1. Introduction

A large body of observational, theoretical, and numerical evidence supports the hypothesis that the climate state is predictable beyond 2 weeks (Shukla 1998; Shukla and Kinter 2006). This evidence is based almost entirely on analysis of time-averaged fields. For instance, climate predictability, such as that due to anthropogenic forcing, is routinely established on the basis of 50-yr means (Hegerl et al. 2007). Seasonal-to-interannual predictability is established almost exclusively on the basis of 3-month averages. Intraseasonal predictability, if any, is often claimed on the basis of 5- to 14-day averages. The reason for using time averages in these cases is that the predictability cannot be detected easily otherwise. That is, if one computes predictability, based on instantaneous states, predictability decays until essentially all predictability is lost after 2 weeks (Simmons and Hollingsworth 2002). Yet, if there is predictability of time averages after 2 weeks, then not all predictability of instantaneous states can be lost after 2 weeks. A plausible explanation for the near vanishing of predictability after 2 weeks is that the predictable part beyond 2 weeks explains relatively little variance and hence is obscured by weather noise.

If the predictability beyond weeks arises from slowly varying structures, then time averaging will reduce the weather noise without appreciably reducing the predictable signal, thereby increasing the signal-to-noise ratio. Nevertheless, there are three obvious problems with using time averaging to diagnose predictability: time averages cannot resolve variations on time scales shorter than the averaging window, time averaging fails to capture structures that are predictable but not persistent, and the ability of time averaging to separate signal from noise is limited by the number of samples in the averaging window. An example of a predictable structure that is not persistent is a propagating wave packet—if the time averaging window exceeds the time during which the packet remains in a domain, then time averaging within the domain removes the packet.

Time averaging is effective in predictability analysis when different predictable phenomena are characterized by different time scales. An equally plausible assumption is that different predictable phenomena are characterized by different spatial structures. The latter assumption implies that spatial filters could be constructed to optimally estimate the amplitude of the structure. Such filters would allow predictable structures to be diagnosed even if they are not persistent and even if they vary on time scales shorter than typical averaging windows. Filters based on optimal projection vectors are a common tool in signal processing theory and are used extensively in climate and seasonal predictability under the names optimal fingerprinting and signal-to-noise empirical orthogonal functions (EOFs). These techniques are now recognized to be generalizations of linear regression theory and emerge naturally in a framework based on information theory (DelSole and Tippett 2007).

There are at least two bases for constructing spatial filters for diagnosing predictability beyond 2 weeks. The first is to identify structures that are highly persistent. For instance, DelSole (2001) proposed a procedure called optimal persistence analysis (OPA) that identifies structures with the longest time scales, as measured by the integral time scale. This approach, however, fails to capture predictability that is not persistent. The second is to identify structures that are highly predictable, relative to a chosen forecast model. Schneider and Griffies (1999) proposed a procedure called predictable component analysis that identifies structures whose forecast spread differs as much as possible from its climatological spread. One drawback of this method is that it optimizes predictability at a single lead time and hence returns structures that depend on lead time.

This paper proposes a new approach to constructing projection vectors for diagnosing predictability. The main idea is to optimize predictability integrated over all time lags, thereby removing the lead time dependence of conventional decomposition approaches (e.g., predictable component analysis, signal-to-noise EOFs, canonical correlation analysis). The integral of predictability, called the average predictability time (APT), was proposed in DelSole and Tippett (2009, hereafter Part I) and shown to posses several attractive properties, as reviewed in the next section. The general procedure for decomposing APT is discussed in section 3, and the specific case of linear regression models is discussed in section 4. The estimation of APT from finite time series is addressed in section 5. The result of applying this method to 1000-hPa zonal velocity fields is discussed in section 6, where the method is shown to seamlessly diagnose predictability on time scales from 6 h to decades. We conclude with a summary and discussion of our results.

