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48 changes: 24 additions & 24 deletions blog/basic-python.html
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Expand Up @@ -915,7 +915,7 @@ <h2>Defining functions in python<a class="headerlink" href="#defining-functions-
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/be5189d9ea9e420e6e307b605c5abfc17d94cc94a4840138184a93cb3c6a5a82.png" src="../_images/be5189d9ea9e420e6e307b605c5abfc17d94cc94a4840138184a93cb3c6a5a82.png" />
<img alt="../_images/c82ad5f4f149f722010224c9bf22c74e45e0d161ead809156d5dd67914ef1e54.png" src="../_images/c82ad5f4f149f722010224c9bf22c74e45e0d161ead809156d5dd67914ef1e54.png" />
</div>
</div>
</section>
Expand Down Expand Up @@ -1042,7 +1042,7 @@ <h2>Advanced function creation<a class="headerlink" href="#advanced-function-cre
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/da56408cdd111fb9cd0c7a825af977190e4be0664334b4e8e4a6dfae1a1e1b50.png" src="../_images/da56408cdd111fb9cd0c7a825af977190e4be0664334b4e8e4a6dfae1a1e1b50.png" />
<img alt="../_images/fb1c5ed136fd0b48780878510d4bb46bbb9ebaedc063915bffd511339908a6ed.png" src="../_images/fb1c5ed136fd0b48780878510d4bb46bbb9ebaedc063915bffd511339908a6ed.png" />
</div>
</div>
<div class="cell docutils container">
Expand All @@ -1056,7 +1056,7 @@ <h2>Advanced function creation<a class="headerlink" href="#advanced-function-cre
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/15325932dc2e1d2b3d9607fa216f2fd844f043ec008f5547ad236ec5fb19ea0f.png" src="../_images/15325932dc2e1d2b3d9607fa216f2fd844f043ec008f5547ad236ec5fb19ea0f.png" />
<img alt="../_images/70735392f4187d29834bd045176d16c6e8097afeb2d2c4215a05c10869caa48e.png" src="../_images/70735392f4187d29834bd045176d16c6e8097afeb2d2c4215a05c10869caa48e.png" />
</div>
</div>
<p>In that example we wrap the matplotlib plotting commands in a function, which we can call the way we want to, with arbitrary optional arguments. In this example, you cannot pass keyword arguments that are illegal to the plot command or you will get an error.</p>
Expand Down Expand Up @@ -1096,7 +1096,7 @@ <h2>Lambda Lambda Lambda<a class="headerlink" href="#lambda-lambda-lambda" title
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;function &lt;lambda&gt; at 0x7f46989c0c20&gt;
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;function &lt;lambda&gt; at 0x7f36faa9a020&gt;
4
</pre></div>
</div>
Expand All @@ -1112,7 +1112,7 @@ <h2>Lambda Lambda Lambda<a class="headerlink" href="#lambda-lambda-lambda" title
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;function &lt;lambda&gt; at 0x7f469ac29bc0&gt;
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;function &lt;lambda&gt; at 0x7f36fab7df80&gt;
5
</pre></div>
</div>
Expand All @@ -1129,7 +1129,7 @@ <h2>Lambda Lambda Lambda<a class="headerlink" href="#lambda-lambda-lambda" title
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;function &lt;lambda&gt; at 0x7f46989c0d60&gt;
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;function &lt;lambda&gt; at 0x7f36faa9a160&gt;
5
5
</pre></div>
Expand All @@ -1150,7 +1150,7 @@ <h2>Lambda Lambda Lambda<a class="headerlink" href="#lambda-lambda-lambda" title
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;function &lt;lambda&gt; at 0x7f46989c1080&gt;
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;function &lt;lambda&gt; at 0x7f36faa9a480&gt;
1
3
6
Expand Down Expand Up @@ -1326,7 +1326,7 @@ <h2>Creating arrays in python<a class="headerlink" href="#creating-arrays-in-pyt
[4 5 6]]
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/tmp/ipykernel_1909/3002503601.py:4: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/tmp/ipykernel_1922/3002503601.py:4: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
print(np.row_stack([a, b]))
</pre></div>
</div>
Expand Down Expand Up @@ -1509,7 +1509,7 @@ <h2>Functions on arrays of values<a class="headerlink" href="#functions-on-array
