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-{"documenter":{"julia_version":"1.10.1","generation_timestamp":"2024-02-21T02:17:12","documenter_version":"1.2.1"}}
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     \mathrm{min}\quad &amp; \frac{1}{2} \Vert G - X \Vert^2 \\
     \mathrm{s.t.}\quad &amp; X_{ii} = 1, \quad i = 1, \ldots , n, \\
     &amp; X \in S_{+}^{n}
-\end{aligned}\]</p><ul><li>Qi, H., &amp; Sun, D. (2006). A quadratically convergent Newton method for computing the nearest correlation matrix. SIAM journal on matrix analysis and applications, 28(2), 360-385.</li></ul><h2 id="Normal-to-Anything-(NORTA)"><a class="docs-heading-anchor" href="#Normal-to-Anything-(NORTA)">Normal to Anything (NORTA)</a><a id="Normal-to-Anything-(NORTA)-1"></a><a class="docs-heading-anchor-permalink" href="#Normal-to-Anything-(NORTA)" title="Permalink"></a></h2><p>Given:</p><ul><li>A target correlation matrix, <span>$\rho$</span></li><li>A list of marginal distributions, <span>$F$</span></li></ul><p>Do:</p><ul><li>Generate <span>$Z_{n \times d} = \mathcal{N}(0, 1)$</span> IID standard normal samples</li><li>Transform <span>$Y = ZC$</span> where <span>$C$</span> is the upper Cholesky factor of <span>$\rho$</span></li><li>Transform <span>$U = \Phi(Y)$</span> where <span>$\Phi(\cdot)$</span> is the CDF of the standard normal distribution</li><li>Transform <span>$X_i = F_{i}^{-1}(U_i)$</span></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../main_functions/">« API Reference</a><a class="docs-footer-nextpage" href="../function_index/">Index »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Wednesday 21 February 2024 02:17">Wednesday 21 February 2024</span>. Using Julia version 1.10.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{aligned}\]</p><ul><li>Qi, H., &amp; Sun, D. (2006). A quadratically convergent Newton method for computing the nearest correlation matrix. SIAM journal on matrix analysis and applications, 28(2), 360-385.</li></ul><h2 id="Normal-to-Anything-(NORTA)"><a class="docs-heading-anchor" href="#Normal-to-Anything-(NORTA)">Normal to Anything (NORTA)</a><a id="Normal-to-Anything-(NORTA)-1"></a><a class="docs-heading-anchor-permalink" href="#Normal-to-Anything-(NORTA)" title="Permalink"></a></h2><p>Given:</p><ul><li>A target correlation matrix, <span>$\rho$</span></li><li>A list of marginal distributions, <span>$F$</span></li></ul><p>Do:</p><ul><li>Generate <span>$Z_{n \times d} = \mathcal{N}(0, 1)$</span> IID standard normal samples</li><li>Transform <span>$Y = ZC$</span> where <span>$C$</span> is the upper Cholesky factor of <span>$\rho$</span></li><li>Transform <span>$U = \Phi(Y)$</span> where <span>$\Phi(\cdot)$</span> is the CDF of the standard normal distribution</li><li>Transform <span>$X_i = F_{i}^{-1}(U_i)$</span></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../main_functions/">« API Reference</a><a class="docs-footer-nextpage" href="../function_index/">Index »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Wednesday 21 February 2024 02:24">Wednesday 21 February 2024</span>. Using Julia version 1.10.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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@@ -10,47 +10,47 @@
 | 114 |    14 |    75 |
 | 115 |    18 |    76 |
 | 116 |    20 |    68 |
-       110 rows omitted</code></pre><p>Let’s look at the joint distribution of the Ozone and Temperature</p><img src="79aecc74.svg" alt="Example block output"/><p>We can see that not all margins are normally distributed; the ozone level is highly skewed. Though we don’t know the true distribution of ozone levels, we can go forward assuming that it is log-normally distributed.</p><p>To simulate observations from this joint distribution, we need to estimate the correlation and the marginal parameters.</p><h2 id="Estimating-Correlation"><a class="docs-heading-anchor" href="#Estimating-Correlation">Estimating Correlation</a><a id="Estimating-Correlation-1"></a><a class="docs-heading-anchor-permalink" href="#Estimating-Correlation" title="Permalink"></a></h2><p>To estimate the correlation, we use <code>cor</code> with an argument specifying the type of correlation to estimate. The options are <code>Pearson</code>, <code>Spearman</code>, or <code>Kendall</code>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; ρ = cor(Matrix(df), Pearson)</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}:
