## 1. Introduction

The Madden–Julian oscillation (MJO; Madden and Julian 1972) and boreal summer intraseasonal oscillation (BSISO; Wang et al. 2009; Kikuchi et al. 2012; Lee et al. 2013), are the two most dominant intraseasonally varying patterns of the tropical atmosphere. The former, an eastward-propagating envelope of convective activity originating in the Indian Ocean during boreal winter, is known to modulate Pacific and Caribbean cyclogenesis, affect rainfall variability on multiple coastal regions, and alter the strength of the ENSO cycle, among other effects (Zhang 2005). The BSISO, meanwhile, is closely related to intraseasonal oscillations of the South Asian monsoon (Goswami 2011, and references therein) and influences its onset and active and dry spells. More broadly speaking, improved understanding of these two intraseasonal modes would go a long way in filling the gap that is beyond the reach of short-term weather forecasts and below the resolution of long-term climate models (Waliser 2011; Zhang et al. 2013). Yet, despite the understood importance of these modes of variability and several decades of research, there are still significant challenges to improving the real-time monitoring and forecasting of these two modes.

Among these challenges is the still-open problem of defining intraseasonal oscillations in a consistent and objective manner. Although the real-time multivariate MJO index (RMM; Wheeler and Hendon 2004) has emerged as the most commonly used standard for MJO measurement, drawbacks such as biased sensitivity toward wind velocity data and overreliance on preprocessed data have meant that the development and testing of alternative indices has yet to cease (Kiladis et al. 2014). No comparably popular measure has emerged for the BSISO, in no small part because of the relatively greater complexity of monsoon dynamics (Lee et al. 2013). One well-recognized common source of these difficulties, among several, is that the majority of current techniques are not ideally suited for multiscale phenomena like organized tropical convection, as they generally require ad hoc data preprocessing to isolate the temporal and spatial scales of interest. The recently developed technique of nonlinear Laplacian spectral analysis (NLSA; Giannakis and Majda 2011, 2012, 2013, 2014) seeks to redress this mismatch by producing indices based on time-lagged embedding and local measures of data similarity that more sensitively capture nonlinear dynamics than standard eigendecomposition techniques, such as empirical orthogonal function (EOF) analysis. This technique has been used to extract families of modes of variability from equatorially averaged (Giannakis et al. 2012; Tung et al. 2014) and two-dimensional (2D) (Székely et al. 2016a,b) brightness temperature (*T*_{b}) data spanning interannual to diurnal time scales without prefiltering the input data. These mode families include representations of the MJO and BSISO with higher temporal coherence (Székely et al. 2016b) and stronger discriminating power between eastward and poleward propagation (Székely et al. 2016a) than patterns extracted through comparable linear approaches. Insofar as improved accuracy of representing tropical intraseasonal oscillations (ISOs) can bolster predictability, we explore in this paper the extent to which NLSA-derived indices can be used as a basis for forecasting the MJO and BSISO.

A second challenge is determining how to incorporate information about both the past and present into predictions of any given tropical ISO pattern’s future evolution. There will always be, of course, some unpredictability to tropical ISOs that cannot be overcome without numerical simulations, especially since tropical ISOs are affected by global warming (Subramanian et al. 2014), the precise future spatiotemporal characteristics and magnitude of which remain uncertain. Nevertheless, the MJO and BSISO have shown enough regularity and consistency over the past 40 yr to suggest that the past can serve as a guide to the future. One of the simplest empirical forecasting techniques is the classical analog forecasting method of Lorenz (1969), which first identifies, via Euclidean distances, a moment in the past that is most similar to the present and then casts the historical evolution from that moment as the forecast for the future. In the context of tropical intraseasonal oscillations, analog techniques have been employed in real-time forecasting of Indian monsoons with skill extending to 20–25 days (Xavier and Goswami 2007). Generalizations of analog forecasting based on modifications, such as taking weighted averages of multiple historical paths, varying the type of distance measure by which analogs are chosen, and iterating multiple times in order to account for multiple scales, have recently been developed in a framework called kernel analog forecasting (KAF; Zhao and Giannakis 2016; Comeau et al. 2017), which will be employed later in this paper.

