- updated optimizeAPA tutorial post

content/post/2020-04-05-optimizeAPA-tutorial/index.Rmd

@@ -27,6 +27,8 @@ library(ggplot2)
 library(optimizeAPA)
 library(gmp)
 library(Rmpfr)
+library(knitr)
+library(rgl)
 ```
 
 <tt>optimizeAPA</tt> is an R package which allows for multi-parameter optimization. That means you can use it to find the maximum (or the minimum) value of a function with many input values. What makes <tt>optimizeAPA</tt> unique? It works with arbitrary precision arithmetic.
@@ -105,8 +107,9 @@ remotes::install_github("mrparker909/optimizeAPA")
 For the folowing examples, we will use these libraries:
 
 ```{r eval=F}
-library(ggplot2)
-library(optimizeAPA)
+library(ggplot2)     # for 2d plots
+library(optimizeAPA) # for functional optimization
+library(rgl)         # for interactive 3d plots
 ```
 
 ### Single Parameter Function
@@ -150,8 +153,8 @@ op1 <- optimizeAPA::optim_DFP_NAPA(starts=3.5,
 
 Let's look at the path to convergence:
 
-```{r fig.cap="Plot of convergence for single parameter function F1 (NAPA).", fig.width=6, fig.height=6, warning=F, message=F}
-optimizeAPA::plotConvergence(op1)
+```{r fig.cap="Plot of convergence for single parameter function F1 (NAPA).", fig.width=10, fig.height=4, warning=F, message=F}
+optimizeAPA::plotConvergence(op1, flip_axes=T)
 ```
 
 So the maximum value of the function (found after 11 iterations of the algorithm) is $F_0 = F1(3.2543) = 4.235$.
@@ -181,8 +184,8 @@ op2 <- optimizeAPA::optim_DFP_APA(starts=3.5,
 
 Let's look at the path to convergence:
 
-```{r fig.cap="Plot of convergence for single parameter function F2 (APA).", fig.width=6, fig.height=6}
-optimizeAPA::plotConvergence(op2)
+```{r fig.cap="Plot of convergence for single parameter function F2 (APA).", fig.width=10, fig.height=4}
+optimizeAPA::plotConvergence(op2, flip_axes = T)
 ```
 
 So the convergence plots look the same, but let's compare the values found by the two algorithms:
@@ -247,10 +250,10 @@ op3 <- optimizeAPA::optim_DFP_NAPA(starts=c(x[29],y[29]),
 
 Again we can look at the convergence plot:
 
-```{r fig.cap="Plot of convergence for multi-parameter function F3 (NAPA).", fig.width=9, fig.height=9}
+```{r fig.cap="Plot of convergence for multi-parameter function F3 (NAPA).", fig.width=10, fig.height=4.5}
 # this time we disable the function labels for legibility
 # using: labels = F
-optimizeAPA::plotConvergence(op3, labels = F)
+optimizeAPA::plotConvergence(op3, labels = F, flip_axes = T)
 ```
 
 The algorithm converged after `r op3$steps` iterations. We can get at the maximum function value and at the values of $x$ and $y$ which jointly maximize the function:
@@ -262,6 +265,40 @@ op3$x[[1]]    # note that "x" here denotes c(x,y)
 
 The maximum value of $F3$ is `r -1*op3$f[[1]]`, and that value is attained by $x=$ `r op3$x[[1]][1]`, and $y=$ `r op3$x[[1]][2]`.
 
+We can also visualize what has happened by plotting the convergence path on the 3D plot of the function:
+
+
+```{r eval=F}
+# here we define an x,y coordinate grid, and we
+# calculate z=F3(x,y) at each point of the grid
+x = seq(3.23,3.29,length=50)
+y = seq(2.23,2.29,length=50)
+z = outer(x,y,FUN = Vectorize(function(x,y) { F3(par=c(x,y)) }))
+
+# using rgl
+persp3d(x,y,z,col="dodgerblue",border="navy",
+        theta=30,phi=22,shade=0.75,
+        xaxs="i",yaxs="i",
+        ltheta=-55,lphi=30, alpha=0.45)
+lines3d(x=unlist(op3$x)[c(T,F)],
+        y=unlist(op3$x)[c(F,T)],
+        z=-1*unlist(op3$f), 
+        col="blue", size=5)
+points3d(x=unlist(op3$x)[c(T,F)],
+         y=unlist(op3$x)[c(F,T)],
+         z=-1*unlist(op3$f), 
+         col="red", size=5)
+rgl.postscript(filename = "img/rgl-snapshot.pdf", fmt = "pdf")
+```
+
+
+```{r echo=F, fig.cap="Plot of multi-parameter function (zoomed near maximum). Visited points shown in red, path of convergence traced in blue."}
+
+knitr::include_graphics("img/convergence-path-zoomed.png")
+```
+
+
+
 Let's make $F3$ APA:
 
