a. General measures of correlation

A common measure of the linear adjustment between two random variables X and Y is the linear or Pearson correlation (Papoulis 1991), represented by c(X, Y). When c(X, Y) ≠ 0, there is common information or statistical redundancy. However, the reverse does not hold, in general, because this measure does not account for nonlinear relationships between variables. Those relations can be taken into account through the Pearson correlation c[A(X), B(Y)] between general nonlinear functions A(X) and B(Y). The correlation is maximized in absolute value when A(x) = E(Y|X = x), B(y) = E(X|Y = y). A particular case of nonlinear correlation is the Spearman or rank correlation (Wilks 1995), where the nonlinear functions are simply the sampling ranks of X and Y, respectively, measuring the degree of monotonic association between both variables. Another nonlinear correlation is hereby denoted as Gaussian correlation and defined as

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where g_X (equivalently for Y) is the standard Gaussian transformation or Gaussian anamorphosis of X, given by

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where ρ_X(u) is the PDF of X and Φ⁻¹ is the inverse of the cumulative standard Gaussian distribution function. This transformation is a common data analysis procedure in geostatistical kriging and climatic data analysis (Biau et al. 1999) that ensures that the marginal distributions are standardized Gaussians. However, while marginal distributions are rendered Gaussian, the joint distribution does not necessarily become Gaussian. That way, if X and Y have a joint Gaussian distribution, c_g(X, Y) = c(X, Y). Both rank and Gaussian correlations are nonlinear correlations that are invariant for the class of monotonous homeomorphisms on X and Y individually (though not necessarily homeomorphisms mixing these variables). Consequently, an advantage of both of these measures over the Pearson correlation is the fact that, unlike the latter, the former are not artificially inflated by the coincidence of large outlier values (Jolliffe and Stephenson 2003). Results will be shown, both for a pair (X, Y) of untransformed variables and for their correspondent Gaussian anamorphosis where X is the standardized (i.e., zero average, unit variance) principal component (PC)-based NAO index (see section 3) and the variable Y is the standardized monthly precipitation.

b. Asymmetric Gaussian correlations

The sensitivity of one variable (Y) to another (X) is not necessarily the same over every subdomain of X. A global correlation measure is not able to account for the sensitivity of one variable to the other within a particular subdomain. To do so, we introduce the conditional correlation c(X, Y|X ∈ I_X), also called asymmetric correlation, between two standard variables X, Y for a certain interval I_X of X. We consider here the partition of X into two complementary intervals, separated by the median M_X of X. In particular, for X being the NAO index, that partition separates the NAO− and NAO+ regimes. The corresponding asymmetric or side correlations are defined as

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and the conditional standard deviations, for each half of the data, are

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where var denotes variance. Because the variables X, Y are centered and have unit variance, it is easy to verify that

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and

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Differences between both asymmetric correlations [(3)] shall lead to a nonlinear asymmetric (X, Y) relationship. For standard variables, the correlation is simply given by the covariance, which can be decomposed, as for any set partition, into intra- and interset covariances. For the referred X partitioned into two halves, we have

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where

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works as the interset covariance of (X, Y), proportional to the difference between the asymmetric conditional Y means. Given the constraints [(5a), (5b)] it is also clear that |c_M| ≤ 1 and that |c_M| tends to increase for low conditional Y variance. The other two terms of (6) are also less than one in absolute value.

Because we are also interested in diagnosing joint non-Gaussianity, we compare the values of c_M, c₊, and c₋ with those that would be obtained if X and Y were jointly Gaussian. In this case, the above quantities are functions of the correlation c,

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The side correlations are thus equal and given by a nonlinear increasing function of the correlation c,

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From (10) we see that the absolute values |c₊| and |c₋| are below the absolute value |c|. This means that, under Gaussian conditions, the global correlation c is always greater in absolute value than both side correlations. Both of their contributions for c are equal to βc in decomposition (6). This leads us to define statistical tests of bi-Gaussianity, hereafter called test central correlation t_M, test positive side correlation t₊, and test negative side correlation t₋, which are directly comparable to the correlation in the Gaussian case,

