We want to find \key{coherent sets}, which are regions in the phase space that keep their geometric integrity to a large extent
We want to find \key{coherent sets}, which are regions in state space that keep their geometric integrity to a large extent
during temporal evolution.\\
during temporal evolution.\\
$\rightarrow$ Find tight bundles of trajectories.
$\rightarrow$ Find tight bundles of trajectories.
%This is equivalent to finding \textcolor{red}{tight bundles of trajectories} if the points forming the trajectories are conceived as samples from such a set.
%This is equivalent to finding \textcolor{red}{tight bundles of trajectories} if the points forming the trajectories are conceived as samples from such a set.
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@@ -67,8 +67,8 @@ $\rightarrow$ Find tight bundles of trajectories.
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@@ -67,8 +67,8 @@ $\rightarrow$ Find tight bundles of trajectories.
%If $\Phi_{\epsilon, t}$ is the deterministic flow map plus some small random pertubation with variance $\epsilon$, \textcolor{red}{coherence implies robustness} in the sense of "$\Phi_{\epsilon, t}^{-1}(\Phi_{\epsilon, t} \mathbb{X}) \approx \mathbb{X}$", where $\mathbb{X} \subset \mathbb{R}^3$ is a coherent set at $t_0$. We make use of this by constructing a \textcolor{red}{diffusion operator on the data points}, whose eigenvectors give a low-dimensional representation which is used to extract coherent sets (\textcolor{red}{spectral clustering}) [1].
%If $\Phi_{\epsilon, t}$ is the deterministic flow map plus some small random pertubation with variance $\epsilon$, \textcolor{red}{coherence implies robustness} in the sense of "$\Phi_{\epsilon, t}^{-1}(\Phi_{\epsilon, t} \mathbb{X}) \approx \mathbb{X}$", where $\mathbb{X} \subset \mathbb{R}^3$ is a coherent set at $t_0$. We make use of this by constructing a \textcolor{red}{diffusion operator on the data points}, whose eigenvectors give a low-dimensional representation which is used to extract coherent sets (\textcolor{red}{spectral clustering}) [1].
QSAS -- a.k.a.~high pressure blockings -- are \key{critical features of mid-latitude weather} and, importantly, associated with extreme events. Forecasting skill is, however, still unsatisfying. Physically, QSAS are characterized by \key{high stability}.
High pressure blocking -- a.k.a.~Quasi-Stationary Atmospheric States(QSAS) -- are \key{critical features of mid-latitude weather} and, importantly, associated with extreme events. Forecasting skill is, however, still unsatisfying. Physically, QSAS are characterized by \key{high stability}.
{\footnotesize Left: Blocking example with PV field (shaded), wind $>30\mathrm{\frac{m}{s}}$ (blue) and blocking region (purple). Right: Point vortex model idealizations of $\Omega$- and High-over-Low-Blockings (from [2]).}
{\footnotesize Left: Blocking example with PV field (shaded), wind $>30\mathrm{\frac{m}{s}}$ (blue) and blocking region (purple). Right: Point vortex model idealizations of $\Omega$- and High-over-Low-Blockings (from [2]).}
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@@ -88,13 +88,13 @@ $\rightarrow$ Find tight bundles of trajectories.
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@@ -88,13 +88,13 @@ $\rightarrow$ Find tight bundles of trajectories.
Assume at time $t$ the points $x_t^i$ are sampled from some manifold $\mathbb{M}_t$. Then, the sum of the diffusion similarities can be interpreted as a \key{Monte Carlo integral approximation} [4]
Assume at time $t$ the points $x_t^i$ are sampled from some manifold $\mathbb{M}_t$. Then, the sum of the diffusion similarities can be interpreted as a \key{Monte Carlo integral approximation} [4]
\begin{align*}S_{\epsilon, t} := \sum_{i,j}\mathbf{K}_{\epsilon, t}(i,j) \approx\frac{m^2}{\mathrm{vol}(\mathbb{M}_t)^2}\int_{\mathbb{M}_t}\int_{\mathbb{M}_t}\exp (-\epsilon^{-1}\|x-y\|^2) \dd x \dd y \approx\frac{m^2}{\mathrm{vol}(\mathbb{M}_t)} (\pi\epsilon)^{d/2}
\begin{align*}S_{\epsilon, t} := \sum_{i,j}\mathbf{K}_{\epsilon, t}(i,j) \approx\frac{m^2}{\mathrm{vol}(\mathbb{M}_t)^2}\int_{\mathbb{M}_t}\int_{\mathbb{M}_t}\exp (-\epsilon^{-1}\|x-y\|^2) \dd x \dd y \approx\frac{m^2}{\mathrm{vol}(\mathbb{M}_t)} (2 \pi\epsilon)^{d(t)/2}
\end{align*}
\end{align*}
%$S_{\epsilon, t}$ is a measure of the \textcolor{red}{density} of points, with $\lim_{{\epsilon \to 0}} S_{\epsilon, t}= m$ and $\lim_{{\epsilon \to \infty}}S_{\epsilon, t} = m^2$. Also $\log(S_{\epsilon, t}) \approx \frac{d}{2}\log(\epsilon) + \log(\frac{(2\pi)^{d/2} m^2}{\mathrm{vol}(\mathbb{M}_t)})$.
%$S_{\epsilon, t}$ is a measure of the \textcolor{red}{density} of points, with $\lim_{{\epsilon \to 0}} S_{\epsilon, t}= m$ and $\lim_{{\epsilon \to \infty}}S_{\epsilon, t} = m^2$. Also $\log(S_{\epsilon, t}) \approx \frac{d}{2}\log(\epsilon) + \log(\frac{(2\pi)^{d/2} m^2}{\mathrm{vol}(\mathbb{M}_t)})$.
For suited $\epsilon$, this yields a linear relationship between $\log(\epsilon)$ and $\log(S_{\epsilon, t})$.
For suited $\epsilon$, this yields a linear relationship between $\log(\epsilon)$ and $\log(S_{\epsilon, t})$.
The dimension $d$ and density $\rho(t)$ can be computed from this graph. We use $\ell(t):=\frac{1}{d}\log(\rho(t))$ as a measure of density. The trajectories tend to be \key{denser after passage} through the QSAS, which is in line with the established notion of blockings as regions characterized by high stability.
The dimension $d(t)$ and density $\rho(t)$ can be computed from this graph. We use $\ell(t):=\frac{1}{d(t)}\log(\rho(t))$ as a measure of density. The trajectories tend to be \key{denser after passage} through the QSAS, which is in line with the established notion of blockings as regions characterized by high stability.
%We obtain $d = \dim (\mathbb{M}_t)$ by maximizing $\frac{\dd \log (S_{\epsilon, t})}{\dd \log (\epsilon)}$ and calculate the point density measure $\rho (t) := \frac{1}{d}\log (\frac{m}{\mathrm{vol}(\mathbb{M}_t)})$.
%We obtain $d = \dim (\mathbb{M}_t)$ by maximizing $\frac{\dd \log (S_{\epsilon, t})}{\dd \log (\epsilon)}$ and calculate the point density measure $\rho (t) := \frac{1}{d}\log (\frac{m}{\mathrm{vol}(\mathbb{M}_t)})$.
