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Chapter 5  第 5 章

Multiresolution Dynamic Mode Decomposition
多分辨率动态模态分解

Abstract  摘要

Modeling of multiscale systems, in both space and time, pervades modern developments in theory and computation across the engineering, biological, and physical sciences. A significant challenge is making effective and efficient connections between microscale and macroscale effects, especially as they are potentially separated by orders of magnitude in space and time. Wavelet-based methods and/or windowed Fourier transforms are ideally structured to perform such multiresolution analyses (MRAs), as they systematically remove temporal or spatial features by a process of recursive refinement of sampling from the data [166, 76, 78]. Typically, MRA is performed in either space or time, but not both simultaneously. In this chapter, we integrate the concept of MRA in time with DMD [167]. This multiresolution DMD (mrDMD) is shown to naturally separate multiscale spatiotemporal features, providing an effective means to uncover multiscale structures in the data.
多尺度系统在空间和时间上的建模贯穿于工程、生物和物理科学领域理论与计算的现代发展之中。一个重大挑战是在微观尺度和宏观尺度效应之间建立有效且高效的联系，尤其是当它们在空间和时间上可能相差几个数量级时。基于小波的方法和/或加窗傅里叶变换具有理想的结构来执行此类多分辨率分析（MRA），因为它们通过对数据采样进行递归细化的过程系统地去除时间或空间特征[166, 76, 78]。通常，MRA 是在空间或时间中进行的，但不是同时在两者中进行。在本章中，我们将时间上的 MRA 概念与动态模态分解（DMD）[167]相结合。这种多分辨率 DMD（mrDMD）被证明能够自然地分离多尺度时空特征，为揭示数据中的多尺度结构提供了一种有效手段。

5.1 - Time-frequency analysis and the Gábor transform
5.1 - 时频分析与伽柏变换

The Fourier transform is one of the most important and foundational methods for the analysis of time-series signals. However, it was realized quite early on that Fourier transform-based methods have severe limitations. Specifically, when transforming a given time signal, the entire frequency content of the signal can be captured with the transform, but the transform fails to capture the moment in time when various frequencies are actually exhibited. Indeed, by definition, the Fourier transform eliminates all time-domain information by integrating over all time.
傅里叶变换是分析时间序列信号最重要且最基础的方法之一。然而，人们很早就意识到基于傅里叶变换的方法存在严重局限性。具体而言，在对给定的时间信号进行变换时，该变换能够捕捉到信号的全部频率成分，但却无法捕捉到各个频率实际出现的时刻。实际上，根据定义，傅里叶变换通过对所有时间进行积分消除了所有时域信息。
The limitations of the direct application of the Fourier transform, and its inability to localize a signal in both the time and the frequency domains, were articulated in the early development of radar and sonar detection. The Hungarian physicist/mathematician/electrical engineer Gábor Dénes (Physics Nobel Prize in 1971 for the discovery of holography in 1947) was the first to propose a formal method for localizing both time and frequency. His method involved a simple modification of the Fourier transform kernel. Gábor introduced the kernel
傅里叶变换直接应用的局限性，以及它无法在时域和频域中同时对信号进行定位，这在雷达和声纳探测的早期发展中就已被阐明。匈牙利物理学家/数学家/电气工程师加博尔·德内斯（因 1947 年发现全息术而于 1971 年获得诺贝尔物理学奖）是第一个提出在时域和频域中同时进行定位的形式化方法的人。他的方法涉及对傅里叶变换核进行简单修改。加博尔引入了该核
g t , ω ( τ ) = e i ω τ g ( τ − t ) g t , ω ( τ ) = e i ω τ g ( τ − t ) g_(t,omega)(tau)=e^(i omega tau)g(tau-t)g_{t, \omega}(\tau)=e^{i \omega \tau} g(\tau-t)
where the new term to the Fourier kernel g ( τ − t ) g ( τ − t ) g(tau-t)g(\tau-t) was introduced with the aim of localizing both time and frequency. 
其中，傅里叶核 g ( τ − t ) g ( τ − t ) g(tau-t)g(\tau-t) 的新项是为了在时间和频率上进行局部化而引入的。
Figure 5.1. Graphical depiction of the Gábor transform for extracting the time-frequency content of a signal S ( t ) S ( t ) S(t)S(t). The time-filtering window g ( τ − t ) g ( τ − t ) g(tau-t)g(\tau-t) is centered at τ τ tau\tau with width a.
图 5.1. 用于提取信号 S ( t ) S ( t ) S(t)S(t) 时频内容的伽柏变换的图形描述。时间滤波窗口 g ( τ − t ) g ( τ − t ) g(tau-t)g(\tau-t) 以 τ τ tau\tau 为中心，宽度为 a。
The Gábor transform, also known as the short-time Fourier transform (STFT) is then defined as
加博尔变换，也称为短时傅里叶变换（STFT），然后定义为
G [ f ] ( t , ω ) = f ~ g ( t , ω ) = ∫ − ∞ ∞ f ( τ ) g ¯ ( τ − t ) e − i ω τ d τ = ( f , g ¯ t , ω ) , G [ f ] ( t , ω ) = f ~ g ( t , ω ) = ∫ − ∞ ∞   f ( τ ) g ¯ ( τ − t ) e − i ω τ d τ = f , g ¯ t , ω , G[f](t,omega)= tilde(f)_(g)(t,omega)=int_(-oo)^(oo)f(tau) bar(g)(tau-t)e^(-i omega tau)d tau=(f, bar(g)_(t,omega)),\mathscr{G}[f](t, \omega)=\tilde{f}_{g}(t, \omega)=\int_{-\infty}^{\infty} f(\tau) \bar{g}(\tau-t) e^{-i \omega \tau} d \tau=\left(f, \bar{g}_{t, \omega}\right),
where the bar denotes the complex conjugate of the function. Thus, the function g ( τ − t ) g ( τ − t ) g(tau-t)g(\tau-t) acts as a time filter for localizing the signal over a specific window of time. The integration over the parameter τ τ tau\tau slides the time-filtering window down the entire signal, picking out the frequency information at each instant of time. Figure 5.1 illustrates the Gábor time-filtering scheme. In this figure, the time-filtering window is centered at τ τ tau\tau with width a a aa. Thus, the frequency content of a window of time is extracted and τ τ tau\tau is modified to extract the frequencies of another window. The definition of the Gábor transform captures the entire time-frequency content of the signal; the Gábor transform is a function of the two variables t t tt and ω ω omega\omega.
其中上划线表示函数的复共轭。因此，函数 g ( τ − t ) g ( τ − t ) g(tau-t)g(\tau-t) 用作时间滤波器，用于在特定的时间窗口内对信号进行定位。对参数 τ τ tau\tau 进行积分会使时间滤波窗口在整个信号上向下滑动，从而在每个时刻提取频率信息。图 5.1 展示了加博尔时间滤波方案。在该图中，时间滤波窗口以 τ τ tau\tau 为中心，宽度为 a a aa 。因此，提取了一个时间窗口的频率内容，并修改 τ τ tau\tau 以提取另一个窗口的频率。加博尔变换的定义捕捉了信号的整个时频内容；加博尔变换是两个变量 t t tt 和 ω ω omega\omega 的函数。
Although the Gábor transform gives a method whereby time and frequency can be simultaneously characterized, there are obvious limitations to the method. Specifically, the method is limited by the time filtering itself. Consider the illustration of the method in Figure 5.1. The time window filters out the time behavior of the signal in a window centered at τ τ tau\tau with width a a aa. It follows that, when considering the spectral content of this window, any portion of the signal with a wavelength longer than the window is completely lost. Indeed, since the Heisenberg uncertainty relationship must hold, the shorter the time-filtering window, the less information there is concerning the frequency content. In contrast, longer windows retain more frequency components, but this comes at the expense of losing the time resolution of the signal. As a consequence of a fixed time-filtering window, there are trade-offs between time and frequency resolution; increased accuracy in one of these parameters comes at the expense of resolution in the other parameter.
尽管伽柏变换给出了一种可以同时表征时间和频率的方法，但该方法存在明显的局限性。具体而言，该方法受到时间滤波本身的限制。考虑图 5.1 中该方法的图示。时间窗口在以 τ τ tau\tau 为中心、宽度为 a a aa 的窗口中滤除信号的时间行为。由此可知，在考虑该窗口的频谱内容时，信号中任何波长比窗口长的部分都会完全丢失。实际上，由于海森堡不确定性关系必须成立，时间滤波窗口越短，关于频率内容的信息就越少。相比之下，较长的窗口保留了更多的频率成分，但这是以牺牲信号的时间分辨率为代价的。由于时间滤波窗口固定，时间分辨率和频率分辨率之间存在权衡；其中一个参数的精度提高是以牺牲另一个参数的分辨率为代价的。

