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Accessing and preparing the data -
<matplotlib.collections.QuadMesh at 0x7f4b475ba9b0>
+
<matplotlib.collections.QuadMesh at 0x7f3d0e1e69b0>
 
../_images/e916d7882e68d2252816f31c236a5711530a38dc2fbc11136cba6a2fbd9f4e16.png @@ -921,7 +921,7 @@

Accessing and preparing the data -
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+
<xarray.plot.facetgrid.FacetGrid at 0x7f3d745c7a90>
 
../_images/792737bbd01dd9ba581e1d39d1e3f9726b7a17e69c5c03f3c161802ed7af6d31.png @@ -934,10 +934,10 @@

Accessing and preparing the data -
<xarray.plot.facetgrid.FacetGrid at 0x7f4b3f191ba0>
+
<xarray.plot.facetgrid.FacetGrid at 0x7f3d05df97e0>
 
-../_images/d0142dab514b5f804a6ab65c9b0a2117f4896110cff3e36d324d05acb535a3e3.png +../_images/d01657b2b32124b20432bc29b4836bdce10779ae336c2002f03b7f6b583f5d90.png
@@ -1318,7 +1318,7 @@

Accessing and preparing the data @@ -1337,7 +1337,7 @@

Accessing and preparing the data -
[<matplotlib.lines.Line2D at 0x7f4b3dfad780>]
+
[<matplotlib.lines.Line2D at 0x7f3d04c15c60>]
 
../_images/1206ca70f269c3e5d11fdd6703af874f779b17cb816423e59b6f906e49a12acc.png @@ -1389,10 +1389,10 @@

Accessing and preparing the data -
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+
<xarray.plot.facetgrid.FacetGrid at 0x7f3d05d36d40>
 
-../_images/8f1db5c298e04c0905796f68679997f86abb339fa742a3cf12d32179564074f5.png +../_images/00df743148a5028e5a7485650fe0d1ee8aeb208995de1cb9c6493e0f3fa7d878.png
@@ -1402,10 +1402,10 @@

Accessing and preparing the data -
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+
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-../_images/b45f41b4b8df7e722bc123afbf5ca9b7910ccbcb8b4d2a2af67ca4bd6f31d7f9.png +../_images/609b8b70a907f50ca94d46f9280547d16d33b28aaf6df8526d2fd627bf3da697.png
@@ -1779,14 +1779,14 @@

Accessing and preparing the data

El Niño Southern Oscillation (ENSO)

@@ -1827,10 +1827,10 @@

El Niño Southern Oscillation (ENSO) -
<xarray.plot.facetgrid.FacetGrid at 0x7f4b3da37c70>
+
<xarray.plot.facetgrid.FacetGrid at 0x7f3d0486ff40>
 
-../_images/a5a36a2c71b2d8fb5e38e21708cbbfd914f9dd87e23641c4c35f991973f5e9e9.png +../_images/54747e18fcf9c109e84ad936956c027a1bc2ef4ae483d275309d1fa71dc614fe.png
@@ -1840,10 +1840,10 @@

El Niño Southern Oscillation (ENSO) -
<xarray.plot.facetgrid.FacetGrid at 0x7f4b3d957a30>
+
<xarray.plot.facetgrid.FacetGrid at 0x7f3d0e065fc0>
 
-../_images/3852676d3354ebb8bbfd20668a6c4d56e1a7ee7fb25f4c165e9ee2c421100f0d.png +../_images/7f72d88f32397dc10069cac7cd8e9eab3d8024f23a52bc57d2a4f2932f78ff9b.png
@@ -2217,14 +2217,14 @@

El Niño Southern Oscillation (ENSO)
@@ -2306,10 +2306,10 @@

Pacific Decadal Oscillation (PDO) -
<xarray.plot.facetgrid.FacetGrid at 0x7f4b3d537490>
+
<xarray.plot.facetgrid.FacetGrid at 0x7f3d041a3a60>
 
