replaced instances of average/mean with median to reflect new figures

tdabench-paper/tdabench.Rmd

@@ -136,7 +136,7 @@ Therefore, we can predict that low dimensional point clouds favor alpha complexe
 \section{Methods}
 
 We use \CRANpkg{readr} v1.3.1 to read rectangular data \citep{readr}, \CRANpkg{ggplot2} v3.2.1 \citep{ggplot2}, \CRANpkg{scatterplot3d} v0.3-41 \citep{scatterplot3d}, \CRANpkg{recexcavAAR} v0.3.0 \citep{recexcavAAR}, \CRANpkg{deldir} v0.1 \citep{deldir}, ggtda v0.1 \citep{ggtda}, and \CRANpkg{magick} v2.2 \citep{magick} to visualize data, \CRANpkg{bench} v1.0.4 to collect benchmark data \citep{bench}, \CRANpkg{TDA} v1.6.9 \citep{TDA} and \CRANpkg{TDAstats} v0.4.1 \citep{TDAstats} to calculate persistent homology of Vietoris-Rips and alpha simplicial complices, and \CRANpkg{pryr} v0.1.4 for calculations involving R objects \citep{pryr}.
-Runtime calculations were averaged over 10 iterations and are visualized as mean $\pm$ standard deviation.
+Median runtime calculations are shown along with the minimum and maximum of 10 iterations per benchmark.
 Datasets were generated by sampling functions in base R to generate points uniformly distributed over circles (dimension = 2), spheres (dimension = 3), filled squares (dimension = 2), filled cubes (dimension = 3, 4), and tori (dimension = 3).
 The number of points per point cloud varied from 25 to 500 along intervals of 25 points, which were empirical limits chosen after considering available computational resources.
 For consistency between software libraries, the minimum and maximum simplicial complex radii were predetermined for each point cloud and provided as parameters to the \CRANpkg{TDA} and \CRANpkg{TDAstats} R packages.
@@ -157,15 +157,15 @@ Video explanations of TDA concepts and reproducing all results in this report ca
 \section{Results}
 
 Computing persistent homology of a canonical torus grants quick insight into efficiency of each library (Figure \ref{fig:tor}).
-Dionysus exhibits the longest average runtime, and, although Ripser and GUDHI have similar runtimes for smaller point clouds, as the number of points increases Ripser eventually has a significant lead.
+Dionysus exhibits the longest median runtime, and, although Ripser and GUDHI have similar runtimes for smaller point clouds, as the number of points increases Ripser eventually has a significant lead.
 Next, we compare library performance with multiple canonical datasets to ensure that the noted pattern generalizes.
 