Analog forecasting methods can preserve many of the attractive qualities of NLSA-derived indices. The compatibility of these otherwise two distinct techniques lies in their mutual reliance on dynamics-dependent geometric measures of data similarity. In particular, NLSA indices are the eigenvectors of a kernel operator (Belkin and Niyogi 2003; Coifman and Lafon 2006; Berry and Sauer 2016), which can be loosely thought of as a local covariance matrix. Much of the character of these indices is due to a specific choice of a smooth, data-dependent Gaussian-like kernel that takes dynamics into account through Takens delay-coordinate embeddings (Takens 1981; Packard et al. 1980; Broomhead and King 1986; Sauer et al. 1991). Meanwhile, KAF methods produce forecasts by taking weighted averages of historical data, with weights determined by a measure of similarity between the present and any prior moment in time. By letting these analog weights be determined by kernels of the same class as that used in the extraction of the NLSA indices, the resulting algorithm is more sensitive to oscillations in the intraseasonal range and can thus produce more faithful forecasts than otherwise (Zhao and Giannakis 2016). As shown here, this pairing of NLSA and kernel analog techniques can yield nearly 6 weeks’ worth of skill in forecasting the MJO and BSISO. This result is comparable to those of the stochastic oscillator models of Chen et al. (2014) and Chen and Majda (2015a) for predicting the NLSA-derived MJO and BSISO modes, respectively.

Some advantages that KAF can be said to have over other recent skillful forecasting methods of tropical ISOs are its nonparametric nature, which helps avoid dynamical model error, and its ability to operate in real time. Dynamical model errors, in particular, have historically been a significant obstacle to forecasting ISOs whether through numerical models or parametric statistical models, with skill of earlier models limited to 10–15 days (Waliser 2011, and references therein). More recently, however, advances in MJO simulation by coupled models have led to skill as high as 27 days for forecasting the RMM indices (Miyakawa et al. 2014; Neena et al. 2014; Vitart 2014; Xiang et al. 2015). Similar improvement of skill has also been attained by recent parametric empirical models (Kondrashov et al. 2013; Chen and Majda 2015b). It is important to note that the higher predictive skill reported in this paper is at least partly due to the higher intrinsic predictability of the NLSA-derived modes compared to the RMM. Forecast skill aside, while both coupled models and parametric empirical models are important for elucidating the physical processes underlying tropical ISOs, such models often require iterative tuning of numerous parameters, which in many ways can introduce significant biases. Kernel analog forecasting, on the other hand, is purely empirical in the sense that none of its parameters rely on any physical model. This nonparametric quality allows the method to both avoid model error and be automatic, at least in the sense that no manual intervention is required after initial data entry. Moreover, KAF can operate in real time, which is a feature that is sometimes absent in statistical models.

The dataset of interest in this paper, to which the KAF technique is applied, is tropical 2D

The plan for the rest of the paper is as follows. An overview of the KAF method is provided in section 2. The specific data of interest are described in section 3. The results of applying KAF to this data, as well as a sensitivity analysis, are presented in sections 4 and 5, respectively. A discussion of these results constitutes section 6, and broader context, possible future research directions, and other concluding remarks are given in section 7.

## 2. Kernel analog forecasting

We provide here a brief overview of ensemble kernel analog forecasting, a more complete description of which can be found in Zhao and Giannakis (2016). The first subsection outlines the general framework of the method, illustrating how the final forecast is the result of multiple iterations of weighted averages. The subsequent subsections describe the key components of the method, including time-lagged embedding, creation of training data via NLSA extraction, and kernel specification. The final subsection describes the ways in which the resulting forecasts are assessed.

### a. Preliminaries

The forecasting problem consists of predicting what the value of some physically meaningful quantity *Y*. For instance, in sections 3 and 4,

It will be useful later on to keep in mind the slightly more abstract interpretation of the forecasting problem in the context of dynamical flows within a hidden state space

### b. General analog forecasting

### c. Laplacian pyramid iteration

We have thus far shown, given a training period

*δ*= 10

^{−6}. The final forecasting function is set to be

A common iterative scheme that rapidly moves through multiple scales, and one that we employ in this paper, is that of a decreasing dyadic sequence of bandwidth parameters: that is, letting each parameter be twice as small as the previous one [i.e.,

### d. Choice of observation map

^{7}). However, since both KAF and NLSA are based on localizing kernels, the performance of these methods is sensitive to the intrinsic dimension of the subset of

An alternative justification for the use of delay embedding is found in dynamical systems theory. In that framework, the obtained data in

A detail that is relatively inconsequential to the overall theory, but important in actual implementation, is that appropriate truncations of training data must follow any time-lagged embedding. For example, as will be discussed in section 3, we take one of our training periods

### e. Choice of forecast observable

*i*th component of the eigenvector

Multiple eigenmodes

By investigating some of the features of corresponding spatially reconstructed modes (Székely et al. 2016a), it can be possible to identify a particular mode *i** = 12). This mode is then used to define the restrictions onto the training data of the forecast observable

