a. Sample time series

The present investigation uses three climate-mode time series: an ENSO, an AO, and an MJO time series. These time series were chosen because of their prominence in climate research, as well as to reinforce the notion that the least squares techniques are applicable to infinite filtering scenarios. ENSO indices are computed in numerous different ways. However, this is ordinarily accomplished by averaging sea surface temperature (SST) anomalies over a given domain box in the tropical Pacific. A commonly used indicator region is the Niño-3.4 region (5°S–5°N, 120°–170°W). Monthly values from the extended reconstructed sea surface temperature (ERSST) dataset (see Smith and Reynolds 2003), binned into 2° by 2° grid boxes, from 1946 to 2000 are utilized to compute the box averaged time series for the Niño-3.4 region (Fig. 1a). The most common filters applied to SST-based ENSO indices are running means of 3 or 5 months. In the present investigation, a 5-month running average is used (the FRF is shown in Fig. 1b).

The AO is defined using empirical orthogonal function (EOF) analysis (see Thompson and Wallace 1998). The AO time series is defined as the leading principal component (PC) time series of National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis sea level pressure (SLP) north of 20°N from 1948 to 2002 (Fig. 2a). Since the AO definition utilizes EOF analysis and does not make any assumptions on the time scale, the time series has spectral energy spread across the frequency domain. Therefore, filtering of the AO time series is utilized to extract the component of AO variability within a particular frequency range. In the present investigation, our AO filter will be a low-pass filter comprising 28 consecutive passes of the 1–2–1 filter (see Von Storch and Zwiers 1999), which is also known as the von Hann filter or simply as a “Hanning.” This results in a 57-point Gaussian filter; the FRF is shown in Fig. 2b.

Unlike the AO and ENSO, the MJO is unique because its time scale is its defining characteristic. Therefore, MJO time series are customarily defined postfiltering. Alternatively, the MJO can be defined using EOF-type analysis of nonfiltered data (see Arguez et al. 2005). The MJO was first observed in station time series (Madden and Julian 1971) using spectral analysis. In the present investigation, a time series must be obtained that reveals intraseasonal variability after bandpass filtering. Analogous to the pioneering work of Madden and Julian (1971), this is accomplished by using a point source in the MJO domain. The time series is obtained from a gridded Quick Scatterometer (QuikSCAT) zonal velocity field (see Pegion et al. 2000) at the equator and 80°E (Fig. 3a). The time series comprises 392 pentad averages from late July 1999 through early December 2004. Based on the results from Arguez et al. (2005), 31 Lanczos weights are utilized (see FRF in Fig. 3b).

b. Generating multiple samples

Investigating the impact of the endpoint schemes on one climate time series may provide anecdotal evidence of its efficacy, but a significantly larger sample size is necessary to increase confidence in the filtering method. On the other hand, using a large number of purely random time series may help in terms of increased confidence, but it would also be beneficial to create daughter time series that, in the spectral average sense, are acceptably similar to the mother time series. To accomplish both goals, a manipulation of white noise is utilized (see Schoof et al. 2005). For each of the three climate indices, the FFT is computed. The resulting function will be utilized as the FRF to be imposed on white-noise time series.

Ten thousand randomly generated, uniformly distributed time series are created, representing white noise. Each new sample is passed through an FFT and multiplied by the FRF before inverse Fourier transforming back into the time domain. Finally, the 10 000 daughter time series are normalized using each individual mean and standard deviation. This results in 10 000 daughter time series that are representative of the original climate series (e.g., the ENSO time series). Note that this protocol is analogous to filtering in spectral space except that, here, the FRF is used to impose, rather than regulate, the spectral characteristics of the output time series. Therefore, the only difference is in scientific intent; the mathematics is completely identical. Although white noise is utilized to create these time series, it must be emphasized that the resulting daughter time series are not representative of white noise, since the “red” spectral properties have been imparted on them. Using these daughter time series, the various endpoint estimate methods are assessed by computing the root-mean-square errors (RMSEs) for each position in the endpoint interval.

c. The least squares filtering schemes

As stated earlier, the problem with filtering in the time domain is that the full convolution cannot be computed at the endpoints of x(t). This leads to the following question: can variable-length filter weights be determined in the endpoint intervals such that the signal extraction replicates that observed in the interior points? In other words, how can an incalculable acausal filter (a filter that depends on past and future values) be replaced with a causal filter (a filter that depends on past and current values only) for endpoint estimation? For this project, a penalty function is constructed to minimize the squared error between the FRF in the interior of the time series, and a new FRF in each point of the endpoint intervals. Following (4) the FRF of interior points are denoted as follows:

