# Discovering spatiotemporal patterns of COVID-19 pandemic in South Korea

### Compatible window-wise dynamic mode decomposition (CwDMD)

In this section, we shall describe the compatible window-wise Dynamic Mode Decomposition (CwDMD), a novel dynamic mode decomposition method that respects the compatibility of the data set. A detailed statement of compatibility will be presented as well. Basically, we present a new observation that the consistent data is a linear data and suggest that DMD has to be applied for the consistent or linear data. A compatibility condition is a way of achieving this consistency or linearity of the data set. We shall show that certain windows of the given time series data has to be selected so that a balance between the spatial and temporal resolution of the data set is made. This balance will then lead to the linearity of the selected windows. The application of DMD for each window is shown to result in accurate data analysis.

Throughout this section, for the sake of convenience, we denote ({mathbb {C}}^{ntimes ell }) by the space of complex matrices of size (ntimes ell ). For (n = 1) or (ell = 1), we shall omit writing it. Namely, for (ell = 1), we set ({mathbb {C}}^n := {mathbb {C}}^{n times 1}), that of which is sets of complex vectors of size *n*. For any element (c in {mathbb {C}}), we shall denote ({overline{c}}) by its complex conjugate. We shall denote (mathop {cdot }limits _{sim }) by the vector and (mathop {cdot }limits _{begin{array}{c} approx end{array}}) by the tensor. For (mathop {M}limits _{begin{array}{c} approx end{array}} in {mathbb {C}}^{ntimes ell }), its null and range will be denoted by ({mathcal {N}}(mathop {M}limits _{begin{array}{c} approx end{array}})) and ({mathcal {R}}(mathop {M}limits _{begin{array}{c} approx end{array}})), respectively. We denote (mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^{*}) by its complex adjoint matrix, and also denote (mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^dag ) by the pseudoinverse of (mathop {M}limits _{begin{array}{c} approx end{array}}). The symbol (mathop {delta }limits _{begin{array}{c} approx end{array}}) denotes the identity matrix. Note that (mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^dag ) satisfies the following conditions:

$$begin{aligned} mathop {M}limits _{begin{array}{c} approx end{array}} mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^dag mathop {M}limits _{begin{array}{c} approx end{array}} = mathop {M}limits _{begin{array}{c} approx end{array}}, ,, mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^dag mathop {M}limits _{begin{array}{c} approx end{array}} mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^dag = mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^dag , ,, (mathop {M}limits _{begin{array}{c} approx end{array}} mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^dag )^* = mathop {M}limits _{begin{array}{c} approx end{array}} mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^dag , ,, text{ and } ,,(mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^dag mathop {M}limits _{begin{array}{c} approx end{array}})^* = mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^dag mathop {M}limits _{begin{array}{c} approx end{array}}. end{aligned}$$

In particular, if (mathop {M}limits _{begin{array}{c} approx end{array}}) has a linearly independent columns, it holds that (mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^dag = (mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^dag mathop {M}limits _{begin{array}{c} approx end{array}})^{-1} mathop {mathop {M}limits _{begin{array}{c} approx end{array}}}nolimits ^*).

#### Dynamic mode decomposition (DMD)

Given a data set in a form of a time series data as follows:

$$begin{aligned} mathop {T}limits _{begin{array}{c} approx end{array}} = { {mathop {mathop {u}limits _{sim }}nolimits _0}, mathop {mathop {u}limits _{sim }}nolimits _1, ldots , mathop {mathop {u}limits _{sim }}nolimits _{m-1}, mathop {mathop {u}limits _{sim }}nolimits _m } in {mathbb {C}}^{n times (m+1)}, end{aligned}$$

where (mathop {mathop {u}limits _{sim }}nolimits _k) stands for the (k)th snapshot of the data set for (k ge 0) with (m+1) being the last entry of the data set, we let (mathop {X}limits _{begin{array}{c} approx end{array}}) and (mathop {Y}limits _{begin{array}{c} approx end{array}}) denote the followings:

