Probabilistic canonical correlation analysis forecasts, with application to tropical Pacific sea-surface temperatures

Transcription

1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 34: (014) Published online 18 June 013 in Wiley Online Library (wileyonlinelibrary.com) DOI: /joc.3771 Probabilistic canonical correlation analysis forecasts, with application to tropical Pacific sea-surface temperatures Daniel S. Wilks* Department of Earth & Atmospheric Sciences, Cornell University, Ithaca, NY, USA ABSTRACT: Canonical correlation analysis (CCA) is a higher-dimensional extension of univariate multiple regression that is often used to construct seasonal and other forecasts in a climatological context. Although its use is widespread, to date it has apparently been used only to produce nonprobabilistic forecasts. Here an analytic result for the prediction covariance matrix of vector CCA forecasts is presented, which is sufficient to define a full forecast probability distribution if a multivariate Gaussian distribution can reasonably be assumed for the forecast errors. The approach is illustrated by computing and verifying probabilistic seasonal forecasts for tropical Pacific sea-surface temperatures. KEY WORDS canonical correlation; probability forecasting; El Niño; Modoki Received 6 July 01; Revised 5 April 013; Accepted May Introduction Canonical correlation analysis (CCA) is a statistical technique that finds best linear relationships between two sets of multivariate (i.e. vector) data. The approach was first developed by Hotelling (1936), and was introduced into the meteorology and climatology literature with an emphasis on forecasting applications by Glahn (1968). Glahn (1968) and Mardia et al. (1979) point out that CCA is a multivariate extension of the more familiar technique, multiple linear regression. CCA has been used extensively in the climate literature both for diagnostic (e.g. Bretherton et al., 199; Graham, 1994; Livezey and Smith, 1999; Gershunov and Cayan, 003; Bowden and Semazzi, 007) and prognostic purposes (e.g. Barnett and Preisendorfer, 1987, Graham et al., 1987; Barnston and Ropelewski, 199; Barnston, 1994; Landman and Mason, 001; Wilks, 008). However, specifications and forecasts produced by CCAs have been nonprobabilistic, meaning that the results have not included quantifications or other expressions of uncertainty. Indeed, apparently the only statistical inferences computed for CCA to date have been simple tests examining the null hypothesis that the underlying data vectors are uncorrelated (e.g. Friederichs and Hense, 003). In forecasting contexts, and especially for forecasts with modest skill, it is important to quantify uncertainty by also providing probability bounds (e.g. Mason and Mimmack, 00). This paper will derive the prediction covariance matrix for CCA-based forecasts, by applying * Correspondence to: D. S. Wilks, Department of Earth & Atmospheric Sciences, Cornell University, Ithaca, NY, USA. dsw5@cornell.edu well-known results from linear regression and multivariate statistics. Assuming that the forecast errors follow a multivariate Gaussian distribution, the prediction covariance matrix is sufficient to define a probability model for the forecast errors, and thus to compute multivariate probability forecasts for the predictands and for functions of the predictands. Derivation of the CCA forecast covariance is not a new result, but its use to produce multivariate probabilistic forecasts apparently is. Section will review CCA, and present the calculation of the prediction error covariance matrix. Section 3 illustrates the method in the contexts of both univariate and multivariate probabilistic seasonal forecasts of tropical Pacific seasurface temperature anomalies (SSTAs), and Section 4 concludes.. CCA and probability forecasts.1. Nonprobabilistic CCA forecasts The purpose of CCA is to find linear combinations in each of two multivariate data sets that are most highly correlated with each other while simultaneously being uncorrelated with the other linear combinations that have been or will be found. When one of the two data sets (the predictors) leads the other (the predictands) in time, then it is natural to use CCA as a linear forecasting procedure. Nonlinear generalizations of CCA have also been proposed (Lai and Fyfe, 1999; Hsieh, 000, 001, 009), although the time averaging that is intrinsic to seasonal forecasting settings renders them at least quasilinear, so that nonlinear statistical methods have not yielded improved forecasts at seasonal timescales (Yuval and Hsieh, 00; Hsieh, 009). 013 Royal Meteorological Society