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Differences between the correlation c and the tests [(11)] are measures of the distribution asymmetry. The vanishing of all of those differences is a necessary but not sufficient condition for joint Gaussianity. To get a measure of asymmetry that is independent from the correlation c, we will consider the pair of uncorrelated variables (X, Y_r), where Y_r is the standardized residue of the linear prediction of Y from X,

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Let us decompose the (X, Y_r) correlation as a combination of expectancies as follows:

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Both terms in (13) are symmetric, vanish under Gaussian conditions, and lead, after a few lines of algebra, to a measure J_c of asymmetry, as

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By subtracting M_X from X and taking the absolute value of the product XY_r in (14), it can be inferred that J_c is proportional to the nonlinear correlation c(|X − M_X|, Y_r), which has, under some conditions, a monotonic relationship with the nonlinear correlation c(|X − M_X|², Y_r).

c. Mutual information and negentropy

Beyond asymmetry, a more complete approach to diagnosing non-Gaussianity and statistical redundancy, based on concepts of information theory (Shannon 1948), is considered here.

In the paper, MI is computed between two continuous random variables: X and Y. Mutual information is nonnegative and measures the reduction of uncertainty of a random variable given the knowledge about the other, and vice versa. Mathematically speaking, MI is defined as

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where h( ) is the differential entropy. Mutual information is the Kullback–Leibler distance (KLD) between the joint PDF ρ_X,Y(X, Y) and the product ρ_X(X)ρ_Y(Y) of the marginal PDFs (Cover and Thomas 1991). Mutual information vanishes iff (if and only if) X and Y are statistically independent or, equivalently, iff all nonlinear correlations are zero for smooth PDFs, thus making MI a stronger measure of independence than the Pearson correlation. Furthermore, MI is invariant for any X and Y single homeomorphisms. In particular, it is invariant when X and Y are replaced by the corresponding Gaussian anamorphosis, say

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By imposing the knowledge of the correlation c, a positive lower MI bound I_g (hereby denoted as Gaussian mutual information) can be found by solving a constrained variational problem of MI minimization (Kraskov et al. 2004), thus leading to the decomposition

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where

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is the MI within the bivariate Gaussian distribution of (X, Y) with correlation c (Cover and Thomas 1991). The upper bound [(18)] can be generalized by replacing the correlation c by any nonlinear (X, Y) correlation. The Gaussian upper bound [(18)] of MI is also used in speech analysis (Abdallah and Plumbey 2003). Mutual information can be compared with the Gaussian correlation by defining a distance between X and Y, hereby denoted as information correlation, and given by

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The equality holds iff the joint distribution of (X, Y) is Gaussian or, equivalently, iff I_ng is null. As it happens with MI, c_inf vanishes in the case of statistical independence. By applying the chain rule of KLD (Cover and Thomas 1991), the non-Gaussian term of MI, I_ng, can be decomposed as

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where J( ) is negentropy, a positive quantity defined as the KLD between the true PDF and the Gaussian PDF with the same first- and second-order statistics. In (20), negentropy J(X, Y) is invariant under a two-dimensional linear homeomorphism of (X, Y). Therefore, without loss of generality, it is equal to the negentropy between the uncorrelated variables X and the prediction residue Y_r [(12)].

If X and Y are previously subjected to Gaussian anamorphosis, the marginal negentropies J(X) and J(Y) vanish.

d. Numerical estimation of MI

The numerical estimation of MI is rather difficult and has no unbiased estimators (Paninski 2003, 2004). It can be dealt with through numerous approaches, such as the plug-in, bin-adaptive networks (Kraskov et al. 2004), and ME (Abramov 2006). Here, we will estimate MI through two independent methods: ME, and another one based on the Edgeworth PDF expansion (EDG-PDF; Edgeworth 1905). This method is only reasonable on a weak non-Gaussianity scenario, contrary to the ME, which, on the other hand, is computationally more costly. A summarized account of both methods is given as follows.