5.2 - Wavelets and MRA
5.2 - 小波与多分辨率分析

The Gábor transform established two key principles for time-frequency analysis: translation of a short-time window and scaling of the short-time window to capture finer time resolution.  
加博尔变换为时频分析确立了两个关键原则：短时窗的平移和短时窗的缩放，以获取更精细的时间分辨率。
Figure 5.2. Illustration of the Mexican bat wavelet ψ 1 , 0 ψ 1 , 0 psi_(1,0)\psi_{1,0} (top left panel), its Fourier transform ψ ^ 1 , 0 ψ ^ 1 , 0 hat(psi)_(1,0)\hat{\psi}_{1,0} (top right panel), and two additional dilations and translations of the basic ψ 1 , 0 ψ 1 , 0 psi_(1,0)\psi_{1,0} wavelet, namely the ψ 3 / 2 , − 3 ψ 3 / 2 , − 3 psi_(3//2,-3)\psi_{3 / 2,-3} and ψ 1 / 4 , 6 ψ 1 / 4 , 6 psi_(1//4,6)\psi_{1 / 4,6} (bottom panel).
图 5.2. 墨西哥蝙蝠小波 ψ 1 , 0 ψ 1 , 0 psi_(1,0)\psi_{1,0} 的图示（左上角面板）、其傅里叶变换 ψ ^ 1 , 0 ψ ^ 1 , 0 hat(psi)_(1,0)\hat{\psi}_{1,0} （右上角面板）以及基本 ψ 1 , 0 ψ 1 , 0 psi_(1,0)\psi_{1,0} 小波的另外两个伸缩和平移，即 ψ 3 / 2 , − 3 ψ 3 / 2 , − 3 psi_(3//2,-3)\psi_{3 / 2,-3} 和 ψ 1 / 4 , 6 ψ 1 / 4 , 6 psi_(1//4,6)\psi_{1 / 4,6} （底部面板）。
A simple modification to the Gábor method is to allow the scaling window (a) to vary and successively extract improvements in the time resolution. In other words, first the low-frequency (poor time resolution) components are extracted using a broad scaling window. The scaling window is subsequently shortened to extract higher frequencies at better time resolution. By keeping a catalog of the extracting process, excellent resolution in both time and frequency of a given signal can be obtained. This principle is fundamental to wavelet theory. The term wavelet means little wave and originates from the processes whereby the scaling window extracts smaller and smaller pieces of waves from the larger signal.
对加博尔方法的一个简单修改是允许缩放窗口（a）变化，并依次提取时间分辨率的改进。换句话说，首先使用宽缩放窗口提取低频（时间分辨率差）分量。随后缩短缩放窗口以提取更高频率且具有更好时间分辨率的分量。通过保存提取过程的目录，可以获得给定信号在时间和频率上的出色分辨率。这一原理是小波理论的基础。术语“小波”意思是小的波，它源于缩放窗口从较大信号中提取越来越小的波片段的过程。
Wavelet analysis begins with the consideration of a function known as the mother wavelet:
小波分析始于对一个称为母小波的函数的考虑：
ψ a , b ( t ) = 1 a ψ ( t − b a ) , ψ a , b ( t ) = 1 a ψ t − b a , psi_(a,b)(t)=(1)/(sqrta)psi((t-b)/(a)),\psi_{a, b}(t)=\frac{1}{\sqrt{a}} \psi\left(\frac{t-b}{a}\right),
where a ≠ 0 a ≠ 0 a!=0a \neq 0 and b b bb are real constants. The effect of these two parameters on the shape of the wavelet is illustrated in Figure 5.2; the a parameter controls scaling and b b bb denotes translation (previously denoted by τ τ tau\tau in Figure 5.1). Unlike Fourier analysis, and very much like Gábor transforms, a vast variety of mother wavelets can be constructed. In principle, the mother wavelet is designed to have certain properties that are somehow beneficial for a given problem. Depending on the application, different mother wavelets may be selected. Ultimately, the wavelet is simply another expansion basis for representing a given signal or function. Historically, the first wavelet was constructed by Haar in 1910 [124], so the concepts and ideas of wavelets are over a 
其中 a ≠ 0 a ≠ 0 a!=0a \neq 0 和 b b bb 为实常数。图 5.2 展示了这两个参数对小波形状的影响；参数 a 控制缩放， b b bb 表示平移（在图 5.1 中先前用 τ τ tau\tau 表示）。与傅里叶分析不同，并且与加博尔变换非常相似，可以构造各种各样的母小波。原则上，母小波被设计为具有某些对给定问题有益的特性。根据应用的不同，可以选择不同的母小波。最终，小波只是表示给定信号或函数的另一种展开基。历史上，第一个小波是由哈尔在 1910 年构造的[124]，所以小波的概念和思想已经存在了很长时间。
However, their use was not widespread until the mid-1980s.
然而，直到 20 世纪 80 年代中期，它们的使用才广泛起来。
The wavelet basis can be accessed via an integral transform of the form
小波基可以通过如下形式的积分变换来获取
( T f ) ( ω ) = ∫ t K ( t , ω ) f ( t ) d t ( T f ) ( ω ) = ∫ t   K ( t , ω ) f ( t ) d t (Tf)(omega)=int_(t)K(t,omega)f(t)dt(T f)(\omega)=\int_{t} K(t, \omega) f(t) d t
where K ( t , ω ) K ( t , ω ) K(t,omega)K(t, \omega) is the kernel of the transform. This is equivalent in principle to the Fourier transform, whose kernel consists of the oscillations given by K ( t , ω ) = K ( t , ω ) = K(t,omega)=K(t, \omega)= exp ⁡ ( − i ω t ) exp ⁡ ( − i ω t ) exp(-i omega t)\exp (-i \omega t). The key idea now is to define a transform that incorporates the mother wavelet as the kernel. We may define the continuous wavelet transform (CWT):
其中 K ( t , ω ) K ( t , ω ) K(t,omega)K(t, \omega) 是变换的核。这在原理上等同于傅里叶变换，其核由 K ( t , ω ) = K ( t , ω ) = K(t,omega)=K(t, \omega)= exp ⁡ ( − i ω t ) exp ⁡ ( − i ω t ) exp(-i omega t)\exp (-i \omega t) 给出的振荡组成。现在的关键思想是定义一种将母小波作为核的变换。我们可以定义连续小波变换（CWT）：
W ψ [ f ] ( a , b ) = ( f , ψ a , b ) = ∫ − ∞ ∞ f ( t ) ψ ¯ a , b ( t ) d t W ψ [ f ] ( a , b ) = f , ψ a , b = ∫ − ∞ ∞   f ( t ) ψ ¯ a , b ( t ) d t W_(psi)[f](a,b)=(f,psi_(a,b))=int_(-oo)^(oo)f(t) bar(psi)_(a,b)(t)dt\mathscr{W}_{\psi}[f](a, b)=\left(f, \psi_{a, b}\right)=\int_{-\infty}^{\infty} f(t) \bar{\psi}_{a, b}(t) d t
Much like the windowed Fourier transform, the CWT is a function of the dilation parameter a a aa and translation parameter b b bb. Parenthetically, a wavelet is admissible if the following property holds:
与加窗傅里叶变换非常相似，连续小波变换是伸缩参数 a a aa 和平移参数 b b bb 的函数。顺便提一下，如果满足以下性质，则小波是容许的：
C ψ = ∫ − ∞ ∞ | ψ ^ ( ω ) | 2 | ω | d ω < ∞ C ψ = ∫ − ∞ ∞   | ψ ^ ( ω ) | 2 | ω | d ω < ∞ C_(psi)=int_(-oo)^(oo)(|( hat(psi))(omega)|^(2))/(|omega|)d omega < ooC_{\psi}=\int_{-\infty}^{\infty} \frac{|\hat{\psi}(\omega)|^{2}}{|\omega|} d \omega<\infty
where the Fourier transform of the wavelet is defined by
小波的傅里叶变换由以下公式定义：
ψ ^ a , b = 1 | a | ∫ − ∞ ∞ e − i ω t ψ ( t − b a ) d t = 1 | a | e − i b ω ψ ^ ( a ω ) ψ ^ a , b = 1 | a | ∫ − ∞ ∞   e − i ω t ψ t − b a d t = 1 | a | e − i b ω ψ ^ ( a ω ) hat(psi)_(a,b)=(1)/(sqrt(|a|))int_(-oo)^(oo)e^(-i omega t)psi((t-b)/(a))dt=(1)/(sqrt(|a|))e^(-ib omega) hat(psi)(a omega)\hat{\psi}_{a, b}=\frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} e^{-i \omega t} \psi\left(\frac{t-b}{a}\right) d t=\frac{1}{\sqrt{|a|}} e^{-i b \omega} \hat{\psi}(a \omega)
Thus, provided the admissibility condition (5.6) is satisfied, the wavelet transform can be well defined.
因此，只要满足容许性条件(5.6)，小波变换就可以被很好地定义。
In the wavelet transform, the signal is first split up into a collection of smaller signals by translating the wavelet with the parameter b b bb over the entire time domain of the signal. Second, the same signal is processed at different frequency bands, or resolutions, by scaling the wavelet window with the parameter a a aa. The combination of translation and scaling allows for processing of the signals at different times and frequencies. Figure 5.3 is a schematic that illustrates the Gábor and wavelet transform concepts in the time-frequency domain. In this figure, the standard time series, Fourier transform, and windowed Fourier transform are represented along with the multiresolution concept of the wavelet transform. In particular, the box illustrating the wavelet transform shows the multiresolution concept in action. Starting with a large Fourier domain window, the entire frequency content is extracted. The time window is then scaled in half, leading to finer time resolution at the expense of worse frequency resolution. This process is continued until a desired time-frequency resolution is obtained. This simple figure is critical for understanding wavelet application to time-frequency analysis.
在小波变换中，首先通过在信号的整个时域上用参数 b b bb 平移小波，将信号分解为一组较小的信号。其次，通过用参数 a a aa 缩放小波窗口，在不同的频带或分辨率下处理相同的信号。平移和缩放的组合允许在不同的时间和频率下处理信号。图 5.3 是一个示意图，展示了时频域中的伽柏变换和小波变换概念。在该图中，标准时间序列、傅里叶变换和加窗傅里叶变换与小波变换的多分辨率概念一起表示。特别是，说明小波变换的方框展示了多分辨率概念的实际应用。从一个大的傅里叶域窗口开始，提取整个频率内容。然后将时间窗口缩小一半，以牺牲较差的频率分辨率为代价获得更精细的时间分辨率。这个过程一直持续，直到获得所需的时频分辨率。 这个简单的图形对于理解小波在时频分析中的应用至关重要。
As an example, consider one of the more common wavelets: the Mexican hat wavelet (Figure 5.2). This wavelet is essentially a second moment of a Gaussian in the frequency domain. The definitions of this wavelet and its transform are as follows:
例如，考虑一种较为常见的小波：墨西哥草帽小波（图 5.2）。这种小波本质上是频域中高斯分布的二阶矩。该小波及其变换的定义如下：
ψ ( t ) = ( 1 − t 2 ) e − t 2 / 2 = − d 2 d t 2 ( e − t 2 / 2 ) = ψ 1 , 0 ψ ^ ( ω ) = ψ ^ 1 , 0 ( ω ) = 2 π ω 2 e − ω 2 / 2 ψ ( t ) = 1 − t 2 e − t 2 / 2 = − d 2 d t 2 e − t 2 / 2 = ψ 1 , 0 ψ ^ ( ω ) = ψ ^ 1 , 0 ( ω ) = 2 π ω 2 e − ω 2 / 2 {:[psi(t)=(1-t^(2))e^(-t^(2)//2)=-(d^(2))/(dt^(2))(e^(-t^(2)//2))=psi_(1,0)],[ hat(psi)(omega)= hat(psi)_(1,0)(omega)=sqrt(2pi)omega^(2)e^(-omega^(2)//2)]:}\begin{aligned} & \psi(t)=\left(1-t^{2}\right) e^{-t^{2} / 2}=-\frac{d^{2}}{d t^{2}}\left(e^{-t^{2} / 2}\right)=\psi_{1,0} \\ & \hat{\psi}(\omega)=\hat{\psi}_{1,0}(\omega)=\sqrt{2 \pi} \omega^{2} e^{-\omega^{2} / 2} \end{aligned}
Figure 5.3. Graphical depiction of the difference between the time-series analysis, Fourier analysis, Gábor analysis, and wavelet analysis of a signal. The wavelet transform starts with a large Fourier domain window so that the entire frequency content is extracted. The time window is then scaled in half, leading to finer time resolution at the expense of worse frequency resolution. This process is continued until a desired time-frequency resolution is obtained.
图 5.3. 信号的时间序列分析、傅里叶分析、伽柏分析和小波分析之间差异的图形描述。小波变换从一个大的傅里叶域窗口开始，以便提取整个频率内容。然后将时间窗口缩小一半，以牺牲较差的频率分辨率为代价获得更精细的时间分辨率。这个过程一直持续，直到获得所需的时间频率分辨率。
The Mexican hat wavelet has excellent localization properties in both time and frequency due to the minimal time-bandwidth product of the Gaussian function. Figure 5.2 (top panels) shows the basic Mexican wavelet function ψ 1 , 0 ψ 1 , 0 psi_(1,0)\psi_{1,0} and its Fourier transform, both of which decay in t ( ω ) t ( ω ) t(omega)t(\omega) like exp ⁡ ( − t 2 ) ( exp ⁡ ( − ω 2 ) ) exp ⁡ − t 2 exp ⁡ − ω 2 exp(-t^(2))(exp(-omega^(2)))\exp \left(-t^{2}\right)\left(\exp \left(-\omega^{2}\right)\right). The Mexican hat wavelet can be dilated and translated easily, as is depicted in Figure 5.2 (bottom panel). Here three wavelets are depicted: ψ 1 , 0 , ψ 3 / 2 , − 3 ψ 1 , 0 , ψ 3 / 2 , − 3 psi_(1,0),psi_(3//2,-3)\psi_{1,0}, \psi_{3 / 2,-3}, and ψ 1 / 4 , 6 ψ 1 / 4 , 6 psi_(1//4,6)\psi_{1 / 4,6}, showing both scaling and translation of the wavelet.
由于高斯函数的时间带宽积最小，墨西哥帽小波在时间和频率上都具有出色的局部化特性。图 5.2（上图）展示了基本的墨西哥小波函数 ψ 1 , 0 ψ 1 , 0 psi_(1,0)\psi_{1,0} 及其傅里叶变换，两者在 t ( ω ) t ( ω ) t(omega)t(\omega) 中都像 exp ⁡ ( − t 2 ) ( exp ⁡ ( − ω 2 ) ) exp ⁡ − t 2 exp ⁡ − ω 2 exp(-t^(2))(exp(-omega^(2)))\exp \left(-t^{2}\right)\left(\exp \left(-\omega^{2}\right)\right) 一样衰减。墨西哥帽小波可以很容易地进行伸缩和平移，如图 5.2（下图）所示。这里描绘了三个小波： ψ 1 , 0 , ψ 3 / 2 , − 3 ψ 1 , 0 , ψ 3 / 2 , − 3 psi_(1,0),psi_(3//2,-3)\psi_{1,0}, \psi_{3 / 2,-3} 和 ψ 1 / 4 , 6 ψ 1 / 4 , 6 psi_(1//4,6)\psi_{1 / 4,6} ，展示了小波的伸缩和平移。