2 1406 D. S. WILKS Consider an I -dimensional vector of centred (i.e. zeromean) predictor variables, x,andaj -dimensional vector of centred predictand variables y. A CCA transforms pairs of these vectors into M = min(i, J )setsofnew variables, called canonical variates, defined by and v m = a T m x w m = b T m y. (1a) (1b) The canonical variates are thus linear combinations of the original predictor and predictand variables, with the weights a m and b m being the unique choices, called the canonical vectors, yielding the properties Corr (v 1, w 1 ) Corr (v, w ) Corr (v M, w M ) 0 (a) { r Cm, l = m Corr (v l, w m ) = (b) 0, l m and Corr (v l, v m ) = Corr (w l, w m ) = 0, l m (c) Var (v m ) = Var (w m ) = 1, m = 1,..., M. (d) That is, each canonical pair exhibits no larger correlation than does the previous (lower-numbered) pair (Equation (a)), and these correlations are called the canonical correlations r C (Equation (b)). Each canonical variate is uncorrelated with all other canonical variates except its counterpart in the mth pair (Equations (b) and (c)), and the lengths of the canonical vectors are defined to yield unit variance for each of the canonical variables (Equation (d)). The mechanics of computing the canonical vectors and canonical correlations from the joint covariance matrix of x and y are explained in standard references (e.g. Mardia et al., 1979; Wilks, 011). These computations are not feasible when one or both of the original vector dimensions are larger than the sample size, n (number of pairs of predictor and predictand vectors in the training data), in which case it is usual for x to contain the I leading principal components (PCs) of the predictors, and/or for y to contain the leading J PCs of the predictands (Barnett and Preisendorfer, 1987; Wilks, 011). Forecasting with CCA is straightforward because of its properties outlined above. Each predictand canonical variate is forecast using the simple linear regression function of its counterpart predictor canonical variate ŵ m = β 0,m + β 1,m v m = r Cm v m, m = 1,..., M, (3) where the intercepts β 0,m are zero because centred data have been used, and the regression slopes β 1,m are equal to the canonical correlations because the canonical variates all have unit variance. The regressions for each of the predictand canonical variates can be specified independently of the others because of the uncorrelatedness property expressed in the second line of Equation (b). Jointly, all M of these regressions can be written as where ŵ = [R C ] v = [R C ][A] T x (4) r C r C 0 0 [R C ] = 0 0 r C r CM is the (M M ) diagonal matrix containing the canonical correlations, and the columns of the (I M ) matrix [A] are the predictor canonical vectors a m,sothatv is the vector of predictor canonical variates. A nonprobabilistic CCA forecast is completed by recovering the predictand anomaly vector ŷ. As, collectively, the predictand canonical variates (Equation (1b)) can be expressed in matrix notation as (5) w = [B] T y (6) where the columns of the (J M ) matrix [B] are the predictand canonical vectors, the predicted anomaly vector is ŷ = ( [B] T) 1 ŵ = ( [B] T ) 1 [RC ] v = ( [B] T ) 1 [RC ][A] T x. (7) In order for the matrix inversion in Equation (7) to be computable, the matrix [B] must be square, implying that the number M of regressions in Equation (4) must be equal to the dimensionality J of the predictand vector y, although the predictor vector x may have any dimensionality I that is at least as large as J. In cases where the leading predictand PCs have been used to compute the CCA, Equation (7) will yield a vector of predicted PCs, and an addition step (detailed in the Appendix) is necessary to recover the physical predictand variables. In either case, Equation (7) indicates the equivalence of CCA to a set of simultaneous multiple regressions, and that the matrix [R C ] of canonical correlations is a diagonalization of the matrix of regression coefficients (e.g. DelSole and Chang, 003)... Probabilistic CCA forecasts The forecast anomaly vector ŷ in Equation (7) will never be without prediction uncertainty. However, it is straightforward to estimate the prediction error covariance matrix, using elementary results from regression and multivariate statistics. Consider first the well-known result (Draper and Smith, 1981; Wilks, 011) for the 013 Royal Meteorological Society Int. J. Climatol. 34: (014)