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where α represents the new filter weights to be determined. The value of b is incrementally decreased to force a symmetric filter at each point. Consider once again a 5-month running average. For the second and penultimate (i.e., second to last) point in the time series (b = 1), three weights are determined. For the first point and the last point of the time series, only one weight is determined (b = 0). The result is a set of filter weights for each point in the endpoint interval. Without constraints, this minimization technique truncates the a priori filter weights (see Bloomfield 2000). In the case of the 5-month running average, the second and penultimate points would be computed using [1/5, 1/5, 1/5] as the three weights, whereas the first and last points in the time series would simply equal one-fifth the input value. A practical constraint is to force the new weights to sum to 1 (see Arguez et al. 2005), thereby preserving the expectation of the mean of the output series. The cost function for a given point L in the endpoint interval is a function of α and λ (the Lagrange multiplier that imposes our constraint):

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The minimum of this penalty function is obtained by taking the partial derivatives with respect to each α and λ and setting them equal to zero:

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Equations (9) and (10) are solved simultaneously using simple matrix operations to arrive at the coefficients. We term this method the equal-mean method. Using the 5-month running mean as an example once again, the three weights for computing the second and penultimate values would each be 1/3, whereas the first and last values remain unchanged (α_o = 1; b = 0). It can be shown algebraically that the resulting weights from the equal-mean constraint for a running-mean filter, regardless of the length of the a priori filter, will always be the average over the filtering interval (e.g., 1/3, 1/5, 1/7, 1/9, etc.).

An alternative, albeit more complicating, constraint is to force the total variance to be preserved. Variance is related to the spectrum via the following relation:

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By definition, time series filtering alters the variance of an input signal, and the change in variance is controlled by the filter weights (i.e., the FRF). Typically, time series filtering (e.g., smoothing) results in a (sometimes drastic) reduction in variance. This is somewhat intuitive, if one considers that filtering is often employed to “smooth out” outliers, in effect displacing values toward the mean, thereby reducing the time series variance. Therefore, there is a natural desire to ensure that endpoint estimates account for the deliberate effect that filtering has on time series variance. Combining (11) with (2) and discretizing, the following equality will be imposed to ensure that total variance in the interior will match the total variance of each point in the endpoint interval:

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The asterisks indicate complex conjugates. Note that the constraint is dependent on the input time series (specifically, its raw spectrum), unlike all other methods employed in the present study. Modifying (8) to impose the new constraint and incorporating the definition in (12), results in the new cost function:

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where T is the variance of interior points and is proportional to the left-hand side of (12). As before, the minimum is found by partial differentiation and setting to 0:

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These equations are not solvable with simple matrix manipulation because of nonlinearities between the Lagrange multiplier and the coefficients in (14) and amongst the coefficients in (15). The equations are solved using the Newton method available in the IDL programming language. We term this method the equal-variance method. Note that, for the terminal points in the series, the output values are computed by scaling the original termini by the standard deviation of the interior points, since the expected variance is preserved. Since the standard deviation depends on the input time series, the weights for the equal-variance method will generally vary from one sample time series to another. For the unconstrained and equal-mean methods, the weights do not depend on the input time series, and they are therefore uniform for each and every input time series.

d. Test methodology

The equal-mean, equal-variance, and unconstrained methods will be applied to subsets of time series of ENSO, the AO, and the MJO. These “estimates” will be compared to the “true” filtered values. Please consult Fig. 4, which shows a contrived 24-point time series that helps explain the test methodology. The top series represents the input time series to be filtered with a 5-month running mean (b = 2). The bottom series shows the output time series. The diamonds represent the estimation zone. The green squares are only used to compute “true” values; these represent the points in time that are not available to compute the full convolution. For example, values 20–24 are averaged to compute the true value for position 22. Positions 23 and 24 represent future data points that cannot be accounted for when estimating the filtered value for position 22, which is indicated by the rightmost blue diamond. The gray circles represent the “interior points” that are not subject to estimation because full convolutions can be computed for these positions. As indicated by line drawings, the output value for position 12 is computed by averaging values 10–14. Note that, as stated earlier, the estimates are computed using filter arrays that are symmetric, resulting in three weights to be determined to compute output values for positions 4 and 21, and one weight only for the terminal ends (positions 3 and 22).