+       110 rows omitted</code></pre><p>Let’s look at the joint distribution of the Ozone and Temperature</p><img src="e08869a4.svg" alt="Example block output"/><p>We can see that not all margins are normally distributed; the ozone level is highly skewed. Though we don’t know the true distribution of ozone levels, we can go forward assuming that it is log-normally distributed.</p><p>To simulate observations from this joint distribution, we need to estimate the correlation and the marginal parameters.</p><h2 id="Estimating-Correlation"><a class="docs-heading-anchor" href="#Estimating-Correlation">Estimating Correlation</a><a id="Estimating-Correlation-1"></a><a class="docs-heading-anchor-permalink" href="#Estimating-Correlation" title="Permalink"></a></h2><p>To estimate the correlation, we use <code>cor</code> with an argument specifying the type of correlation to estimate. The options are <code>Pearson</code>, <code>Spearman</code>, or <code>Kendall</code>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; ρ = cor(Matrix(df), Pearson)</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}:
  1.0      0.69836
  0.69836  1.0</code></pre><h2 id="Defining-Marginal-Distributions"><a class="docs-heading-anchor" href="#Defining-Marginal-Distributions">Defining Marginal Distributions</a><a id="Defining-Marginal-Distributions-1"></a><a class="docs-heading-anchor-permalink" href="#Defining-Marginal-Distributions" title="Permalink"></a></h2><p>Next we can estimate the marginal parameters. Assuming that the <code>Temperature</code> is normally distributed, it has parameters:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; μ_Temp = mean(df.Temp)</code><code class="nohighlight hljs ansi" style="display:block;">77.87068965517241</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; σ_Temp = std(df.Temp)</code><code class="nohighlight hljs ansi" style="display:block;">9.48548563759966</code></pre><p>and assuming that <code>Ozone</code> is log-normally distributed, it has parameters:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; μ_Ozone = mean(log.(df.Ozone))</code><code class="nohighlight hljs ansi" style="display:block;">3.418515100812007</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; σ_Ozone = sqrt(mean((log.(df.Ozone) .- mean(log.(df.Ozone))).^2))</code><code class="nohighlight hljs ansi" style="display:block;">0.8617359690270703</code></pre><p>Finally we take the parameters and put them into a vector of margins:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; margins = [Normal(μ_Temp, σ_Temp), LogNormal(μ_Ozone, σ_Ozone)]</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{Distribution{Univariate, Continuous}}:
  Normal{Float64}(μ=77.87068965517241, σ=9.48548563759966)
  LogNormal{Float64}(μ=3.418515100812007, σ=0.8617359690270703)</code></pre><h2 id="Correlation-Bounds"><a class="docs-heading-anchor" href="#Correlation-Bounds">Correlation Bounds</a><a id="Correlation-Bounds-1"></a><a class="docs-heading-anchor-permalink" href="#Correlation-Bounds" title="Permalink"></a></h2><p>Given a vector of margins, the theoretical lower and upper correlation coefficients can be estimated using simulation:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; lower, upper = cor_bounds(margins, Pearson);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; lower</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}:
-  1.0       -0.807527
- -0.807527   1.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; upper</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}:
- 1.0       0.808883
- 0.808883  1.0</code></pre><p>The <code>pearson_bounds</code> function uses more sophisticated methods to determine the theoretical lower and upper Pearson correlation bounds. It also requires more computational time.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; lower, upper = pearson_bounds(margins);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; lower</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}:
+  1.0       -0.827695
+ -0.827695   1.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; upper</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}:
+ 1.0       0.826898
