# Clustered heatmaps with dendrograms

In biology it's common to cluster both dimensions of a matrix, display it as a heatmap (MatrixPlot, ArrayPlot) and show dendrograms on the X and Y axes indicating the clustering. There's a nice blog post by @verebia showing this (image below is from that post), but it uses empirically determined PlotPadding and ImageSize options specific to the input data set to align the elements, which makes it a bit of pain to work with. Is there a way to combine DendrogramPlot and MatrixPlot automatically instead? Internally, top-oriented DendrogramPlot lines have X coordinates equal to the index of the Cluster and Y coordinates equal to the dissimilarity (d in Cluster[c1, c2, d, n1, n2]). The opposite is true for left-oriented plots. This makes it easy to combine with a grid of Rectangles by using Show: (The code is a bit long because I added support for tick marks. Scroll down to the Show part.)

Options[ClusteredHeatMap] = {
ColorFunction -> Automatic,
"LeftDendrogramRatio" -> 0.4, "TopDendrogramRatio" -> 0.4,
"BottomLabels" -> None, "RightLabels" -> None}~Join~Options[Graphics];

(* data: original matrix *)
(* hClustering: result of e.g. Agglomerate[data -> Range[Length[data]]] *)
(* vClustering: result of e.g. Agglomerate[Transpose[data] -> Range[Length[Transpose[data]]]] *)
ClusteredHeatMap[data_, hClustering_, vClustering_, opts : OptionsPattern[]] :=
Module[{
hClustOrder = ClusterFlatten[hClustering],
vClustOrder = ClusterFlatten[vClustering],
orderedData,
left = DendrogramPlot[vClustering, Orientation -> Left],
leftXcoords,
leftDomain,
top = DendrogramPlot[hClustering],
topYcoords,
topDomain,
xLength = Length@data,
yLength = Length@data[],
$defaultColorFunction,$colorFunction,
$leftDendrogramRatio = OptionValue["LeftDendrogramRatio"],$topDendrogramRatio = OptionValue["TopDendrogramRatio"],
rightLabels = OptionValue["RightLabels"],
bottomLabels = OptionValue["BottomLabels"],
$xTicks,$xTickLocations, $xTickLabels,$yTicks
},

orderedData = data[[hClustOrder, vClustOrder]];

$defaultColorFunction[x_] := Blend[ { {-1, Blue}, {0, Black}, {1, Yellow} }, x];$colorFunction = OptionValue[ColorFunction] /. Automatic -> $defaultColorFunction; leftXcoords = Cases[left, Line[{x : {_, _} ..}] :> x, Infinity][[All, 1]]; leftDomain = {Min[leftXcoords], Max[leftXcoords]}; topYcoords = Cases[top, Line[{x : {_, _} ..}] :> x, Infinity][[All, 2]]; topDomain = {Min[topYcoords], Max[topYcoords]}; (* X ticks are normally drawn below the axis, which interferes with the dendrogram. Draw manually. *)$xTickLocations =
Table[{i,
Rescale[i, leftDomain, {-$leftDendrogramRatio*xLength, 0}]}, {i, Select[FindDivisions[leftDomain, 3], leftDomain[] < # < 0 &]}];$xTicks = Table[{i[], ""}, {i, $xTickLocations}];$xTickLabels =
Table[{GrayLevel[0.4],
Text[i[], Offset[{0, 10}, {i[], 0}]]}, {i,
If[OptionValue[Axes] == True, $xTickLocations, {}]}];$yTicks =
Table[{Rescale[i, topDomain, {0, $topDendrogramRatio*yLength}], i}, {i, FindDivisions[topDomain, 5]}]; Show[ (* Dendrograms *) (* Shift the left dendrogram to go from {0, 0} to {0, negativeValue}. *) (* Rescale both dendrograms to be half the width (left) or height (top) of the heatmap. *) left /. (Line[x : {{_, _} ..}] :> Line[({Rescale[#[], leftDomain, {-$leftDendrogramRatio*xLength, 0}], #[] - yLength - 0.5}) & /@ x]),
top /. (Line[x : {{_, _} ..}] :>
Line[({#[] - 0.5, Rescale[#[], topDomain, {0, $topDendrogramRatio*yLength}]}) & /@ x]), (* Heatmap *) Graphics[ Table[{$colorFunction[orderedData[[x, y]]],
Rectangle[{x - 1, y - yLength}, {x, y - yLength - 1}]}, {x, xLength}, {y, 1, yLength}]],
(* Bottom labels *)
Graphics[
Table[Text[Rotate[bottomLabels[[hClustOrder[[i]]]], 90 Degree],
Offset[{0, -5}, {i - 0.5, -yLength}], {0, 1}], {i, 1,
Length@bottomLabels(*, optional skipping increment*)}]],
(* Right labels *)
Graphics[
Table[Text[rightLabels[[vClustOrder[[i]]]],
Offset[{5, 0}, {xLength, i - yLength - 0.5}], {-1, 0}], {i, 1,
Length@rightLabels(*, optional skipping increment*)}]],
(* X tick labels *)
Graphics[$xTickLabels], Ticks -> {$xTicks, \$yTicks}, PlotRangePadding -> 0,
FilterRules[{opts}, Options[Graphics]]
]
]


Notes:

• Set Axes -> True|False to show or hide the top-left corner axes, which are the dissimilarity value scales. Neat!
• The above does not rescale the data. MatrixPlot by default rescales to {0,1}. I think it's hard to have a good default for biology -- probably half the time you want {-1, 1} -- so I left this out. Use Rescale as needed.
• Because this example heat map has a ton of columns, I'm only displaying every 4th label by replacing the comment optional skipping increment with 4.
• The *DendrogramRatio options control what fraction of the image is occupied by the respective dendrogram.

This code is also available in GitHub, here. (Eventually I'll make it into a package.)