1) Edgeworth expansion method

Our purpose is to estimate the joint PDF of (X, Y), or any transformed pair such as (g_X, g_Y), and then its MI. The fact of taking rotated standard variables U = X and W = Y_r [(12)] from (X, Y) will considerably simplify the proposed PDF expansion. The joint PDF ρ_U,W(u, w) can be approximated by

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where φ( ) is the standard Gaussian PDF and υ_U,W(u, w) is a truncated fitting polynomial vanishing if the joint PDF is Gaussian. As far as the truncation error is concerned, the variables U and W are assumed to be arithmetic averages of an equivalent number n_eq of independent and identically distributed (iid) variables, and l is positive and increases with the truncation order (Comon 1994). The greater n_eq, the smaller the truncation error of the expression and the closer to Gaussianity the joint (U, W) PDF, due to the central limit theorem. The function υ_U,W is expanded in terms of Hermite orthogonal polynomials [(A2), (A3)] and cumulants k^(p,q) [(A4)] of order p in U and of order q in W, which are appropriate joint polynomial expectancies of U and W. The cumulants k^(p,q) are scaled as O[n^−(p+q)/2+1_eq]. The function υ_U,W for l = −3/2 is given by (A1) in the appendix. Nonzero cumulants of order p + q, higher than or equal to three, reveal non-Gaussianity. In particular, if X is rendered Gaussian the self U cumulants k^(p,0), p ≥ 3 vanish. The EDG-PDF expansion converges in L², needing a large truncation l in cases of high non-Gaussianity. It has the drawback that errors in tail regions of the distribution may be comparable to the PDF itself and may even present negative values. To verify the positivity and normalization of the EDG-PDF, we numerically compute the integrals P_pos and P_neg of the EDG-PDF, respectively, in the domain of positive and negative values of the estimated truncated density [(21)]. The integrals are estimated by bivariate Gaussian quadrature, mapping the open intervals ]−∞, ∞[ into ]−1, 1[ through the transformation x → f (x) = x/(1 + |x|). The use of 50 weighting quadrature points has been sufficient for the convergence of integrals with an accuracy of ∼10⁻⁴. A satisfactory condition for (21) to be a density is |P_neg| ≪ P_pos ∼ 1, which holds if cumulants in (A1) are sufficiently small. A nongeneral rule for approaching the referred condition is to perform Gaussian anamorphosis of the original variables. A reduced Edgeworth truncation is still valid considering a “tilted” variable whose modal region is nearer to the neighborhood where we wish to approximate, using the saddle approximation technique (Daniels 1954). However, this approach will be not followed herein. An independent test of validity of the EDG-PDF is obtained after comparing it with the ME-obtained density ME-PDF.

After assessing the validity of EDG-PDF ρ_U,W(u, w), the PDF ρ_X,Y(x, y) of the unrotated variables can be retrieved through the following expression:

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To compute the non-Gaussian MI [(20)] of (X, Y) one must obtain the joint negentropy J(X, Y_r) = J(U, W), which is simply the KLD distance between ρ_U,W(u, w) and the product φ_U(u)φ_W(w), reducing to

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The marginal negentropies J(X) and J(Y) are estimated through the equivalent equation to (23) for single variables. Two ways of computing (23) are followed. First, it is numerically computed in the domain (u, w): 1 + υ_U,W(u, w) ≥ 0, in the same way as P_neg and P_pos. The non-Gaussian MI obtained through the estimated integral is denoted as I_ng(EI). Then, we consider the Taylor expansion

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where the logarithm expansion error O(υ³) is O(n^−3/2_eq) because υ ∼ n^−1/2_eq. By noting that the function υ_U,W(u, w) is orthogonal to the product of Gaussian PDFs ϕ(u)ϕ(w), the negentropy becomes the following positive quantity:

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By using the norms and orthogonality properties of the Hermite polynomials, (25) is expanded in terms of powers of cumulants. For the truncature l = 3/2, we have

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The above estimation is valid for standard uncorrelated U, W variables. Equation (26) is a sum of the quadratic positive contributions from cumulants of order higher than two, while also resulting from a truncated expansion of the logarithm function. Furthermore, these are a simplification of what had been obtained by Comon (1994), where a truncation of up to O(n⁻²_eq) had been considered. While simplifying the order of truncation, a generalization has been performed as to obtain the Edgeworth expansion of the joint negentropy, whereas Comon (1994) had studied the one-dimensional case. The non-Gaussian MI obtained through (26) is denoted by I_ng(EF). By having an explicit estimation of the PDF and taking into account the integral properties of Hermite polynomials, we can derive an analytic expression for the conditional expectancy of Y given X, as

