# Visualization: elliptical insights into data sets…

I was recently reading the following paper on visualization techniques using ellipses to gain statistical insights. Elliptical visualization for looking at correlations has briefly been touched on this board as well.

There are a couple of plots in the paper on quickly plotting correlations between complex data that I found quite useful and would like to replicate them. Having a function to visualize complex correlations would be useful to a lot of folks… However, my skills in MMA are lacking a bit...

• I believe you'll have to construct these manually using overlays. ListPlot will create the scatters, and LinearModelFit will regress the data and give you the error information you need to calculate the ellipses. Can you take a stab at implementing the ellipse method described in the paper, or at least spell out the steps? Here's a nudge as far as splitting the data goes: groups = Map[Tuples[Most@#, {2}] &, GatherBy[irisData, Last], {2}]; ListPlot[groups[[All, All, #]]] & /@ Range@16 // Partition[#, 4] & // TableForm Jun 20, 2014 at 0:57
• Look at this link. You can find fantastic techniques related to Confidence ellipses.
– user9660
Jun 20, 2014 at 5:26
• Pam, if you've come up with some code that answers the question, please post it as an answer (it's allowed, in fact positively encouraged, to answer your own question). Jun 24, 2014 at 14:33
• tnx. just posted it as an answer.
– Pam
Jun 24, 2014 at 14:46

I don't have time to do the full-monty on the question, but perhaps this little-known functionality might be of use:

Needs["MultivariateStatistics"]

(* fake some data *)
data = RandomVariate[BinormalDistribution[{20, 20}, {5, 5}, .75], 500];

Show[{ListPlot[data, PlotRange -> Automatic, AspectRatio -> 1],
Graphics[{Red, EllipsoidQuantile[data, .95],
Green, EllipsoidQuantile[data, .99]}]},
PlotRange -> {{0, 40}, {0, 40}}]


• this is great… simplifies stuff…
– Pam
Jun 20, 2014 at 13:13

Updated with working code (tnx @rasher @mfvonh)

Let’s start by importing Fisher’s classic dataset on Iris flower measurements… Fisher’s classic paper can be found here….

Needs["MultivariateStatistics"]

(*Import Data*)
irisData = Import["http://aima.cs.berkeley.edu/data/iris.csv", "CSV"];
plotLabels = {"Sepal.Length", "Sepal.Width", "Petal.Length", "Petal.Width", "Type"};
T= Transpose;
(*Parse Data and Regress, thanks @mfvonh *)

groups = Map[Tuples[Most@#, {2}] &, GatherBy[irisData, Last], {2}];
pairs = (Dimensions@groups)[[3]];
lm = Table[LinearModelFit[groups[[All, All, i]][[#]], {x}, x] & /@ Range[3], {i, 1, pairs}];
plotLabels =Flatten@
ConstantArray[{"Sepal.Length", "Sepal.Width", "Petal.Length", "Petal.Width”},Sqrt@pairs];


Setting up plotting options:

(*Set up plot options *)

DodgerBlue = RGBColor[0.117603, 0.564699, 1.];
CrimsonRed = RGBColor[0.889996, 0.149998, 0.209998];
SeaGreen = RGBColor[0.180395, 0.545106, 0.341197];

SetOptions[{ListPlot, SmoothHistogram}, AspectRatio -> 1,
Frame -> True, ImageSize -> 150,
PlotStyle -> {CrimsonRed, SeaGreen, DodgerBlue},
FrameTicks -> {Automatic, Automatic, None, None},
BaseStyle -> {FontFamily -> "Myriad Pro", FontTracking -> "SemiCondensed",
FontWeight ->"Thin", FontSize -> 10}];


Let’s create some helper functions for the individual plots:

(*Elliptical Insights*)

Clear[regPlot, data, regressions, MinMax];

MinMax[x_] := Flatten[{Min[x], Max[x]}];

ellipseInsight[data_, regressions_, colors_: {CrimsonRed, SeaGreen, DodgerBlue}, ci_: 0.68] :=

Show[ {

ListPlot[data[[1]], PlotStyle -> Lighter@colors[[1]]]
, ListPlot[data[[2]], PlotStyle -> Lighter@colors[[2]] ]
, ListPlot[data[[3]], PlotStyle -> Lighter@colors[[3]] ]

, Plot[regressions[[1]][x], {x, Min@(T@data[[1]])[[1]], Max@(T@data[[1]])[[1]]}, PlotStyle -> colors[[1]] ]
, Plot[regressions[[2]][x], {x, Min@(T@data[[2]])[[1]], Max@(T@data[[2]])[[1]]}, PlotStyle -> colors[[2]] ]
, Plot[regressions[[3]][x], {x, Min@(T@data[[3]])[[1]], Max@(T@data[[3]])[[1]]}, PlotStyle -> colors[[3]] ]

, Graphics[{colors[[1]] , Quiet@EllipsoidQuantile[data[[1]], ci]}]
, Graphics[{colors[[2]] , Quiet@EllipsoidQuantile[data[[2]], ci]}]
, Graphics[{colors[[3]] , Quiet@EllipsoidQuantile[data[[3]], ci]}]

}
, PlotRange -> Automatic
, FrameTicks -> {False, True, False, False}
, FrameStyle -> Directive[Thin, Gray]
, Axes -> False
, AspectRatio -> 1]


Let’s generate the plots:

(* Generate Regression Plots *)

plots = Table[ellipseInsight[groups[[All, All, i]], lm[[i]]], {i, 1, pairs}];

(* Generate Histogram Plots For the Diagonal *)

diags = Table[i (1 + Sqrt@pairs) + 1, {i, 0, Sqrt@pairs - 1} ];

histogramsPlots =Table[
SmoothHistogram[(T@groups[[All, All, i]][[#1]]),
AspectRatio -> 1, PlotStyle -> #2] &
, {Range[3], {CrimsonRed, SeaGreen, DodgerBlue}}]
, PlotRange -> {MinMax @ groups[[All, All, i, 1]], All}
, Frame -> True, FrameStyle -> Directive[Thin, Gray]
, FrameTicks -> {True, False, False, False}, ImageSize -> 150
, FrameLabel -> {"", plotLabels[[i]]}, Axes -> False], {i, diags}];

(*Merge Plots*)

Do[plots[[i (1 + Sqrt@pairs) + 1]] = histogramsPlots[[i + 1]], {i, 0, Sqrt@pairs - 1}];

(*Draw the plots*)
plots // Partition[#, 4] & // Grid
`

And here’s the sample output:

• +1 - very cool you took the time to flesh this out!
– ciao
Jun 24, 2014 at 20:56
• tnx, was a way of learning to plot on MMA, something I wanted to do for a while…
– Pam
Jun 24, 2014 at 21:10