3 PROBABILISTIC CANONICAL CORRELATION ANALYSIS FORECASTS 1407 prediction error variance sŵ m of the mth simple linear regression in Equation (3): sŵ m = se m n + (v 0,m v m) n ( ) = vi,m v m s e m i= n + v n 0,m vi,m i=1. (8) Here v 0,m is the value of the mth predictor canonical variate, the v i,m are the n training-data values for this canonical variate, and se m is the residual variance (i.e. regression MSE) for the mth regression. The prediction error variance is larger than the residual variance because of the second and third terms in the square brackets, which relate respectively to uncertainty in estimating the regression intercept (here, the sample mean used to centre the predictand data), and the uncertainty in estimating the regression slope (i.e. the mth canonical correlation). The second equality holds because the predictor data have been centred so that the means of the predictor canonical variates in the training data will also be zero. Next, recall that the fraction of variance of the canonical variate w m accounted for by the mth regression in Equation (3) is simply rc m, so that the fraction of variance represented by the corresponding regression residual is 1 rc m. However, because the predictand canonical variate has unit variance by construction (Equation (d)), se m = Var (v m ) ( 1 rc ) ( ) m = 1 r Cm (9) so that Equation (8) becomes sŵ m = ( 1 rc ) m n + v n 0,m vi,m i=1 (10) Because each of the M regressions in Equation (3) is independent of the others, the joint error covariance for the vector of predictand canonical variates ŵ is the matrix s ŵ sŵ [Sŵ] 0 =. (11) sŵ M where each of the diagonal elements are as specified by Equation (10). Finally, as Equation (7) recovers the physical predictand anomalies ŷ through the M linear transformations held as the rows of the matrix ([B] T ) 1, their prediction covariance matrix can be obtained from Equation (11) using the standard result in multivariate statistics (e.g. Mardia et al., 1979; Wilks, 011), [ ] ( Sŷ = [B] 1 ) T [Sŵ] [B] 1 (1) where the identity ([B] T ) 1 = ([B] 1 ) T has also been used. If the forecast errors can be reasonably approximated by a multivariate Gaussian distribution (e.g. Mardia et al., 1979; Wilks, 011), then a joint probability model for the forecast ŷ is fully defined, with mean vector and covariance matrix given by Equations (7) and (1), respectively. In cases where the leading predictand PCs have been used to compute the CCA, an additional step beyond Equation (1) is required to compute the predictand error covariance matrix, as detailed in the Appendix. 3. Tropical Pacific SST forecasts 3.1. Data, predictors, and predictands The results from Section will be illustrated by constructing joint probability forecasts for average January February March (JFM) SSTAs for the Niño 1 + (80 W 90 W, 10 S 0 S), Niño3(90 W 150 W, 5 S 5 N), Niño 3.4 (10 W 170 W, 5 S 5 N), and Niño 4 (150 W 160 E, 5 S 5 N) regions of the tropical Pacific ocean. These have been abstracted from the HadISST data set (Rayner et al., 003) for , with forecasts to be computed for the years The predictor variables are the I = 4 leading PCs of seasonal (3-month averaged) gridded Indo-Pacific SSTAs for the region ranging from 51 E 81 W to 13 S 55 N, after the original 1 1 SSTA data have been averaged to a 4 4 grid. Prediction lead times range from 0 months (predictor SSTAs for October November December) through 9 months (predictor SSTAs for JFM of the previous year). A separate principal component analysis (PCA) was computed for each of the ten predictor seasons, and these were updated with an additional year included in the training data set for each of the prediction years For example, lead forecasts for JFM 1981 were trained using predictor PCs for September October November (SON) 1950 through SON 1979, with the PCA having been computed using the 31 years Lead forecasts for JFM 198 were trained using predictor PCs for SON 1950 through SON 1980, with the PCA computed using The protocol thus emulates what could have been achieved operationally during the forecast period Because there are only J = 4 predictand variables, the four Niño-region SSTAs have not been represented by their leading PCs, so that Equations (7) and (1) pertain directly to the predictand temperature anomalies. As was also done in Wilks (008), canonical correlations not large enough to be significantly different from zero at the 1% level (values ranging from r C < 0.46 for the n = 30 forecasts for 1981, to r C < 0.3 for the n = 60 forecasts for 011) have been set to zero in Equations (7) and (10). 013 Royal Meteorological Society Int. J. Climatol. 34: (014)

4 1408 D. S. WILKS 0.6 Nino Nino 3 Nino CRPS skill 0. Count Nino Lead time, months Cumulative Probability of Observation Figure 1. CRPS skill scores (Equation (13)) for univariate Gaussian probability forecasts for the four SSTA regions, , as functions of lead time. 3.. Univariate probability forecasts Equations (7) and (1) are sufficient to define four univariate Gaussian forecast distributions, with means given by the J = 4 elements of the vector ŷ and variances given by the corresponding diagonal elements of [ Sŷ ].A convenient generalized, univariate measure for evaluating the accuracy of these forecasts is the continuous ranked probability score (CRPS) (Matheson and Winkler, 1976; Wilks, 011), which is the