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which highlights the effects of non-Gaussianity. The first rhs term of (27) is the linear prediction of Y from X, also represented by Y(lin), whereas the second rhs term is the additive correction resulting from non-Gaussianity, thus yielding the full nonlinear prediction Y(nolin). Given the Edgeworth expansions of both the joint and the marginal probability distributions, the conditional expectancy of the variable W given the variable U can be approximated by

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where H_i( ) are single Hermite polynomials with the standard Gaussian kernel [(A2), (A3)]. The correctional polynomial γ_U(x) at truncation l = −3/2 of the X marginal EDG-PDF is given by

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The cumulants k^(3,0) and k^(4,0) are, respectively, the skewness and the kurtosis (relative to that of the normal distribution) of the probability distribution of X = U [see (A4)]. Equations (27) and (28) provide an easy nonlinear downscaling relationship of a Y from X.

2) Maximum entropy method

The bivariate and single negentropies, necessary to compute the non-Gaussian MI I_ng(X, Y) [(20)], are estimated using the maximum entropies H_M(U = X, W = Y_r), H_M(X), H_M(Y), constrained under the set of known cumulants or, equivalently, the involved expectancies, following the maximum entropy principle of Shannon (1948) and Jaynes (1982). Let us consider the constraints on N_c expectancies, up to fourth joint order,

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The maximum entropy H_M(U, W) of the ME-PDF ρ_M(U,W)(u, w), satisfying (30), is the solution of the unconstrained minimization problem

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where the function Γ is a globally convex function with a global minimum at certain values of the Lagrangian parameters λ₁, . . . , λ_Nc. The function Γ is given by

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The support set of the ME-PDF is the (U, W) domain D.

The ME-PDF is expressed by

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Here, we follow a modified version of the ME algorithm of Rockinger and Jondeau (2002) by considering a finite square set D = [−L, L] ⊗ [−L, L] in (U, W) and then increasing L in order to asymptotically reach the maximum entropy H_M(U, W) for L = ∞. By using the Leibnitz derivation rule, it is easy to obtain the derivative of H_M(U, W) with respect to L,

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where the line integral is always positive, and computed around the boundary line δD of D. The bounding of (33) when |u|, |w| → ∝ leads to the scaling of the logarithm of the ME-PDF as O(−L^m), with m taking the largest exponents p_i, q_i in (30). Therefore, given the range of moments [(30)], we can extend the limits of D sufficiently further so as to get negligible bound effects on the entropy and the ME-PDF. Furthermore, in order to get integrands of the order exp[O(1)] during the optimization process, we solve the ME problem for the scaled variables (U/L, W/L) in the square S = [−1, 1] ⊗ [−1, 1] taking the appropriate scaled constraints. Afterward, we apply the scaling entropy relationship

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The minimization problem is solved by the quasi-Newton method starting at λ_i = 0 for all i. The integrals giving the function Γ and its λ derivatives are approximated by the bivariate Gauss truncation rule with N_p weighting factors in the interval [−1, 1]. To get full resolution during the minimization, and to avoid not a number (NAN) and infinite (INF) numbers in computation, we subtract the polynomials in the arguments of exponentials by the correspondent maximum in D. Finally, the function Γ is multiplied by a sufficiently high factor F in order to emphasize the gradient.

Considering the range of constraint moments in our cases, several experiments have led to the reasonable values of L = 10, N_p = 50, F = 1000, for which convergence is obtained after ∼60 optimization iterates for an accuracy of 10⁻⁶ of the gradient of Γ. Larger values of L require larger values of N_p, thus decreasing the convergence rate.

