deploy: bde5207

_sources/notebooks/climate-modes-xeofs.ipynb

@@ -275,7 +275,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Now we can see some modes of variability! EOF1 looks like ENSO or IPO, and EOF2 probably picking up a pattern of the recent temperature trend where the Southern Ocean and southeastern Pacific are slightly cooling. EOF3 and EOF4 appear to be showing some decadal modes of variability (PDO and maybe AMO), among other things. There is a lot going on in each of these maps, so to get a clearer index of some modes, we can restrict our domain. "
+    "Now we can see some modes of variability! EOF1 looks like ENSO or IPO, and EOF2 is probably picking up a pattern of the recent temperature trend where the Southern Ocean and southeastern Pacific are slightly cooling. EOF3 and EOF4 appear to be showing some decadal modes of variability (PDO and maybe AMO), among other things. There is a lot going on in each of these maps, so to get a clearer index of some modes, we can restrict our domain. "
    ]
   },
   {

notebooks/eof-intro.html

@@ -510,13 +510,13 @@ <h2>Deriving the eigenvalue problem<a class="headerlink" href="#deriving-the-eig
 </tbody>
 </table>
 <p>This is now an optimization problem. To maximize the alignment between the basis and observations, we will maximize the projection of the vectors <span class="math notranslate nohighlight">\(\mathbf f_n\)</span> onto each basis vector using the sum of squares:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-be6e0868-bab0-4574-95b6-976c6e079f3d">
-<span class="eqno">(1)<a class="headerlink" href="#equation-be6e0868-bab0-4574-95b6-976c6e079f3d" title="Permalink to this equation"><i class="fas fa-link"></i></a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-f93a4f61-d48e-406d-b238-0130ae72a50a">
+<span class="eqno">(1)<a class="headerlink" href="#equation-f93a4f61-d48e-406d-b238-0130ae72a50a" title="Permalink to this equation"><i class="fas fa-link"></i></a></span>\[\begin{equation}
 \sum_{n=1}^N(\mathbf f_n \cdot \mathbf e_m)^2
 \end{equation}\]</div>
 <p>for <span class="math notranslate nohighlight">\(m=1, 2, \dots, M\)</span>. We also require mutual orthonormality for the basis: <span class="math notranslate nohighlight">\(\mathbf e_i\cdot\mathbf e_j=\delta_{ij}\)</span>. Now, let’s define this quantity as <span class="math notranslate nohighlight">\(\psi(\mathbf e_m)\)</span>:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-02d840c5-8cea-4c0d-b289-bc369a1ef79f">
-<span class="eqno">(2)<a class="headerlink" href="#equation-02d840c5-8cea-4c0d-b289-bc369a1ef79f" title="Permalink to this equation"><i class="fas fa-link"></i></a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-3ffcfe62-7ec4-4f71-81f2-a74fea60ecbc">
+<span class="eqno">(2)<a class="headerlink" href="#equation-3ffcfe62-7ec4-4f71-81f2-a74fea60ecbc" title="Permalink to this equation"><i class="fas fa-link"></i></a></span>\[\begin{equation}
 \begin{aligned}
 \psi(\mathbf e_m)\equiv\sum_{n=1}^N(\mathbf f_n \cdot \mathbf e_m)^2
 &amp;=\sum_{n=1}^N(\mathbf e_m^\mathrm T \mathbf f_n)(\mathbf f_n^\mathrm T\mathbf e_m)\\
@@ -527,8 +527,8 @@ <h2>Deriving the eigenvalue problem<a class="headerlink" href="#deriving-the-eig
 \end{equation}\]</div>
 <p>where <span class="math notranslate nohighlight">\(\mathbf R=\mathbf F\mathbf F^\mathrm T\)</span> is the covariance matrix of <span class="math notranslate nohighlight">\(\mathbf F\)</span>.</p>
 <p>We now seek the extrema of <span class="math notranslate nohighlight">\(\psi(\mathbf e)\)</span> (dropping the subscript <span class="math notranslate nohighlight">\(m\)</span> for convenience), which satisfy <span class="math notranslate nohighlight">\(\psi(\mathbf e+\delta\mathbf e)=\psi(\mathbf e)\)</span> for a small change <span class="math notranslate nohighlight">\(\delta\mathbf e\)</span> in the direction of <span class="math notranslate nohighlight">\(\mathbf e\)</span>. Then</p>
-<div class="amsmath math notranslate nohighlight" id="equation-cd3b83a3-932a-4c35-9f0c-8aa7a638f305">
-<span class="eqno">(3)<a class="headerlink" href="#equation-cd3b83a3-932a-4c35-9f0c-8aa7a638f305" title="Permalink to this equation"><i class="fas fa-link"></i></a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-6bc17aad-b0d4-4a43-a436-200badf42c07">
+<span class="eqno">(3)<a class="headerlink" href="#equation-6bc17aad-b0d4-4a43-a436-200badf42c07" title="Permalink to this equation"><i class="fas fa-link"></i></a></span>\[\begin{equation}
 \begin{aligned}
 \psi(\mathbf e+\delta\mathbf e)&amp;=(\mathbf e+\delta\mathbf e)^\mathrm T\mathbf R(\mathbf e+\delta\mathbf e)\\
 &amp;=\mathbf e^\mathrm T\mathbf R\mathbf e+2(\delta\mathbf e)^\mathrm T\mathbf R\mathbf e + (\delta\mathbf e)^\mathrm T\mathbf R\delta\mathbf e\\
@@ -537,13 +537,13 @@ <h2>Deriving the eigenvalue problem<a class="headerlink" href="#deriving-the-eig
 \end{aligned}
 \end{equation}\]</div>
 <p>where we only keep the terms first order in <span class="math notranslate nohighlight">\(\delta\mathbf e\)</span>. The optimization condition implies</p>
-<div class="amsmath math notranslate nohighlight" id="equation-563376db-1451-47fd-807e-1ca781f6961d">
-<span class="eqno">(4)<a class="headerlink" href="#equation-563376db-1451-47fd-807e-1ca781f6961d" title="Permalink to this equation"><i class="fas fa-link"></i></a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-15f46f49-eb67-4fdb-8098-9e9d9a3077d7">
+<span class="eqno">(4)<a class="headerlink" href="#equation-15f46f49-eb67-4fdb-8098-9e9d9a3077d7" title="Permalink to this equation"><i class="fas fa-link"></i></a></span>\[\begin{equation}
 (\delta\mathbf e)^\mathrm T\mathbf R\mathbf e=0
 \end{equation}\]</div>
 <p>By the orthonormality of the basis, we also require that these variations only change the direction of <span class="math notranslate nohighlight">\(\mathbf e\)</span>, so <span class="math notranslate nohighlight">\((\mathbf e+\delta\mathbf e)^\mathrm T(\mathbf e+\delta\mathbf e)=1\Rightarrow(\delta\mathbf e)^\mathrm T\mathbf e=0\)</span>, again only keeping terms first order in <span class="math notranslate nohighlight">\(\delta\mathbf e\)</span>. We can combine these two conditions, provided that we multiply the orthonormality condition by a constant that has the same units as the entries of <span class="math notranslate nohighlight">\(\mathbf R\)</span>:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-50f61f70-f62c-4570-b6c4-212a2ea4ffd2">
-<span class="eqno">(5)<a class="headerlink" href="#equation-50f61f70-f62c-4570-b6c4-212a2ea4ffd2" title="Permalink to this equation"><i class="fas fa-link"></i></a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-70f44d9d-b16d-469b-bad6-8e18625f73d7">
+<span class="eqno">(5)<a class="headerlink" href="#equation-70f44d9d-b16d-469b-bad6-8e18625f73d7" title="Permalink to this equation"><i class="fas fa-link"></i></a></span>\[\begin{equation}
 (\delta\mathbf e)^\mathrm T\mathbf R\mathbf e-\lambda(\delta\mathbf e)^\mathrm T\mathbf e=(\delta\mathbf e)^\mathrm T[\mathbf R\mathbf e-\lambda\mathbf e]=0
 \end{equation}\]</div>
 <p>Since <span class="math notranslate nohighlight">\(\delta\mathbf e\)</span> is arbitrary, we must have <span class="math notranslate nohighlight">\(\mathbf R\mathbf e-\lambda\mathbf e=0\)</span>, the eigenvalue problem for <span class="math notranslate nohighlight">\(\mathbf R\)</span>, or for nontrivial solutions, <span class="math notranslate nohighlight">\(|\mathbf R-\lambda\mathbf I|=0\)</span>.</p>