5.3 - Formulating mrDMD
5.3 - 制定 mrDMD

The mrDMD model is inspired by the observation that the slow and fast modes can be separated for such applications as foreground/background subtraction in video streams, as discussed in Chapter 4. The mrDMD approach is a recursive computation of DMD to remove low-frequency, or slowly varying, features from a given collection of snapshots. This process is illustrated in Figure 5.4. In DMD, the number of snapshots m m mm is chosen so that DMD modes provide an approximately full-rank approximation of the dynamics observed. Thus, m m mm is chosen so that all high- and low-frequency content is present. In mrDMD, m m mm is chosen initially to allow extraction of the lowest-frequency modes. At each resolution level, the slowest modes are removed, the time domain is divided into two segments with m / 2 m / 2 m//2m / 2 snapshots each, and DMD is once again performed on each  snapshot sequence. 
mrDMD 模型的灵感来自于这样一个观察结果：对于视频流中的前景/背景减法等应用，可以分离慢模式和快模式，如第 4 章所述。mrDMD 方法是对 DMD 进行递归计算，以从给定的快照集合中去除低频或缓慢变化的特征。此过程如图 5.4 所示。在 DMD 中，选择快照数量 m m mm ，以便 DMD 模式提供所观察到的动力学的近似满秩近似。因此，选择 m m mm ，以便同时存在所有高频和低频内容。在 mrDMD 中，最初选择 m m mm 以允许提取最低频率模式。在每个分辨率级别，去除最慢的模式，将时域分成两个段，每个段有 m / 2 m / 2 m//2m / 2 个快照，然后再次对每个 快照序列执行 DMD。
Figure 5.4. The m r D M D m r D M D mrDMDm r D M D approach takes successive samples of the data, initially with m m mm snapshots and decreasing by a factor of two at each resolution level. The DMD spectrum is shown in the middle panel, where there are m 1 m 1 m_(1)m_{1} (blue dots) slow-dynamic modes at the slowest level, m 2 m 2 m_(2)m_{2} (red) modes at the next level, and m 3 m 3 m_(3)m_{3} (green) modes at the fastest time scale shown. The shaded region represents the modes that are removed at that level. The bottom panel shows the wavelet-like time frequency decomposition of the data color-coded with the snapshots and DMD spectral representations [167].
图 5.4。 m r D M D m r D M D mrDMDm r D M D 方法对数据进行连续采样，最初有 m m mm 个快照，并在每个分辨率级别上以二分之一的因子递减。中间面板显示了 DMD 频谱，在最慢的级别上有 m 1 m 1 m_(1)m_{1} 个（蓝色点）慢动态模式，在下一个级别上有 m 2 m 2 m_(2)m_{2} 个（红色）模式，在所示的最快时间尺度上有 m 3 m 3 m_(3)m_{3} 个（绿色）模式。阴影区域表示在该级别上被去除的模式。底部面板显示了数据的小波状时频分解，用快照和 DMD 频谱表示进行颜色编码[167]。
 This recursive process continues until a desired termination. Because we are only interested in the lowest-frequency modes at each level, it is possible to subsample the snapshots, reducing m m mm and increasing the computational efficiency.
这个递归过程会一直持续，直到达到期望的终止条件。由于我们只对每一层的最低频率模式感兴趣，因此可以对快照进行子采样，减少 m m mm 并提高计算效率。
Mathematically, mrDMD separates the DMD approximate solution (1.24) at the first level as follows: 
在数学上，mrDMD 按如下方式在第一级分离 DMD 近似解(1.24)：
x miDMD ( t ) = ∑ k = 1 m b k ( 0 ) ϕ k ( 1 ) exp ⁡ ( ω k t ) = ∑ k = 1 m 1 b k ( 0 ) ϕ k ( 1 ) exp ⁡ ( ω k t ) + ∑ k = m 1 + 1 m b k ( 0 ) ϕ k ( 1 ) exp ⁡ ( ω k t ) , (flow modes) (fast modes) x miDMD  ( t ) = ∑ k = 1 m   b k ( 0 ) ϕ k ( 1 ) exp ⁡ ω k t = ∑ k = 1 m 1   b k ( 0 ) ϕ k ( 1 ) exp ⁡ ω k t + ∑ k = m 1 + 1 m   b k ( 0 ) ϕ k ( 1 ) exp ⁡ ω k t ,  (flow modes)   (fast modes)  {:[x_("miDMD ")(t)=sum_(k=1)^(m)b_(k)(0)phi_(k)^((1))exp(omega_(k)t)],[=sum_(k=1)^(m_(1))b_(k)(0)phi_(k)^((1))exp(omega_(k)t)+sum_(k=m_(1)+1)^(m)b_(k)(0)phi_(k)^((1))exp(omega_(k)t)","],[quad" (flow modes) "quad" (fast modes) "]:}\begin{aligned} \mathbf{x}_{\text {miDMD }}(t)= & \sum_{k=1}^{m} b_{k}(0) \boldsymbol{\phi}_{k}^{(1)} \exp \left(\omega_{k} t\right) \\ = & \sum_{k=1}^{m_{1}} b_{k}(0) \boldsymbol{\phi}_{k}^{(1)} \exp \left(\omega_{k} t\right)+\sum_{k=m_{1}+1}^{m} b_{k}(0) \boldsymbol{\phi}_{k}^{(1)} \exp \left(\omega_{k} t\right), \\ & \quad \text { (flow modes) } \quad \text { (fast modes) } \end{aligned}
where the ϕ k ( 1 ) ϕ k ( 1 ) phi_(k)^((1))\boldsymbol{\phi}_{k}^{(1)} represent the DMD modes computed from the full m m mm snapshots.
其中 ϕ k ( 1 ) ϕ k ( 1 ) phi_(k)^((1))\boldsymbol{\phi}_{k}^{(1)} 表示从完整的 m m mm 快照计算得到的DMD模式。
The first sum in expression (5.9) represents the slow-mode dynamics, whereas the second sum is everything else. Thus, the second sum can be computed to yield the matrix
表达式(5.9)中的第一项表示慢模动力学，而第二项则是其他所有项。因此，可以计算第二项以得到矩阵
X m / 2 = ∑ k = m 1 + 1 m b k ( 0 ) ϕ k ( 1 ) exp ⁡ ( ω k t ) X m / 2 = ∑ k = m 1 + 1 m   b k ( 0 ) ϕ k ( 1 ) exp ⁡ ω k t X_(m//2)=sum_(k=m_(1)+1)^(m)b_(k)(0)phi_(k)^((1))exp(omega_(k)t)\mathbf{X}_{m / 2}=\sum_{k=m_{1}+1}^{m} b_{k}(0) \boldsymbol{\phi}_{k}^{(1)} \exp \left(\omega_{k} t\right)
The DMD analysis outlined in the previous section can now be performed once again on the data matrix X m / 2 X m / 2 X_(m//2)\mathbf{X}_{m / 2}. However, the matrix X m / 2 X m / 2 X_(m//2)\mathbf{X}_{m / 2} is now separated into two matrices,
上一节中概述的 DMD 分析现在可以再次对数据矩阵 X m / 2 X m / 2 X_(m//2)\mathbf{X}_{m / 2} 进行。然而，矩阵 X m / 2 X m / 2 X_(m//2)\mathbf{X}_{m / 2} 现在被分成了两个矩阵，
X m / 2 = X m / 2 ( 1 ) + X m / 2 ( 2 ) X m / 2 = X m / 2 ( 1 ) + X m / 2 ( 2 ) X_(m//2)=X_(m//2)^((1))+X_(m//2)^((2))\mathbf{X}_{m / 2}=\mathbf{X}_{m / 2}^{(1)}+\mathbf{X}_{m / 2}^{(2)}
where the first matrix contains the first m / 2 m / 2 m//2m / 2 snapshots and the second matrix contains the remaining m / 2 m / 2 m//2m / 2 snapshots. The m 2 m 2 m_(2)m_{2} slow-DMD modes at this level are given by ϕ k ( 2 ) ϕ k ( 2 ) phi_(k)^((2))\boldsymbol{\phi}_{k}^{(2)}, where they are computed separately in the first or second interval of snapshots.
其中第一个矩阵包含前 m / 2 m / 2 m//2m / 2 个快照，第二个矩阵包含其余的 m / 2 m / 2 m//2m / 2 个快照。此级别下的 m 2 m 2 m_(2)m_{2} 个慢 DMD 模式由 ϕ k ( 2 ) ϕ k ( 2 ) phi_(k)^((2))\boldsymbol{\phi}_{k}^{(2)} 给出，其中它们是在快照的第一个或第二个区间中分别计算的。
The mrDMD process works by recursively removing slow-frequency components and building the new matrices X m / 2 , X m / 4 , X m / 8 , … X m / 2 , X m / 4 , X m / 8 , … X_(m//2),X_(m//4),X_(m//8),dots\mathbf{X}_{m / 2}, \mathbf{X}_{m / 4}, \mathbf{X}_{m / 8}, \ldots until a desired/prescribed multiresolution decomposition has been achieved. The approximate DMD solution can then be constructed as follows:
mrDMD 过程的工作方式是通过递归地去除低频分量并构建新的矩阵 X m / 2 , X m / 4 , X m / 8 , … X m / 2 , X m / 4 , X m / 8 , … X_(m//2),X_(m//4),X_(m//8),dots\mathbf{X}_{m / 2}, \mathbf{X}_{m / 4}, \mathbf{X}_{m / 8}, \ldots ，直到实现所需/规定的多分辨率分解。然后，可以按如下方式构建近似 DMD 解：
x mrDMD ( t ) = ∑ k = 1 m 1 b k ( 1 ) ϕ k ( 1 ) exp ⁡ ( ω k ( 1 ) t ) + ∑ k = 1 m 2 b k ( 2 ) ϕ k ( 2 ) exp ⁡ ( ω k ( 2 ) t ) + ∑ k = 1 m 3 b k ( 3 ) ϕ k ( 3 ) exp ⁡ ( ω k ( 3 ) t ) + ⋯ x mrDMD  ( t ) = ∑ k = 1 m 1   b k ( 1 ) ϕ k ( 1 ) exp ⁡ ω k ( 1 ) t + ∑ k = 1 m 2   b k ( 2 ) ϕ k ( 2 ) exp ⁡ ω k ( 2 ) t + ∑ k = 1 m 3   b k ( 3 ) ϕ k ( 3 ) exp ⁡ ω k ( 3 ) t + ⋯ {:[x_("mrDMD ")(t)=sum_(k=1)^(m_(1))b_(k)^((1))phi_(k)^((1))exp(omega_(k)^((1))t)+sum_(k=1)^(m_(2))b_(k)^((2))phi_(k)^((2))exp(omega_(k)^((2))t)],[+sum_(k=1)^(m_(3))b_(k)^((3))phi_(k)^((3))exp(omega_(k)^((3))t)+cdots]:}\begin{aligned} \mathbf{x}_{\text {mrDMD }}(t)= & \sum_{k=1}^{m_{1}} b_{k}^{(1)} \boldsymbol{\phi}_{k}^{(1)} \exp \left(\omega_{k}^{(1)} t\right)+\sum_{k=1}^{m_{2}} b_{k}^{(2)} \boldsymbol{\phi}_{k}^{(2)} \exp \left(\omega_{k}^{(2)} t\right) \\ & +\sum_{k=1}^{m_{3}} b_{k}^{(3)} \boldsymbol{\phi}_{k}^{(3)} \exp \left(\omega_{k}^{(3)} t\right)+\cdots \end{aligned}
where at the evaluation time t t tt, the correct modes from the sampling window are selected at each level of the decomposition. Specifically, ϕ k ( ℓ ) ϕ k ( ℓ ) phi_(k)^((ℓ))\boldsymbol{\phi}_{k}^{(\ell)} and ω k ( ℓ ) ω k ( ℓ ) omega_(k)^((ℓ))\omega_{k}^{(\ell)} are the DMD modes and DMD eigenvalues at the ℓ ℓ ℓ\ell th level of decomposition, the b k ( ℓ ) b k ( ℓ ) b_(k)^((ℓ))b_{k}^{(\ell)} are the initial projections of the data onto the time interval of interest, and the m k m k m_(k)m_{k} are the number of slow modes retained at each level. The advantage of this method is apparent: different spatiotemporal DMD modes are used to represent key multiresolution features. There is not a single set of modes that dominates the SVD and potentially obscures features at other time scales.
在评估时刻 t t tt ，从采样窗口中选择分解各层级的正确模式。具体而言， ϕ k ( ℓ ) ϕ k ( ℓ ) phi_(k)^((ℓ))\boldsymbol{\phi}_{k}^{(\ell)} 和 ω k ( ℓ ) ω k ( ℓ ) omega_(k)^((ℓ))\omega_{k}^{(\ell)} 是分解的 ℓ ℓ ℓ\ell 层级处的 DMD 模式和 DMD 特征值， b k ( ℓ ) b k ( ℓ ) b_(k)^((ℓ))b_{k}^{(\ell)} 是数据在感兴趣时间间隔上的初始投影， m k m k m_(k)m_{k} 是各层级保留的慢模式数量。此方法的优势显而易见：使用不同的时空 DMD 模式来表示关键的多分辨率特征。不存在一组主导奇异值分解且可能掩盖其他时间尺度特征的模式。
Figure 5.4 illustrates the mrDMD process schematically. In the figure, a three-level decomposition is performed, with the slowest scale represented in blue (eigenvalues and snapshots), the midscale in red, and the fast scale in green. The connection to multiresolution wavelet analysis is also evident from the bottom panels, as one can see that the mrDMD method successively pulls out time-frequency information in a principled way.
图 5.4 示意性地展示了 mrDMD 过程。在图中，进行了三级分解，最慢的尺度用蓝色表示（特征值和快照），中间尺度用红色表示，最快的尺度用绿色表示。从底部的图中也可以明显看出与多分辨率小波分析的联系，因为可以看到 mrDMD 方法以一种有原则的方式依次提取时频信息。
The sampling strategy and algorithm can be further optimized since only the slow modes at each decomposition level need to be accurately computed. Thus, one can modify the algorithm so the sampling rate increases as the decomposition proceeds from one level to the next. This assures that the lowest levels of the mrDMD are not highly sampled, since the cost of the SVD would be greatly increased by such a fine sampling rate.
由于在每个分解级别仅需精确计算慢模式，因此采样策略和算法可以进一步优化。因此，可以修改算法，使得随着分解从一个级别推进到下一个级别，采样率增加。这确保了 mrDMD 的最低级别不会被高度采样，因为如此精细的采样率会大大增加奇异值分解（SVD）的成本。