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/0dde0c9d9606e3f45603dff776772f72953e1a858d79bc9be55a4c662ea26006.png" src="../_images/0dde0c9d9606e3f45603dff776772f72953e1a858d79bc9be55a4c662ea26006.png" />
<img alt="../_images/8a9d660ddaaa4737db9d130b3f6931574bd728be7e0250294cef3233e9954a07.png" src="../_images/8a9d660ddaaa4737db9d130b3f6931574bd728be7e0250294cef3233e9954a07.png" />
</div>
</div>
<p>This figure illustrates graphically what the numbers above show. The function crosses zero at approximately <span class="math notranslate nohighlight">\(x = 1.5\)</span>. To get a more precise value, we must actually solve the function numerically. We use the function func:scipy.optimize.fsolve to do that. More precisely, we want to solve the equation <span class="math notranslate nohighlight">\(f(x) = \cos(x) = 0\)</span>. We create a function that defines that equation, and then use func:scipy.optimize.fsolve to solve it.</p>
Expand Down Expand Up @@ -1779,7 +1779,7 @@ <h2>Indexing vectors and arrays in Python<a class="headerlink" href="#indexing-v
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>-1.7950016288086892
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/tmp/ipykernel_1909/518394352.py:3: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/tmp/ipykernel_1922/518394352.py:3: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
print(np.trapz( x[x &gt; 2], y[x &gt; 2]))
</pre></div>
</div>
Expand Down Expand Up @@ -1876,17 +1876,17 @@ <h3>3D arrays<a class="headerlink" href="#id2" title="Link to this heading">#</a
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[[0.75563895 0.18498477 0.77047058]
[0.26274517 0.89107749 0.34583296]
[0.84546459 0.65605378 0.48609064]]
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[[0.2227254 0.19726277 0.99153215]
[0.97311734 0.38142852 0.56638625]
[0.48335471 0.51800664 0.87188098]]

[[0.61448601 0.16467032 0.45140894]
[0.50526787 0.78154638 0.79511797]
[0.61749088 0.41062892 0.5587581 ]]
[[0.37090501 0.88329327 0.93826256]
[0.25215415 0.65693695 0.03286427]
[0.97872232 0.19969579 0.79929687]]

[[0.24758141 0.21205917 0.49050752]
[0.27125643 0.22989573 0.68153965]
[0.35608903 0.97072653 0.08109636]]]
[[0.69161839 0.70311344 0.87964521]
[0.99408152 0.88432325 0.46235968]
[0.2289108 0.37483611 0.78532766]]]
</pre></div>
</div>
</div>
Expand All @@ -1900,11 +1900,11 @@ <h3>3D arrays<a class="headerlink" href="#id2" title="Link to this heading">#</a
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[0.75563895 0.26274517 0.84546459]
[0.61448601 0.50526787 0.61749088]
[0.24758141 0.27125643 0.35608903]]
[0.75563895 0.61448601 0.24758141]
[0.45140894 0.79511797 0.5587581 ]
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[0.2227254 0.97311734 0.48335471]
[0.37090501 0.25215415 0.97872232]
[0.69161839 0.99408152 0.2289108 ]]
[0.2227254 0.37090501 0.69161839]
[0.93826256 0.03286427 0.79929687]
</pre></div>
</div>
</div>
Expand Down
28 changes: 14 additions & 14 deletions blog/data-analysis.html
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Expand Up @@ -623,7 +623,7 @@ <h2>Fit a line to numerical data<a class="headerlink" href="#fit-a-line-to-numer
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/9cab5f9733126ae4bfbea465ec1662fc1d035ac8145f6083eef298e826e67eb0.png" src="../_images/9cab5f9733126ae4bfbea465ec1662fc1d035ac8145f6083eef298e826e67eb0.png" />
<img alt="../_images/78fdc6defabc7a3280fc782a33aa4feb2e1250a448940c4378e1d478d0b8a436.png" src="../_images/78fdc6defabc7a3280fc782a33aa4feb2e1250a448940c4378e1d478d0b8a436.png" />
</div>
</div>
</section>
Expand Down Expand Up @@ -670,7 +670,7 @@ <h2>Linear least squares fitting with linear algebra<a class="headerlink" href="
0.000624573378839699 -0.3145221843003413
</pre></div>
</div>