a. Data

We use National Centers for Environmental Prediction–National Center for Atmospheric Research reanalysis CD-ROMs (Kistler et al. 2001) to extract DJF monthly data of SLP and corrected precipitation, from 1951 to 2003 over the NAE domain (30°–70°N, 80°W–40°E), with grid size of 2.5° in latitude and longitude. We consider X to be the standardized (zero average and unit variance) NAO index given by the first principal component of the detrended SLP monthly data over the NAE in DJF, over the above-mentioned period. This PC-based NAO index (X) is related to more traditional indexes based on SLP differences (Osborn et al. 1999). The PDF of the PC-based NAO index is slightly platykurtic [kur(X) = E(X′³) − 3 = −0.7, where kur indicates the kurtosis] and bimodal with the presence of two regimes: NAO+ and NAO−. The grid-point-standardized monthly detrended precipitation is assigned to the variable Y. The distribution of monthly DJF precipitation over oceanic areas is closer to the Gaussian than the one over land. It is positively skewed and some locations exhibit outliers with large positive anomalies, with high kurtosis values, especially over Greenland [kur(Y) ≈ 5] Canadian Arctic [kur(Y) ≈ 5] and some deserted areas over North Africa [kur(Y) ≥ 10]. Those extreme precipitation values inflate joint (X, Y) cumulants, thus rendering inapplicable the Edgeworth formalism of estimating mutual information and non-Gaussianity. To obtain, in general, smaller cumulants, we apply the Gaussian anamorphosis both to X and Y. For that purpose, in practice, we start by sorting data within X in ascending order. The kth value of the Gaussian variable X_g of X in ascending order is given by

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where N = 53 × 3 is the total number of months in the sample, and Φ⁻¹( ) is defined as in (2). The same procedure applies to Y. The transformation [(36)] assumes that data uniformly cover the true probability distribution, and consequently it suffers in practice from sampling errors.

To illustrate the relevance of both non-Gaussianity and asymmetry of the precipitation response to NAO, we choose the following six grid points: 1) central Atlantic (ATL; 37.5°N, 25°W); 2) northwest Scotland (SCO; 60°N, 12.5°W); 3) Balearic Islands (BAL; 37.5°N, 2.5°E); 4) Greenland (GRE; 62.5°N, 45°W); 5) east United States (EUS; 40°N, 60°W); and 6) Russia (RUS; 62.5°N, 22.5°E).

b. Statistical tests

To verify whether the test-side correlations [(11)] and the non-Gaussianity measures are significantly different from zero, we apply the Monte Carlo technique in two versions: the generation of Gaussian noise (MCG), and the random reordering of working series (MCR). In MCG we generate an ensemble of 10 000 bivariate uncorrelated time series of standard Gaussian white noise with N_df temporal degrees of freedom. The estimated N_df of the pair (monthly NAO index X, monthly precipitation Y) uses the 1- and 2-month-lag X, Y auto correlations over the DJF period (Livezey and Chen 1983),

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The N_df is fairly uniform over NAE and close to its spatial average N_df ∼ 0.95N = 151, both for Gaussianized and original variables. In the MCR version we consider 300 different proxy NAO series by randomly permuting the 53 analyzed years while keeping the DJF monthly sequence. Then the statistical tests of the randomized NAO and precipitation are computed in each of 833 NAE grid points and then collected altogether in an ensemble of 833 × 300 realizations. Both for MCG and MCR, the values of the statistical tests are sorted so as to compute quantiles giving the 90%, 95%, and 99% significance level intervals (summarized in Table 1) of rejection of the null hypothesis H_o of (X, Y) independence. Rejection of H_o is easier for test-side correlations than c because they deal with half of the data, below or above the median of X. The thresholds of H_o rejection for I_ng(ME) are slightly larger than those for I_ng(EF) (Table 1). The confidence regions of the information correlation are obtained from I_ng(ME) and I_g. Thresholds for k^(2,1) and k^(3,1) are also computed. The MCR tests are more appropriate than the MCG tests because they preserve the marginal distribution of working variables and their spatial correlations. This leads to less conservative criteria of H₀ rejection in MCR, especially for the non-Gaussianity measures I_ng(ME), I_ng(EF), and J_c. In all statistical maps we shade the 90% MCR statistical significant regions. The fraction of statistically significant area (FS) must be larger than that occurring by mere chance (10%). The values of FS for MCR and MCG are denoted, respectively, as FS-MCR and FS-MCG.