5.3.1 • Formal mrDMD expansion
5.3.1 • 形式上的 mrDMD 展开

The solution (5.12) can be made more precise. Specifically, one must account for the number of levels ( L ) ( L ) (L)(L) of the decomposition, the number of time bins ( J ) ( J ) (J)(J) for each level, and the number of modes retained at each level ( m L ) m L (m_(L))\left(m_{L}\right). This can be easily seen in Figure 5.4. Thus, the solution is parameterized by the following three indices:
解（5.12）可以更精确。具体来说，必须考虑分解的层数 ( L ) ( L ) (L)(L) 、每层的时间区间数 ( J ) ( J ) (J)(J) 以及每层保留的模式数 ( m L ) m L (m_(L))\left(m_{L}\right) 。这在图 5.4 中很容易看出。因此，该解由以下三个指标参数化：
ℓ = 1 , 2 , … , L : number of decomposition levels, j = 1 , 2 , … , J : number time bins per level ( J = 2 ( ℓ − 1 ) ) , k = 1 , 2 , … , m ℓ : number of modes extracted at level L . ℓ = 1 , 2 , … , L :  number of decomposition levels,  j = 1 , 2 , … , J :  number time bins per level  J = 2 ( ℓ − 1 ) , k = 1 , 2 , … , m ℓ :  number of modes extracted at level  L . {:[ℓ=1","2","dots","L:" number of decomposition levels, "],[j=1","2","dots","J:" number time bins per level "(J=2^((ℓ-1)))","],[k=1","2","dots","m_(ℓ):" number of modes extracted at level "L.]:}\begin{aligned} & \ell=1,2, \ldots, L: \text { number of decomposition levels, } \\ & j=1,2, \ldots, J: \text { number time bins per level }\left(J=2^{(\ell-1)}\right), \\ & k=1,2, \ldots, m_{\ell}: \text { number of modes extracted at level } L . \end{aligned}
To formally define the series solution for x mPDMD ( t ) x mPDMD  ( t ) x_("mPDMD ")(t)\mathbf{x}_{\text {mPDMD }}(t), we define the indicator function
为了正式定义 x mPDMD ( t ) x mPDMD  ( t ) x_("mPDMD ")(t)\mathbf{x}_{\text {mPDMD }}(t) 的级数解，我们定义指示函数
f ℓ , j ( t ) = { 1 , t ∈ [ t j , t j + 1 ] 0 , elsewhere , with j = 1 , 2 , … , J and J = 2 ( ℓ − 1 ) f ℓ , j ( t ) = 1 ,      t ∈ t j , t j + 1 0 ,       elsewhere  ,  with  j = 1 , 2 , … , J  and  J = 2 ( ℓ − 1 ) f^(ℓ,j)(t)={[1",",t in[t_(j),t_(j+1)]],[0","," elsewhere "]," with "j=1,2,dots,J" and "J=2^((ℓ-1)):}f^{\ell, j}(t)=\left\{\begin{array}{ll} 1, & t \in\left[t_{j}, t_{j+1}\right] \\ 0, & \text { elsewhere } \end{array}, \text { with } j=1,2, \ldots, J \text { and } J=2^{(\ell-1)}\right.
which is only nonzero in the interval, or time bin, associated with the value of j j jj.
它仅在与 j j jj 的值相关联的区间或时间间隔内非零。
The three indices and indicator function (5.14) give the mrDMD solution expansion
这三个指标和指示函数（5.14）给出了 mrDMD 解的展开式
x mrDMD ( t ) = ∑ ℓ = 1 L ∑ j = 1 J ∑ k = 1 m L f ℓ , j ( t ) b k ( ℓ , j ) ϕ k ( ℓ , j ) exp ⁡ ( ω k ( ℓ , j ) t ) . x mrDMD ( t ) = ∑ ℓ = 1 L   ∑ j = 1 J   ∑ k = 1 m L   f ℓ , j ( t ) b k ( ℓ , j ) ϕ k ( ℓ , j ) exp ⁡ ω k ( ℓ , j ) t . x_(mrDMD)(t)=sum_(ℓ=1)^(L)sum_(j=1)^(J)sum_(k=1)^(m_(L))f^(ℓ,j)(t)b_(k)^((ℓ,j))phi_(k)^((ℓ,j))exp(omega_(k)^((ℓ,j))t).\mathbf{x}_{\operatorname{mrDMD}}(t)=\sum_{\ell=1}^{L} \sum_{j=1}^{J} \sum_{k=1}^{m_{L}} f^{\ell, j}(t) b_{k}^{(\ell, j)} \boldsymbol{\phi}_{k}^{(\ell, j)} \exp \left(\omega_{k}^{(\ell, j)} t\right) .
This concise definition of the mrDMD solution includes the information on the level, time bin location, and number of modes extracted. Figure 5.5 demonstrates mrDMD in terms of the solution (5.15). In particular, each mode is represented in its respective time bin and level. An alternative interpretation of this solution is that it yields the least-square fit, at each level ℓ ℓ ℓ\ell of the decomposition, to the discrete-time linear dynamical system
mrDMD 解的这个简洁定义包含了关于水平、时间间隔位置以及提取的模式数量的信息。图 5.5 根据解 (5.15) 展示了 mrDMD。特别地，每个模式在其各自的时间间隔和水平中表示。这个解的另一种解释是，在分解的每个水平 ℓ ℓ ℓ\ell 处，它产生对离散时间线性动态系统的最小二乘拟合
x t + 1 ( ℓ , j ) = A ( ℓ , j ) x t ( ℓ , j ) , x t + 1 ( ℓ , j ) = A ( ℓ , j ) x t ( ℓ , j ) , x_(t+1)^((ℓ,j))=A^((ℓ,j))x_(t)^((ℓ,j)),\mathbf{x}_{t+1}^{(\ell, j)}=\mathbf{A}^{(\ell, j)} \mathbf{x}_{t}^{(\ell, j)},
where the matrix A ( ℓ , j ) A ( ℓ , j ) A^((ℓ,j))\mathbf{A}^{(\ell, j)} captures the dynamics in a given time bin j j jj at level ℓ ℓ ℓ\ell.
其中矩阵 A ( ℓ , j ) A ( ℓ , j ) A^((ℓ,j))\mathbf{A}^{(\ell, j)} 在级别 ℓ ℓ ℓ\ell 的给定时间间隔 j j jj 内捕获动态。
The indicator function f ℓ , j ( t ) f ℓ , j ( t ) f^(ℓ,j)(t)f^{\ell, j}(t) acts as a sifting function for each time bin. Interestingly, this function acts as the Gábor window of a windowed Fourier transform [166].
指示函数 f ℓ , j ( t ) f ℓ , j ( t ) f^(ℓ,j)(t)f^{\ell, j}(t) 对每个时间区间起到筛选函数的作用。有趣的是，该函数充当加窗傅里叶变换的伽柏窗[166]。
Since our sampling bin has a hard cutoff of the time series, it may introduce some artificial high-frequency oscillations. Time-series analysis, and wavelets in particular, introduce various functional forms that can be used in an advantageous way. Thinking more broadly, one can imagine using wavelet functions for the sifting operation, thus allowing the time function f ℓ , j ( t ) f ℓ , j ( t ) f^(ℓ,j)(t)f^{\ell, j}(t) to take the form of one of the many potential wavelet bases, e.g., Haar, Daubechies, Mexican hat, etc. [76, 78]. For the present, we simply use the sifting function introduced in (5.14).
由于我们的采样区间对时间序列有硬截断，它可能会引入一些人为的高频振荡。时间序列分析，特别是小波分析，引入了各种可以以有利方式使用的函数形式。从更广泛的角度考虑，可以想象使用小波函数进行筛选操作，从而使时间函数 f ℓ , j ( t ) f ℓ , j ( t ) f^(ℓ,j)(t)f^{\ell, j}(t) 采用许多潜在小波基之一的形式，例如哈尔小波、多贝西小波、墨西哥草帽小波等[76, 78]。目前，我们仅使用(5.14)中引入的筛选函数。