<img alt="../_images/5f47d6ccf5b0e9d3efeea70505c18b5cc75e117b51ffe30dd6ed47e02df4e495.png" src="../_images/5f47d6ccf5b0e9d3efeea70505c18b5cc75e117b51ffe30dd6ed47e02df4e495.png" />
<img alt="../_images/a42c3bb59cc2c093bf6b9be4cef11ca9d5730d46661aaeb77ffebfcf23e02eb1.png" src="../_images/a42c3bb59cc2c093bf6b9be4cef11ca9d5730d46661aaeb77ffebfcf23e02eb1.png" />
</div>
</div>
<p>This method can be readily extended to fitting any polynomial model, or other linear model that is fit in a least squares sense. This method does not provide confidence intervals.</p>
Expand Down Expand Up @@ -768,7 +768,7 @@ <h2>Linear regression with confidence intervals.<a class="headerlink" href="#lin
R^2 = 0.9999869672459537
</pre></div>
</div>
<img alt="../_images/0c7ceeb2db5947cacc509a929757156f9712aaa9e2693936045a1a82171bd07f.png" src="../_images/0c7ceeb2db5947cacc509a929757156f9712aaa9e2693936045a1a82171bd07f.png" />
<img alt="../_images/c2fcc135cc0c7607f4a583b24f4e95675d723267656ca42013e030cddad2149a.png" src="../_images/c2fcc135cc0c7607f4a583b24f4e95675d723267656ca42013e030cddad2149a.png" />
</div>
</div>
<p>A fourth order polynomial fits the data well, with a good R^2 value. All of the parameters appear to be significant, i.e. zero is not included in any of the parameter confidence intervals. This does not mean this is the best model for the data, just that the model fits well.</p>
Expand Down Expand Up @@ -820,7 +820,7 @@ <h2>Nonlinear curve fitting<a class="headerlink" href="#nonlinear-curve-fitting"
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Fitted parameters = [-56.46839641 0.57233217 2.7407944 16.55905648]
</pre></div>
</div>
<img alt="../_images/1f7130eb4df4056f7c7d7e9fa936f170c751bf140ebad1f6007d9d9d56d84e3d.png" src="../_images/1f7130eb4df4056f7c7d7e9fa936f170c751bf140ebad1f6007d9d9d56d84e3d.png" />
<img alt="../_images/24eb55b923a124faacb7313ff1c228340e9260844b0fc762c5d67b5eca5d2991.png" src="../_images/24eb55b923a124faacb7313ff1c228340e9260844b0fc762c5d67b5eca5d2991.png" />
</div>
</div>
<p>See additional examples at \url{<a class="reference external" href="http://docs.scipy.org/doc/scipy/reference/tutorial/optimize.html">http://docs.scipy.org/doc/scipy/reference/tutorial/optimize.html</a>}.</p>
Expand Down Expand Up @@ -879,7 +879,7 @@ <h2>Nonlinear curve fitting by direct least squares minimization<a class="header
parameters = [-56.46932645 0.59141447 1.9044796 16.59341303]
</pre></div>
</div>
<img alt="../_images/a38ede221cd66330d205227d728ef49b083fec2bee4969ab291d4b9af8d5d322.png" src="../_images/a38ede221cd66330d205227d728ef49b083fec2bee4969ab291d4b9af8d5d322.png" />
<img alt="../_images/777ca1e2dd8aa60b9cbd6adcc4f1e992a8b64aafe6bac269136764841f1821ad.png" src="../_images/777ca1e2dd8aa60b9cbd6adcc4f1e992a8b64aafe6bac269136764841f1821ad.png" />
</div>
</div>
</section>
Expand All @@ -901,7 +901,7 @@ <h2>Parameter estimation by directly minimizing summed squared errors<a class="h
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/cdf5aefcbe99ce74ac9a7a9ad2146a8107f711df445488fb7e05df0a1b5b0fc2.png" src="../_images/cdf5aefcbe99ce74ac9a7a9ad2146a8107f711df445488fb7e05df0a1b5b0fc2.png" />
<img alt="../_images/0b31ff53625e2d9582b199e5b7ea5280e0d0c3233b3e7bb91948276e0b03b691.png" src="../_images/0b31ff53625e2d9582b199e5b7ea5280e0d0c3233b3e7bb91948276e0b03b691.png" />
</div>
</div>
<p>We are going to fit the function <span class="math notranslate nohighlight">\(y = x^a\)</span> to the data. The best <span class="math notranslate nohighlight">\(a\)</span> will minimize the summed squared error between the model and the fit.</p>
Expand All @@ -923,7 +923,7 @@ <h2>Parameter estimation by directly minimizing summed squared errors<a class="h
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/2eacd9e5b13ba5d4c83a9be7e22292d0bf5268feb1c2f4812fd8fa67ce5b0926.png" src="../_images/2eacd9e5b13ba5d4c83a9be7e22292d0bf5268feb1c2f4812fd8fa67ce5b0926.png" />