a. Gaussian and asymmetric correlations

The correlation maps over NAE of the global correlation c, the Gaussian correlation c_g, and the rank correlation (not shown) between X and Y are rather similar over NAE. They only differ by no more than approximately ±10% at certain regions. The map of c_g shown in Fig. 1a is mainly related to the northward (southward) shift of synoptic storm tracks in the NAO+ (NAO−), thus producing positive (negative) precipitation anomalies in the range 50°N–70°N, east of 40°W, and negative (positive) precipitation anomalies in the range 35°N–45°N, east of 60°W (Hurrell 1995). Other negative correlation regions are the southeastern part of Greenland, the Canadian Arctic, and Labrador, Canada. There are some differences between the map of c_g and those of test-side correlations of Gaussian data, thus revealing asymmetry. This means that the response of precipitation to NAO is asymmetric and thus non-Gaussian. The gross contribution of c_g (Fig. 1a) is due to the test central correlation t_M(X_g, Y_g) (Fig. 1b), which comes from the alignment of NAO+ and NAO− centroids. Consequently, these maps are rather similar.

The differences between t₋(X_g, Y_g) and t₊(X_g, Y_g) reveal differences in sensitivities to the NAO+ and NAO− regimes. This is highlighted in Figs. 2a–f, which show for the six target locations 1) the distribution of the (X_g, Y_g) data, 2) the contours of the joint PDF obtained with the expansion [(A1)], 3) the linear and nonlinear prediction of Y_g [(27)], and 4) the smoothed graphics of the X_g conditional mean square error (MSE) of the linear Y_g(lin) and nonlinear Y_g(nolin) [(27)] predictions, obtained in full cross-validation mode over the N = 159 data.

For all six cases, the largest data concentration is seen near the origin (X_g = 0, Y_g = 0), where PDF contours are elliptic. The farthest contours are deformed because of extreme values. In the distribution, the negative and positive sides (NAO− and NAO+) can be quite different. Near the central Atlantic, where a large area of strong negative correlations is seen, the sensitivity of precipitation to the NAO index is negatively stronger in the NAO− (wetter) regime than in the NAO+ (drier) regime. This is in accordance with the values of t₊ = −0.30 and t₋ = −0.64 for the ATL point (see Table 2). Therefore, as seen in Fig. 2a, those data and the PDF spread over a larger domain in the NAO+ regime than in the NAO− regime.

The improvement due the nonlinear predictions is visible from the reduction of the cross-validated MSE of the NAO-downscaled precipitation, especially for the negative precipitation anomalous values (top of Fig. 2a and Table 2).

The larger sensitivity of precipitation in the wetter NAO regime is also verified north of Scotland, near the location of most positive correlation extremes, in accordance with t₋(X_g, Y_g) = 0.43 and t₊(X_g, Y_g) = 0.82 for the SCO point (see Fig. 2e and Table 2). This behavior occurs because near the average storm tracking of NAO− and NAO+ regimes, the sensitivity of the precipitation response must be enhanced. The higher the sensitivity, the closer the phenomenon, that is, a local source produces higher sensitivity than remote sources.

There are regions where the precipitation sensitivity is nearly restricted to one of the regimes, that is, where only one of the test-side correlations (t₋ or t₊) is significantly different from zero. In particular, the negative test-side correlation is particularly strong over the Ukraine, Romania, and former Yugoslavia (t₋ ≈ −0.6), whereas the corresponding positive one is quite small (t₊ ≈ −0.2). In the Mediterranean region, south of 40°N, the positive test-side correlation t₊ is negative, whereas the corresponding one on the negative side practically vanishes. This means that, in the south Mediterranean, while in the NAO− regime the statistical mean response of precipitation to NAO is not significant, in the NAO+ regime it is favorable to strong extreme drought events. This is consistent with the values of t₋ = 0.19 and t₊ = −0.65 for the BAL point. This is also apparent from the shape of contours (Fig. 2b) and from the nonlinear prediction graphic. This situation occurs especially on the second half of the analyzed time interval (i.e., 1978–2003), and may have a connection with the increasing desertification over Mediterranean regions during that same period. These results agree with the positive NAO trend over the last two decades (Hurrell et al. 2004), with higher positive extremes in the corresponding index.