2. Review of average predictability time

This section briefly summarizes average predictability time (APT) and its most important properties (see Part I for further details). APT is derived from a measure of predictability called the Mahalanobis signal, which is defined as

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where Σ_τ is the forecast covariance matrix at lead time τ, Σ_∞ is the climatological covariance matrix, and K is the state dimension; also, tr [·] denotes the trace operator. In one dimension, the Mahalanobis signal reduces to 1 minus the normalized error variance

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which in turn is the explained variance of a linear regression model. Therefore, the Mahalanobis signal can be interpreted as the multivariate generalization of a familiar measure of predictability. The Mahalanobis signal equals unity for a perfect deterministic forecast (i.e., Σ_τ = 0) and vanishes for a forecast that is no better than a randomly drawn state from the climatological distribution (i.e., Σ_τ = Σ_∞). Because in this paper only discrete time is considered, the APT S is defined to be twice the sum of the Mahalanobis signal over all positive lead times:

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The factor of 2 makes APT agree with the usual e-folding time in the univariate case. Important properties of APT include the following: 1) It is invariant to nonsingular, linear transformations of the variables and hence is independent of the arbitrary basis set used to represent the state; 2) upper and lower bounds on the APT of linear stochastic models can be derived from the dynamical eigenvalues; and 3) it is related to the shape of the power spectra and hence clarifies the relation between predictability and power spectra.

3. Decomposition of average predictability time

The APT (3) for a single component is

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A component is derived from a state vector x through the inner product q^Tx, where q is a projection vector and the superscript T denotes the transpose operation. We seek the projection vector that optimizes APT. The component q^Tx has forecast and climatological variances

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Substituting (5) into (4) gives

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Thus, the problem of finding the projection vector q that optimizes APT is equivalent to finding the projection vector that optimizes (6). This optimization problem can be solved by noting that (6) is a Rayleigh quotient. Specifically, a standard theorem in linear algebra (Noble and Daniel 1988) states that the vectors that optimize a Rayleigh quotient of the form (6) are given by the eigenvectors of the generalized eigenvalue problem

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Because 𝗚 and Σ_∞ are symmetric, the eigenvectors of (8) produce components that are uncorrelated. The eigenvalue λ gives the value of APT associated with each eigenvector. It is convention to order the eigenvectors by decreasing order of eigenvalue, in which case the first eigenvector maximizes APT, the second maximizes APT subject to being uncorrelated with the first, and so on.

The above description of the components that maximize APT is not complete because the eigenvalue problem (8) gives only projection vectors for deriving the time series. The spatial pattern associated with the component can be derived by recognizing that the component time series are uncorrelated, in which case the spatial pattern p can be obtained simply by projecting the component time series q^Tx on the original data, which gives

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where the angle brackets denote an average over the climatological distribution; also, we have assumed without loss of generality that the climatological mean vanishes and that the component has unit variance—that is, σ_∞² = q^TΣ_∞q = 1. With this notation, the state x can be decomposed into components that optimize APT as

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where the projection patterns q and spatial patterns p have been ordered in decreasing order of APT. The above decomposition is analogous to the manner in which principal component (PC) analysis decomposes data by variance, except that here we are decomposing APT.

Because the component time series are uncorrelated, the total variance is the sum of the variance explained by each component. Because the components have unit variance, if p represents the state vector on a grid, then the absolute value of each element of the vector p is the standard deviation at that grid point of the component in question. Furthermore, the total variance due to the component is given by p^Tp. The fraction of variance explained by the component is therefore p^Tp divided by the total variance 〈x^Tx〉.

4. Decomposition of APT for linear regression models

In this section we discuss the decomposition of APT for a linear regression model, which can be derived from data without a prior forecast model. A linear regression prediction of the future state x_t+τ based on the present state x_t is of the form

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where 𝗟_τ is a regression operator and the hat denotes a predicted quantity. The regression operator that minimizes the mean square prediction error is a standard problem with solution

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where 𝗖_τ is the time-lagged covariance matrix, defined as

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for zero mean stationary processes, 𝗖₀ = Σ_∞. The covariance matrix of forecast error is therefore given by

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where we have used (12). Substituting this forecast covariance matrix into the generalized eigenvalue problem (8) gives

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Thus, the components that maximize the APT of a linear regression model are obtained by solving the generalized eigenvalue problem (15). Note that this solution depends only on time-lagged covariances, which can be estimated from data.