 
@@ -286,10 +323,10 @@ op4 <- optimizeAPA::optim_DFP_APA(starts=c(x[29],y[29]),
 
 This time the convergence plot is slightly different, and the convergence took `r op4$steps` steps to achieve:
 
-```{r fig.cap="Plot of convergence for multi-parameter function F4 (APA).", fig.width=9, fig.height=9}
+```{r fig.cap="Plot of convergence for multi-parameter function F4 (APA).", fig.width=10, fig.height=4.5}
 # this time we disable the function labels for legibility
 # using: labels = F
-optimizeAPA::plotConvergence(op4, labels = F)
+optimizeAPA::plotConvergence(op4, labels = F, flip_axes = T)
 ```
 
 ```{r}

content/post/2020-04-05-optimizeAPA-tutorial/index.html

@@ -74,8 +74,9 @@ <h2>Install <tt>optimizeAPA</tt></h2>
 <div id="examples" class="section level2">
 <h2>Examples</h2>
 <p>For the folowing examples, we will use these libraries:</p>
-<pre class="r"><code>library(ggplot2)
-library(optimizeAPA)</code></pre>
+<pre class="r"><code>library(ggplot2)     # for 2d plots
+library(optimizeAPA) # for functional optimization
+library(rgl)         # for interactive 3d plots</code></pre>
 <div id="single-parameter-function" class="section level3">
 <h3>Single Parameter Function</h3>
 <p>First, let’s define a regular function <tt>F1</tt> to optimize:</p>
@@ -107,9 +108,9 @@ <h3>Single Parameter Function</h3>
                                    keepValues = T, 
                                    tolerance=10^-3)</code></pre>
 <p>Let’s look at the path to convergence:</p>
-<pre class="r"><code>optimizeAPA::plotConvergence(op1)</code></pre>
+<pre class="r"><code>optimizeAPA::plotConvergence(op1, flip_axes=T)</code></pre>
 <div class="figure"><span id="fig:unnamed-chunk-9"></span>
-<img src="/post/2020-04-05-optimizeAPA-tutorial/index_files/figure-html/unnamed-chunk-9-1.png" alt="Plot of convergence for single parameter function F1 (NAPA)." width="576" />
+<img src="/post/2020-04-05-optimizeAPA-tutorial/index_files/figure-html/unnamed-chunk-9-1.png" alt="Plot of convergence for single parameter function F1 (NAPA)." width="960" />
 <p class="caption">
 Figure 2: Plot of convergence for single parameter function F1 (NAPA).
 </p>
@@ -130,9 +131,9 @@ <h3>Single Parameter Function</h3>
                                   tolerance=10^-3, 
                                   precBits = 64)</code></pre>
 <p>Let’s look at the path to convergence:</p>
-<pre class="r"><code>optimizeAPA::plotConvergence(op2)</code></pre>
+<pre class="r"><code>optimizeAPA::plotConvergence(op2, flip_axes = T)</code></pre>
 <div class="figure"><span id="fig:unnamed-chunk-12"></span>
-<img src="/post/2020-04-05-optimizeAPA-tutorial/index_files/figure-html/unnamed-chunk-12-1.png" alt="Plot of convergence for single parameter function F2 (APA)." width="576" />
+<img src="/post/2020-04-05-optimizeAPA-tutorial/index_files/figure-html/unnamed-chunk-12-1.png" alt="Plot of convergence for single parameter function F2 (APA)." width="960" />
 <p class="caption">
 Figure 3: Plot of convergence for single parameter function F2 (APA).
 </p>
@@ -207,9 +208,9 @@ <h3>Multi-Parameter Function</h3>
 <p>Again we can look at the convergence plot:</p>
 <pre class="r"><code># this time we disable the function labels for legibility
 # using: labels = F
-optimizeAPA::plotConvergence(op3, labels = F)</code></pre>
+optimizeAPA::plotConvergence(op3, labels = F, flip_axes = T)</code></pre>
 <div class="figure"><span id="fig:unnamed-chunk-18"></span>
-<img src="/post/2020-04-05-optimizeAPA-tutorial/index_files/figure-html/unnamed-chunk-18-1.png" alt="Plot of convergence for multi-parameter function F3 (NAPA)." width="864" />
+<img src="/post/2020-04-05-optimizeAPA-tutorial/index_files/figure-html/unnamed-chunk-18-1.png" alt="Plot of convergence for multi-parameter function F3 (NAPA)." width="960" />