-../_images/432d0f79084a170620914ffb4c7ba2edb41dbe0ae1ac3bf30ae85fa45adf6402.png +../_images/c913472cbfca54962dd3a44c479a8c2e6e67b59c180915296c88ffb55be5af5e.png
diff --git a/notebooks/eof-intro.html b/notebooks/eof-intro.html index 706129e..5b0df57 100644 --- a/notebooks/eof-intro.html +++ b/notebooks/eof-intro.html @@ -510,13 +510,13 @@

Deriving the eigenvalue problem\(\mathbf f_n\) onto each basis vector using the sum of squares:

-
-(1)\[\begin{equation} +
+(1)\[\begin{equation} \sum_{n=1}^N(\mathbf f_n \cdot \mathbf e_m)^2 \end{equation}\]

for \(m=1, 2, \dots, M\). We also require mutual orthonormality for the basis: \(\mathbf e_i\cdot\mathbf e_j=\delta_{ij}\). Now, let’s define this quantity as \(\psi(\mathbf e_m)\):

-
-(2)\[\begin{equation} +
+(2)\[\begin{equation} \begin{aligned} \psi(\mathbf e_m)\equiv\sum_{n=1}^N(\mathbf f_n \cdot \mathbf e_m)^2 &=\sum_{n=1}^N(\mathbf e_m^\mathrm T \mathbf f_n)(\mathbf f_n^\mathrm T\mathbf e_m)\\ @@ -527,8 +527,8 @@

Deriving the eigenvalue problem\(\mathbf R=\mathbf F\mathbf F^\mathrm T\) is the covariance matrix of \(\mathbf F\).

We now seek the extrema of \(\psi(\mathbf e)\) (dropping the subscript \(m\) for convenience), which satisfy \(\psi(\mathbf e+\delta\mathbf e)=\psi(\mathbf e)\) for a small change \(\delta\mathbf e\) in the direction of \(\mathbf e\). Then

-
-(3)\[\begin{equation} +
+(3)\[\begin{equation} \begin{aligned} \psi(\mathbf e+\delta\mathbf e)&=(\mathbf e+\delta\mathbf e)^\mathrm T\mathbf R(\mathbf e+\delta\mathbf e)\\ &=\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 @@

Deriving the eigenvalue problem\(\delta\mathbf e\). The optimization condition implies

-
-(4)\[\begin{equation} +
+(4)\[\begin{equation} (\delta\mathbf e)^\mathrm T\mathbf R\mathbf e=0 \end{equation}\]

By the orthonormality of the basis, we also require that these variations only change the direction of \(\mathbf e\), so \((\mathbf e+\delta\mathbf e)^\mathrm T(\mathbf e+\delta\mathbf e)=1\Rightarrow(\delta\mathbf e)^\mathrm T\mathbf e=0\), again only keeping terms first order in \(\delta\mathbf e\). 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 \(\mathbf R\):

-
-(5)\[\begin{equation} +
+(5)\[\begin{equation} (\delta\mathbf e)^\mathrm T\mathbf R\mathbf e-\lambda(\delta\mathbf e)^\mathrm T\mathbf e=(\delta\mathbf e)^\mathrm T[\mathbf R\mathbf e-\lambda\mathbf e]=0 \end{equation}\]

Since \(\delta\mathbf e\) is arbitrary, we must have \(\mathbf R\mathbf e-\lambda\mathbf e=0\), the eigenvalue problem for \(\mathbf R\), or for nontrivial solutions, \(|\mathbf R-\lambda\mathbf I|=0\).

diff --git a/notebooks/notebook-template.html b/notebooks/notebook-template.html index c45859f..b6f2e98 100644 --- a/notebooks/notebook-template.html +++ b/notebooks/notebook-template.html @@ -514,8 +514,8 @@
of further and further
header levels

as well \(m = a * t / h\) text! Similarly, you have access to other \(\LaTeX\) equation functionality via MathJax (demo below from link),

-
-()\[\begin{align} +
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