SmoothDensityHistogram hiding data structure

Question

I have data which shows very clear structure in a ListPlot.

xf = Import["https://pastebin.com/raw/r5Aiq03E", "List"];
zf = Import["https://pastebin.com/raw/7bvzMuQV", "List"];

a0 = 5.29177*^-11; λ = 800*^-9/a0;
V2aGrid = Partition[Riffle[zf/λ, xf/λ], 2]; v2a = {{33, 37}, {-2, 2}};

ListPlot[V2aGrid, ImageSize -> Large, PlotRange -> v2a]

However, when I try to represent this data with SmoothDensityHistogram, the structure is hidden. Having used:

 SmoothDensityHistogram[V2aGrid, Automatic, "PDF", ColorFunction->
 "AvocadoColors", PlotRangePadding->None, PlotPoints->200]

Clearly, the structure has vanished. Is there a way to represent the data as a continuum, yet not lose the critical shape of the plot? Increasing PlotPoints doesn't seem to help.

Edit: Lowing the bandwidth helps a little, but there is still a lot of structure that isn't represented in the density plot. It seems to stop changing at a bandwidth lower than 0.01, so I'll post the image at that value.

I tried all of the pre-defined kernel options along with bw in espec, and none of them seemed to make a difference.

Edit: Minimal working example here, or raw data here and here.

Answer 1

Use the bandwidth option for SmoothDensityHistogram (where you have Automatic).

First generate some data (and I've literally stolen this from @ybeltukov in MathematicaStackExchange: Generate random points in a region...):

n = 10000000;
r2 = 0.1;
choise = RandomChoice[{\[Pi] r2, 1} -> {0, 1}, n];
box = RandomReal[1, {n, 2}];
circle = Transpose@{0.5 + # Cos@#2, 0.5 + # Sin@#2} &[
   Sqrt@RandomReal[r2, n], RandomReal[2 \[Pi], n]];
pts = box choise + circle (1 - choise);

Plot data and SmoothDensityHistogram at various bandwidths:

Grid[{{ListPlot[pts[[;; ;; 300]], AspectRatio -> Automatic, 
    PlotLabel -> "Data"],
   SmoothDensityHistogram[pts[[;; ;; 300]], Automatic, 
    PlotRange -> {{0, 1}, {0, 1}},
    PlotLabel -> "Bandwidth = Automatic"]},
  {SmoothDensityHistogram[pts[[;; ;; 300]], .1, 
    PlotRange -> {{0, 1}, {0, 1}},
    PlotLabel -> "Bandwidth = 0.1"],
   SmoothDensityHistogram[pts[[;; ;; 300]], .02, 
    PlotRange -> {{0, 1}, {0, 1}},
    PlotLabel -> "Bandwidth=0.02"]}}]

Update:

One of the reasons for a poor representation is that what is shown above is just a small portion of the complete data set. Having a single bandwidth with the complexity of the data (multiple peaks, multiple voids, etc.) is not likely to reproduce the pattern seen in the points. Here is a figure showing all of the data and the red box delineates the subset shown above:

ListPlot[{V2aGrid, {{33, -2}, {33, 2}, {37, 2}, {37, -2}, {33, -2}}}, 
 ImageSize -> Medium, PlotRange -> All, PlotRangePadding -> None, Joined -> {False, True}, 
 AspectRatio -> 1, PlotStyle -> {Darker[Blue], {Thickness[0.01], Red}},
 PlotRangeClipping -> False]

An alternative approach is to use an "adaptive" kernel which increases in size in low density areas and decreases in size in high density areas. SmoothDensityHistogram offers this but with the number of points for this example, it did not complete after 20 minutes.

But if one just looks at the subset of data shown above, then a better match with the data points is achieved. And here I use SmoothKernelDistribution followed by a ContourPlot to have a bit more control (i.e., convenient manipulation):

V2aGrid = Select[V2aGrid, -2 < #[[2]] < 2 &];
ListPlot[V2aGrid, ImageSize -> Medium, PlotRangePadding -> None]
skd = SmoothKernelDistribution[V2aGrid, 0.075];
ContourPlot[(1 - PDF[skd, {z, x}])^10, {z, 33, 37}, {x, -2, 2}, 
 PlotRange -> All,
 PlotPoints -> 100, ColorFunction -> "AvocadoColors", Contours -> 30,
 AspectRatio -> 0.65, ContourShading -> True, ContourStyle -> None]

I used (1 - PDF[skd, {z, x}])^10 rather than PDF[skd, {z, x}] to separate the colors to better match the figure with the data points.

2nd update:

Rather than messing with the PDF it's probably better to do a bit of rescaling of the "AvocadoColors" (again using a subset of the complete dataset):

data = Select[V2aGrid, -2 < #[[2]] < 2 &];
SmoothDensityHistogram[data, 0.05, "PDF",
 ColorFunction -> Function[x, ColorData["AvocadoColors"][x^0.5]],
 PlotRangePadding -> None, AspectRatio -> 0.65,
 PlotRange -> All]

And here's the whole dataset:

SmoothDensityHistogram[V2aGrid, Automatic, "PDF", 
 InterpolationPoints -> 200,
 ColorFunction -> Function[x, ColorData["AvocadoColors"][x^0.5]],
 PlotRangePadding -> None, AspectRatio -> 0.65,
 PlotRange -> All, PlotPoints -> 200]

So some of the issue seems to be that the color scheme needs a bit of rescaling to show the detail.