There are important connections between the APT of regression models and standard statistical metrics. The APT for a component of the regression model can be written as

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In regression theory, the term R_τ² is called the coefficient of multiple determination; in predictability theory, this term is called the explained variance. This term measures the amount of variation in the predictand that is accounted for by the variation in the predictors. In the present case, the predictors are the state variables at the initial time and the predictand is a component of the state vector at the final time. The positive square root of the coefficient of multiple determination, R_τ, is called the multiple correlation coefficient. The multiple correlation coefficient is the correlation between the predictand and the predicted value of the predictand by the regression equation. In this sense, the multiple correlation can be interpreted as a multivariate generalization of the familiar correlation coefficient. Because (18) is a Rayleigh quotient, it follows immediately that the q that maximizes the multiple correlation at fixed τ is given by the leading eigenvector of

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This eigenvalue problem is precisely that which arises in canonical correlation analysis (CCA). CCA finds components in two datasets that are maximally correlated; in the present case, the two datasets are x_t+τ and x_t. Thus, the decomposition of APT is closely related to CCA, the main difference being that CCA maximizes the multiple correlation at one lag, whereas the decomposition of APT maximizes the sum of squared multiple correlations at all lags. This connection shows that the procedure we propose is an extension of a familiar procedure in predictability analysis.

It is instructive to compare APT to the quantity

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where ρ_τ is the autocorrelation function of a component and τ is the lag, because T₂ often is used as a measure of the time scale of a process. For instance, DelSole (2001) used this form to define optimal persistence patterns and showed that this expression is preferable to the traditional integral time scale if the autocorrelation function oscillates between positive and negative values. This expression also arises in statistical theory, as will be explained in section 5. We now show that

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To prove (21), we first recall the extended Cauchy–Schwartz inequality (Johnson and Wichern 1982)

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Substituting x = q and y = 𝗖_τ^Tq gives

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which states the simple fact that ρ_τ² ≤ R_τ²; that is, the autocorrelation is always less than or equal to the multiple correlation, for a given component. This inequality is consistent with the principle that adding predictors to a regression equation can never decrease the explained variance (recall that ρ_τ² is the explained variance of a prediction of a component based on its time-lagged value alone, whereas R_τ² is the explained variance of a prediction of a component based on time-lagged values of all the components). Because inequality (23) holds for each lead time τ, it also holds for the sum over all lead times, which proves (21).

Inequality (21) reveals a key distinction between optimal persistence analysis, which maximizes T₂, and maximization of APT, which maximizes S. In particular, the persistence time of a component bounds the APT of a component. This fact follows naturally from the fact that univariate prediction is less accurate than multivariate prediction (or at most, equally accurate). This difference becomes especially important if a system is characterized by propagating structures. In particular, a propagating structure can be captured by a regression model if the model contains several predictors, which allows different components to interact with each other. In contrast, a single component cannot capture a propagating structure. Put another way, OPA requires the initial and final patterns to be the same, whereas predictable components allow these patterns to differ.

5. Estimation of APT

Two practical difficulties arise when attempting to decompose predictability by solving the eigenvalue problem (15) based on finite time series: 1) the time lag covariances cannot be estimated at arbitrarily large lags, and 2) the covariance matrices are not full rank if the dimension of the state space exceeds the number of samples. This second difficulty is a reflection of the more general problem that supervised learning methods such as regression are vulnerable to overfitting. We discuss methods for dealing with these issues in this section.

a. Lag window

For finite time series, only a finite number of lagged covariances can be evaluated. One obvious approach then is to approximate (16) by using only a finite number of time lags. However, simply truncating the sum at some maximum time lag can produce components with negative APT. A source of this problem is the fact that the covariances at large time lags are estimated with fewer samples than covariances at small lags. Therefore, the accuracy of the covariances decreases with time lag and so weighting the covariances equally is inappropriate. At this point it is helpful to recall the connection between APT and power spectra discussed in Part I. As is well known, the power spectrum matrix is related to the covariance matrix by a Fourier transform. However, the Fourier transform of a sample covariance matrix is not a consistent estimator for the power spectrum because it does not uniformly converge as the sample size becomes large (Jenkins and Watts 1968). Instead, a consistent estimator for the power spectrum often is obtained by computing a truncated weighted sum of sample covariances. By analogy, then, we suggest that the APT should be computed as