5.4 - The mrDMD algorithm
5.4 - mrDMD 算法

The mrDMD is a recursive, hierarchical application of the basic DMD algorithm introduced in Chapter 1. However, the sampling windows (snapshot collection) are now
mrDMD 是第 1 章中介绍的基本 DMD 算法的递归分层应用。然而，采样窗口（快照收集）现在
Figure 5.5. Illustration of the m r D M D m r D M D mrDMDm r D M D bierarchy. Represented are the modes ϕ k ℓ , j ϕ k ℓ , j phi_(k)^(ℓ,j)\boldsymbol{\phi}_{k}^{\ell, j} and their position in the decomposition structure. The triplet of integer values ℓ , j ℓ , j ℓ,j\ell, j, and k k kk uniquely expresses the time level, bin, and mode of the decomposition.
图 5.5. m r D M D m r D M D mrDMDm r D M D 层次结构的图示。表示了 ϕ k ℓ , j ϕ k ℓ , j phi_(k)^(ℓ,j)\boldsymbol{\phi}_{k}^{\ell, j} 模式及其在分解结构中的位置。整数值三元组 ℓ , j ℓ , j ℓ,j\ell, j 和 k k kk 唯一地表示分解的时间级别、区间和模式。
adjusted and only the slowest modes are extracted at each level of the decomposition. We highlight again the DMD algorithm but now in the context of the mrDMD innovation. As before, when the state dimension n n nn is large, the matrix A may be intractable to analyze directly. Instead, DMD circumvents the eigendecomposition of A A A\mathbf{A} by considering a rank-reduced representation in terms of a POD-projected matrix Ã. The mrDMD algorithm proceeds as follows:
进行调整，并且仅在分解的每个级别提取最慢的模式。我们再次强调 DMD 算法，但现在是在 mrDMD 创新的背景下。和以前一样，当状态维度 n n nn 很大时，直接分析矩阵 A 可能很棘手。相反，DMD 通过考虑基于 POD 投影矩阵Ã的降秩表示来规避 A A A\mathbf{A} 的特征分解。mrDMD 算法的步骤如下：
  1. Consider a sampling window in time at level ℓ ℓ ℓ\ell and time bin j j jj of the decomposition. In this sampling window, construct the data matrices X X X\mathbf{X} and X ′ X ′ X^(')\mathbf{X}^{\prime}.
    考虑在分解的 ℓ ℓ ℓ\ell 级别的时间和 j j jj 时间仓中的一个采样窗口。在这个采样窗口中，构建数据矩阵 X X X\mathbf{X} 和 X ′ X ′ X^(')\mathbf{X}^{\prime} 。
  2. Take the SVD of X X X\mathbf{X} :
    对 X X X\mathbf{X} 进行奇异值分解：
X = U Σ V ∗ X = U Σ V ∗ X=USigmaV^(**)\mathbf{X}=\mathbf{U} \Sigma \mathbf{V}^{*}
where ∗ ∗ *** denotes the conjugate transpose, U ∈ C n × r , Σ ∈ C r × r U ∈ C n × r , Σ ∈ C r × r UinC^(n xx r),Sigma inC^(r xx r)\mathbf{U} \in \mathbb{C}^{n \times r}, \boldsymbol{\Sigma} \in \mathbb{C}^{r \times r}, and V ∈ C m × r V ∈ C m × r VinC^(m xx r)\mathbf{V} \in \mathbb{C}^{m \times r}. Here r r rr is the rank of the reduced SVD approximation to X X X\mathbf{X}. The left singular vectors U U U\mathbf{U} are POD modes.
其中 ∗ ∗ *** 表示共轭转置， U ∈ C n × r , Σ ∈ C r × r U ∈ C n × r , Σ ∈ C r × r UinC^(n xx r),Sigma inC^(r xx r)\mathbf{U} \in \mathbb{C}^{n \times r}, \boldsymbol{\Sigma} \in \mathbb{C}^{r \times r} 和 V ∈ C m × r V ∈ C m × r VinC^(m xx r)\mathbf{V} \in \mathbb{C}^{m \times r} 。这里 r r rr 是对 X X X\mathbf{X} 的约简奇异值分解（SVD）近似的秩。左奇异向量 U U U\mathbf{U} 是本征正交分解（POD）模式。
3. Next, compute A ~ ( ℓ , j ) A ~ ( ℓ , j ) tilde(A)^((ℓ,j))\tilde{\mathbf{A}}^{(\ell, j)}, the r × r r × r r xx rr \times r projection of the full matrix A ( ℓ , j ) A ( ℓ , j ) A^((ℓ,j))\mathbf{A}^{(\ell, j)} onto POD modes:
3. 接下来，计算 A ~ ( ℓ , j ) A ~ ( ℓ , j ) tilde(A)^((ℓ,j))\tilde{\mathbf{A}}^{(\ell, j)} ，即完整矩阵 A ( ℓ , j ) A ( ℓ , j ) A^((ℓ,j))\mathbf{A}^{(\ell, j)} 在本征正交分解（POD）模式上的 r × r r × r r xx rr \times r 投影：
⇒ A ( ℓ , j ) = X ′ V Σ − 1 U ∗ A ~ ( ℓ , j ) = U ∗ A ( ℓ , j ) U = U ∗ X ′ V Σ − 1 . ⇒ A ( ℓ , j ) = X ′ V Σ − 1 U ∗ A ~ ( ℓ , j ) = U ∗ A ( ℓ , j ) U = U ∗ X ′ V Σ − 1 . =>quad{:[A^((ℓ,j))=X^(')VSigma^(-1)U^(**)],[ tilde(A)^((ℓ,j))=U^(**)A^((ℓ,j))U=U^(**)X^(')VSigma^(-1).]:}\Rightarrow \quad \begin{aligned} & \mathbf{A}^{(\ell, j)}=\mathbf{X}^{\prime} \mathbf{V} \boldsymbol{\Sigma}^{-1} \mathbf{U}^{*} \\ & \tilde{\mathbf{A}}^{(\ell, j)}=\mathbf{U}^{*} \mathbf{A}^{(\ell, j)} \mathbf{U}=\mathbf{U}^{*} \mathbf{X}^{\prime} \mathbf{V} \boldsymbol{\Sigma}^{-1} . \end{aligned}
  1. Compute the eigendecomposition of A ~ ( ℓ , j ) A ~ ( ℓ , j ) tilde(A)^((ℓ,j))\tilde{\mathbf{A}}^{(\ell, j)} :
    计算 A ~ ( ℓ , j ) A ~ ( ℓ , j ) tilde(A)^((ℓ,j))\tilde{\mathbf{A}}^{(\ell, j)} 的特征分解：
A ~ ( ℓ , j ) W = W Λ A ~ ( ℓ , j ) W = W Λ tilde(A)^((ℓ,j))W=WLambda\tilde{\mathbf{A}}^{(\ell, j)} \mathbf{W}=\mathbf{W} \boldsymbol{\Lambda}
where the columns of W W W\mathbf{W} are eigenvectors and Λ Λ Lambda\boldsymbol{\Lambda} is a diagonal matrix containing the corresponding eigenvalues λ k λ k lambda_(k)\lambda_{k}.
其中 W W W\mathbf{W} 的列是特征向量， Λ Λ Lambda\boldsymbol{\Lambda} 是一个对角矩阵，包含相应的特征值 λ k λ k lambda_(k)\lambda_{k} 。
5. Identify and collect the slow eigenvalues, if any, where ‖ λ j ‖ < ρ λ j < ρ ||lambda_(j)|| < rho\left\|\lambda_{j}\right\|<\rho, and use them to construct the slow modes at this level of decomposition.
5. 识别并收集慢特征值（如果存在，其中 ‖ λ j ‖ < ρ λ j < ρ ||lambda_(j)|| < rho\left\|\lambda_{j}\right\|<\rho ），并使用它们在此分解级别构建慢模式。
6. Finally, reconstruct the eigendecomposition of A ( ℓ , j ) A ( ℓ , j ) A^((ℓ,j))\mathbf{A}^{(\ell, j)} from W W W\mathbf{W} and Λ Λ Lambda\boldsymbol{\Lambda}. In particular, the eigenvalues of A ( ℓ , j ) A ( ℓ , j ) A^((ℓ,j))\mathbf{A}^{(\ell, j)} are given by Λ Λ Lambda\boldsymbol{\Lambda} and the eigenvectors of A ( ℓ , j ) A ( ℓ , j ) A^((ℓ,j))\mathbf{A}^{(\ell, j)} (DMD modes) are given by the columns of Φ ( ℓ , j ) Φ ( ℓ , j ) Phi^((ℓ,j))\boldsymbol{\Phi}^{(\ell, j)} :
6. 最后，根据 W W W\mathbf{W} 和 Λ Λ Lambda\boldsymbol{\Lambda} 重构 A ( ℓ , j ) A ( ℓ , j ) A^((ℓ,j))\mathbf{A}^{(\ell, j)} 的特征分解。具体而言， A ( ℓ , j ) A ( ℓ , j ) A^((ℓ,j))\mathbf{A}^{(\ell, j)} 的特征值由 Λ Λ Lambda\boldsymbol{\Lambda} 给出， A ( ℓ , j ) A ( ℓ , j ) A^((ℓ,j))\mathbf{A}^{(\ell, j)} （DMD 模式）的特征向量由 Φ ( ℓ , j ) Φ ( ℓ , j ) Phi^((ℓ,j))\boldsymbol{\Phi}^{(\ell, j)} 的列给出：
Φ ( ℓ , j ) = X ′ V Σ − 1 W Φ ( ℓ , j ) = X ′ V Σ − 1 W Phi^((ℓ,j))=X^(')VSigma^(-1)W\boldsymbol{\Phi}^{(\ell, j)}=\mathbf{X}^{\prime} \mathbf{V} \Sigma^{-1} \mathbf{W}
The power at level ℓ ℓ ℓ\ell and time bin j j jj may now be computed, following the approach described in Chapter 8.
现在可以按照第 8 章中描述的方法来计算 ℓ ℓ ℓ\ell 级和 j j jj 时间区间的功率。
7. Divide the original sampling window in half, and repeat all of the above steps for each of the first and second halves of the sampling window at level ℓ + 1 ℓ + 1 ℓ+1\ell+1.
7. 将原始采样窗口分成两半，并在 ℓ + 1 ℓ + 1 ℓ+1\ell+1 级别对采样窗口的前半部分和后半部分分别重复上述所有步骤。