<img alt="../_images/89fc0f5aacff6472047e29b90385979d36f6e56768e97daf355aa71e8e9f55ef.png" src="../_images/89fc0f5aacff6472047e29b90385979d36f6e56768e97daf355aa71e8e9f55ef.png" />
</div>
</div>
<p>Based on the graph above, you can see a minimum in the summed squared error near <span class="math notranslate nohighlight">\(a = 2.1\)</span>. We use that as our initial guess. Since we know the answer is bounded, we use scipy.optimize.fminbound</p>
Expand All @@ -948,7 +948,7 @@ <h2>Parameter estimation by directly minimizing summed squared errors<a class="h
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>2.0900483893314967
</pre></div>
</div>
<img alt="../_images/0a437456b591008802a35ce5c58c38960f2b08801526795aa2eb42409647ea31.png" src="../_images/0a437456b591008802a35ce5c58c38960f2b08801526795aa2eb42409647ea31.png" />
<img alt="../_images/e647cd165d4193d18c6e3a656f25867f94b4262f084d94bb3e3d16f15bf6776a.png" src="../_images/e647cd165d4193d18c6e3a656f25867f94b4262f084d94bb3e3d16f15bf6776a.png" />
</div>
</div>
<p>We can do nonlinear fitting by directly minimizing the summed squared error between a model and data. This method lacks some of the features of other methods, notably the simple ability to get the confidence interval. However, this method is flexible and may offer more insight into how the solution depends on the parameters.</p>
Expand Down Expand Up @@ -1008,7 +1008,7 @@ <h2>Nonlinear curve fitting with parameter confidence intervals<a class="headerl
p1: 0.026461556970080666 [0.023607653829234403 0.02931546011092693]
</pre></div>
</div>
<img alt="../_images/4fc18791ae2e3c887cf310b19379a0dee56dbda6675557f6fbf35ccdcb1447b6.png" src="../_images/4fc18791ae2e3c887cf310b19379a0dee56dbda6675557f6fbf35ccdcb1447b6.png" />
<img alt="../_images/b9035cad5be313768c8516100330a097323a9eb6c9cebf7b21e48c3500c8ec02.png" src="../_images/b9035cad5be313768c8516100330a097323a9eb6c9cebf7b21e48c3500c8ec02.png" />
</div>
</div>
<p>You can see by inspection that the fit looks pretty reasonable. The parameter confidence intervals are not too big, so we can be pretty confident of their values.</p>
Expand Down Expand Up @@ -1064,7 +1064,7 @@ <h2>Nonlinear curve fitting with confidence intervals<a class="headerlink" href=
c1: 2.1099511246584144 [1.7671162238028324 2.4527860255139964]
</pre></div>
</div>
<img alt="../_images/7ec782d86cff42f160b56f649ca49452b54f1b071db22ed7312387d577b87a71.png" src="../_images/7ec782d86cff42f160b56f649ca49452b54f1b071db22ed7312387d577b87a71.png" />
<img alt="../_images/ef4bf682b11c7286704710cf714511aee9e12334706dbdc859bb7bcfaf90f535.png" src="../_images/ef4bf682b11c7286704710cf714511aee9e12334706dbdc859bb7bcfaf90f535.png" />
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</div>
</section>
Expand Down Expand Up @@ -1096,7 +1096,7 @@ <h2>Graphical methods to help get initial guesses for multivariate nonlinear reg
</div>
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<div class="cell_output docutils container">
<img alt="../_images/c6861fa3f7459ae33c59dab3aafe595e094c4f942ce4a7647cf631f31b750208.png" src="../_images/c6861fa3f7459ae33c59dab3aafe595e094c4f942ce4a7647cf631f31b750208.png" />
<img alt="../_images/1dc50d5bb826b2e7e0fd005cacbb350fbfc0e298482bf7f8d8797c8ab36ec47c.png" src="../_images/1dc50d5bb826b2e7e0fd005cacbb350fbfc0e298482bf7f8d8797c8ab36ec47c.png" />
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Expand Down Expand Up @@ -1145,7 +1145,7 @@ <h2>Graphical methods to help get initial guesses for multivariate nonlinear reg
[ 3.25873623 6.59792994 10.29473657 13.68011436 17.29161001 23.62366445]
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<p>It can be difficult to figure out initial guesses for nonlinear fitting problems. For one and two dimensional systems, graphical techniques may be useful to visualize how the summed squared error between the model and data depends on the parameters.</p>
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