integrated squared difference between the cumulative distribution function (cdf) of the forecast and the step function that increases from 0 to 1 at the observation, and which has a convenient analytical representation when the forecast probability distribution is Gaussian (Gneiting et al., 005; Wilks, 011). Figure 1 shows CRPS for the four predictand regions as functions of lead time, expressed as the skill score SS CRPS = 1 CRPS (13) CRPS clim where the overbars indicate averages over the prediction period, and the climatological reference consists of the Gaussian distributions fit to the SSTAs for the respective regions over the common training period. Forecasts for the Niño 3.4, Niño 4, and Niño 3 regions are reasonably skilful according to this measure at the shorter lead times, with accuracy declining to essentially zero skill at lead times of 7 months and longer. In contrast, the probability forecasts for Niño 1+ are notably less accurate, which has likely resulted from only the leading I = 4 predictor PCs being used in the CCA calculations, which PCs represent the larger scale features of the SST variations that the other three Niño regions have been specifically designed to capture. It is also of interest to evaluate the calibration (or reliability ) of the probability forecasts. This can be done using the probability-integral-transform (PIT) histogram Figure. PIT histogram aggregated over all regions, lead times, and years, with cumulative probabilities for the observations binned at a resolution of The horizontal dashed line shows the ideal level of uniformity for the histogram bars. (Gneiting et al., 005; Wilks, 011). The PIT histogram is the extension, for continuous forecast distributions, of the more familiar verification rank histogram (Hamill and Colucci, 1998; Wilks, 011) used to evaluate the calibration of discrete ensemble forecasts, and can be interpreted using the same diagnostics (Hamill, 001; Wilks, 011). The idea behind the PIT histogram is that, if the forecast probability distributions are well calibrated, then each observation is equally likely to correspond to any cumulative probability with respect to its forecast distribution. Figure shows the PIT histogram aggregated over all 4 (predictand regions) 10 (lead times) 31 (years) = 140 forecasts, with cumulative probabilities for the observations binned at a resolution of The horizontal dashed line shows the ideal level of PIT uniformity for the histogram bars, located at 140/0 = 6 forecasts per bin. The character of the histogram bars in the first 19 bins is for a small positive slope, which is diagnostic for a slight systematic underforecasting, or cold bias. That is, observed SSTAs were slightly less likely to occur among the lowest quantiles of their forecast distributions during This feature of the forecasts has almost certainly resulted from the gradual warming of the tropical Pacific during this period. The most prominent feature of Figure is the over-representation, by more than a factor of, of observations falling above the 95th percentiles of their forecast distributions. The overwhelming majority of these correspond to the very warm SSTAs of El Niño years, several of which during the forecast period were unusually warm relative to the common training period. Overall, deviations from forecast calibration in Figure are attributable to the nonstationarity of tropical Pacific SSTs during the period considered, and apparently do not reflect shortcomings in the probability calculations from Section. 013 Royal Meteorological Society Int. J. Climatol. 34: (014)

5 PROBABILISTIC CANONICAL CORRELATION ANALYSIS FORECASTS : (0.53) 1998 (0.09) 199 (0.8) 010 (0.59) 1987 (0.45) Observed Anomaly, C (empirical) 011 (0.34) 1989 (0.14) 1995 (0.11) 003 (0.11) 1993 (0.19) 1988 (0.36) 1991 (0.11) 005 (0.08) 1990 (0.31) 007 (0.1) 004 (0.10) 00 (0.47) 198 (0.09) 1994 (0.34) 1997 (0.11) 1981 (0.6) 001 (0.10) 1984 (0.40) 1986 (0.11) 1996 (0.10) 009 (0.09) 006 (0.17) 1985 (0.6) 1999 (0.10) 000 (0.14) 008 (0.10) (Gaussian) Forecast Anomaly, C Figure 3. Box plots for individual Gaussian forecast distributions, for JFM Niño 3.4 SSTAs at 1-month lead time, plotted at the vertical level of the corresponding observed value. Whiskers delineate central 90% forecast intervals, and CRPS for each forecast is indicated parenthetically. Horizontal climatological box plot at figure bottom shows Gaussian fit to the common training period in the same format. Vertical climatological distribution is the corresponding empirical box plot, with whiskers extending to the warmest and coldest SSTAs during Figure 3 looks in more detail at individual forecasts for the Niño 3.4 SSTAs, at the 1-month lead time. The thin horizontal box plots in the main portion of the figure sketch the Gaussian forecast distributions