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where y_τ are weights, called the lag window, that decrease with the absolute value of the time lag, M is an upper bound called the truncation point, and primes denote sample estimates. The numerous desirable properties of estimators based on lag windows have been discussed by Jenkins and Watts (1968). A particularly attractive lag window for our purposes is the Parzen window, defined as

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An important property of the Parzen window is that it never produces negative spectra. This property implies that the Parzen window never produces negative APT.

c. Statistical significance

There are at least two types of significance tests that are relevant here: a test for the total APT and a test for individual components. A test for the total APT has appeared in the statistics literature. Specifically, if we substitute (14) into (3), we obtain

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If we replace expectation quantities by sample quantities, and the infinite sum by a finite sum, then the resulting statistic is similar to the statistic used to test the whiteness of residuals of a vector autoregression model (Lütkepohl 2005, p. 169). This test is not useful in the present context because significant predictability is expected at least on short time scales, so disproving the null hypothesis of no predictability (S = 0) is not very interesting. However, in other contexts in which the predictability is in doubt, such as perhaps decadal predictability of annual averages, this test may be appropriate.

To test the significance of the predictability of a single component, we first derive the regression model and predictable components with one dataset and assess predictability with another dataset. This separation of training and assessment samples eliminates the artificial skill expected when the same data are used to both train and validate the model. The fundamental question is whether the forecast error is systematically larger than the error of a prediction based on the climatological mean. As we shall see, the required significance test for the Mahalanobis signal does not appear to be standard. Therefore, we develop a new significance test appropriate to this question.

Let υ_i be the verification of the ith forecast, and assume that it is independent and normally distributed with mean μ and variance σ², a relation we denote as

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Let be the mean of υ_i in the assessment sample, assumed to be of size N. The sample mean under these assumptions is distributed as

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Let υ̂_i be the prediction of υ_i. An appropriate null hypothesis is that the forecast υ̂_i is drawn independently from the same distribution as , namely (28). In this case, the forecast error is distributed as

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In essence, the forecast error variance is the sum of the climatological variance and the error in estimating the climatological mean. The sample Mahalanobis signal for this forecast is defined as

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where the sum square error (SSE) and the sum square anomaly (SSA) are defined as

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Standard theorems in statistics show that

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where χ_ν² denotes a chi-squared distribution with ν degrees of freedom. However, the variance ratio appearing in (30) does not have an F-type distribution because the SSE and SSA are not independent, as required for F-type distributions. On the other hand, the distribution of the ratio SSE/SSA is independent of the mean and variance of the normal distribution and hence depends only on the sample size N. Therefore, the sampling distribution of S′ can be estimated by Monte Carlo methods as follows: First, N independent random numbers are drawn from the distribution (27), representing verifications, and an additional N independent random numbers are drawn from the distribution (28), representing forecasts. The sum square difference between these two sets of random variables defines SSE, while the sum square difference between the first sample and its sample mean defines SSA. This procedure is repeated for 10 000 trials, after which the resulting variance ratios are sorted in ascending order and the 100th element is selected, corresponding to the 1% significance level.

The number of independent samples N used in the significance test procedure is not simply the number of time steps in the assessment sample (because the time series are correlated). We address this issue by using the effective number of independent samples for N. To estimate the effective number of independent samples, we invoke the fact that the variance of the sample autocorrelation of moving average model of any order is

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where ρ̂_s is the sample autocorrelation at lag s, ρ_τ is the autocorrelation function of the moving average model, and N is the number of samples used to estimate the sample autocorrelation (Brockwell and Davis 1991, p. 223). Note the similarity between (33) and APT (20). This similarity suggests that a reasonable value for the effective number of independent samples is N/T₂. However, T₂ is not readily available from the results of the analysis, whereas the APT value S is. Therefore, for convenience, we estimate the effective number of degrees of freedom as

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Inequality (21) implies that the effective number of degrees of freedom defined in (34) is always smaller than would be estimated from N/T₂. It follows that this choice leads to a more conservative significance test, in the sense that it tends to accept the null hypothesis of no predictability when it should reject it.