+ 0.826898  1.0</code></pre><p>The <code>pearson_bounds</code> function uses more sophisticated methods to determine the theoretical lower and upper Pearson correlation bounds. It also requires more computational time.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; lower, upper = pearson_bounds(margins);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; lower</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}:
   1.0       -0.821122
  -0.821122   1.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; upper</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}:
  1.0       0.821122
  0.821122  1.0</code></pre><h2 id="Simulating-Multivariate-Data"><a class="docs-heading-anchor" href="#Simulating-Multivariate-Data">Simulating Multivariate Data</a><a id="Simulating-Multivariate-Data-1"></a><a class="docs-heading-anchor-permalink" href="#Simulating-Multivariate-Data" title="Permalink"></a></h2><p>Let’s now simulate 10,000 observations from the joint distribution using <code>rvec</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; x = rvec(10_000, ρ, margins)</code><code class="nohighlight hljs ansi" style="display:block;">10000×2 Matrix{Float64}:
- 73.5149   33.4932
- 87.8056   28.5674
- 68.7403   12.457
- 61.3621    7.22837
- 74.7645   70.4748
- 75.4706   20.6238
- 69.9828   10.0259
- 56.979     6.31042
- 82.285    45.5811
- 75.3011   37.3477
+ 87.4109   22.3591
+ 75.7104   19.7585
+ 61.4875    7.91311
+ 87.2304   93.3604
+ 80.5391   20.9102
+ 74.0107   42.8532
+ 92.2595   76.5276
+ 59.7821   20.2999
+ 73.4545    4.69385
+ 77.3004   18.7691
   ⋮
- 74.3841   33.5517
- 76.7599   29.6025
- 93.9216  136.805
- 87.9215   91.6839
- 84.193    56.5316
- 71.6759   21.7275
- 58.5467    8.13902
- 80.2676   41.4696
- 69.3834   10.5428</code></pre><h2 id="Visualizing-Bivariate-Data"><a class="docs-heading-anchor" href="#Visualizing-Bivariate-Data">Visualizing Bivariate Data</a><a id="Visualizing-Bivariate-Data-1"></a><a class="docs-heading-anchor-permalink" href="#Visualizing-Bivariate-Data" title="Permalink"></a></h2><pre><code class="language-julia hljs">df_sim = DataFrame(x, [:Temp, :Ozone]);
+ 82.8761   34.0147
+ 75.0262   41.8732
+ 67.5626   53.1339
+ 64.4427    7.12912
+ 81.6657   47.3555
+ 63.7596    7.44514
+ 88.6181   44.4193
+ 83.7085  134.4
+ 71.7335   35.387</code></pre><h2 id="Visualizing-Bivariate-Data"><a class="docs-heading-anchor" href="#Visualizing-Bivariate-Data">Visualizing Bivariate Data</a><a id="Visualizing-Bivariate-Data-1"></a><a class="docs-heading-anchor-permalink" href="#Visualizing-Bivariate-Data" title="Permalink"></a></h2><pre><code class="language-julia hljs">df_sim = DataFrame(x, [:Temp, :Ozone]);
 
 histogram2d(df_sim.:Temp, df_sim.:Ozone, nbins=250, legend=false,
 			xlims=extrema(df.:Temp) .+ (-10, 10),
-			ylims=extrema(df.:Ozone) .+ (0, 20))</code></pre><img src="413288e4.svg" alt="Example block output"/><p><strong>Compared to Uncorrelated Samples</strong></p><p>We can compare the bivariate distribution above to one where no correlation is taken into account.</p><pre><code class="language-julia hljs">df_sim2 = DataFrame(
+			ylims=extrema(df.:Ozone) .+ (0, 20))</code></pre><img src="29851180.svg" alt="Example block output"/><p><strong>Compared to Uncorrelated Samples</strong></p><p>We can compare the bivariate distribution above to one where no correlation is taken into account.</p><pre><code class="language-julia hljs">df_sim2 = DataFrame(
 	Temp  = rand(margins[1], 10000),
 	Ozone = rand(margins[2], 10000)
 );
 
 histogram2d(df_sim2.:Temp, df_sim2.:Ozone, nbins=250, legend=false,
 			xlims=extrema(df.:Temp) .+ (-10, 10),
-			ylims=extrema(df.:Ozone) .+ (0, 20))</code></pre><img src="cc250c52.svg" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Bigsimr.jl</a><a class="docs-footer-nextpage" href="../pearson_matching/">Pearson Matching »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Wednesday 21 February 2024 02:17">Wednesday 21 February 2024</span>. Using Julia version 1.10.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+			ylims=extrema(df.:Ozone) .+ (0, 20))</code></pre><img src="15ad7dd8.svg" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Bigsimr.jl</a><a class="docs-footer-nextpage" href="../pearson_matching/">Pearson Matching »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Wednesday 21 February 2024 02:24">Wednesday 21 February 2024</span>. Using Julia version 1.10.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>