for the indicated years, with the whiskers spanning the 90% central prediction intervals. The vertical plotting levels locate the corresponding observed values, and the CRPS for each forecast is shown parenthetically, after the year label. With the exception of the warmest (El Niño) years, these forecast distributions exhibit reasonably good calibration overall, consistent with the results in Figure. The individual forecast distributions also differ substantially from the Gaussian fit to JFM Niño 3.4 SSTAs for the common training period, shown as the larger horizontal box plot at the bottom of the figure, again with whiskers extending to the 5th and 95th percentiles. The individual forecasts are both sharper (narrower) than this climatological distribution, and are often centred well away from the zero anomaly, reflecting substantial forecast resolution. The large vertical box plot on the left indicates the empirical distribution for JFM Niño 3.4 SSTAs, with the ends of the serifed whiskers locating the maximum and minimum SSTAs during Three of the El Niño years during the forecast period can be seen to have been warmer than any of the years during Comparing the two climatological box plots suggests that the Gaussian assumption for this predictand is a reasonable one. Figure 4 shows the corresponding results for the 6-month lead time. These forecasts are clearly less sharp than the 1-month lead forecasts in Figure 3, with wider box-plots, but are nevertheless still shaper than the climatological distribution shown at the bottom of the figure. The forecast distributions also have a noticeably smaller tendency to be centred on the dashed 1:1 diagonal than do their counterparts in Figure 3, reflecting their weaker resolution. CRPS values are in most cases substantially larger as well, but in aggregate these forecast do exhibit positive CRPS skill relative to the climatological distribution (compare Figure 1) Multivariate probability forecasts The primary advantage of CCA-based forecasts over their multiple-regression counterparts is the potential to better capture covariability among multiple predictands. In particular, the results in Section can be used to formulate multivariate Gaussian forecast distributions, which may be of interest in their own right, and may also be used to calculate probability forecasts for predictands that are functions of the multiple CCA predictands. 013 Royal Meteorological Society Int. J. Climatol. 34: (014)

6 1410 D. S. WILKS 3 1: (1.39) 1998 (0.65) 199 (1.30) 010 (0.96) 1987 (0.8) (0.51) Observed Anomaly, C (empirical) 00 (0.86) 001 (0.4) 009 (0.1) 005 (0.3) 007 (0.38) 004 (0.40) 198 (0.11) 1997 (0.19) 011 (0.45) 1986 (0.16) 1985 (0.34) 1999 (0.44) 003 (0.57) 1988 (0.5) 1991 (0.14) 1990 (0.6) 1994 (0.31) 1981 (0.4) 1996 (0.35) 006 (0.53) 1993 (0.44) 1984 (1.97) 1989 (0.15) 000 (0.15) 008 (0.47) (Gaussian) Forecast Anomaly, C Figure 4. Box plots for individual Gaussian forecast distributions, for JFM Niño 3.4 SSTAs at 6-month lead time, plotted at the vertical level of the corresponding observed value. Whiskers delineate central 90% forecast intervals, and CRPS for each forecast is indicated parenthetically. Horizontal climatological box plot at figure bottom shows Gaussian fit to the common training period in the same format. Vertical climatological distribution is the corresponding empirical box plot, with whiskers extending to the warmest and coldest SSTAs during Figure 5 shows 90% forecast probability ellipses for joint Niño3andNiño 4 forecasts for JFM 1995, at 9-, 5-, -, and 1-month lead times. The corresponding numerals locate the mean vectors for each of these forecasts, and the joint observed JFM 1995 SSTAs are located by the solid dot. The 9-month lead forecast differs little from the joint climatological distribution for the Niño 3 and Niño 4 SSTAs. Forecasts at progressively shorter lead times are centred closer to the observation, and forecasts for the intermediate lead times (not shown in order to improve readability) also occur along this trajectory. The observed SSTAs fall within each of the 90% forecast ellipses, but these are progressively sharper (more compactly concentrating probability) for the shorter lead times. The character (or so-called flavor ) of the 1995 El Niño differed from the conventional or canonical (Rasmusson and Carpenter, 198) event, in the sense that the strongest warming occurred in the Niño 4 rather than the Niño 3 region. Events of this type have been called variously Modoki (Ashok et al., 007; Takahashi et al., 011), dateline (Larkin and Harrison, 005), central Pacific (Kao and Yu, 009), or warm pool (Kug et al., 009, Ren and Jin, 011) El Niño events. Ren and Jin (011) proposed the index = N 4 αn 3 (14) to represent warm-pool El Niño events, in which N 4 is the