5.4.1 - Parameters of the mrDMD algorithm
5.4.1 - mrDMD 算法的参数

This algorithm has a few parameters. In step 1, at each level ℓ ℓ ℓ\ell, the number of snapshots in each sampling window may be a subsample of the full-resolution data based on the desired slow eigenvalue threshold in step 5. This subsampling enables a substantial increase in efficiency of execution, especially at lower levels of recursion with larger windows, without sacrificing the ability to extract the lowest-amplitude (i.e., slowest) modes. In step 2, the rank r r rr of the reduced SVD approximation may be chosen to exploit low-rank truncation of the data. As noted in Chapters 1 and 8 , this r r rr may be chosen in a principled way.
该算法有几个参数。在步骤 1 中，在每个级别 ℓ ℓ ℓ\ell ，每个采样窗口中的快照数量可以是基于步骤 5 中所需的慢特征值阈值的全分辨率数据的子样本。这种子采样能够大幅提高执行效率，特别是在具有较大窗口的较低递归级别，而不会牺牲提取最低幅度（即最慢）模式的能力。在步骤 2 中，可以选择简化奇异值分解（SVD）近似的秩 r r rr ，以利用数据的低秩截断。如第 1 章和第 8 章所述，可以以一种有原则的方式选择此 r r rr 。
In step 5, the parameter ρ ρ rho\rho is selected to extract only slow modes. There is some freedom in the choice of ρ ρ rho\rho, and this value may be selected based on the duration of the sampling window. To be specific, the implementation in the example code in Algorithm 5.3 below chooses ρ ρ rho\rho to select modes with fewer than two cycles of oscillations within the sampling window.
在步骤 5 中，选择参数 ρ ρ rho\rho 仅提取慢模式。在 ρ ρ rho\rho 的选择上有一定的自由度，该值可根据采样窗口的持续时间来选择。具体而言，下面算法 5.3 中的示例代码实现选择 ρ ρ rho\rho 来选择在采样窗口内振荡周期少于两个的模式。
Note that the above algorithm also allows construction of the best-fit dynamical system at each level (5.16). After decomposing at a specified number of levels, the solution may be reconstructed using (5.15).
请注意，上述算法还允许在每个级别构建最佳拟合动力系统(5.16)。在指定数量的级别上进行分解后，可以使用(5.15)重建解。

5.5 - Example code and decomposition
5.5 - 示例代码与分解

The first example we consider for application of mrDMD is illustrated in Figure 5.6. This example consists of a synthetic movie of m = 1000 m = 1000 m=1000m=1000 frames where each frame has n = 80 × 80 = 6400 n = 80 × 80 = 6400 n=80 xx80=6400n=80 \times 80=6400 pixels, sampled at d t = 0.01 sec d t = 0.01 sec dt=0.01secd t=0.01 \mathrm{sec}. The three underlying modes of the data have different, overlapping spatial distributions. Mode 1 oscillates at 5.55 Hz for the first half of the movie. Mode 2 oscillates at 0.9 Hz from seconds 3 to 7 of the movie. Mode 3 has a slow, stationary oscillation of 0.15 Hz for the entire duration of the movie. For this demonstration, Gaussian white sensor noise was added to every pixel so that the signal-to-noise ratio is approximately 3 . The following section of code constructs an instance of this example data.
我们考虑的第一个应用 mrDMD 的示例如图 5.6 所示。此示例由一个包含 m = 1000 m = 1000 m=1000m=1000 帧的合成视频组成，其中每一帧有 n = 80 × 80 = 6400 n = 80 × 80 = 6400 n=80 xx80=6400n=80 \times 80=6400 个像素，采样频率为 d t = 0.01 sec d t = 0.01 sec dt=0.01secd t=0.01 \mathrm{sec} 。数据的三种潜在模式具有不同的、重叠的空间分布。模式 1 在视频的前半部分以 5.55 赫兹振荡。模式 2 在视频的第 3 秒到第 7 秒以 0.9 赫兹振荡。模式 3 在整个视频持续时间内有一个缓慢的、固定的 0.15 赫兹振荡。为了进行这个演示，高斯白噪声被添加到每个像素上，使得信噪比约为 3。以下代码段构建了这个示例数据的一个实例。

ALGORITHM 5.1. Construct example movie (mrDMD_demo.m).
算法 5.1。构建示例影片（mrDMD_demo.m）。

% parameters
nx = 80; % horizontal pixels
ny = 80; % vertical pixels
n = nx * ny;
T = 10; % sec
dt = 0.01;
t = dt:dt:T;
sig = 0.1; % magnitude of gaussian noise added
% 3 underlying modes
[Xgrid, Ygrid] = meshgrid(1:nx, 1:ny);
mode1.u = exp(-((Xgrid-40).^2/250+(Ygrid-40).^2/250));
mode1.f = 5.55; % cycles/sec
mode1.A = 1;
mode1.lambda = exp(1i * mode1.f*2*pi*dt);
mode1.range = [0, 5];
mode2.u = zeros(nx, ny);
mode2.u(nx-40:nx-10, ny-40:ny-10) = 1;
mode2.f = 0.9; % cycles/sec
mode2.A = 1;
mode2.lambda = exp(1i * mode2.f*2*pi*dt);
mode2.range = [3, 7];
mode3.u = zeros(nx, ny);
mode3.u(1:nx-20, 1:ny-20) = 1;
mode3.f = 0.15; % cycles/sec
mode3.A = 0.5;
mode3.lambda = exp(1i * mode3.f*2*pi*dt);
mode3.range = [0, T];
modes(1) = mode1;
modes(2) = mode2;
modes(3) = mode3;
% make the movie
Xclean = zeros(n, numel(T));
for ti = 1:numel(t),
    Snap = zeros(nx, ny);
    for mi = 1:numel(modes),
        mymode = modes(mi);
        if ti > round(mymode.range(1)/dt) && ...
            ti < round(mymode.range(2)/dt),
                Snap = Snap + mymode.A * mymode.u * ...
                    real (mymode.lambda^ti);
        end;
    end;
    Xclean(:, ti) = Snap(:);
end;
% add noise
Noise = sig * randn(size(Xclean));
Figure 5.6. Modes of example data to demonstrate m r D M D m r D M D mrDMDm r D M D. The data is n = 6400 n = 6400 n=6400n=6400 pixels by m = 1000 m = 1000 m=1000m=1000 snapshots, sampled at d t = 0.01 d t = 0.01 dt=0.01d t=0.01 sec. The three modes have overlapping spatial distributions and oscillate at 5.55 , 0.9 5.55 , 0.9 5.55,0.95.55,0.9, and 0.15 Hz , respectively. The first two modes are only active for a part of the movie.
图 5.6. 用于演示 m r D M D m r D M D mrDMDm r D M D 的示例数据模式。数据为 n = 6400 n = 6400 n=6400n=6400 像素乘以 m = 1000 m = 1000 m=1000m=1000 个快照，采样时间间隔为 d t = 0.01 d t = 0.01 dt=0.01d t=0.01 秒。这三种模式具有重叠的空间分布，分别以 5.55 , 0.9 5.55 , 0.9 5.55,0.95.55,0.9 和 0.15 赫兹振荡。前两种模式仅在电影的一部分中活跃。
|| X = Xclean + Noise;
|| X = Xclean + 噪声;
To visualize the data constructed above as a movie, we use the following code.
为了将上述构建的数据可视化为一部影片，我们使用以下代码。

ALGORITHM 5.2. Visualize example as a movie (mrDMD_demo.m).
算法 5.2。将示例可视化为电影（mrDMD_demo.m）。

| figure;
for ti = 1:numel(t),
    Pic = reshape(X(:, ti), nx, ny);
    imagesc(Pic, range(X(:))/2*[-1 1]);
    axis square; axis off;
    pause(dt);
end;
We seek a method to decompose this data and characterize its dynamics in space and time within the sampling window. The code in Algorithm 5.3 calls the function mrDMD.m, which implements the algorithm described in this chapter. Figure 5.7
我们寻求一种方法来分解这些数据，并在采样窗口内表征其在空间和时间上的动态。算法 5.3 中的代码调用了函数 mrDMD.m，该函数实现了本章中描述的算法。图 5.7
Figure 5.7. Results of mrDMD for the spatiotemporal dynamics example in Figure 5.6. The multiresolution view of power as it varies in time and in frequency is shown as an image. The absolute values of spatial DMD modes at a few levels ϕ ( ℓ , j ) ϕ ( ℓ , j ) phi^((ℓ,j))\boldsymbol{\phi}^{(\ell, j)} are shown on the right, and these closely resemble the underlying spatial modes.
图 5.7. 图 5.6 中时空动力学示例的 mrDMD 结果。功率随时间和频率变化的多分辨率视图以图像形式显示。右侧显示了几个级别 ϕ ( ℓ , j ) ϕ ( ℓ , j ) phi^((ℓ,j))\boldsymbol{\phi}^{(\ell, j)} 处空间 DMD 模式的绝对值，这些与基础空间模式非常相似。
shows the resulting output as a wavelet-inspired view of the multiresolution structures present in the data, following the axes illustrated in Figure 5.4, as well as the associated spatial modes at each level of description.
按照图 5.4 所示的轴，将结果输出显示为数据中存在的多分辨率结构的小波启发视图，以及每个描述级别上的相关空间模式。
Comparing these modes extracted at levels 1,4 , and 6 to the generative modes in Figure 5.6, it is clear that mrDMD is a natural description of the data. Note that where the time course of the mode does not match well with the halving windows of the algorithm, such as for the middle mode in Figure 5.7 in its first nonzero time window, our ability to estimate the mode amplitude is poor.
将在第 1、4 和 6 级提取的这些模式与图 5.6 中的生成模式进行比较，可以清楚地看到 mrDMD 是对数据的自然描述。请注意，在模式的时间进程与算法的减半窗口不太匹配的情况下，例如图 5.7 中间模式的第一个非零时间窗口，我们估计模式幅度的能力很差。
ALGORITHM 5.3. Compute mrDMD of example movie (mrDMD_demo.m).
算法 5.3。计算示例影片的 mrDMD（mrDMD_demo.m）。
L = 6; % number of levels
r = 10; % rank of truncation
mrdmd = mrDMD(X, dt, r, 2, L);
% compile visualization of multires mode amplitudes
[map, low_f] = mrDMD_map(mrdmd);
[L, J] = size(mrdmd);
%%
figure;
imagesc(-map);

| set(gca, 'YTick', 0.5:(L+0.5), 'YTickLabel', floor(low_f*10)/10)
The function mrDMD. m used in Algorithm 5.3 is shown below. Note that this function recurses, terminating only when the parameter L is 1 .
算法 5.3 中使用的函数 mrDMD.m 如下所示。请注意，此函数是递归的，仅当参数 L 为 1 时才会终止。