 \begin{figure}
   \centering
   \includegraphics[height=2.5in]{../Figures/Final_Figures/fig1.png}
   \caption{
     \textbf{Calculating persistent homology of a torus with three TDA libraries.}
-    Average runtime (mean $\pm$ SD, $n=10$ iterations per data point) for each TDA library (denoted by color) is plotted against point cloud size.
+    Median runtime (min to max, $n=10$ iterations per data point) for each TDA library (denoted by color) is plotted against point cloud size.
     Homological features up to 2 dimensions were calculated.
     Time complexity follows a power law for all three libraries (see GitHub repo for regression details).
     Although the libraries have similar runtimes for smaller point clouds, Dionysus has a clear disadvantage when the number of points exceeds 100.
@@ -189,7 +189,7 @@ Interestingly, all curves plotted in Figure \ref{fig:cir} grow polynomially with
   \includegraphics[height=2in]{../Figures/Final_Figures/fig2.png}
   \caption{
     \textbf{Calculating persistent homology of round point clouds of varying dimensions with three TDA libraries.}
-    Average runtime (mean $\pm$ SD, $n=10$ iterations per data point) for each TDA library (denoted by color) is plotted against point cloud size and faceted by data dimension.
+    Median runtime (min to max, $n=10$ iterations per data point) for each TDA library (denoted by color) is plotted against point cloud size and faceted by data dimension.
     Left panel compares library performance for a 2-dimensional circular point cloud; center panel for a 3-dimensional spherical point cloud; and right panel for a 4-dimensional hyperspherical point cloud.
     Maximum feature dimensions (one less than data dimension) were calculated in each case.
   }
@@ -206,7 +206,7 @@ It is unclear whether runtime for each library grows polynomially or exponential
   \includegraphics[height=2in]{../Figures/Final_Figures/fig3.png}
   \caption{
     \textbf{Comparison of Vietoris-Rips complex persistent homology calculation as a function of feature dimension.}
-    Average runtime (mean $\pm$ SD, $n=10$ iterations per data point) for various point cloud sizes (denoted by color) is plotted against calculated feature dimension and faceted by TDA libary.
+    Median runtime (min to max, $n=10$ iterations per data point) for various point cloud sizes (denoted by color) is plotted against calculated feature dimension and faceted by TDA libary.
     Persistent homology was calculated on a uniformly distributed random sample of points contained within a 1 unit, 8-dimensional cube.
     Computational limitations of calculating persistent homology for a large number of feature dimensions restricted point clouds to relatively small sizes.}
 \label{fig:box}
@@ -227,7 +227,7 @@ Although unconcerning for a data dimension up to 3, failure to run any alpha com
   \includegraphics[height=2in]{../Figures/Final_Figures/fig4.png}
   \caption{
     \textbf{Comparing persistent homology calculation between Vietoris-Rips and alpha complices.}
-    Average runtime (mean $\pm$ SD, $n=10$ iterations per data point) for various data dimensions (denoted by color) are plotted against point cloud size and faceted by type of simplicial complex.
+    Median runtime (min to max, $n=10$ iterations per data point) for various data dimensions (denoted by color) are plotted against point cloud size and faceted by type of simplicial complex.
     Maximum of feature dimension was kept constant at 1.
     Alpha complex runtimes are linear, in contrast to polynomial Vietoris-Rips runtimes (see Supplement for regression details).}
 \label{fig:ann}
@@ -248,7 +248,7 @@ Table \ref{tbl:limit} lists the maximum point cloud size for which persistent ho
   \includegraphics[height=2.5in]{../Figures/Final_Figures/fig5a.png}
   \caption{
     \textbf{Comparing memory usage of persistent homology calculation using a Vietoris-Rips versus an alpha complex.}
-    Mean boundary matrix size, a proxy for memory usage, plotted against point cloud size varying by point cloud structure (facet) and simplicial complex type (color, shape).
+    Boundary matrix size, a proxy for memory usage, plotted against point cloud size varying by point cloud structure (facet) and simplicial complex type (color, shape).
     Data dimension and feature dimension were held constant at 3 and 2, respectively.
     Point cloud structures for which memory usage was calculated were 3-dimensional annulus (top-left), 3-dimensional sphere (top-right), torus (bottom-left), and 3-dimensional cube (bottom-right).
     Similar to runtime results, alpha complex calculation showed linear growth and Vietoris-Rips complex calculation showed polynomial growth (see Supplement for regression details).}
@@ -260,7 +260,7 @@ Table \ref{tbl:limit} lists the maximum point cloud size for which persistent ho
   \includegraphics[height=2.5in]{../Figures/Final_Figures/fig5t.png}
   \caption{
     \textbf{Factors affecting boundary matrix size in Vietoris-Rips and alpha complices.}
-    Mean boundary matrix size plotted against point cloud size varying by point cloud structure (right, faceted), simplicial complex type (left, right), and feature dimension (color).
+    Boundary matrix size plotted against point cloud size varying by point cloud structure (right, faceted), simplicial complex type (left, right), and feature dimension (color).
     Whereas the boundary matrix for a Vietoris-Rips complex (left) is independent of point cloud structure, the same is not true for an alpha complex (right, faceted).
     For the Vietoris-Rips complex, the decreasing number of data points plotted for successive feature dimensions reflects the computational requirements of increasing point cloud size.}
 \label{fig:mem2}
@@ -307,7 +307,7 @@ Vietoris-Rips complex calculations consistently had a polynomial growth for both
   \includegraphics[height=2.5in]{../Figures/Final_Figures/fig6.png}
   \caption{
     \textbf{Runtime comparison of persistent homology calculation between Ripser's Vietoris-Rips and GUDHI's alpha complex functionality.}
-    Average runtime (mean $\pm$ SD, $n=10$ iterations per data point) for various 3-dimensional point cloud structures (facet) plotted against point cloud size for each library (color).
+    Median runtime (min to max $n=10$ iterations per data point) for various 3-dimensional point cloud structures (facet) plotted against point cloud size for each library (color).
     Benchmarking was conducted on an annulus (top-left), a sphere (top-right), a torus (bottom-left), and a cube (bottom-right).
     Data was not collected for data dimensions greater than 3 due to computational limitations of calculating alpha complices.}
 \label{fig:fin}