### f. Choice of kernel

The choice of

Besides the bandwidth function

### g. Out-of-sample extension

The efficacy of the produced forecast

### h. Error assessment

Having defined the true signal

Because of the recentering and normalization steps in its formula, pattern correlation gives a good measure of how well

## 3. Application to global brightness temperature data

### a. NLSA-derived MJO and BSISO modes

Our primary object of study is infrared brightness temperature data recorded under the CLAUS project over 26 yr, from 1 July 1983 to 30 June 2009, and used in Giannakis et al. (2012), Tung et al. (2014), and Székely et al. (2016a,b). This dataset is often considered to be a reasonable proxy for convective activity in the tropics, with positive *n*_{lon} = 360 longitudinal and *n*_{lat} = 31 latitudinal grid points. Each *n* = *n*_{lon} × *n*_{lat} = 11 160. These observations are collected at an interval of Δ*t* = 3 h, for a total sample size of *s*_{total} = 75 796 over the 26 yr of the CLAUS record.

To examine the effects of differently sized training datasets, our study in this section is broken into a long training period set *s*_{long} = 67 208 samples. The period *s*_{short} = 26 304. Note that no preprocessing, such as bandpass filtering, seasonal partitioning, or equatorial averaging, is performed.

The testing period

For any given period *n* × *s*. The time-lagged embedding procedure described in section 2d is applied to construct a separate matrix *nq* by *q* = 512. This further means that the embedded vectors *N* = *nq* ≈ 2.3 × 10^{7}. As noted in section 2d, NLSA and KAF are well-suited techniques for data analysis and modeling in such high-dimensional spaces, as their performance is governed by the intrinsic dimension of the dataset

The forecast observables for *k*_{nm} = 5000 (corresponding to ~10% of the dataset); these are the same parameter values used by Giannakis et al. (2012), Tung et al. (2014), and Székely et al. (2016a,b). Using these parameters, we compute the first 50 eigenmodes,

Phase composites of

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

Phase composites of

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

Phase composites of

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

Selecting which of the NLSA eigenmodes correspond to the MJO or BSISO is done through a one-time qualitative assessment of their associated spatially reconstructed patterns. In particular, the MJO forecast observables were chosen as those corresponding to eastward-propagating wave trains of enhanced and suppressed convection during the boreal winter, initiating over the Indian Ocean, traversing the Maritime Continent and western Pacific warm pool, and eventually decaying in the central Pacific Ocean near the date line. Meanwhile, the BSISO forecast observable was chosen as the one matching boreal summer convective activity initiating in the Indian Ocean and moving north toward the Indian subcontinent. We refer the reader to Székely et al. (2016a,b) for additional discussions on the properties of these modes, including coarse-grained predictability properties and initiation and termination statistics. Besides the MJO and BSISO modes studied here, the NLSA spectrum recovered from CLAUS

To retain the real-time applicability of kernel analog forecasting, the forecast observables corresponding to the short training period

### b. Comparison with the RMM index

That there is no unambiguously correct measure of intraseasonal variability is part of the rationale for constructing alternative indices based on new techniques. Nevertheless, the RMM index (Wheeler and Hendon 2004) is a commonly accepted and used measure of the MJO. Therefore, for completeness, we include in our analysis correlations between the RMM and our proposed NLSA-derived indices. Figure 3 illustrates what the amplitude of the RMM’s two modes looks like, both before and after performing a 64-day-running-mean smoothing. The correlation of the NLSA MJO amplitude with that of the raw RMM index is small at 0.20 but becomes a more significant 0.46 after the RMM is smoothed. This correlation does not change significantly upon restriction to the DJF period, when the MJO is most active. Unlike the MJO, the year-round BSISO data do not at all correlate with the raw RMM amplitude and, moreover, barely exhibit any increase after smoothing. Upon restriction to the JJA active BSISO period, however, the correlation with the smoothed RMM is also found to be 0.46.

Visualization of the RMM amplitude, as discussed in section 3b: (a) amplitude of the Wheeler–Hendon RMM modes within the period from 1 Jul 1983 to 30 Jun 2006. (b) The 64-day moving average of the RMM amplitude and the amplitude of the NLSA-derived MJO modes.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

Visualization of the RMM amplitude, as discussed in section 3b: (a) amplitude of the Wheeler–Hendon RMM modes within the period from 1 Jul 1983 to 30 Jun 2006. (b) The 64-day moving average of the RMM amplitude and the amplitude of the NLSA-derived MJO modes.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