6. Example: 1000-hPa zonal velocity

A natural observational dataset for diagnosing predictability is the National Centers for Environmental Prediction National Center for Atmospheric Research (NCEP–NCAR) reanalysis set (Kalnay et al. 1996), which provides more than 50 yr of continuous, global data at 6-hourly time intervals. Because APT is invariant to linear transformations, variables with different units or natural variability can be included in the state vector without normalization. Nevertheless, we found 1000-hPa zonal velocity alone to be sufficient for our purpose because this variable contains signals from ENSO, the Madden–Julian oscillation (MJO), and weather. We refer to this variable as U1000.

Before computing principal components, we subtract the time mean and the first two harmonics of the annual and diurnal periods from each grid point. In addition, the time series at each grid point was standardized by dividing by its standard deviation. This standardization is necessary because U1000 variance in the tropics is small relative to midlatitudes, so truncating principal components with weak variances effectively removes tropical variability. Also, each grid point was multiplied by the square root of the cosine of latitude to conform to an area-weighted measure.

The U1000 field was sampled every 6 h during the 50-yr period 1 January 1956 to 31 December 2005, which corresponds to 73 052 time steps. Furthermore, the U1000 field is represented on a 2.5° × 2.5° grid, corresponding to a 10 368-dimensional state vector. These dimensions are too large to permit numerical solution of principal components. Accordingly, the principal components were computed only for data sampled every 3 days and 6 h, corresponding to 5619 time steps; this sampling interval was chosen to produce a subset as large as numerically feasible while also avoiding sampling the same time of day. After computing principal components, the spatial part of the component—the empirical orthogonal function—was projected onto the original data to compute all 73 052 time steps for each component. We verified that the principal component values of the full set and subsampled set coincided at the appropriate times. Furthermore, the amount of variance computed from the full data was approximately the same as the amount of variance computed from the subset data, indicating that the subsampling method introduced little bias.

As discussed in the previous section, we use the first 50 PCs to represent the state space. Predictable components and regression models were derived using PCs from the earlier 25-yr period 1956–80; the PCs in the later 25-yr period 1981–2005 were reserved for assessment. In computing these quantities, time-lagged covariances were derived for each lead time starting at 6 h and increasing in increments of 6 h until 180 days was reached (results are not sensitive to the choice of upper limit as long as the limit exceeds 15 days).

The total predictability of the regression models, as measured by the Mahalanobis signal (1), is shown in Fig. 1. The Mahalanobis signal also measures forecast skill because it measures the ratio of prediction error to climatological variance. We emphasize that these are legitimate forecasts: the regression models were derived from the training period 1956–80 and then used without modification to generate forecasts in the independent period 1981–2005. As expected, the total predictability decreases monotonically with lead time. The traditional interpretation of the top panel in Fig. 1 is that predictability essentially disappears after about 2 weeks, consistent with the time scale often invoked for weather predictability. If this interpretation were true, then the predictability of individual components also should disappear after 2 weeks. The actual predictability of individual components is shown in the bottom panel of Fig. 1. In contrast to the top panel, several components have significant predictability well after 2 weeks. In plotting the values for individual components, we mask out results that are not significant at the 1% level (according to the procedure discussed in the previous section); thus, some curves stop abruptly near the bottom of the panel.

Recall that the multiple correlation coefficient gives the fraction of variance explained by the regression model for the component in question. Thus, Fig. 1 shows that at 2 weeks about 1% of the total variance can be explained by the model; however, about 30% of the variance of one component can be explained by the model.

If predictability and the regression models were computed from the same sample, then the average of the bottom curves should equal the top curve in Fig. 1. In the present case, however, this correspondence is not ensured because these items were computed from different samples. Nevertheless, the difference between the total predictability and the average of individual components is always less than 0.01, indicating acceptable consistency of results derived from separate samples.