In south Greenland and Baffin Bay the driest conditions are again especially favored in the NAO+ regime, whereas the NAO index in the negative regime practically does not have any average statistical influence on precipitation. Consistent values of t₋ = 0.14 and t₊ = −0.44 are given at the GRE point. This is visible from the completely different shape of PDF contours in the positive and negative regimes (Fig. 2d). This may be due to a nonlinear influence of NAO on the systems influencing the precipitation in Greenland, which must be synoptically analyzed in further studies. The precipitation in Greenland is also correlated with the presence of other regimes, such as the Greenland–Scandinavian regime, influencing the strength of the Icelandic low (Serreze et al. 1997). We have studied another particular situation where t₋ and t₊ have the same signal, opposite to that of t_M. This holds at the EUS point (Fig. 2c) with t₋ = 0.24, t₊ = 0.35, and t_M = −0.28. Unlike other points of strong correlation, the (X_g, Y_g) distribution in the RUS point is rather close to bi-Gaussianity, as is clear from Fig. 2f and Table 2. At this point c_g = 0.69.

The asymmetry measure J_c(X_g, Y_g) for Gaussian data is presented in Fig. 3a. The largest values are significant at ∼95% significance level (see Table 1). Some coherent regions of significant J_c are visible in the map over the Mediterranean and the central Atlantic, and near 40°N, South Greenland, and the Canadian Arctic coast.

b. Cumulants and Edgeworth PDF applicability

The contribution to non-Gaussianity comes from high-order cumulants, easily expressed in terms of nonlinear correlations, leading to nonzero cumulant terms in the Edgeworth expansion of the joint PDF [(A1)] and the negentropy [(26)]. We show maps of the main cumulant terms of the Gaussian variables k^(2,1), k^(3,1) (Figs. 3b–3c) intervening the most in (A1) and (26). They are proportional to the nonlinear correlations cor(X²_g, Y_r) and cor(X³_g, Y_r) respectively. The other cumulants—k^(1,2), k^(1,3), and k^(2,2)—are only residual (not shown).

The first and second mentioned correlations express the correlation between the residues of the precipitation linear prediction and, respectively, the squares (X²_g) and cubes (X³_g) of the Gaussian NAO index X_g. These correlations express, respectively, how a quadratic or a cubic function of NAO fits those residues. The cumulant k^(2,1) is dominant over Greenland, the Mediterranean, and the Atlantic Central Basin. The corresponding spatial dependence is closely related to the asymmetry test J_c map (Fig. 3a), with a map correlation of 0.93 over the NAE. This is explained because the nonlinear correlations cor(X²_g, Y_r) and cor(|X_g|, Y_r), related with J_c (see section 2b), behave in the same way.

The cumulants k^(3,1) and k^(2,1) also contribute to the nonlinear prediction [(28)]. Note, in particular for the central Atlantic, that the stronger correlation on the negative side (NAO−) is consistent with a fitting predictive curve formed from a negative slope straight line [resulting from the negative correlation c(X_g, Y_g)], a positive concavity elliptic curve [resulting from the positive cor(X²_g, Y_r), Fig. 3b] and cubic curve dependent on X³_g [resulting from the positive cor(X³_g, Y_r), Fig. 3c]. To verify how the EDG-PDF differs from a probability density, we compute P_neg (Fig. 3d). The maxima of |P_neg| are reached in the central Atlantic region (∼0.014) and south of Iceland (∼0.008). These small values constitute a necessary albeit not a sufficient condition for the EDF-PDF to be a good representation of the real PDF.