Visualization of the RMM amplitude, as discussed in section 3b: (a) amplitude of the Wheeler–Hendon RMM modes within the period from 1 Jul 1983 to 30 Jun 2006. (b) The 64-day moving average of the RMM amplitude and the amplitude of the NLSA-derived MJO modes.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

Overall, these results indicate that, while the two sets of modes are by no means equivalent, the NLSA-based MJO and BSISO modes are related to the RMM modes when restricted on the boreal winter and summer periods, respectively, and smoothed to remove high-frequency spectral content. In particular, that NLSA represents the two dominant ISOs through distinct modes with moderately narrowband spectra likely plays an important role in the higher predictability of these modes compared to the RMM modes (which represent the two distinct ISOs as a single pair of modes). We will discuss the sensitivity of the correlation results presented in this section on the NLSA embedding window and the RMM smoothing window in section 5a.

## 4. Hindcast results

In this section, we present the results of applying the KAF method to the NLSA-based MJO and BSISO modes described in section 3. The main result is that the pattern correlations of forecasts remain above 0.6 for 50 days for both the MJO and BSISO when 23 yr of training data are used. Another result is that the RMSE stays below one standard deviation of the historical variability. Other results, such as the relatively worse predictability at the beginning of an ISO event than toward the end and the specifics of individual years, are also presented.

### a. MJO 2006–09

Figure 4 shows running forecasts and corresponding RMSE and PC scores for lead times of up to 60 days, corresponding to the application of KAF in predicting one of the two MJO modes during the 2006–09 testing period, using the 1983–2006 training period. The relatively large amplitude of the monitored MJO signal around January 2007 and January 2008 corresponds to the fact that the winters of 2007 and 2008 contained significant MJO activity, and the relatively small amplitude around January 2009 corresponds to a season of weak MJO in 2009. A key question to ask of any forecasting method is if it can capture these periods of relatively increased and decreased activity. From a strictly visual inspection of Figs. 4a–d, this can be said to be true of KAF with 15- and 30-day lead times and to be false in the case of a 60-day lead time. That the ability to qualitatively discern periods of greater MJO activity is only lost well after 30 days is already an improvement over some of the existing methods discussed in the introduction. One qualitative feature that deteriorates faster than the general ability to discern activity is the ability to detect initial activity. More specifically, the forecast fails to capture the full amplitude of the first spike of the MJO season, occurring in December 2007 and December 2008. The difficulty of predicting initiations, however, is a challenge that is not unique to this method.

Kernel analog forecasting of the NLSA-based MJO mode over the 2006–09 testing period using the 1983–2006 training period, as discussed in section 4a. (a)–(d) Running forecasts (orange) with lead times of 15, 30, 45, and 60 days, respectively, along with the true signal (blue). (e) RMSE and (f) PC error metrics for individual years as well as for the entire testing period. The amount of time spent above the 0.6-PC threshold is listed in the legend of (f) for each grouping. The PC and RMSE skill scores are calculated by excluding JJA, as discussed in section 2h.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

Kernel analog forecasting of the NLSA-based MJO mode over the 2006–09 testing period using the 1983–2006 training period, as discussed in section 4a. (a)–(d) Running forecasts (orange) with lead times of 15, 30, 45, and 60 days, respectively, along with the true signal (blue). (e) RMSE and (f) PC error metrics for individual years as well as for the entire testing period. The amount of time spent above the 0.6-PC threshold is listed in the legend of (f) for each grouping. The PC and RMSE skill scores are calculated by excluding JJA, as discussed in section 2h.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

Kernel analog forecasting of the NLSA-based MJO mode over the 2006–09 testing period using the 1983–2006 training period, as discussed in section 4a. (a)–(d) Running forecasts (orange) with lead times of 15, 30, 45, and 60 days, respectively, along with the true signal (blue). (e) RMSE and (f) PC error metrics for individual years as well as for the entire testing period. The amount of time spent above the 0.6-PC threshold is listed in the legend of (f) for each grouping. The PC and RMSE skill scores are calculated by excluding JJA, as discussed in section 2h.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

A quantitative analysis of these results is obtained through RMSE and PC evaluation. A typically used threshold for separating skillful and unskillful forecasts is a PC score of 0.6. As such, the 0.97 and 0.86 scores for the 15- and 30-day lead forecasts in Figs. 4e and 4f reflect the qualitatively good nature of those forecasts, whereas the 0.41 pattern correlation of the 60-day lead reflects the natural decrease in forecast skill at long lead times.

The RMSE and PC of forecasts with lead times ranging from 0 to 60 days are also displayed. The decrease in pattern correlation score for the entire testing period is very slight up to 20 days. The decrease is slightly more modest between 20 and 50 days of lead time. The skill dips below 0.6, and is thus said to become unskillful, when the lead time exceeds 50 days.