ALGORITHM 5.4. Recursive function to compute mrDMD (mrDMD.m).
算法 5.4。用于计算多分辨率动态模态分解（mrDMD.m）的递归函数。

function tree = mrDMD(Xraw, dt, r, max_cyc, L)
% function tree = mrDMD(Xraw, dt, r, max_cyc, L)
% Inputs:
% Xraw n by m matrix of raw data,
% n measurements, m snapshots
% dt time step of sampling
% r rank of truncation
% max_cyc to determine rho, the freq cutoff, compute
% oscillations of max_cyc in the time window
% L number of levels remaining in the recursion
T = size(Xraw, 2) * dt;
rho = max_cyc/T; % high freq cutoff at this level
sub = ceil(1/rho/8/pi/dt); % 4x Nyquist for rho
%% DMD at this level
Xaug = Xraw (:, 1:sub:end); % subsample
Xaug = [Xaug(:, 1:end-1); Xaug(:, 2:end)];
X = Xaug(:, 1:end-1);
Xp = Xaug(:, 2:end);
[U, S, V] = svd(X, 'econ');
r = min(size(U,2), r);
U_r = U(:, 1:r); % rank truncation
S_r = S(1:r, 1:r);
V_r = V(:, 1:r);
Atilde = U_r' * Xp * V_r / S_r;
[W, D] = eig(Atilde);
lambda = diag(D);
Phi = Xp * V(:,1:r) / S(1:r,1:r) * W;
%% compute power of modes
Vand = zeros(r, size(X, 2)); % Vandermonde matrix
for k = 1:size(X, 2),
    Vand(:, k) = lambda.^(k-1);
end;

% the next 5 lines follow Jovanovic et al, 2014 code:
G = S_r * V_r';
P = (W'*W).*conj(Vand*Vand');
q = conj (diag (Vand*G'*W));
Pl = chol(P,'lower');
b = (Pl')\(Pl\q); % Optimal vector of amplitudes b
%% consolidate slow modes, where abs(omega) < rho
omega = log(lambda)/sub/dt/2/pi;
mymodes = find(abs(omega) <= rho);
thislevel.T = T;
thislevel.rho = rho;
thislevel.hit = numel(mymodes) > 0;
thislevel.omega = omega(mymodes);
thislevel.P = abs(b(mymodes));
thislevel.Phi = Phi(:, mymodes);
%% recurse on halves
if L > 1,
    sep = floor(size(Xraw,2)/2);
    nextlevel1 = mrDMD(Xraw(:, 1:sep),dt, r, max_cyc, L-1);
    nextlevel2 = mrDMD(Xraw(:, sep+1:end),dt, r, max_cyc, L-1);
else
    nextlleve1 = cell(0);
    nextlevel2 = cell(0);
end;
%% reconcile indexing on output
% (because MATLAB does not support recursive data structures)
tree = cell(L, 2^(L-1));
tree{1,1} = thislevel;
for l = 2:L,
    col = 1;
    for j = 1:2^(1-2),
        tree{l, col} = nextlevel1{l-1, j};
        col = col + 1;
    end;
    for j = 1:2^(l-2)
        tree{l, col} = nextlevel2{l-1, j};
        col = col + 1;
    end;
end;

5.5.1 - Global temperature data and El Niño
5.5.1 - 全球气温数据与厄尔尼诺现象

In this example, we use mrDMD on data measuring global surface temperature over the ocean. The data is open source from the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA. The NOAA_OI_SST_V2 data set considered can be downloaded at http://www.esrl.noaa.gov/psd/. The data spans a 20-year period from 1990 to 2010.
在这个例子中，我们将 mrDMD 应用于测量海洋全球表面温度的数据。该数据由美国科罗拉多州博尔德市的美国国家海洋和大气管理局/海洋和大气研究办公室/物理科学部（NOAA/OAR/ESRL PSD）开源提供。所考虑的 NOAA_OI_SST_V2 数据集可从 http://www.esrl.noaa.gov/psd/下载。数据涵盖了从 1990 年到 2010 年的 20 年时间。
Figure 5.8. Application of m r D M D m r D M D mrDMDm r D M D on SST data from 1990 to 2010. The left panel illustrates the process for a four-level decomposition. At each level, modes slower than a specific cutoff are extracted. At a given level, the zero-mode component has period T = ∞ T = ∞ T=ooT=\infty. Illustrated are the (a) level-1 mrDMD mode with period T = ∞ T = ∞ T=ooT=\infty and (b) Level- 1 mrDMD mode with period T = 52 T = 52 T=52T=52 weeks. Further on in the decomposition we can extract the © level-4 mrDMD mode of 1997 and (d) level-4 mrDMD mode of 1999. Mode © clearly shows the El Niño mode of interest that develops in the central and east-central equatorial Pacific. The El Niño mode was not present in 1999, as is clear from mode (d) [167].
图 5.8. m r D M D m r D M D mrDMDm r D M D 在 1990 年至 2010 年海表面温度（SST）数据上的应用。左图展示了四级分解的过程。在每一级，提取比特定截止频率慢的模态。在给定级别，零模态分量的周期为 T = ∞ T = ∞ T=ooT=\infty 。图中展示了（a）周期为 T = ∞ T = ∞ T=ooT=\infty 的一级多分辨率动态模态分解（mrDMD）模态和（b）周期为 T = 52 T = 52 T=52T=52 周的一级 mrDMD 模态。在进一步分解中，我们可以提取（c）1997 年的四级 mrDMD 模态和（d）1999 年的四级 mrDMD 模态。模态（c）清楚地显示了在赤道中太平洋和东中太平洋发展的感兴趣的厄尔尼诺模态。从模态（d）可以明显看出，1999 年不存在厄尔尼诺模态[167]。
Figure 5.8 shows the results of the mrDMD algorithm. Specifically, a four-level decomposition was performed with the slow spatiotemporal modes pulled at each level, as suggested in Figure 5.4. At the first level of the decomposition, two modes are extracted: the zero mode ( T = ∞ ) ( T = ∞ ) (T=oo)(T=\infty) depicted in Figure 5.8(a) and a yearly cycle ( T = 52 T = 52 T=52T=52 weeks) shown in Figure 5.8(b). The yearly cycle is the slowest mode extracted at level 1. Note that the ω = 0 ω = 0 omega=0\omega=0 mode component of level 1 , ϕ 1 ( 1 , 1 ) 1 , ϕ 1 ( 1 , 1 ) 1,phi_(1)^((1,1))1, \boldsymbol{\phi}_{1}^{(1,1)}, corresponds to the average ocean temperature over the entire 20 -year data set.
图 5.8 展示了 mrDMD 算法的结果。具体而言，按照图 5.4 中的建议，进行了四级分解，在每一级提取缓慢的时空模式。在分解的第一级，提取了两种模式：图 5.8(a)所示的零模式 ( T = ∞ ) ( T = ∞ ) (T=oo)(T=\infty) 和图 5.8(b)所示的年周期（ T = 52 T = 52 T=52T=52 周）。年周期是在第一级提取的最慢模式。请注意， 1 , ϕ 1 ( 1 , 1 ) 1 , ϕ 1 ( 1 , 1 ) 1,phi_(1)^((1,1))1, \boldsymbol{\phi}_{1}^{(1,1)} 级的 ω = 0 ω = 0 omega=0\omega=0 模式分量对应于整个 20 年数据集上的平均海洋温度。
We continue the mrDMD analysis through to L = 4 L = 4 L=4L=4. At the fourth level, the data-driven method of mrDMD discovers the 1997 El Niño mode generated from the well-known El Niño, Southern Oscillation (ENSO). El Niño is the warm phase of the ENSO cycle and is associated with a band of warm ocean water that develops in the central and east-central equatorial Pacific (between approximately the International Date Line and 120 ∘ W 120 ∘ W 120^(@)W120^{\circ} \mathrm{W} ), including off the Pacific coast of South America. ENSO refers to the cycle of warm and cold temperatures, as measured by sea surface temperature, SST, of the tropical central and eastern Pacific Ocean. El Niño is accompanied by high air pressure in the western Pacific and low air pressure in the eastern Pacific. 
我们继续进行 mrDMD 分析，直至 L = 4 L = 4 L=4L=4 。在第四层，mrDMD 的数据驱动方法发现了由著名的厄尔尼诺-南方涛动（ENSO）产生的 1997 年厄尔尼诺模式。厄尔尼诺是 ENSO 循环的暖期，与赤道中太平洋和中东部太平洋（大约在国际日期变更线和 120 ∘ W 120 ∘ W 120^(@)W120^{\circ} \mathrm{W} 之间）形成的一带温暖海水有关，包括南美洲太平洋沿岸海域。ENSO 指的是热带中东部太平洋海表温度（SST）所测量的冷暖温度循环。厄尔尼诺现象伴随着西太平洋的高气压和东太平洋的低气压。
The cool phase of ENSO is called La Niña, with SST in the eastern Pacific below average and air pressures high in the eastern and low in the western Pacific. The ENSO cycle, both El Niño and La Niña, causes global changes of both temperature and rainfall. 
厄尔尼诺-南方涛动（ENSO）的冷相位被称为拉尼娜，东太平洋的海表温度低于平均水平，东太平洋气压高，西太平洋气压低。ENSO 循环，包括厄尔尼诺和拉尼娜，都会导致全球气温和降雨的变化。
Indeed, 1997 is known to have been a strong El Niño year, as verified by its strong modal signature in the ℓ = 4 ℓ = 4 ℓ=4\ell=4 level decomposition of mrDMD. In contrast, the same sampling window shifted down to 1999 produces no El Niño mode, which is in keeping with known ocean patterns that year. The mrDMD mode clearly shows this band of warm ocean water as a spatiotemporal mode in 1997 in Figure 5.8©.
事实上，1997 年是众所周知的强厄尔尼诺年，这一点已通过其在 mrDMD 的 ℓ = 4 ℓ = 4 ℓ=4\ell=4 层分解中的强模态特征得到验证。相比之下，将相同的采样窗口下移到 1999 年则没有产生厄尔尼诺模式，这与当年已知的海洋模式相符。图 5.8©中 mrDMD 模式清楚地显示了 1997 年这片温暖海水带作为一种时空模式。
These results could not have been produced with DMD unless the correct sampling windows were chosen ahead of time, requiring a supervised learning step not required by mrDMD. Thus mrDMD provides a principled, algorithmic approach that is capable of data-driven discovery in data from complex systems, such as ocean/atmospheric data.
除非提前选择了正确的采样窗口，否则使用直接模态分解（DMD）无法得到这些结果，这需要一个监督学习步骤，而多分辨率直接模态分解（mrDMD）则不需要。因此，mrDMD 提供了一种有原则的算法方法，能够在来自复杂系统的数据（如海洋/大气数据）中进行数据驱动的发现。