A similar procedure is followed for the 31-point Lanczos weights (b = 15) in the MJO project, as well as the 57-point Gaussian weights (b = 28) in the AO project. For the MJO project, the series are extended by 15 points on each end (there would be 15 green squares on each side of the time series). For the AO project, the series are extended by 28 points on both ends. Note that the parameter b determines the number of extra points needed in one direction (left or right end), and that 2b + 1 is the length of the filter weights array.

To assess the viability of the least squares filtering schemes, the RMSE is computed at each point in the estimation zones. Since the methodology does not impart any preference to the left or the right estimation zones, the RMSE values should be symmetric between the left and right zones. In addition, the RMSE values will also be computed for the spectral filtering method (the standard of comparison) and the three techniques used in Ma04. Separate RMSE figures are computed for each of the filtering scenarios: the AO, the MJO, and ENSO projects. To infer whether the various results are accurately appropriating variance, the variance of the output array (the 10 000 simulations) will be computed for each point in the estimation zones, and then compared to the “true” variances (the variance of the 10 000 “true” values for each point in the estimation zone). By definition, the equal-variance constraint should produce output whose overall variances are close to the true variances. Note that the filtering results are not shown for the mother time series, since results from such a small sample size (N = 1) are not robust.

The errors due to the least squares methods increase closer to the termini due to two primary estimation errors. First, fewer data are available for the computation (data beyond the terminal ends are blindly disregarded). These random errors are not absolutely quantifiable, since there are no restrictions on the particular values a time series can take beyond the endpoints. Second, the ability to reproduce the FRF is hampered by a reduced number of allowed filter weights. This second source of error is especially problematic for bandpass filters, such as the Lanczos filter used in the MJO project, since the weights must include both positive and negative weights to extract salient time scales (Arguez et al. 2005).

As a final analysis, empirical weights are determined for the ENSO project. The entire 1946–2000 period, save a few months near the termini, is used to determine the weights that minimize the RMSE as a filter of the following form proceeds throughout the period of record:

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Note that (16) applies only to the right endpoint. For the penultimate point, the following filter is used:

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The coefficients are determined to the thousandths place. The resulting empirically determined filters represent a lower bound on the RMSE that the theoretically based filtering schemes (the least squares methods, the spectral method, and the Ma04 constraints) can attain.

a. ENSO

In this project, domain-averaged SST is used to define the ENSO time series. Unlike other ENSO indicators, such as the noisy Southern Oscillation index (a pressure difference between Darwin, Australia; and Tahiti), SST-based indices have disproportionately lesser amounts of high-frequency variability, attributable to the high heat capacity and inertia of the ocean. Therefore, SST indices have much longer decorrelation times than atmospheric indices; longer decorrelation times, in turn, are linked to potential predictability.

For the ENSO project, the equal-variance method performs the best, followed closely by the slope, equal-mean, and rough methods (Fig. 5). The RMSE for the equal-variance method’s penultimate and terminal points are 0.118 and 0.242, respectively. The three second-tier performers all slightly overestimate the variance (Fig. 6), whereas the equal-variance method results have a variance close to that of the “true” results. The three remaining schemes (spectral, norm, and unconstrained) have even larger RMSE values and underestimate the variance. The rough and equal-mean methods perform as well as they do because the ENSO time series has a large decorrelation time and the filter width is very small (b = 2). On the other hand, the norm and unconstrained methods underperform because of their tendencies to approach zero. Note the level to which the equal-variance method outperforms the spectral method, which is utilized in this study as the standard of comparison. The equal-variance method generally outperforms the spectral method for the interior points of the estimation zone, but outperforms for all points in the estimation zone when the filter width is not very large.

In this project, only two estimates are computed per times series end for each filtering method tested. For the unconstrained case, all the weights remain at 0.2 since this method simply truncates the weights. For the equal mean case, the weights become 1/3 when computing the second and penultimate points, and 1 for computing the terminal ends. Therefore, the unconstrained case tends to result in values that veer toward 0 near the terminal ends, as implied in Fig. 6. On the other hand, many of the other methods remain close to the original prefiltered values, explaining the relative success of the equal-mean scheme over the unconstrained scheme.