Niño 4 SSTA, N 3 is the Niño 3 SSTA, and α = /5 Niño 4 Anomaly, C 1 0 =3/ =1 =1/ =0 =1/ =0 Lead=9 9 Lead= Lead= Lead=1 Niño 3 Anomaly, C =0 =/ =/ = =-3/ Figure 5. Ninety percent forecast probability ellipses for joint Niño 3 and Niño 4 forecasts for JFM 1995, at the 1-, -, 5-, and 9-month lead times. Numerals show respective forecast means, and the solid dot locates the observed 1995 value. Grey background contours are for the Ren and Jin (011) index. if N 4 N 3 > 0, with α = 0 otherwise. A warm-pool El Niño event is indicated for > 0.46, which is +1 standard deviation above the mean of its climatological distribution. Similarly Ren and Jin (011) also proposed the cold-tongue index N CT = N 3 αn 4 (15) 013 Royal Meteorological Society Int. J. Climatol. 34: (014)

7 PROBABILISTIC CANONICAL CORRELATION ANALYSIS FORECASTS 1411 Probability Density (a) (b) (c) (d) 1.5 Pr{ 0.46} Pr{ 0.46} Pr{ 0.46} =0.56 =0.43 = Pr{ 0.46} = Index Index Index Index Figure 6. Forecast probability density functions for the JFM 1995 index, at lead times of (a) 1 month, (b) months, (c) 5 months, and (d) 9 months. Probability Density (a) (b) (c) (d) 1.5 Pr{N CT 0.70} Pr{N CT 0.70} Pr{N CT 0.70} =0.4 =0.09 = Pr{N CT 0.70} = N CT Index N CT Index N CT Index N CT Index Figure 7. Forecast probability density functions for the JFM 1995 N CT index, at lead times of (a) 1 month, (b) months, (c) 5 months, and (d) 9 months. where the symbols are as defined previously. A conventionalelniño is indicated for N CT > 0.70, which again is 1 standard deviation above its mean. The heavy grey contours in Figure 5 show isolines of, which locate the value of this index as 0.74 for JFM Equation (14) is a nonlinear function of the two prognostic variables in Figure 5. In particular, it is discontinuous at N 4 = 0 from below when N 3 < 0and discontinuous from above when N 3 > 0, as suggested by the gaps in the contours for =±1/. Probability forecasts for can nevertheless be computed from the joint Gaussian forecasts sketched by the ellipses in Figure 5, by numerically integrating Pr { <η} = I ( η) f ( ) N 3, N 4 dn 3 dn 4 (16) to obtain the forecast cdf for. Here f (N 3, N 4) is the forecast bivariate Gaussian probability density function (pdf) for the Niño 3 and Niño 4 SSTAs, the mean and variance parameters for which are taken from the appropriate elements of Equations (7) and (1), respectively; the indicator function I ( ) = 1 if its argument is true, and is zero otherwise; and η represents a generic value for the index. Figure 6 shows resulting forecast pdfs for the JFM 1995 index, corresponding to the four lead times shown in Figure 5. The vertical dashed lines show the threshold above which a warm-pool El Niño is deemed to occur, which is forecast as progressively more likely at the shorter lead times. Figure 7 shows the corresponding forecast pdfs for the JFM 1995 N CT index, which show much lower probabilities for the occurrence of a conventional El Niño at all lead times. The realized value of N CT in JFM 1995 was Summary and conclusions CCA continues to be a popular multivariate statistical seasonal forecasting approach. However, to date it has been used only to produce nonprobabilistic multivariate forecasts, although nonprobabilistic CCA forecasts for scalar predictand elements have been post-processed to yield probabilities based on cross-validated error estimates (Landman et al., 01). This paper has presented an analytic method for calculating the forecast covariance matrix pertaining to the conventionally forecast mean vector. Together, this mean vector and covariance matrix define a multivariate Gaussian forecast distribution that can be used to evaluate probabilities of outcomes corresponding to, or derived from, the prognostic variables. The approach has been illustrated for seasonal forecasts of JFM tropical Pacific SSTAs, which exhibit approximately Gaussian distributions, but non-gaussian predictands could also be forecast after suitable variable transformations. Computation of prediction covariance matrices could likely be extended to the related multivariate technique, maximum covariance analysis (MCA, Bretherton et al., 199; Wilks, 011), which is sometimes confusingly called singular value decomposition (SVD) analysis. The primary difference between MCA and CCA is the fact that covariances, rather than correlations, are maximized between linear combinations of predictor and predictand vectors, so that the independence conditions expressed in Equations (b) and (c) do not hold for MCA. Therefore 013 Royal Meteorological Society Int. J. Climatol. 34: (014)