$$begin{aligned} mathop {X}limits _{begin{array}{c} approx end{array}} = { mathop {mathop {u}limits _{sim }}nolimits _0, mathop {mathop {u}limits _{sim }}nolimits _1, ldots , mathop {mathop {u}limits _{sim }}nolimits _{m-1}} quad text{ and } quad mathop {Y}limits _{begin{array}{c} approx end{array}} = { mathop {mathop {u}limits _{sim }}nolimits _1, mathop {mathop {u}limits _{sim }}nolimits _1, ldots , mathop {mathop {u}limits _{sim }}nolimits _{m}}. end{aligned}$$

We shall briefly review the general description of the dynamic mode decomposition (DMD) applied for (mathop {T}limits _{begin{array}{c} approx end{array}}). For clarity, we assume an ordered sequence of data separated by a constant sampling time (Delta t). The idea of DMD lies at the assumption that there exists a linear operator (mathop {A}limits _{begin{array}{c} approx end{array}}) that connects at least, approximately two data (mathop {mathop {u}limits _{sim }}nolimits _k) and its subsequent data (mathop {mathop {u}limits _{sim }}nolimits _{k+1}) for all (k ge 0), that is

$$begin{aligned} mathop {mathop {u}limits _{sim }}nolimits _{k+1} approx mathop {A}limits _{begin{array}{c} approx end{array}} mathop {mathop {u}limits _{sim }}nolimits _k, quad forall k ge 0 quad text{ equivalently } quad mathop {Y}limits _{begin{array}{c} approx end{array}} approx mathop {A}limits _{begin{array}{c} approx end{array}}mathop {X}limits _{begin{array}{c} approx end{array}}. end{aligned}$$

(1)

The ambiguity in the approximation (approx ) will be clarified by defining (mathop {A}limits _{begin{array}{c} approx end{array}} = mathop {Y}limits _{begin{array}{c} approx end{array}}mathop {mathop {X}limits _{begin{array}{c} approx end{array}}}nolimits ^dag ) or as the solution to the following optimization problem:

$$begin{aligned} mathop {A}limits _{begin{array}{c} approx end{array}} = mathop {hbox {arg min}}limits _{mathop {C}limits _{begin{array}{c} approx end{array}}} Vert mathop {Y}limits _{begin{array}{c} approx end{array}} – mathop {C}limits _{begin{array}{c} approx end{array}} mathop {X}limits _{begin{array}{c} approx end{array}}Vert _F, end{aligned}$$

(2)

where (Vert cdot Vert _F) is the Frobenius norm. We note that the operator (mathop {A}limits _{begin{array}{c} approx end{array}}) is a type of dynamic operator that relates two consecutive data set. The goal of the dynamic mode decomposition is to extract the dynamic characteristic of (mathop {A}limits _{begin{array}{c} approx end{array}}), not directly to construct the mapping (mathop {A}limits _{begin{array}{c} approx end{array}}). More precisely, DMD obtains spectrums or spatial–temporal characteristics of the dynamical process described by (mathop {A}limits _{begin{array}{c} approx end{array}}). We note that the spectrums can be used to completely construct the action of the operator (mathop {A}limits _{begin{array}{c} approx end{array}}) if needs arise.

The essential algorithmic background lies in singular value decomposition of data, (mathop {X}limits _{begin{array}{c} approx end{array}}) and the relationship between eigen-pairs of (mathop {A}limits _{begin{array}{c} approx end{array}}) and its representation in principal component modes (see Lemma 1 and Lemma 2, in Supplementary note for Method). These are used to obtain the standard dynamic mode decomposition algorithm, as provided in Algorithm 1^{51}.