5.6-Overcoming translational and rotational invariances
5.6 - 克服平移和旋转不变性

The final application of mrDMD is to an example that is notoriously difficult for SVD-based methods to characterize, namely, when translational and/or rotational structures are present. Indeed, such invariances completely undermine our ability to compute low-rank embeddings of the data as driven by correlated structures, or POD/PCA modes. This has been the Achilles heel of many SVD-based methods in applications such as PCA-based face recognition, where well-cropped and centered faces are required for reasonable performance, so that translation and rotation are removed in an expensive preprocessing procedure.
mrDMD 的最终应用是针对一个基于奇异值分解（SVD）的方法极难刻画的示例，即存在平移和/或旋转结构的情况。确实，这种不变性完全削弱了我们根据相关结构或本征正交分解（POD）/主成分分析（PCA）模式来计算数据的低秩嵌入的能力。这一直是许多基于 SVD 的方法在诸如基于 PCA 的人脸识别等应用中的致命弱点，在这类应用中，为了获得合理的性能，需要裁剪良好且居中的面部图像，以便在一个昂贵的预处理过程中消除平移和旋转。
In a dynamical systems setting, a simple traveling wave will appear to be a highdimensional object in POD space despite the fact that it is only constructed from two modes, one associated with translational invariance. For dynamical cases exhibiting translation and/or rotation, Rowley and Marsden [234] formulated one of the only mathematical strategies to date to extract the low-dimensional embeddings. In particular, they developed a template-matching technique to first remove the invariance before applying SVD. Although effective, it is not suited for cases where multiple objects and time scales are present in the data.
在动力系统环境中，尽管简单行波仅由两种模式构成，其中一种与平移不变性相关，但在本征正交分解（POD）空间中它看起来却是一个高维对象。对于表现出平移和/或旋转的动力学情形，罗利和马斯登[234]制定了迄今为止仅有的数学策略之一来提取低维嵌入。具体而言，他们开发了一种模板匹配技术，以便在应用奇异值分解（SVD）之前先消除不变性。尽管该方法有效，但它不适用于数据中存在多个对象和时间尺度的情况。
The mrDMD approach is able to handle invariances such as translation and rotation. Consider once again the case of a simple traveling wave. Standard DMD applied to the traveling wave would result in a solution approximation requiring many DMD modes due to the slow fall-off of the singular values (1.18) in step 1 of the DMD algorithm. Further, the eigenvalues ω k ω k omega_(k)\omega_{k} in (1.24) would also be bounded away from the origin as there is no background mode for such translating data.
mrDMD 方法能够处理诸如平移和旋转等不变性。再次考虑一个简单行波的情况。由于 DMD 算法步骤 1 中奇异值（1.18）的缓慢衰减，应用于行波的标准 DMD 会导致一个需要许多 DMD 模式的解近似。此外，由于这种平移数据没有背景模式，（1.24）中的特征值 ω k ω k omega_(k)\omega_{k} 也会远离原点。
In the mrDMD architecture, the fact that the eigenvalues are bounded away from the origin (and typically O ( 1 ) O ( 1 ) O(1)O(1) ) in the initial snapshot window X X X\mathbf{X} would simply allow the mrDMD method to ignore the traveling wave at the first level of mrDMD. Once the domain is divided into two for the next level of analysis, the traveling wave is now effectively moving at half the speed in this new domain, i.e., the eigenvalues have migrated toward the origin and the traveling wave is now reevaluated. The recursive procedure eventually advances to a sampling window where the traveling wave looks sufficiently stationary and low rank to be extracted in the multiresolution analysis. The level at which it is extracted also characterizes the speed of the traveling wave. Specifically, the higher the level in the decomposition where the traveling wave is extracted, the faster its speed.
在 mrDMD 架构中，在初始快照窗口 X X X\mathbf{X} 中特征值远离原点（通常为 O ( 1 ) O ( 1 ) O(1)O(1) ）这一事实，会使 mrDMD 方法在 mrDMD 的第一级忽略行波。一旦在下一级分析中将域划分为两个部分，行波在这个新域中的有效移动速度就变为原来的一半，即特征值已向原点迁移，现在要重新评估行波。递归过程最终会推进到一个采样窗口，在行波在多分辨率分析中看起来足够平稳且低秩以便提取。提取行波的级别也表征了行波的速度。具体而言，在行波被提取的分解级别越高，其速度就越快。
Figure 5.9. The mrDMD approach separates moving objects. Two modes (top panel labeled “Fast Mode” and “Slow Mode”) are combined in the data snapshots (top right panel). The “Fast Mode” moves rightward at speed 10 v 10 v 10 v10 v while the “Slow Mode” moves upward at speed v. We take v = 1 / 40 v = 1 / 40 v=1//40v=1 / 40 without loss of generality. Thus the fast and slow mode speeds differ by approximately an order of magnitude. In mrDMD (bottom left panel), the “Slow Mode” is extracted at level 2, as represented by panels (a) and (b). The “Fast Mode” is extracted at level 13, as represented in the three representative panels ©-(e). Although there is some shadow (residual) of the slow mode on the fast mode and vice versa, m r D M D m r D M D mrDMDm r D M D is remarkably effective at extracting the correct modes. Moreover, the level at which they are extracted can allow for a reconstruction of the velocity and direction. To our knowledge, this is the best performance achieved to date with an SVD-based method with multiple time-scale objects.
图 5.9. mrDMD 方法可分离运动物体。在数据快照（右上角面板）中结合了两种模式（顶部面板标记为“快速模式”和“慢速模式”）。“快速模式”以速度 10 v 10 v 10 v10 v 向右移动，而“慢速模式”以速度 v 向上移动。我们不失一般性地取 v = 1 / 40 v = 1 / 40 v=1//40v=1 / 40 。因此，快速模式和慢速模式的速度相差约一个数量级。在 mrDMD（左下角面板）中，“慢速模式”在第 2 级被提取，如面板(a)和(b)所示。“快速模式”在第 13 级被提取，如三个代表性面板©-(e)所示。尽管快速模式上有慢速模式的一些影子（残余），反之亦然，但 m r D M D m r D M D mrDMDm r D M D 在提取正确模式方面非常有效。此外，提取它们的级别可以实现速度和方向的重建。据我们所知，这是迄今为止基于奇异值分解的多时间尺度物体方法所取得的最佳性能。
Figure 5.9 demonstrates the application of the mrDMD method to a simple example in which there are two moving objects, one moving slowly and the other moving faster. Specifically, the example constructed results from the dynamical sequence
图 5.9 展示了 mrDMD 方法在一个简单示例中的应用，该示例中有两个移动物体，一个移动较慢，另一个移动较快。具体而言，该示例构建的结果来自动态序列
x ― = ψ ¯ 1 ( x − 10 v t , y ) + ψ ¯ 2 ( x , y − v t ) x ¯ = ψ ¯ 1 ( x − 10 v t , y ) + ψ ¯ 2 ( x , y − v t ) bar(x)= bar(psi)_(1)(x-10 vt,y)+ bar(psi)_(2)(x,y-vt)\overline{\mathbf{x}}=\bar{\psi}_{1}(x-10 v t, y)+\bar{\psi}_{2}(x, y-v t)
where the two modes used to construct the true solution are ψ ¯ j ( x , y ) ψ ¯ j ( x , y ) bar(psi)_(j)(x,y)\bar{\psi}_{j}(x, y) for j = 1 , 2 j = 1 , 2 j=1,2j=1,2. The two modes are Gaussians of the form ψ ¯ j = exp ⁡ ( − σ ( x − x 0 ) 2 − σ ( y − y 0 ) 2 ) ψ ¯ j = exp ⁡ − σ x − x 0 2 − σ y − y 0 2 bar(psi)_(j)=exp(-sigma(x-x_(0))^(2)-sigma(y-y_(0))^(2))\bar{\psi}_{j}=\exp \left(-\sigma\left(x-x_{0}\right)^{2}-\sigma\left(y-y_{0}\right)^{2}\right) with σ = 0.1 σ = 0.1 sigma=0.1\sigma=0.1, and ( x 0 , y 0 ) = ( − 18 , 20 ) x 0 , y 0 = ( − 18 , 20 ) (x_(0),y_(0))=(-18,20)\left(x_{0}, y_{0}\right)=(-18,20) and ( x 0 , y 0 ) = ( − 20 , − 9 ) x 0 , y 0 = ( − 20 , − 9 ) (x_(0),y_(0))=(-20,-9)\left(x_{0}, y_{0}\right)=(-20,-9) for the fast and slow modes, respectively. Note that the first mode is translating rightward at speed 10 v 10 v 10 v10 v and the second mode is translating upward at speed v v vv. Without loss of generality, v v vv can be set to any value. It is chosen to be v = 1 / 40 v = 1 / 40 v=1//40v=1 / 40 for the domain and Gaussians considered.
用于构建真实解的两种模式是 ψ ¯ j ( x , y ) ψ ¯ j ( x , y ) bar(psi)_(j)(x,y)\bar{\psi}_{j}(x, y) 用于 j = 1 , 2 j = 1 , 2 j=1,2j=1,2 。这两种模式是形式为 ψ ¯ j = exp ⁡ ( − σ ( x − x 0 ) 2 − σ ( y − y 0 ) 2 ) ψ ¯ j = exp ⁡ − σ x − x 0 2 − σ y − y 0 2 bar(psi)_(j)=exp(-sigma(x-x_(0))^(2)-sigma(y-y_(0))^(2))\bar{\psi}_{j}=\exp \left(-\sigma\left(x-x_{0}\right)^{2}-\sigma\left(y-y_{0}\right)^{2}\right) 的高斯函数， σ = 0.1 σ = 0.1 sigma=0.1\sigma=0.1 ，并且 ( x 0 , y 0 ) = ( − 18 , 20 ) x 0 , y 0 = ( − 18 , 20 ) (x_(0),y_(0))=(-18,20)\left(x_{0}, y_{0}\right)=(-18,20) 和 ( x 0 , y 0 ) = ( − 20 , − 9 ) x 0 , y 0 = ( − 20 , − 9 ) (x_(0),y_(0))=(-20,-9)\left(x_{0}, y_{0}\right)=(-20,-9) 分别用于快速模式和慢速模式。请注意，第一种模式以速度 10 v 10 v 10 v10 v 向右平移，第二种模式以速度 v v vv 向上平移。不失一般性， v v vv 可以设置为任何值。对于所考虑的域和高斯函数，选择为 v = 1 / 40 v = 1 / 40 v=1//40v=1 / 40 。
In this case, the straightforward template-matching procedure would fail due to the two distinct time scales of the objects. As shown in the figure, mrDMD is capable of pulling out the two modes at levels 2 (slow) and 13 (fast) of the analysis. 
在这种情况下，由于对象的两个不同时间尺度，直接的模板匹配过程将会失败。如图所示，mrDMD 能够提取出分析中第 2 级（慢）和第 13 级（快）的两种模式。
Although there is a residual in both extracted modes, slow and fast, mrDMD does a reasonable job of extracting the fast and slow modes. To our knowledge, this is the only SVD-based method capable of such unsupervised performance.
尽管在慢速和快速这两种提取模式中都存在残差，但 mrDMD 在提取快速和慢速模式方面做得相当不错。据我们所知，这是唯一一种能够实现这种无监督性能的基于奇异值分解（SVD）的方法。
The motivation for this example is quite clear when considering video processing and surveillance. In particular, for many applications in this area, background subtraction is only one step in the overall video assessment. Identification of the foreground objects is the next, and ultimately more important, task. Consider the example of a video of a street scene in which there is a pedestrian walking (slow translation) along with a vehicle driving down the road (fast translation). Not only would we want to separate the foreground from the background, but also we would want to separate the pedestrian from the vehicle. More precisely, we would want to make a separate video and identify different objects in the video. The results of Figure 5.9 show that mrDMD may be effective at this kind of task.
在考虑视频处理和监控时，这个示例的动机相当明确。特别是，对于该领域的许多应用而言，背景减除只是整个视频评估中的一个步骤。识别前景物体是接下来的、最终也更为重要的任务。以一个街道场景的视频为例，其中有一名行人在行走（慢速移动），同时有一辆车辆在路上行驶（快速移动）。我们不仅希望将前景与背景分离，还希望将行人与车辆分离。更确切地说，我们希望制作一个单独的视频并识别视频中的不同物体。图 5.9 的结果表明，mrDMD 在这类任务中可能是有效的。