## Abstract

We present and apply a novel method of describing and modeling complex multivariate datasets in the geosciences and elsewhere. Data-adaptive harmonic (DAH) decomposition identifies narrow-banded, spatio-temporal modes (DAHMs) whose frequencies are not necessarily integer multiples of each other. The evolution in time of the DAH coefficients (DAHCs) of these modes can be modeled using a set of coupled Stuart-Landau stochastic differential equations that capture the modes’ frequencies and amplitude modulation in time and space. This methodology is applied first to a challenging synthetic dataset and then to Arctic sea ice concentration (SIC) data from the US National Snow and Ice Data Center (NSIDC). The 36-year (1979–2014) dataset is parsimoniously and accurately described by our DAHMs. Preliminary results indicate that simulations using our multilayer Stuart-Landau model (MSLM) of SICs are stable for much longer time intervals, beyond the end of the twenty-first century, and exhibit interdecadal variability consistent with past historical records. Preliminary results indicate that this MSLM is quite skillful in predicting September sea ice extent.

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## Acknowledgements

The authors would like to acknowledge Andreas Groth for developing the synthetic dataset in the SSA-MTM Toolkit example of varimax-rotated M-SSA (http://www.atmos.ucla.edu/tcd/ssa/guide/mssa/mssarot.html); it is this dataset that was utilized in Sect. 2. Preliminary results of this research were reported at “30 Years of Nonlinear Dynamics in Geosciences” conference in Rhodes, Greece, July 2017. The design of this study and the development of the DAH-MSLM techniques were supported by ONR’s Multidisciplinary Research Initiative (MURI) grants N00014-12-1-0911 and N00014-16-1-2073, and by the National Science Foundation grants OCE-1243175 and DMS-1616981. Analysis of Arctic sea ice data was also supported by Government of Russian Federation (Agreement No. 14.Z50.31.0033 with the Institute of Applied Physics of RAS).

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## Appendices

### Appendix 1: Details on the DAH Decomposition

The DAH modes (DAHMs) are obtained as follows. First, we estimate from a given *d*-channel time series **X**(*t*
_{
n
}) = (*X*
_{1}(*t*
_{
n
}), *…*, *X*
_{
d
}(*t*
_{
n
})), *n* = 1, *…*, *N*, the *cross-correlation coefficient* (CCF) *ρ*
_{
τ
}
^{(p, q)} at lag *τ* between channels *p* and *q*, where − *M* + 1 ≤ *τ* ≤ *M* − 1. In spectral analysis, it is common to refer to *M* as the window width.

Next, we form the following Hankel matrix:

Equivalently, this matrix can be viewed as a left circulant matrix formed from the (2*M* − 1)-dimensional row *r* = (*ρ*
_{−M+1}
^{(p, q)}, *…*, *ρ*
_{0}
^{(p, q)}, *…*, *ρ*
_{
M−1}
^{(p, q)}), i.e.:

in other words, the rows of **H**
^{(p, q)} are obtained by successive shifts to the left by one position, starting from *r* as a first row. Finally, we consider the block-Hankel matrix \(\mathfrak{C}\) formed by *d*
^{2} blocks of size (2*M* − 1) × (2*M* − 1), each given according to

Note that \(\mathfrak{C}\) is symmetric by construction due to symmetry of its building blocks **H**
^{(p, q)}, i.e., \(\mathfrak{C}^{(p,q)} = \mathfrak{C}^{(q,p)}\), and hereafter we use *M*
^{′} = 2*M* − 1 for concision, reindexing the string { − *M* + 1, *…*, *M* − 1} from 1 to *M*
^{′} as necessary.

The DAH eigenpairs (*λ*
_{
j
}, **E**
^{j}), with 1 ≤ *j* ≤ *dM*
^{′}, reveal useful information about the variability contained in the multivariate time series. In contrast to other data-adaptive methods built from cross-correlations, each of the DAH eigenvectors **E**
^{j} represents a data-adaptive spatio-temporal pattern naturally associated with a Fourier frequency *ω*
_{
l
} given by

These frequencies are equally spaced within the Nyquist interval [0, 0. 5] with a resolution of 1∕(*M*
^{′}− 1), essentially given by the embedding dimension *M*.

Each temporal frequency *ω*
_{
ℓ
} is associated with *d* pairs of DAH eigenvalues ±*λ*
_{
j
} that are opposite in sign but equal in absolute value, except at zero frequency, where there is only one eigenvector per eigenvalue, for a total of 2*d*(*M* − 1) + *d* eigenvalues. The association between a particular frequency and a given DAHM is obtained by counting zero-crossings *δ*
_{
j
} across the window width *M* for all channels:

One can thus assign a frequency that is in one-to-one correspondence to *δ*
_{
j
}. In Eq. (12), **E**
_{
k
}
^{j} denotes the *k*th spatial component of the DAHM, **E**
^{j}. One can then rank the DAHMs from the lowest to the highest frequency by simply looking at their number of sign changes. As shown in Chekroun and Kondrashov (2017), the corresponding fraction of the energy they capture is given by | *λ*
_{
j
} |, up to a scaling factor.