Generally, the data analysis can be accomplished through the dynamic modes and eigenvalues, as given as ((lambda _i, mathop {mathop {phi }limits _{sim }}nolimits _i, )_{i=1,cdots ,n}). We remark that ({mathop {mathop {phi }limits _{sim }}nolimits _i}_{i=1,cdots ,n})’s are called the DMD modes or mode vectors and they provide a rich set of information, especially spatial information about the data set^{25}. For example, the modulus of the element of the mode vector provides measure of the spatial region’s participation for that mode. On the other hand, the eigenvalues ({lambda _i}_{i=1,cdots ,n}) are relevant to the time evolution of the data sets and thus, they contain temporal information.

#### Linearity, consistency, and CwDMD

A loophole in DMD lies in that DMD spectrums are found for an approximate dynamic operator (mathop {A}limits _{begin{array}{c} approx end{array}}) for the data set (mathop {T}limits _{begin{array}{c} approx end{array}}). It is very much ambiguous and completely unknown theoretically how much the error observed in Eq. (1) results in misleading data interpretation from DMD spectrums. This has been elaborated in Fig. 8 for further clarity. The desired DMD is then not to start with constructing DMD-spectrums for (mathop {A}limits _{begin{array}{c} approx end{array}}) that satisfies (1), but, to build DMD spectrums based on (mathop {A}limits _{begin{array}{c} approx end{array}}) that satisfies the following relationship:

$$begin{aligned} mathop {mathop {u}limits _{sim }}nolimits _{k+1} = mathop {A}limits _{begin{array}{c} approx end{array}} mathop {mathop {u}limits _{sim }}nolimits _k, quad forall 0 le k le m, quad text{ equivalently } quad mathop {Y}limits _{begin{array}{c} approx end{array}} = mathop {A}limits _{begin{array}{c} approx end{array}}mathop {X}limits _{begin{array}{c} approx end{array}}. end{aligned}$$

(3)

Thus, we investigate the condition for the existence of an operator (mathop {A}limits _{begin{array}{c} approx end{array}}) that satisfies the Eq. (3). This is in fact dependent on the data set (mathop {T}limits _{begin{array}{c} approx end{array}}). Namely, there must be a condition for (mathop {T}limits _{begin{array}{c} approx end{array}}), which leads to the existence of such an operator (mathop {A}limits _{begin{array}{c} approx end{array}}). Therefore, we introduce a notion of the linearity. Basically, we say that the data (mathop {T}limits _{begin{array}{c} approx end{array}}) is linear if and only if there exists an operator (mathop {A}limits _{begin{array}{c} approx end{array}} in {mathbb {C}}^{ntimes n}) such that (mathop {Y}limits _{begin{array}{c} approx end{array}} = mathop {A}limits _{begin{array}{c} approx end{array}} mathop {X}limits _{begin{array}{c} approx end{array}}) (see the notion of linearity precisely defined for (mathop {T}limits _{begin{array}{c} approx end{array}}) in Definition 1 of Supplementary note). The compatibility condition is basically the condition for which the data (mathop {T}limits _{begin{array}{c} approx end{array}}) is linear. We remark that a relevant notion that states the Eq. (3) for a particular (mathop {A}limits _{begin{array}{c} approx end{array}}) of the form (mathop {A}limits _{begin{array}{c} approx end{array}} = mathop {Y}limits _{begin{array}{c} approx end{array}}mathop {mathop {X}limits _{begin{array}{c} approx end{array}}}nolimits ^dag ) for the data (mathop {T}limits _{begin{array}{c} approx end{array}}) has been provided by Tu et al. in^{32}, i.e., a notion of linear consistency, stating that the null space of (mathop {X}limits _{begin{array}{c} approx end{array}}) is contained in that of (mathop {Y}limits _{begin{array}{c} approx end{array}}) (({mathcal {N}}(mathop {X}limits _{begin{array}{c} approx end{array}}) subset {mathcal {N}}(mathop {Y}limits _{begin{array}{c} approx end{array}}))) (see the notion of linear consistency defined for (mathop {T}limits _{begin{array}{c} approx end{array}}) in Definition 2 and also Theorem 1 of Supplementary note). We remark that the linearity is much more intuitive and general than the linear consistency. The notion of the linearity is a certain extension of the existence of line connecting two points in two dimensional Euclidean space consisting of one spatial dimension and one temporal dimension. On the other hand, we observe that these two concepts; linearity and linear consistency are in fact equivalent. Namely, the linear consistency of (mathop {T}limits _{begin{array}{c} approx end{array}}) holds if and only if the linearity of (mathop {T}limits _{begin{array}{c} approx end{array}}) holds (see Theorem 2 in Supplementary note for detailed proof). In another words, nonlinear data is inconsistent and inconsistent data is nonlinear. This equivalency is remarkable since these two concepts can be used to derive so-called the compatibility condition, which can be used to easily verify the linearity of (mathop {T}limits _{begin{array}{c} approx end{array}}). Note that the linear consistency condition provides an important algebraic condition for the data being linear. However, authors find it difficult to verify that condition in general.