The RMSE and PC plots also show scores over three shorter time periods (July 2006–July 2007, July 2007–June 2008, and July 2008–June 2009), which we refer to by the year in which the period begins. Although the scores of the 2006 and 2007 periods are similar to those over the entire testing period, the 2008 scores have notable differences. First, the RMSE of the 2008 period diverges from that of the other periods at around *τ* = 10 days of lead time and stays significantly below the others. Second, the PC of the 2008 period dips faster than that of the other periods after about 40 days of lead time. It may seem paradoxical at first how a period can simultaneously have better RMSE and a worse PC. However, both aspects are explained by the fact that the 2008 period contains less activity than the others; that is, it contains few large peaks, if any. Thus, the relatively small RMSE of the 2008 period is because there is only modest deviation from the mean.

### b. BSISO 2006–09

Figure 5 shows the results of KAF applied to predicting one of the BSISO modes during the 2006–09 period, using the same 1983–2006 training data as before. The significant BSISO events during this time occurred in the summers of 2007 and 2008, as shown in the plots, and their durations are longer than those of MJO events. As in the previous two cases, the 15- and 30-day-lead-time forecasts are qualitatively good, while the 60-day-lead-time forecast is not. Although the skillful PC score extends slightly farther than for the MJO, not falling below 0.6 until about 50 days, there is more variance, with the 2006 PC score dipping below 0.6 at 45 days, and the 2007 and 2008 scores doing so after 50 days. This may be explained by the seeming property of KAF forecasts to be markedly better when locked onto a regular, oscillatory event. Therefore, the fact that the KAF method produces a slightly better PC score for the BSISO mode is probably due to the longer and more sustained BSISO events.

As in Fig. 4, but for the NLSA-based BSISO mode, and discussed in section 4b. The RMSE and PC skill scores are calculated by excluding DJF, as discussed in section 2h.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

As in Fig. 4, but for the NLSA-based BSISO mode, and discussed in section 4b. The RMSE and PC skill scores are calculated by excluding DJF, as discussed in section 2h.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

As in Fig. 4, but for the NLSA-based BSISO mode, and discussed in section 4b. The RMSE and PC skill scores are calculated by excluding DJF, as discussed in section 2h.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

### c. MJO 1992–95

Figure 6 shows the results of applying KAF to predict values of one of the MJO modes during the 1992–95 testing period, using the 1983–92 training period. As stated in section 3, the reason this testing period was chosen is that it contains the well-documented large MJO events that occurred during the TOGA COARE IOP (Yanai et al. 2000). The plots show that the true signal captures the succession of these two events, as well as a similarly large MJO event in the winter of 1995 and a couple of smaller ones in the winter of 1994. The goals of applying KAF with these testing and training periods are twofold: to determine the effects on the KAF of 1) a shorter training period and 2) a testing period with well-documented MJO events. Note that the shortened training period impacts the KAF skill in two distinct ways: namely, through poorer-quality MJO indices in the NLSA step and fewer available analogs in the forecast step. We will return to this point in section 6.

As in Fig. 4, but for the short 1983–92 training period and the 1992–95 testing period. Discussed in section 4c.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

As in Fig. 4, but for the short 1983–92 training period and the 1992–95 testing period. Discussed in section 4c.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

As in Fig. 4, but for the short 1983–92 training period and the 1992–95 testing period. Discussed in section 4c.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

Qualitatively speaking, many of the results in Fig. 2 are similar to those of Fig. 1: the KAF forecasts perform reasonably well with lead times of 15 and 30 days, but not so well with a lead time of 60 days. That being said, the initial detection of MJO events is more difficult because of the smaller training sample size. Quantitatively, although the overall RMSE scales similarly as in the previous case, there is more variance, with a low RMSE for 1993 and high RMSE for 1995. There is also more variance in the PC scores, as the 1994 forecast remains well above a score of 0.5 beyond 50 days. Most significantly, however, is that the overall PC score dips below 0.6 after about 37 days, which is 13 days earlier than in the previous case.