8 141 D. S. WILKS the predictand covariance matrix in Equation (11) will not in general be diagonal, and its elements would need to be estimated using the n-member training data. Other potential applications derive from the capacity to statistically simulate from (i.e. generate random vectors consistent with) the forecast multivariate Gaussian distributions (e.g. Wilks, 011). For example, CCA can be used to downscale climate-change projections (e.g. Karl et al., 1990; von Storch et al., 1993; Huth, 00), effectively predicting smaller-scale fields from largerscale climate model simulations. This approach could be extended to include generation of ensembles of stochastic realizations centred on downscaled CCA means, by drawing randomly from the multivariate Gaussian distributions with covariance matrices given by Equation (1). Similarly, in modelling exercises involving a dynamical ocean forced by a statistical atmosphere (Barnett et al., 1993; Syu et al., 1995) or a dynamical atmosphere forced by a statistical ocean (Goddard and Graham, 1999; Mason et al., 1999), stochastic ensembles of the forcing could also be generated in this way. It has long been known that probability forecasts convey more information, and are more valuable economically to forecast users, than nonprobabilistic forecasts (e.g. Thompson, 196; Murphy, 1977; Krzysztofowicz, 1983; Katz and Murphy, 1997). The results presented here allow calculation of CCA-based forecasts that include the probability information necessary to enhance their information content and thus to extend and improve their applicability. Acknowledgements This research was supported by the US National Science Foundation under grant AGS1100. SST data were provided by the UK Met Office through Appendix: Prediction variance when the predictands are leading principal components The dimensionality K of the predictand vector is often too large for direct computation of CCA in climatological applications. For K n, inversion of singular matrices would be required, so that the computations are not defined. Even when K < n the sampling properties of CCA tend to be poor unless K n, so that often it is the J K leading PCs of the predictand vector that will be forecast using Equation (7). Define y* asthek-dimensional vector of predictand anomalies. Often K will be the number of gridpoints in a predictand field. Their J leading PCs will be defined by y = [E] T y (A1) where each of the J columns of [E] contain one of the leading K -dimensional eigenvectors of the covariance matrix of y*, so that ŷ [E] ŷ (A) expresses the truncated synthesis of the K -dimensional predictand field from its predicted J leading PCs. Substituting into Equation (7) yields the K - dimensional vector prediction ŷ = [E] ( [B] T) 1 ŵ = [E] ( [B] T ) 1 [RC ] v = [E] ( [B] T) 1 [RC ][A] T x (A3) Similarly, the (K K ) prediction covariance matrix for this forecast vector is obtained by extending Equation (1): [ ] [ Sŷ = [E] Sŷ ] [E] T = [E] ( [B] 1) T [Sŵ] [B] 1 [E] T (A4) References Ashok K, Behera SK, Rao SA, Weng H, Yamagata T El Niño Modoki and its possible teleconnection. Journal of Geophysical Research C11: C DOI: /006JC Barnett TP, Preisendorfer RW Origins and levels of monthly and seasonal forecast skill for United States air temperatures determined by canonical correlation analysis. Monthly Weather Review 115: Barnett TP, Latif M, Graham N, Flügel M, Pazan S, White W ENSO and ENSO-related predictability. Part I: prediction of equatorial Pacific sea surface temperature with a hybrid coupled ocean atmosphere model. Journal of Climate 6: Barnston AG Linear statistical short-term climate predictive skill in the northern hemisphere. Journal of Climate 7: Barnston AG, Ropelewski CF Prediction of ENSO episodes using canonical correlation analysis. Journal of Climate 5: Bowden JH, Semazzi FHM Empirical analysis of intraseasonal climate variability over the Greater Horn of Africa. Journal of Climate 0: Bretherton CS, Smith JM, Wallace JM An intercomparison of methods for finding coupled patterns in climate data. Journal of Climate 5: DelSole T, Chang P Predictable component analysis, canonical correlation analysis, and autoregressive models. Journal of the Atmospheric Sciences 60: Draper NR, Smith H Applied Regression Analysis. Wiley: New York; 709. Friederichs P, Hense A Statistical inference in canonical correlation analyses exemplified by the influence of North Atlantic SST on European climate. Journal of Climate 16: Gershunov A, Cayan DR Heavy daily precipitation frequency over the contiguous United States: sources of climatic variability and seasonal predictability. Journal of Climate 16: Glahn HR Canonical correlation and its relationship to discriminant analysis and multiple regression. Journal of the Atmospheric Sciences 5: Gneiting T, Raftery AE, Westveld AH, Goldman T Calibrated probabilistic forecasting using ensemble model output statistics and minimum CRPS estimation. Monthly Weather Review 133: Goddard L, Graham NE The importance of the Indian Ocean for simulating rainfall anomalies over eastern and southern Africa. Journal of Geophysical Research 104: Graham NE Decadal-scale climate variability in the tropical and North Pacific during the 1970s and 1980s: observations and model results. Climate Dynamics 10: Graham NE, Michaelsen J, Barnett TP Investigation of the El Niño-southern oscillation cycle with statistical models. Model results. Journal of Geophysical Research C9: Hamill TM Interpretation of rank histograms for verifying ensemble forecasts. Monthly Weather Review 19: Hamill TM, Colucci SJ Evaluation of Eta-RSM ensemble probabilistic precipitation forecasts. Monthly Weather Review 16: Royal Meteorological Society Int. J. Climatol. 34: (014)