The concept of compatibility is based on the observation that the data (mathop {T}limits _{begin{array}{c} approx end{array}}) being linear is relevant to the balance between spatial and temporal resolutions. As mentioned, for example, in one spatial dimension, only two points (two temporal data) can be connected in general by a line, unless data consisting of more than two points are collinear. Its extension for higher dimensional case can be understood as a simple inequality: (m le n). More precisely, the compatibility condition can be stated as follows:

### **Definition**

(*Compatibility Condition*) Compatibility condition is the balance between to the balance between temporal and spatial resolutions, i.e., a data set (mathop {T}limits _{begin{array}{c} approx end{array}}) with the temporal resolution (m+1) and spatial resolution *n* have the relationship that (m le n).

Note that for (m > n), (mathop {T}limits _{begin{array}{c} approx end{array}}) will be in general inconsistent unless it is linear. The compatibility condition is stated to cover very general situations for which DMD can have a meaningful usage. We can show that under the compatibility condition, DMD will provide meaningful results with probability one. To be more precise, we note that the consistency can be easily understood in terms of the linear independency of the data (mathop {X}limits _{begin{array}{c} approx end{array}}), i.e., the linear independency of (mathop {X}limits _{begin{array}{c} approx end{array}}) implies the consistency of (mathop {T}limits _{begin{array}{c} approx end{array}}) and this can in particular, remove the trivial case that any column of (mathop {X}limits _{begin{array}{c} approx end{array}}) is the zero vector. Theoretically, it is established that if (mathop {T}limits _{begin{array}{c} approx end{array}}) satisfies the compatibility condition, then almost all (mathop {X}limits _{begin{array}{c} approx end{array}} in {mathbb {C}}^{ntimes m}) with (m le n) will consist of columns which are linearly independent^{52,53}. This means that ({mathcal {N}}(mathop {X}limits _{begin{array}{c} approx end{array}}) = {mathop {0}limits _{begin{array}{c} approx end{array}}}). Therefore, the data set (mathop {T}limits _{begin{array}{c} approx end{array}}) is linear. The compatibility condition thus implies the consistency with probability one. Thus, the compatibility condition implies that the linearity of the data (mathop {T}limits _{begin{array}{c} approx end{array}}) is almost always guaranteed in case (m le n), which then leads to the meaningful DMD results.

In a very much rare case, when the consistency breaks under the compatibility condition, one can provide a small (arbitrarily small) perturbation to obtain (mathop {T_varepsilon }limits _{begin{array}{c} approx end{array}} in {mathbb {C}}^{ntimes (m+1)}), which is proven to result in a linear data^{54}. Namely, for (m le n), let (mathop {X_varepsilon }limits _{begin{array}{c} approx end{array}} in {mathbb {C}}^{n times m}) consist of first *m* columns of (mathop {T_varepsilon }limits _{begin{array}{c} approx end{array}}). Then we consider (widetilde{mathop {X_varepsilon }limits _{begin{array}{c} approx end{array}}} in {mathbb {C}}^{mtimes m}) obtained from (mathop {X_varepsilon }limits _{begin{array}{c} approx end{array}}) by chopping off all rows underneath (m)th row of (mathop {X_varepsilon }limits _{begin{array}{c} approx end{array}}). This square matrix can be proven to be diagonalizable^{52,54}, i.e., it consists of linear independent columns and thus the columns of (mathop {X_varepsilon }limits _{begin{array}{c} approx end{array}}) is linearly independent. In view of the spatio-temporal analysis of the data, arbitrarily small perturbation will not change the result significantly. Furthermore, theoretically, such arbitrarily small perturbation will not affect the computation of the DMD-spectrums if they are in particular, Gaussian^{55,56}. We remark that our data is generally very nice, i.e., whenever we choose (m le n), the data set (mathop {T}limits _{begin{array}{c} approx end{array}}) is always linear consistent and so, no perturbation was needed.