The training sample APTs of the first 30 components are shown in Fig. 2. We see that the first two or three components are separated from the others. The leading component, however, fails to be sustained in the assessment sample, as will be discussed in more detail in section 6b. Only the first five components have APTs exceeding 2 weeks. The fraction of variance explained by the individual components is shown in the bottom panel of Fig. 2. We see that predictable components generally explain about 1% of the variance.

a. Explained variance

The fact that regression models explain about 1% of the total variance at 2 weeks and that individual components themselves explain about 1% of the variance might lead one to conclude that the predictability beyond 2 weeks is too small to be physically significant. However, one should recognize that the predictable components are computed for 6-hourly data and represented on a global domain. Any predictability that exists after 2 weeks, such as that due to climate change, ENSO, or intraseasonal oscillations, is expected to explain relatively little variance on 6-h time scales. Thus, it is presumptuous to dismiss the predictable components derived beyond 2 weeks simply because they explain relatively little variance on 6-hourly time scales. The question is whether the predictability identified here after 2 weeks relates to the predictability believed to occur on long time scales. The usual method for identifying predictability on longer time scales is to filter out short time-scale weather fluctuations by time averaging and then attempt to identify predictability in the time average states. We will quantify this predictability as described next.

By construction, the time series are uncorrelated; thus, decomposing total variance into variance explained by individual components is meaningful. However, if one smoothes the components in time, then orthogonality no longer holds. Nevertheless, one can always quantify the amount of variance explained by a single component simply by computing the variance of the component projected onto the original time series (as shown below). In doing this, though, one should recognize that the sum of variances explained by a complete set of components does not add up to the total if the components are correlated in time. To compute the variance explained on different time scales, let v be the time series of a predictable component obtained by the projection

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where q is the projection vector for the predictable component and 𝗫 is a data matrix organized such that the nth column vector specifies the state at the nth time step. The time mean of 𝗫 has been removed and q is normalized such that the time series has unit variance (i.e., such that v^Tv = 1). To compute the variance explained by time series v, we consider a least squares problem in which we seek a vector p that comes as close to 𝗫 as possible, in the sense that it minimizes

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The above distance measure is called the Frobenius norm. The vector p that minimizes (36) is

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which can be interpreted as the projection of time series v onto 𝗫. The variance explained by time series v is defined as

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A similar reasoning shows that the variance explained at point i is the ith diagonal element of 𝗫vv^T𝗫^T/N. To compute relative variances, these two variances should be normalized by different factors. Specifically, for given time series v, the fraction of global variance is v^T𝗫^T𝗫v/tr[𝗫𝗫^T], while the fraction of variance explained at point i is (𝗫vv^T𝗫^T)_ii/(𝗫𝗫^T)_ii. Potentially, a time series may explain very little global variance but a large fraction of local variance.

Now consider time averages of length w. Let v_w be a time series in which each consecutive interval of length w is replaced by the mean of v in that interval. This time series can be written as

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and where 1 is a column vector of ones of dimension w. Similarly, let 𝗫_w be the data matrix in which each consecutive set of w columns are replaced by the mean of the column vectors in that section. This time-averaged data matrix can be written as

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It follows that the variance explained by v_w is v_w^T𝗫_w^T𝗫_wv_w/N. The latter expression can be interpreted as simply projecting the time-filtered time series v_w onto the time-filtered data matrix 𝗫_w. Note, however, that 𝗙_w𝗙_w = 𝗙_w; that is, 𝗙_w is idempotent. It follows that

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which implies that only the filtered time series v_w need be computed. This fact allows us to compute the explained variance without computing 𝗫_w, which results in significant computational savings. To compute the fraction of variance explained by v_w, the diagonal elements of 𝗫_w𝗫_w^T are computed once for each w and archived.