9 PROBABILISTIC CANONICAL CORRELATION ANALYSIS FORECASTS 1413 Hotelling H Relations between two sets of variates. Biometrika 8: Hsieh WW Nonlinear canonical correlation analysis by neural networks. Neural Networks 13: Hsieh WW Nonlinear canonical correlation analysis of the tropical Pacific climate variability using a neural network approach. Journal of Climate 14: Hsieh WW Machine Learning in the Environmental Sciences. Cambridge University Press: Cambridge; 349. Huth R. 00. Statistical downscaling of daily temperature in central Europe. Journal of Climate 15: Kao H-Y, Yu J-Y Contrasting eastern-pacific and central-pacific types of ENSO. Journal of Climate : Karl TR, Wang W-C, Schlesinger ME, Knight RW, Portman D A method of relating general circulation model simulated climate to the observed local climate. Part I: seasonal statistics. Journal of Climate 3: Katz RW, Murphy AH Economic Value of Weather and Climate Forecasts. Cambridge University Press: Cambridge;. Krzysztofowicz R Why should a forecaster and a decision maker use Bayes theorem? Water Resources Research 19: Kug J-S, Jin F-F, An S-I Two tpyes of El Niño events: cold tongue El Niño and warm pool El Niño. Journal of Climate : Lai PL, Fyfe C A neural implementation of canonical correlation analysis. Neural Networks 1: Landman WA, Mason SJ Forecasts of near-global sea surface temperatures using canonical correlation analysis. Journal of Climate 14: Landman WA, DeWitt D, Lee D-E, Beraki A, Lötter D. 01. Seasonal rainfall prediction skill over South Africa: one- versus two-tiered forecasting systems. Weather and Forecasting 7: Larkin NK, Harrison DE On the definition of El Niño and associated seasonal average U.S. weather anomalies. Geophysical Research Letters 3: L13705, DOI:10.109/005GL0738. Livezey RE, Smith TM Covariability of aspects of North American climate with global sea surface temperatures on interannual and interdecadal timescales. Journal of Climate 1: Mardia KV, Kent JT, Bibby JM Multivariate Analysis. Academic Press: San Diego; 518. Mason SJ, Mimmack GM. 00. Comparison of some statistical methods of probabilistic forecasting of ENSO. Journal of Climate 15: 8 9. Mason SJ, Goddard L, Graham NE, Yulaeva E, Sun L, Arkin PA The IRI seasonal climate prediction system and the 1997/98 El Niño event. Bulletin of the American Meteorological Society 80: Matheson JE, Winkler RL Scoring rules for continuous probability distributions. Management Science : Murphy AH The value of climatological, categorical, and probabilistic forecasts in the cost-loss ratio situation. Monthly Weather Review 105: Rasmusson EM, Carpenter TH Variation in tropical sea surface temperature and surface wind fields associated with Southern Oscillation/El Niño. Monthly Weather Review 110: Rayner NA, Parker DE, Horton EB, Folland CK, Alexander LV, Rowell DP, Kent EC, Kaplan A Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century Journal of Geophysical Research 108(D14): 4407 DOI: /00JD Ren H-L, Jin F-F Niño indices for two types of ENSO. Geophysical Research Letters 38: L DOI: /010GL von Storch H, Zorita E, Cubasch U Downscaling of global climate change estimates to regional scales: an application to Iberian rainfall in wintertime. Journal of Climate 6: Syu H-H, Neelin JD, Gutzler D Seasonal and interannual variability in a hybrid coupled GCM. Journal of Climate 8: Takahashi K, Montecinos A, Goubanova K, Dewitte B ENSO regimes: reinterpreting the canonical and Modoki El Niño. Geophysical Research Letters 38: L DOI: /011GL Thompson JC Economic gains from scientific advances and operational improvements in meteorological prediction. Journal of Applied Meteorology 1: Wilks DS Improved statistical seasonal forecasts using extended training data. International Journal of Climatology 8: Wilks DS Statistical Methods in the Atmospheric Sciences, 3rd edn. Academic Press: Amsterdam; 676. Yuval, Hsieh WW. 00. The impact of time-averaging on the detectability of nonlinear empirical relations. The Quarterly Journal of the Royal Meteorological Society 18: Royal Meteorological Society Int. J. Climatol. 34: (014)