We are in a position to introduce our new algorithm, so-called a compatible window-wise dynamic mode decomposition (CwDMD). Our observation is that for (m > n), (mathop {T}limits _{begin{array}{c} approx end{array}}) will be in general inconsistent unless it is linear. As such, the direct and reliable DMD analysis of large time series data is not feasible in general. The strategy is to choose an adequate set of representative subdomains called windows, each containing a moderate size of time-series data that satisfies the compatibility. The total size-times duration of all the windows serving a given system depends only on local situations that can arise in the full time series data. For example, Fig. 2, A shows a class of windows for the COVID-19 data in South Korea. Namely, given a data set ({ mathop {mathop {u}limits _{sim }}nolimits _0, mathop {mathop {u}limits _{sim }}nolimits _1, ldots , mathop {mathop {u}limits _{sim }}nolimits _k, ldots , mathop {mathop {u}limits _{sim }}nolimits _m }), we consider the following windows that are consistent:

$$begin{aligned} (mathop {X_k}limits _{begin{array}{c} approx end{array}}, mathop {Y_k}limits _{begin{array}{c} approx end{array}}), text{ with } mathop {X_k}limits _{begin{array}{c} approx end{array}} := { mathop {u_{_{k_s}}}limits _{sim }, ldots , mathop {u_{_{k_e-1}}}limits _{sim } } quad text{ and } quad mathop {Y_k}limits _{sim } := { mathop {u_{_{k_s +1}}}limits _{sim }, ldots , mathop {u_{_{k_e}}}limits _{sim } }. end{aligned}$$

for which (mathop {X_k}limits _{begin{array}{c} approx end{array}}) and (mathop {Y_k}limits _{begin{array}{c} approx end{array}}) are consistent for (k = 0, 1, ldots , ell ). The compatible window-wise dynamic mode decomposition is to apply the dynamic mode decomposition locally for each compatible window ((mathop {X_k}limits _{begin{array}{c} approx end{array}}, mathop {Y_k}limits _{begin{array}{c} approx end{array}})). Note that these windows can be constructed so that they may overlap or non-overlap depending on the situations. Therefore, choices of window can be made without too much restriction other than the condition of compatibility. This can be summarized as in the Algorithm 2.