## 5. Sensitivity analysis

The main factors affecting the robustness of the hindcast results presented in section 4 are the choice of NLSA and KAF parameters, the sampling frequency, and the length of the training and test intervals. The NLSA parameter values, sampling frequency, and training interval influence the properties of the extracted ISO modes (i.e., the truth signal), the KAF parameters affect the predictive skill of the forecast model, and the length of the test interval affects the robustness of the skill scores computed for the given choice of NLSA parameters, training interval, and KAF parameters. In this section, we present assessments of the sensitivity of our hindcast results on these factors, focusing on the influence of the embedding window size (section 5a) and length of the training and test time series (section 5b). To manage the computational cost of this study, we reduce the frequency of the time sampling of the raw CLAUS data from eight times a day to four times a day; this downsampling cuts the overall computational cost of the pairwise kernel evaluations (which scale quadratically with the number of samples) by a factor of 4. Aside from a moderate decrease in forecast skill (by approximately 10 days of PC > 0.6 lead time), the resulting MJO and BSISO modes are qualitatively very similar to those presented in section 3.

### a. Sensitivity to lagged embedding window length

The parameters of the combined NLSA–KAF algorithm described in section 2 are the number of embedding lags

Figure 7 displays the truth signals and RMSE and PC scores for the MJO, evaluated using the

(a) Out-of-sample extensions of MJO signals to the testing period from 1 Jul 2006 to 30 Jun 2009, with sizes of embedding window varying from 48 to 96 days. (b) RMSE and (c) PC for each experiment (with JJA excluded, as per section 2h).

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

(a) Out-of-sample extensions of MJO signals to the testing period from 1 Jul 2006 to 30 Jun 2009, with sizes of embedding window varying from 48 to 96 days. (b) RMSE and (c) PC for each experiment (with JJA excluded, as per section 2h).

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

(a) Out-of-sample extensions of MJO signals to the testing period from 1 Jul 2006 to 30 Jun 2009, with sizes of embedding window varying from 48 to 96 days. (b) RMSE and (c) PC for each experiment (with JJA excluded, as per section 2h).

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

As in Fig. 7, but for the BSISO mode (and thus excluding DJF from the calculation of the RMSE and PC scores, as per section 2).

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

As in Fig. 7, but for the BSISO mode (and thus excluding DJF from the calculation of the RMSE and PC scores, as per section 2).

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

As in Fig. 7, but for the BSISO mode (and thus excluding DJF from the calculation of the RMSE and PC scores, as per section 2).

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

The effect of changing the size of the embedding window is further analyzed in Fig. 9 by computing correlations of the NLSA-derived MJO and BSISO amplitudes with the RMM amplitude for the NLSA embedding windows examined above and different values of (backward) running-average smoothing of the RMM amplitude. For both MJO and BSISO, the RMM smoothing window for maximum correlation is an increasing function of the NLSA embedding window, but that relationship is not proportional and appears to saturate at the larger (80 and 96 day) Δ*t* values examined. In the case of the MJO (Fig. 9a) the maximum correlation between NLSA and RMM is 0.51 and occurs for Δ*t* = 80 days and a ~65-day RMM smoothing window. As stated in section 3b, the correlation between NLSA–BSISO and RMM (Fig. 9b) is significantly higher when conditioned on JJA. In particular, the highest correlation is 0.53 and occurs for Δ*t* = 96 days and an ~80-day RMM smoothing window.

Correlations between NLSA-derived modes and RMM, for lengths of RMM averaging. Discussed in both sections 3b and 5a.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

Correlations between NLSA-derived modes and RMM, for lengths of RMM averaging. Discussed in both sections 3b and 5a.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

Correlations between NLSA-derived modes and RMM, for lengths of RMM averaging. Discussed in both sections 3b and 5a.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

### b. Sensitivity to training size

As with any statistical method, the size of training dataset is a critical ingredient of KAF, as it influences the quality of the extracted eigenfunctions (the truth signal) and the availability of adequate analogs for prediction given previously unseen initial data. The size of the test dataset is also important, as it influences the robustness of skill scores. In particular, besides being important for accurately assessing the performance of the method in a hindcast setting, the availability of accurate skill scores is also important in operational forecast settings, where the parameters of the method would be tuned in a validation stage (analogous to the hindcasts performed here) prior to its deployment in actual forecasts.

In this section, we examine the robustness of the forecast skill results presented in section 3 by comparing PC and RMSE scores from multiple hindcast experiments of the MJO and BSISO with different sizes of training and test data. In all cases, we work with 6-hourly sampled data, a 64-day embedding window, and the same

The RMSE and PC scores for the MJO and BSISO hindcasts from these experiments are shown in Figs. 10 and 11, respectively. There it is evident that the scores have noticeable spread (particularly at large leads); they are largely consistent with those in Figs. 4 and 5, respectively. In particular, even for the shortest training period examined in each case, both MJO and BSISO have PC scores greater than 0.6 out to ~40-day leads. Moreover, the skill scores computed for 2006–09 and the longest possible test periods are mutually consistent. This suggests that in an operational setting it should be possible to tune the parameters of the method using a modest, ~3-yr, validation period and utilizing the rest of the available samples to obtain high-quality eigenfunctions and analogs. Note that the RMSE and PC scores in Figs. 10 and 11 are not monotonic functions of the training or test period size. This is likely due to both lack of optimality of NLSA/KAF parameters and variance of the skill scores (particularly at large leads).

