Extracting Interpretable Multi-Dimensional MJO Patterns Using Tensor Decompositions

The traditional methods for identifying and quantifying the Madden-Julian oscillation (MJO) use Empirical Orthogonal Functions (EOFs) or its equivalent Principal Components Analysis (PCA). Typically, outgoing longwave radiation (OLR) and the zonal component of the wind in the upper and lower troposphere are the variables considered in a domain confined to a limited range of tropical latitudes. The data is extensively preprocessed to remove seasonal patterns, long-term trends, and an ENSO signature (e.g. by regression on sea-surface temperature). EOF requires the data to be arranged as a matrix, stacking the different variables into a single column, but such a reshaping can obscure connections and patterns between independent dimensions. Moreover, there is wide disagreement about the choice of variables that best identify MJO through PCA, with discrepancies between different definitions. The orthogonality and limited dimensionality constraints of PCA adds to the difficulty of fully characterizing the MJO, as well as hampering the ability to uncover the nonlinear relationship between MJO and other atmospheric modes, particularly ENSO.

We explore tensor decompositions to alleviate the potential ambiguities and shortcomings of traditional methods used for MJO identification. Tensors are multi-way arrays and tensor decompositions represent higher-order extensions of matrix linear algebra. Tensor decompositions can effectively leverage and reveal naturally occurring higher-order interactions in the data and likely improve the understanding and characterization of MJO. Specifically, we adopt the Canonical Polyadic (CP) decomposition, which approximates a data tensor as sum of outer products.

For our MJO analysis, we consider MERRA-2 reanalysis data for the period of 1979-2022 using daily averages of three variables: outgoing longwave radiation (OLR) and wind at two altitudes (200 hPa and 850 hPa), within a domain spanning 15°N-15°S. This is preprocessed by subtracting the time mean and first three harmonics of the annual cycle. A mean of the previous 120 days is also subtracted to remove slow time evolving signatures such as ENSO. The resulting preprocessed data—a three-way tensor in space, time, and variables of size 35136 ×15586 ×3—is analyzed using the CP decomposition. CP approximates the data as a sum of factors, each factor being an outer product between a spatial (lat×lon) vector, a time vector, and a variables vector (analogous to a pair of left and right singular vectors of PCA).

To facilitate comparison against existing techniques, we define an MJO index similar to the Real-time Multivariate MJO (RMM) index. For a single day, the index is computed by projecting a single day’s data (after appropriate preprocessing) onto the linear subspace implied by the spatial and variable factors of the decomposition, then computing the norm of this projected vector. By repeating this process for the 1979-2022 period, a timeseries of our MJO index is created. A fair comparison of this timeseries requires aligning it with, for example, that of the RMM; this is done by finding the threshold which achieves the maximum Peirce skill score, then using this threshold to shift and scale our timeseries appropriately.

Our results show that, the best matching CP factors—#1 and #2—define an MJO index which performs nearly as well as the RMM index (Peirce skill score of 0.76), in terms of indicating the occurrence of an MJO event. Furthermore, the phase progression predicted by the temporal vectors of these factors is in near perfect agreement with that of RMM EOFs. The next two best CP factors, in terms of Pierce skill score matching the WH index, are #5 and #6 whose temporal vectors also indicate a phase progression that agrees well with RMM. However, the CP factors provide more information beyond merely a classification and phase. Analyzing the projections into the space defined by CP factors #1-6 allows for quantifying variations in characteristic wavelength, north-south shifts, and seasonal variability in MJO events.