#### Data fitting, dimensional reduction, frequency and phase analysis

In this section, we discuss the data fitting using the DMD operator and choice of modes for the dimensional reduction and their uses for the phase analysis of each window. Throughout this section, we assume that (mathop {T}limits _{begin{array}{c} approx end{array}} in {mathbb {C}}^{n times (m+1)}) is consistent and the DMD operator (mathop {A}limits _{begin{array}{c} approx end{array}}) is given in terms of eigen-pairs ((lambda _i,mathop {mathop {phi }limits _{sim }}nolimits _i)_{i=1,cdots ,n}). We would also like to mention that the precise action of the operator (mathop {A}limits _{begin{array}{c} approx end{array}}) may not be found solely from these eigenspectrums. Namely, the data (mathop {X}limits _{begin{array}{c} approx end{array}}) has to be represented in terms of DMD modes, which requires to solve certain optimization problem. In a prior work, this has been accomplished by taking into account the whole data (mathop {X}limits _{begin{array}{c} approx end{array}}). We shall show that this can be done taking into account any single snapshot data in (mathop {X}limits _{begin{array}{c} approx end{array}}) under the consistency condition, thereby achieving a significant computational reduction. We begin our discussion with the fact that almost all complex matrices over complex fields are diagonalizable^{52,54}. Namely, geometric and algebraic multiplicities of almost all complex matrices over complex fields are identical. This means that the DMD modes make a full set of eigenvectors for almost all data set satisfying the compatibility. Some list of a couple of equivalent conditions to the fact that algebraic and geometric multiplicities agree for a matrix (mathop {A}limits _{begin{array}{c} approx end{array}} in {mathbb {C}}^{ntimes n}) can be found at^{57} and Theorem 3 in Supplementary note. Therefore, in general, we have that ({mathbb {C}}^{n} = {mathrm{span}} { mathop {mathop {phi }limits _{sim }}nolimits _i }_{i=1,cdots ,n}). Having a full set of eigenvectors of (mathop {A}limits _{begin{array}{c} approx end{array}}), we can represent for example, the data (mathop {mathop {u}limits _{sim }}nolimits _eta ) of (mathop {T}limits _{begin{array}{c} approx end{array}}) with (0 le eta le m+1), as follows:

$$begin{aligned} mathop {mathop {u}limits _{sim }}nolimits _eta = sum _{i = 1}^n alpha _i mathop {mathop {phi }limits _{sim }}nolimits _i quad text{ or } quad mathop {alpha }limits _{sim } = mathop {mathop {Phi }limits _{begin{array}{c} approx end{array}}}nolimits ^{-1} mathop {u_eta }limits _{sim }, end{aligned}$$

where (mathop {Phi }limits _{begin{array}{c} approx end{array}} = [mathop {mathop {phi }limits _{sim }}nolimits _1 , ldots , mathop {mathop {phi }limits _{sim }}nolimits _n]). With (mathop {alpha }limits _{sim }) given above, we can obtain the action of the DMD operator (mathop {A}limits _{begin{array}{c} approx end{array}}) as follows: for (-eta le k le -eta + m + 1),

$$begin{aligned} mathop {mathop {u}limits _{sim }}nolimits _k = sum _{i=1}^n alpha _i e^{k , mathfrak {R}{(log (lambda _i))}} e^{ hat{i} k mathfrak {I}{(log (lambda _i))}} mathop {mathop {phi }limits _{sim }}nolimits _i, end{aligned}$$

(4)

where (hat{i}) is the pure imaginary number such that (hat{i}^2 = -1). We remark that it is standard to choose (eta = 0), which is also our choice. Oftentimes DMD is argued to be biased to the initial data^{24}, our observation is that it is not really the case, for the consistent data. We recall that the framework of the optimized DMD^{22} is also designed to obtain the same (mathop {alpha }limits _{sim }) for fitting, (mathop {X}limits _{begin{array}{c} approx end{array}}), by solving the following optimization problem:

$$begin{aligned} mathop {alpha }limits _{sim } = mathop {hbox {arg min}}limits _{mathop {mu }limits _{sim } = (mu _i)_{i=1,cdots ,n}} left| mathop {X}limits _{begin{array}{c} approx end{array}} – mathop {Phi }limits _{begin{array}{c} approx end{array}} mathop {D_{mu }}limits _{begin{array}{c} approx end{array}} mathop {V_{m-1}}limits _{begin{array}{c} approx end{array}} right| _{F}, end{aligned}$$

where

$$begin{aligned} mathop {D_mu }limits _{begin{array}{c} approx end{array}} = {mathrm{diag}} (mu ) quad text{ and } quad mathop {V_m}limits _{begin{array}{c} approx end{array}} = left( begin{array}{ccccc} 1 &{} lambda _1 &{} lambda _1^2 &{} cdots &{} lambda _1^m 1 &{} lambda _2 &{} lambda _2^2 &{} cdots &{} lambda _2^m vdots &{} vdots &{} vdots &{} ddots &{} vdots 1 &{} lambda _n &{} lambda _n^2 &{} cdots &{} lambda _n^m. end{array} right) end{aligned}$$

It is clear that the consistency of data leads to a significant reduction of the computational effort.