By analogy with M-SSA (Ghil et al., 2002), the multivariate dataset **X** can be projected onto the orthogonal set formed by the **E**
^{j}’s, to obtain the DAH expansion coefficients (DAHCs):

where *t* varies from 1 to *N*
^{′} = *N* − *M*
^{′} + 1.

Although the DAHCs are not formally orthogonal in time, they also exhibit a phase–quadrature relationship that depends on whether the window *M* is sufficiently large to resolve the decay of temporal correlations of a given dataset. Typically, the larger *M* (subject to the length of the record), the more apparent is the phase quadrature between a pair of DAHCs associated with the same frequency.

Furthermore, any subset **B** ⊂ **A** of DAHCs, as well as the full set **A**, can be convolved with associated **E**
_{
j
}’s, for partial or full reconstruction of the original data, respectively. The transformation between **X** and **A** is unitary, i.e., there is no loss of variance. Thus, the *j*th RC at time *t* for channel *k* is given by:

The normalization factor *M*
_{
t
} equals *M*
^{′}, except near the ends of the time series (Ghil et al., 2002), and the sum of all the RCs recovers the original time series.

It is also useful to consider harmonic reconstruction components (HRCs), namely a sum of *d* RC pairs corresponding to a particular frequency *ω*
_{
ℓ
} ≠ 0:

where \(\mathcal{J}_{\ell}\) denotes the set of all the indices *j* associated with the frequency *ω*
_{
ℓ
}. By construction, for each nonzero frequency, this set is constituted by 2*d* elements.

### Appendix 2: Details on the MSLM Modeling

As discussed in Sect. 4, the DAHMs extract harmonic components of variability that allow for a reduction of the data-driven modeling effort to a simple class of elemental multilayer stochastic models [MSMs: Kondrashov et al. (2015)]; these MSMs are stacked by frequency and only coupled at different frequencies by the same noise realization.

In the simplest case of one layer for the modeled noise, this construction leads to stochastic models of the form:

In (MSLM), the index *j* varies in the set of indices \(\mathcal{J}_{f}\) associated with a single frequency *f*, determined by the zero-crossings of the corresponding **E**
^{j}’s. When *f* ≠ 0, this set consists of *d* elements. In practice *f* = *ω*
_{
ℓ
}∕(2*π*) is determined by a Fourier frequency *ω*
_{
ℓ
} given in Eq. (11). The *W*
_{
k
}
^{j}’s with *k* in {1, 2} and *j* in {1, *…*, *d*} form 2*d* independent Brownian motions.

We call these models *multilayer stochastic Stuart-Landau models* (MSLM). At a given frequency *f*, the *d* pairs are linearly coupled as indicated by the terms in the sums apparent in the *x*
_{
j
}- and *y*
_{
j
}-equations. In (MSLM) and for a given pair indexed by *j*, the noise term (*ɛ*
_{
j
}
^{x}, *ɛ*
_{
j
}
^{y}) is modeled by means of linear dependencies involving only (*ɛ*
_{
j
}
^{x}, *ɛ*
_{
j
}
^{y}), on the one hand, and the *j*th pair (*x*
_{
j
}, *y*
_{
j
}), on the other.

Obviously, for a given pair, and following Kondrashov et al. (2015), more layers can be added as needed to (MSLM), when the noise term (*ɛ*
_{
j
}
^{x}, *ɛ*
_{
j
}
^{y}) at the first level is not white. In this case, the extra layers will depend linearly on the *j*th pair (*x*
_{
j
}, *y*
_{
j
}), and on the noise residuals from the previous layers. The sums in the *ɛ*
_{
j
}
^{x}- and *ɛ*
_{
j
}
^{y}-equations take into account “spatial” correlations between the pairs, at the level of the noise. Note that for the null frequency, *f* ≡ 0, there are exactly *d* modes that are not paired, and they are modeled by a linear multilayer stochastic model as in Kondrashov et al. (2015).

Note that Eq. (MSLM) can be generalized further by allowing coupling of (*x*
_{
j
}, *y*
_{
j
}) pairs at neighboring frequencies, which can be useful for certain applications where cross-frequency interactions are important. Equations (MSLM) are discretized in time and integrated numerically forward from initial conditions that respect the initialization procedure described in Kondrashov et al. (2015, Appendix B).

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Kondrashov, D., Chekroun, M.D., Yuan, X., Ghil, M. (2018). Data-Adaptive Harmonic Decomposition and Stochastic Modeling of Arctic Sea Ice. In: Tsonis, A. (eds) Advances in Nonlinear Geosciences. Springer, Cham. https://doi.org/10.1007/978-3-319-58895-7_10

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