The fraction of global variance explained by individual components as a function of averaging window w is shown in Fig. 3. We see that although the leading predictable components explain only about 1% of the variance on 1-day time scales, they explain several times more variance at 90-day time scales. Note also that components 4–9 explain more variance at 2-week time scales than the leading components. The spatial distribution of the fraction of variance explained by the sum of variances due to components 2–7 is shown in Fig. 4. Summing these variances is appropriate because the components are quasi-independent—the absolute correlation between all pairwise 90-day mean time series for these components is at most 0.26, with the vast majority being less than 0.1. We see that the leading predictable components explain as much as 70% of the variance in the central equatorial Pacific, and 50% of the southern midlatitude variances in the Indian and Pacific Oceans. These results lead us to conclude that the leading predictable components explain a significant amount of variance on seasonal time scales and in certain geographic locations. These results also reveal the error of dismissing components simply because they explain small amounts of variance on short time scales or on global space scales. To be clear, we emphasize that time averaging was not used in any way in our technique; it was used only to compare the results of the analysis to observations.

The following sections examine individual predictable components in closer detail.

7. Summary and discussion

This paper proposes a new method for diagnosing predictability on multiple time scales. The method is to find projection vectors that maximize the average predictability time (APT), as measured by the integral of predictability. If predictability is measured by the Mahalanobis signal, then this optimization problem can be reduced to a standard eigenvalue problem. If the prediction model is based on linear regression, then the general decomposition is closely related to canonical correlation analysis, except that instead of maximizing the correlation at a single time lag, the method maximizes the multiple correlation over many time lags. The solution to the optimization problem leads to a complete set of components that are uncorrelated in time and can be ordered by their contribution to the APT, analogous to the way in which principal component analysis decomposes variance. Specifically, the first component maximizes APT, the second component maximizes APT time subject to being uncorrelated with the first, and so on. The state at any time can be represented as a linear combination of predictable components. Furthermore, the variance at any point can be expressed in terms of the variance of the individual components because the corresponding time series are orthogonal. Importantly, the results are invariant to nonsingular linear transformations and hence allow variables with different units and natural variances to be mixed in the same state vector.

In practice, the decomposition method requires the time dimension to exceed the state dimension. This requirement can be satisfied by projecting the data onto the leading principal components. A simple truncated sum of predictability values leads to an inconsistent measure of predictability time, just as a Fourier transform of a covariance function leads to an inconsistent estimate of the power spectrum. Drawing on spectral estimation theory, we suggest that lag windows used in spectral estimation theory can be applied to the estimation of APT. A critical question is the statistical significance of the results. Unfortunately, conventional cross validation techniques are computationally prohibitive if the number of lags is large. In this study, we took advantage of the long time series by computing the components using the first half of the data and assessing them in the second half. The significance test for the resulting APT is not standard but can be straightforwardly computed by Monte Carlo methods. This test accounts for the fact that the effective number of degrees of freedom depends on the APT of the component.

We applied the decomposition method to 1000-hPa zonal velocity (U1000). The data were sampled every 6 h for a total of 50 yr, giving 73 052 time steps, and projected onto the leading 50 principal components. The forecast model for this study was a linear regression model, which was estimated for each lag from 6 h to 180 days using only the first 25 yr of data. The resulting models then were used to make forecasts in the second 25-yr dataset. The predictability of this model, as measured by the Mahalanobis signal, decreased monotonically with lead time, reaching about 1% after 2 weeks. However, at least four components, as derived from the first half of the data, still were predictable in the second half after 2 weeks in a statistically significant sense. On the other hand, none of the components explained more than 1.5% of the variance of the data. Although this level of explained variance may seem small, this fact is misleading because the variance is compared to the variability on 6-hourly time scales and on global space scales. For instance, the leading components were shown to explain a large fraction of variance (e.g., > 70%) on 90-day time scales in certain geographic locations. Thus, the leading components can be identified with seasonally predictable structures. The first and third predictable components have trends and thus were identified with structures associated with climate predictability. Other leading predictable components were predictable on 2-week time scales and identified with structures associated with intraseasonal predictability. Indeed, a reconstruction of equatorial U1000 revealed eastward-propagating structures reminiscent of the MJO.

Importantly, the predictability on climate, seasonal, intraseasonal, and weather time scales found here was detected in 6-hourly data without time averaging. Furthermore, this predictability was detected in a single framework that allowed the separate structures to be isolated.