We now can consider a discrete to continuous extension of the action of DMD operator. We remark that from the discrete represent of (mathop {mathop {u}limits _{sim }}nolimits _k) in (4), a continuous extension can be achieved as follows: for all (t ge t_0 = 0),

$$begin{aligned} mathop {u}limits _{sim }(t) := sum _{i=1}^n alpha _i (lambda _i)^{t-t_0} mathop {mathop {phi }limits _{sim }}nolimits _i = e^{(t – t_0) , mathfrak {R}{(log (lambda _i))}} e^{ hat{i} (t -t_0) mathfrak {I}{(log (lambda _i))}} mathop {mathop {phi }limits _{sim }}nolimits _i. end{aligned}$$

(5)

We now discuss the mode choice for the phase analysis, which will be used to obtain the dimensional reduction of the data. The most natural guide to choose the important DMD mode is to find the DMD mode which contributes most significantly to the data both temporally and spatially. This leads us to choose the index of DMD mode for which the following quantity, product of the temporal and spatial contribution in each window is maximized:

$$begin{aligned} {mathrm{arg}} left{ max _k { |lambda _k|^p Vert alpha _k mathop {mathop {phi }limits _{sim }}nolimits _kVert _F, 1 le k le n. } right} , end{aligned}$$

(6)

where *p* is the temporal resolutions for the window. We call the quantity (|lambda _k|^p Vert alpha _k mathop {mathop {phi }limits _{sim }}nolimits _kVert _F) the power of the (k)th DMD mode and observe that in general one or two dominant powers exist. These are then chosen to form a dimensionally reduced data. For example, (mathop {mathop {phi }limits _{sim }}nolimits _k) is the DMD mode whose power is the largest. Then it is used to form a dimensionally reduced data: for all (t ge t_0 = 0),

$$begin{aligned} mathop {{widetilde{u}}}limits _{sim }(t) = alpha _k (lambda _k)^{t-t_0} mathop {mathop {phi }limits _{sim }}nolimits _k = e^{(t – t_0) , mathfrak {R}{(log (lambda _k))}} e^{ hat{i} (t -t_0) mathfrak {I}{(log (lambda _k))}} mathop {mathop {phi }limits _{sim }}nolimits _k, end{aligned}$$

(7)

which is used for the data interpretation such as phases and magnitudes. In literature, DMD modes are chosen based on their norms or weighted norm by the corresponding DMD eigenvalues^{32}. For example, the use of weighted norm by DMD eigenvalues, can be interpreted as to penalize spurious modes with large norms but quickly decaying contributions to the dynamics^{29}. In our choice, we incorporate (mathop {alpha }limits _{sim }), the coordinate of data in the frame of DMD modes as a special scale for DMD modes. These measurements are meaningful especially for highly nonlinear data, since coordinates given in terms of DMD modes can much affect the dynamics of data. We remark that the frequency of the solution for the mode *k*, can be defined through (mathfrak {I}{(log (lambda _k))}/2pi ) and thus the period is given by the reciprocal of the frequency. The identified DMD mode can be categorized as periodic, growing or decaying modes depending on the magnitude of (lambda _k). Namely, for eigenvalues on (or close), outside or inside the unit circle, the corresponding modes are considered as oscillatory, growing, and decaying modes, respectively. In the present work, we give a tolerance (epsilon = 5.E{-}2) and denote (N_o = { i : ||lambda _i| – 1| le epsilon }), (N_g = { i : |lambda _i| > 1 + epsilon }), (N_d = { i : |lambda _i| < 1 – epsilon }) by the set of oscillatory modes, the set of growing modes, and the set of decaying modes, respectively. We first select the DMD modes of large powers, and then measure the magnitude of its eigenvalues and determine whether they are oscillatory, growing or decaying mode.