 <p class="caption">
 Figure 5: Plot of convergence for multi-parameter function F3 (NAPA).
 </p>
@@ -222,6 +223,33 @@ <h3>Multi-Parameter Function</h3>
 ## [1,] 3.254288
 ## [2,] 2.254288</code></pre>
 <p>The maximum value of <span class="math inline">\(F3\)</span> is 17.9352205, and that value is attained by <span class="math inline">\(x=\)</span> 3.2542878, and <span class="math inline">\(y=\)</span> 2.2542877.</p>
+<p>We can also visualize what has happened by plotting the convergence path on the 3D plot of the function:</p>
+<pre class="r"><code># here we define an x,y coordinate grid, and we
+# calculate z=F3(x,y) at each point of the grid
+x = seq(3.23,3.29,length=50)
+y = seq(2.23,2.29,length=50)
+z = outer(x,y,FUN = Vectorize(function(x,y) { F3(par=c(x,y)) }))
+
+# using rgl
+persp3d(x,y,z,col=&quot;dodgerblue&quot;,border=&quot;navy&quot;,
+        theta=30,phi=22,shade=0.75,
+        xaxs=&quot;i&quot;,yaxs=&quot;i&quot;,
+        ltheta=-55,lphi=30, alpha=0.45)
+lines3d(x=unlist(op3$x)[c(T,F)],
+        y=unlist(op3$x)[c(F,T)],
+        z=-1*unlist(op3$f), 
+        col=&quot;blue&quot;, size=5)
+points3d(x=unlist(op3$x)[c(T,F)],
+         y=unlist(op3$x)[c(F,T)],
+         z=-1*unlist(op3$f), 
+         col=&quot;red&quot;, size=5)
+rgl.postscript(filename = &quot;img/rgl-snapshot.pdf&quot;, fmt = &quot;pdf&quot;)</code></pre>
+<div class="figure"><span id="fig:unnamed-chunk-21"></span>
+<img src="img/convergence-path-zoomed.png" alt="Plot of multi-parameter function (zoomed near maximum). Visited points shown in red, path of convergence traced in blue."  />
+<p class="caption">
+Figure 6: Plot of multi-parameter function (zoomed near maximum). Visited points shown in red, path of convergence traced in blue.
+</p>
+</div>
 <p>Let’s make <span class="math inline">\(F3\)</span> APA:</p>
 <pre class="r"><code>F4 &lt;- function(par, precBits=53) {
   x = par[1]
@@ -239,11 +267,11 @@ <h3>Multi-Parameter Function</h3>
 <p>This time the convergence plot is slightly different, and the convergence took 38 steps to achieve:</p>
 <pre class="r"><code># this time we disable the function labels for legibility
 # using: labels = F
-optimizeAPA::plotConvergence(op4, labels = F)</code></pre>
-<div class="figure"><span id="fig:unnamed-chunk-22"></span>
-<img src="/post/2020-04-05-optimizeAPA-tutorial/index_files/figure-html/unnamed-chunk-22-1.png" alt="Plot of convergence for multi-parameter function F4 (APA)." width="864" />
+optimizeAPA::plotConvergence(op4, labels = F, flip_axes = T)</code></pre>
+<div class="figure"><span id="fig:unnamed-chunk-24"></span>
+<img src="/post/2020-04-05-optimizeAPA-tutorial/index_files/figure-html/unnamed-chunk-24-1.png" alt="Plot of convergence for multi-parameter function F4 (APA)." width="960" />
 <p class="caption">
-Figure 6: Plot of convergence for multi-parameter function F4 (APA).
+Figure 7: Plot of convergence for multi-parameter function F4 (APA).
 </p>
 </div>
 <pre class="r"><code>knitr::kable(data.frame(algorithm = c(&quot;F3: NAPA&quot;,   &quot;F4: APA&quot;),

layouts/_default/baseof.html

@@ -0,0 +1,24 @@
+<!DOCTYPE html>
+<html lang="{{ site.LanguageCode | default "en-us" }}">
+
+{{ partial "site_head.html" . }}
+<body id="top" data-spy="scroll" data-offset="70" data-target="{{ if or .IsHome (eq .Type "widget_page") }}#navbar-main{{else}}#TableOfContents{{end}}" {{ if not (.Scratch.Get "light") }}class="dark"{{end}}>
+
+  {{ partial "search" . }}
+
+  {{ partial "navbar.html" . }}
+
+  {{ block "main" . }}{{ end }}
+
+  {{ partial "site_js.html" . }}
+
+  {{ if ne .Type "docs" }}
+  <div class="container">
+    {{ partial "site_footer.html" . }}
+  </div>
+  {{ end }}
+
+  {{ partial "citation.html" . }}
+
+</body>
+</html>