c. Mutual information

The map of the Gaussian MI I_g(X_g, Y_g) (Fig. 4a) is obtained from that of c_g, with two cores of maxima reaching 0.4 nats near 60°N, 10°W, and 0.30 nats near 35°N, 15°W, where nat is the MI unit when natural logarithms are used. The non-Gaussian MI is computed using the following three proposed estimators: the maximum entropy estimator I_ng(ME) (Fig. 4b), the Edgeworth estimator of (26) I_ng(EF) (Fig. 4c), and that obtained with the integral (23), I_ng(MI) (practically identical to that of Fig. 4c). There are some regions with I_ng(ME)(X_g, Y_g) above the corresponding 95% significance level (0.039 nats). These regions are the central and west Atlantic, I_ng ≈ 0.06 nats, southeast Iceland, and south Greenland, where I_ng reaches maxima of ∼0.06 nats. The value of I_ng over the Mediterranean, approximately 0.02–0.04 nats, appears to be significant at the 80% level. There are also some regions of non-Gaussianity in central Europe (Fig. 4b) and around 42°N, 48°W, with I_ng ≈ 0.04 nats. As expected, I_ng and the asymmetry measure J_c, share some common regions, because the asymmetry contributes to non-Gaussianity. Contrary to the EDG estimator, the ME estimator has no fundamental limitations as far as the size of cumulants is concerned. To assess the effect of the Taylor approximation of the logarithm of the EDG-PDF [(24)], we have sorted, in ascending order, all the values of the integral value I_ng(MI) in the 833 grid points of the NAE and plotted them against the correspondent values of I_ng(EF) (Fig. 5a). Both estimators agree within the error of the Gaussian quadrature (∼10⁻⁴ nats) up to I_ng ≈ 0.02 nats, followed by a slight overestimation by the Edgeworth equation [(26)] of MI [I_ng(EF)]. The map correlation between the two estimators is 0.98. By comparing the sorted values of I_ng(ME) with the correspondent I_ng(EI) values (Fig. 5b), we have an idea about the effect of Edgeworth truncation error in (23). Their correlation is 0.76, while the correlation with I_ng(EF) is expectedly lower: 0.69. By collecting all the 833 values, the I_ng(EI) estimator exhibits a negative increasing bias in comparison with the maximum entropy estimator, especially for I_ng(ME) > 0.01 nats. This justifies the missing of some “non-Gaussian” regions in central Europe in the map of I_ng(EF) (Fig. 4c). The central Atlantic, south Iceland, and Greenland non-Gaussian regions are retrieved by I_ng(EF), where excessively “spiky” values occur and P_neg reaches the maximum values. Overall, the I_ng(ME) is a smoother field than that of I_ng(EF). As far as non-Gaussianity maps are concerned, the fraction of statistically significant area (FS) is slightly larger than that obtained by mere chance (10%), contrary to FS for the correlation maps. This is a consequence of the intrinsically low non-Gaussianity values resulting from temporal averaging.

d. The effect of Gaussian anamorphosis

Near the centroids of maximum Gaussian correlation, 40°N, 20°W and 60°N, 10°W, we have verified that the side correlation is more intense in the wet NAO phase than in the dry NAO phase. If no Gaussian anamorphosis (GA) is performed, there is an enhancement of the side correlation over the positively skewed half of the precipitation PDF in comparison with the marginally Gaussian variables. This can be seen through the larger intensity of the asymmetry measure J_c (Fig. 6a) as compared with the Gaussian case (Fig. 3a). The GA is a nonlinear transformation that can either increase or decrease the absolute correlation |c| between variables, and thus the Gaussian MI. For example, GA, when performed with appropriate mixtures of two bivariate Gaussian PDFs (one with zero correlation and the other with correlation ∼1), can convert a nearly zero correlation into a nearly 100% correlation and vice versa. In our case, the change of correlation resulting from GA is not very high (maximum of ∼10%). However, given that the derivative of the Gaussian MI (I_g) grows from zero to infinity when the absolute correlation |c| tends to one, a small change of |c| can make a large difference in I_g. The I_g difference between original and Gaussian data is plotted in Fig. 6b with a positive maximum of ∼0.05 nats near 40°N, 20°W where |c| grows from ∼0.64 to ∼0.69. Comparing with the case of marginally Gaussian data, the I_ng over the Mediterranean is preserved in the original data (Fig. 6c) and another large area appears around the White Sea and Kola Peninsula at approximately 60°N, 35°E.

The invariance of MI, for the original and Gaussian data (16), holds for the ME estimator with an accuracy of ∼0.01 nats. Beyond the numerical accuracy, this is also because cumulants of order larger than 4, not taken into account in the ME estimator of MI, have a slightly different effect between the original and Gaussian data. The invariance [(16)] of MI is hardly verified for the estimator I_ng(EF), except for low values of the joint negentropy. The original highly skewed precipitation values render the EDF-PDF inapplicable for much of the NAE because of the much higher values of |P_neg| (Fig. 6d) (above 0.01), as compared with those of “Gaussinized” variables (Fig. 3d).