notebooks/notebook-template.html

@@ -514,8 +514,8 @@ <h5>of further and further<a class="headerlink" href="#of-further-and-further" t
 <section id="header-levels">
 <h6>header levels<a class="headerlink" href="#header-levels" title="Link to this heading"><i class="fas fa-link"></i></a></h6>
 <p>as well <span class="math notranslate nohighlight">\(m = a * t / h\)</span> text! Similarly, you have access to other <span class="math notranslate nohighlight">\(\LaTeX\)</span> equation <a class="reference external" href="https://jupyter-notebook.readthedocs.io/en/stable/examples/Notebook/Typesetting%20Equations.html"><strong>functionality</strong></a> via MathJax (demo below from link),</p>
-<div class="amsmath math notranslate nohighlight" id="equation-b18cf3f6-28f4-4961-a475-5df7abcaecb5">
-<span class="eqno">()<a class="headerlink" href="#equation-b18cf3f6-28f4-4961-a475-5df7abcaecb5" title="Permalink to this equation"><i class="fas fa-link"></i></a></span>\[\begin{align}
+<div class="amsmath math notranslate nohighlight" id="equation-fd37418f-0ef0-431f-acbb-2a00b3943044">
+<span class="eqno">()<a class="headerlink" href="#equation-fd37418f-0ef0-431f-acbb-2a00b3943044" title="Permalink to this equation"><i class="fas fa-link"></i></a></span>\[\begin{align}
 \dot{x} &amp; = \sigma(y-x) \\
 \dot{y} &amp; = \rho x - y - xz \\
 \dot{z} &amp; = -\beta z + xy