The empirically determined weights are presented in Table 1. The associated RMSE values are 0.155 for the termini and 0.073 for the second/penultimate points. This compares to RMSE values of 0.242 and 0.118, respectively, for the best performing theoretically based scheme (the equal-variance method). The “true” filtered values result in a variance reduction of 14.1% (compared to the input time series’ variance). The empirical filters (16) and (17) reduce variance by 17.1% and 14.7%, respectively. Note that the variance ratios are closer to unity than all other schemes except the equal-variance method (Fig. 6), suggesting that the proper appropriation of variance is a key component in accurate endpoint estimation.

b. AO

The RMS errors for the AO filtering project suggest that all seven filtering schemes work very well in the interior of the estimation zone (Fig. 7). The expected RMSE between filtered values of randomly selected input series (using the same procedures as above on scaled white noise series) is about 0.4, representing a liberal skill threshold. In other words, the liberal skill threshold is an upper bound of skill based on purely random time series (as opposed to the daughter time series utilized for determining RMSE). Except for the equal-mean and the roughness methods, the filter schemes show skill for all endpoint positions. The top performers (in terms of RMSE) are the slope and norm schemes. Intermediate performers include the equal-variance, spectral (within a quarter filter width of the termini), and unconstrained methods. The poorest performers are the equal-mean and rough schemes.

The norm scheme tends toward the mean value at the boundary. This results in a tendency toward zero for climate oscillation anomalies, yielding reduced variance (damping) at the endpoints (Fig. 8). The slope scheme tends toward a constant value at the boundary, resulting in overestimated variance near the termini for the AO case. This is undoubtedly due primarily to instances where the constant value approached is large in an absolute value sense (e.g., when the original series endpoints are near absolute extrema). The unconstrained method suffers a similar fate. Recall that the unconstrained filters are truncated—the outer weights are replaced with zeroes, resulting in sharply reduced variance. In the AO case, the unconstrained method produced respectable results in an RMSE sense, but as shown in the ENSO case, the relative success is not robust, but due to the nature of the filtering problem in the AO case. In summary, although the norm, slope, and unconstrained methods have some success in an RMSE sense for this case, they are unreliable methods if representative output time series are desirable.

Both the spectral and equal-variance methods show skill for all grid points, with the equal-variance method outperforming for most of the interior points in the estimation zone, and the spectral method outperforming slightly within one-eighth filter width of the termini. Both methods do a respectable job in appropriating variance as well. The rough and equal-mean constraints perform very poorly for the AO case. The equal-mean case is handicapped by forcing the weights to sum to unity, causing gross overestimates (much larger variance) near the termini. The rough constraint is not designed for trend-free climate oscillations (i.e., stationary time series), and it is therefore not surprising that it underperforms in all three cases.

It is important to comment on the relative success and failure of the unconstrained and equal-mean filters near the terminal ends, respectively. The equal-mean filter suffers in part because the sum of the filter weights used to compute all three interior FRF functions (H) was set to 1 (this can be changed and will be considered in future work). This implies the two terminal values do not change because they are simply multiplied by 1. As a result, when the input subset ends at a series maximum or minimum, the errors with the equal-mean method are disproportionately large. The unconstrained method, on the other hand, performs very well (as shown in the ENSO project, this is not a robust result). However, here we have somewhat of the opposite issue occurring. Instead of constraining the weights to equal a certain value or maintaining a variance, these weights are simply truncated. This results in decreased variance close to the terminal ends. In the case of the AO project, this reduction in variance proved to be to an appropriate level to allow strong performance—this is no more than a coincidence based on the specifics of the filtering parameters. However, if the level of variance reduction is not appropriate, large errors can result (as in the ENSO project).

c. MJO

Filter performance in the MJO project is erratic (Fig. 9). As in the AO case, the lowest RMSE values are found in the norm and slope schemes, yet the variance ratios are neither stable nor close to unity for all endpoint positions (Fig. 10). In fact, the variance ratios appear to be affected by the MJO’s 30–60-day time scale. Variance ratios close to unity throughout the estimation range are only evident in the equal-variance and the spectral method. RMSE values of the equal-variance method are lower than those of the spectral method except for the four points closest to the termini (inclusive). The liberal skill threshold in this case is about 70, suggesting that all methods are useful for all time series positions. Looking at typical results (not shown), the issue of partitioning variance at the terminal ends proves to be problematic for the equal-mean and unconstrained methods, with both evincing outputs that tend to veer toward zero at the terminal points. In other words, both methods are plagued by reduced variance toward the terminal points. The equal-variance method’s RMSE are similar near the endpoints to the equal-mean and unconstrained methods, but it does not exhibit the veering to zero characteristic: the equal-variance method results are neither partial to overshooting nor undershooting the true values.