RMSE and PC of forecasts created by training data spanning the range from 1 Jul 1983 to 30 Jun of the year specified by the legend, excluding DJF as per section 2h. (a) RMSE and (c) PC over the testing range from 1 Jul of the specified year up to 30 Jun 2009; (b) RMSE and (d) PC over a fixed range from 1 Jul 2006 to 30 Jun 2009. See section 5b.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

RMSE and PC of forecasts created by training data spanning the range from 1 Jul 1983 to 30 Jun of the year specified by the legend, excluding DJF as per section 2h. (a) RMSE and (c) PC over the testing range from 1 Jul of the specified year up to 30 Jun 2009; (b) RMSE and (d) PC over a fixed range from 1 Jul 2006 to 30 Jun 2009. See section 5b.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

RMSE and PC of forecasts created by training data spanning the range from 1 Jul 1983 to 30 Jun of the year specified by the legend, excluding DJF as per section 2h. (a) RMSE and (c) PC over the testing range from 1 Jul of the specified year up to 30 Jun 2009; (b) RMSE and (d) PC over a fixed range from 1 Jul 2006 to 30 Jun 2009. See section 5b.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

As in Fig. 10, but for the BSISO (and thus excluding DJF from the calculation of the RMSE and PC scores, as per section 2). See section 5b.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

As in Fig. 10, but for the BSISO (and thus excluding DJF from the calculation of the RMSE and PC scores, as per section 2). See section 5b.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

As in Fig. 10, but for the BSISO (and thus excluding DJF from the calculation of the RMSE and PC scores, as per section 2). See section 5b.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

Figure 12 shows more precisely the increase in training data dependency with respect to forecast time by plotting ratio of the standard deviation of the skill scores at given lead times, across all available experiments, to the mean amount of skill deterioration by that time. Roughly speaking, for each of the plots in Figs. 10 and 11, we are plotting the ratio of the vertical spread (as measured by the standard deviation) to the mean increase in RMSE (or mean decrease in PC). This metric provides a way of approximating how much of the change in skill is due to choice of training data. Overall, Fig. 12 shows that this ratio remains bounded between 0% and 23%.

For both the MJO and BSISO, ratio of the standard deviation of skill across all available training data to the mean amount of (a),(b) RMSE gained and (c),(d) PC lost by a given lead time. See section 5b.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

For both the MJO and BSISO, ratio of the standard deviation of skill across all available training data to the mean amount of (a),(b) RMSE gained and (c),(d) PC lost by a given lead time. See section 5b.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

For both the MJO and BSISO, ratio of the standard deviation of skill across all available training data to the mean amount of (a),(b) RMSE gained and (c),(d) PC lost by a given lead time. See section 5b.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0147.1

## 6. Discussion

Although it is tempting to directly compare the PC scores of KAF to those of other methods, it is important to note that it is often not the case that the same MJO and BSISO definitions are used in different methods. While KAF could be used to predict other commonly defined ISO indices (e.g., the RMM index), investing too much in making such comparisons risks missing the important point that all indices are representations of physical phenomena that depend on a choice of data analysis technique and that much of the appeal of KAF is that it incorporates the same class of kernel operators for both ISO definition and prediction in a unified scheme. Instead, a true comparison with other methods would require assessing how each method fares in predicting physical observables of interest (e.g., precipitation on intraseasonal time scales) conditioned on the predicted values of the indices. Such comparisons are outside the scope of this work, but we believe that the high predictability and coherent spatiotemporal structure of the NLSA-based ISO modes are encouraging properties for future predictability studies of physical observables.

Despite the difficulty of making objective comparisons to other methods, it is nevertheless valuable to place our results in context with existing ISO forecasting techniques. For instance, when efforts to forecast the MJO first began in the 1990s, global climate models (GCMs) were frequently unable to achieve more than 6 days of predictability (Chen and Alpert 1990; Lau and Chang 1992; Slingo et al. 1996; Jones et al. 2000; Hendon et al. 2000). Much of the difficulty of these early GCMs stemmed from their inability to sufficiently represent organized convection, which eventually led to a shift in research focus to empirical methods that are not affected by model error in GCMs. That said, the switch in focus to empirical methods did not yield immediate benefits, as many of the earliest attempts, such as applying principal oscillation pattern (POP) analysis to 200-hPa equatorial velocity potentials (von Storch and Jinsong 1990), still did not attain predictability beyond 1 week.

