead times pass the field-significance test, thus permitting only seven lead combinations shown in Fig. 7c. Despite this rather minor difference from the results with the full NP domain, the results are only slightly weaker than those based on the whole NP. This indicates that, to the extent that the NP holds some predictive value over the NA, the pattern of NP climate variability identified and indexed by Trenberth and Hurrell (1994) is a principal contributor.

6. Conclusions

The significant skill with which one can use North Pacific SLPAs to predict future phases on the NAO supports the notion that the NAO may be a local manifestation of a broader phenomenon [possibly Thompson and Wallace's (2000) annular mode]. That the NAO is not exclusively an NA phenomenon is further corroborated by my failure to forecast the NAOI using an identical methodology with predictors derived from SLPAs or sea surface temperature anomalies in the North Atlantic. The more fundamental questions regarding the dynamics of the hypothesized Pacific–Atlantic teleconnection, and its possible links to the annular mode and ENSO, remain unanswered.

The forecasts can certainly be further refined. However, the method described is simple enough to render unlikely the possibility that the results reported are spurious. The method provides preliminary information about the state of the NAO ∼2 yr later, with further refinement in the preceding spring. Given the enormous socioeconomic impact of NAO-related climate variability, the scheme may prove practically useful.

Acknowledgments

The author wishes to thank the United States Environmental Protection Agency for their generous support of this work through STAR Cooperative Agreement R-82940201-0 to The University of Chicago.

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Fig. 1.
Fig. 1.

The correlation patterns rl for the four field-significant leads. For clarity, the shown fields include all correlations whose magnitude exceeds 0.15, not the derived local significance threshold (which is ∼0.2). The contour interval is 0.05, and the minima–maxima are indicated by plus (+) symbols. Below the color bar, each panel's lead is indicated by L, and the temporal correlation between the 1925–2002 SLPA projection time series on the pattern and NAOI L months later is given by r

Citation: Monthly Weather Review 131, 5; 10.1175/1520-0493(2003)131<1018:FTNAOU>2.0.CO;2

Fig. 2.
Fig. 2.

Results of recalculating the correlation patterns after sequential 3-yr withholding. Since the full time series comprise 78 yr (1925–2002), there are Nw = 76 possible sequential combinations [in which yr {1, 2, 3} {2, 3, 4} ⋯ {76, 77, 78} are withheld]. Each correlation field comprises Ng = 267 grid points. (a) Ratio of the computed field-significance criterion to the corresponding threshold. For each field-significant lead, the 76-experiment mean is given by the open circle, while the vertical bar shows the minima–maxima range of the 76 experiments. (b) Spatial correlations of the patterns derived from the deficient datasets (after withholding) and those derived from the full datasets (rl)

Citation: Monthly Weather Review 131, 5; 10.1175/1520-0493(2003)131<1018:FTNAOU>2.0.CO;2

Fig. 3.
Fig. 3.

The dependence of the predictive patterns on the particular choice of analysis period. The datasets are split into five different periods, given in the lowermost panel. (a) Correlation patterns are recalculated based on the information of that period only. For each lag–subperiod combination, the spatial correlation of the partial pattern with the full pattern (1925–2002, on which the paper is based) is reported. (b) Lagged temporal correlations between the NAOI time series, and the time series that results from projecting the partial SLPA data on the original (1925–2002) pattern. (c) The t-test significance of these temporal correlations, taking note of lost degrees of freedom due to serial correlations. In (c) the four dashed horizontal lines correspond, from the top downward, to p = 0.05, p = 0.01, p = 0.005, and p = 0.001

Citation: Monthly Weather Review 131, 5; 10.1175/1520-0493(2003)131<1018:FTNAOU>2.0.CO;2

Fig. 4.
Fig. 4.

The vertical solid lines divide the panels into four subareas, corresponding to the number of predictors used [K in Eqs. (3) and (4) and in (b)]. As stated in (a), the correlated time series comprise 78 yr, with, taking note of serial correlations, ∼37 degrees of freedom. In (a) the h symbols report hindcasting skill [solving Eq. (3)] for each combination. The 3-yr random (sequential) withholding cross-validation skills are given in (a) by the open circles (plus symbols). (b) The F-test significance (times 100) of the cross-validation skills reported in (a), using dfK and K for the error and regression degrees of freedom, respectively. (c) Various leads participating in the combinations of (a) and (b) (omitting the “0.5” for brevity)

Citation: Monthly Weather Review 131, 5; 10.1175/1520-0493(2003)131<1018:FTNAOU>2.0.CO;2

Fig. 5.
Fig. 5.

Results for two of the most skillful lead combinations (numbers 10 and 13, with lead-time combinations shown in each panel).

Citation: Monthly Weather Review 131, 5; 10.1175/1520-0493(2003)131<1018:FTNAOU>2.0.CO;2

Fig. 6.
Fig. 6.

Autoregressive test of all 15 predictor combinations [1 and 2 in (a) through combination 15 in (h)]. All panels show the cumulative probability density function of 600 trials in which the entire cross-validation procedure is repeated, each time with a different “NAOI-like” time series generated from an AR(2) fit to the actual NAOI time series. The solid (dashed) line corresponds to the first (second) combination shown in the panel. The numbers in the lower left give the number of trials whose resultant cross-validation skills equaled or exceeded that of the same predictor combination with the actual (rather than synthetic) NAOI time series. Those actual skills are given by the vertical solid or dashed bars. For each combination two actual skills are shown (very near each other), for the random and sequential cross-validation calculations.

Citation: Monthly Weather Review 131, 5; 10.1175/1520-0493(2003)131<1018:FTNAOU>2.0.CO;2

Fig. 7.
Fig. 7.

The results with the NP domain including only Trenberth and Hurrell's (1994) NP region, 30°–65°N, 160°E–140°W. The vertical solid lines divide the panels into three subareas, corresponding to the number of predictors used [K in Eq. (3) and in (b)]. As stated in (a), the correlated time series comprise 78 yr, and, taking note of serial correlations, ∼37 degrees of freedom. In (a) the h symbols report hindcasting skill [solving Eq. (3)] for each combination. The 3-yr random withholding cross-validation skill is given in (a) by the various dot-encompassing open symbols (circles for one predictor, diamonds for two predictors, and a triangle for all three). (b) The F-test significance (times 100) of the cross-validation skills reported in (a), using dfK and K for the error and regression degrees of freedom, respectively. (c) Various leads participating in the combinations of (a) and (b) (omitting the “0.5” for brevity).

Citation: Monthly Weather Review 131, 5; 10.1175/1520-0493(2003)131<1018:FTNAOU>2.0.CO;2

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