Multiweek MJO predictability with empirical methods was eventually attained through improvements to both ISO definition and forecasting methods. For instance, multiple-field EOF analysis that includes outgoing longwave radiation data, as first recommended by Kousky and Kayano (1993), was performed by Waliser et al. (1999) to obtain MJO predictability for up to 15–20 days. Time-lagged embedding, meanwhile, was used in a singular spectrum analysis of similar data by Mo (2001) to consistently obtain 20-day predictability. That KAF uses elements of these techniques, in particular the use of cloudiness data (in this case

As with any empirical method, two important factors affecting the skill of KAF are the length of the training time series and its relevance to the future behavior of the system. As discussed in sections 4c and 5b, the length of the training time series affects prediction skill in terms of both the quality of the extracted ISO indices and the availability of analogs matching the current initial data. Those results illustrate that decreasing the length of the training dataset generally leads to a degradation of the quality of the NLSA ISO modes, in the sense that the modes of interest become mixed with modes that are unrelated to ISOs, with the effect more pronounced for the BSISO than for the MJO. For the MJO, 35-day predictability is still within reach when the training size is reduced to 9 yr, and for the BSISO, 30-day predictability is available when the training size decreases to 17 yr. While these experiments may appear somewhat artificial (since there is no reason why a forecaster would not use all of the available CLAUS data in practice), they nevertheless illustrate some of the long-term impacts of forced climate variability (as well as low-frequency natural variability) on future analog ISO forecasts. In particular, GCM simulations suggest that climate change on decadal time scales will have statistically significant impacts on the characteristics of ISOs (Subramanian et al. 2014), and such changes would limit the effective time span of training data available for extracting faithful ISO indices. Similarly, the effective number of analogs would be limited to the latter portions of the training data commensurate with the characteristic time scale of climate change. For tropical variability, that time scale is expected to be in the interdecadal range (e.g., Deser et al. 2012), suggesting that the useful length of available training data is comparable (and possibly longer) than the 23-yr training interval employed in sections 4a and 4b, which was sufficient for skillful MJO and BSISO forecasts. Thus, KAF methods should be useful for ISO forecasts even in the presence of climate change.

## 7. Conclusions

In this paper, we have demonstrated that qualitative features of tropical ISOs can be forecasted, in an empirical and nonparametric manner, on a scale of 5–7 weeks with appropriate kernel algorithms for ISO index definition and analog forecasting. In particular, using kernels developed in the context of NLSA algorithms, it is possible to obtain indices from unprocessed CLAUS

The overall robustness of the KAF method applied to MJO and BSISO forecasting should continue to be investigated. Varying both the type of kernels used and the size and type of training data is important. One class of kernels to be tried is the so-called cone kernel family (Giannakis 2015), which takes into account not only the speed at which data varies, but also the direction in which it changes. Relaxing or tightening certain regularity conditions is another potentially interesting approach, as well as incorporating additional predictor variables (e.g., circulation) in a multivariate kernel analysis. Equally importantly, KAF should be assessed in forecasts of physical variables, such as intraseasonal precipitation.

Combining aspects of KAF with other methods has the potential to extend overall tropical ISO predictability even further than what has already been shown. Several numerical models, such as the European Centre for Medium-Range Weather Forecasts model (Vitart 2014), a 10-petaflop “K” supercomputer (Miyakawa et al. 2014), and a coupled GFDL model (Xiang et al. 2015) have recently obtained MJO predictability of up to 27 days with the RMM as the baseline definition of the MJO. It should be explored whether using NLSA indices in these models would lead to greater predictability. Furthermore, an ensemble of numerical simulations and empirical KAF forecasting can be combined to create an optimized method that produces forecasts more accurately when initialized with novel conditions and yet more quickly when given familiar conditions.

## Acknowledgments

The authors gratefully acknowledge the financial support given by the Earth System Science Organization, Ministry of Earth Sciences, Government of India (Grant/Project MM/SERP/CNRS/2013/INT-10/002) to conduct this research under the Monsoon Mission. D. Giannakis, E. Székely, and Z. Zhao acknowledge support from ONR Grant N00014-14-0150 and ONR MURI Grant 25-74200-F7112. D. Giannakis and Z. Zhao also acknowledge support from NSF Grant DMS-1521775. The authors thank Nan Chen and Andrew Majda for stimulating discussions on low-order modeling of intraseasonal oscillations.

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