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Peter Perkins (view profile)


11 Dec 2006 (Updated )

Plot a smoothed histogram of bivariate data

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A MATLAB implementation of the plot described in Eilers, P.H.C. and Goeman, J.J (2004) "Enhancing scaterplots with smoothed densities", Bioinformatics 20(5):623-628.

This plot is useful when a 2D scatterplot of your data would result in an uninterpretable mass of overlaid points. The smoothed histogram displays the density of points as greyscale intensity in a 2D image plot.

Optionally, you can also plot outliers as individual points, or plot the image as a surface in 3D.


This file inspired Xy 3 D Density Plot (For Two Class Data).

MATLAB release MATLAB 7 (R14)
Other requirements The help example uses MVNRND from the Statistics Toolbox, but it is not required for the function itself.
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Comments and Ratings (11)
17 Feb 2015 mimi

mimi (view profile)

Hi, I am very much benefited with this code; exactly what I wanted. However, though I can get pretty good results (used in Monte carlo simulations for photon transport in tissue), I need to add in my thesis how exactly I got the density plot. In the code, the function smooth1D is not very clear to me, especially the equation P = lambda.^2 .* D2'*D2 + 2.*lambda .* D1'*D1;
Z = (E + P) \ Y; I will be highly obliged if someone please elaborate this equations/ give some links to the origin of it. I am eagerly waiting for some respond from any of you. Thanks in advance.

06 Jun 2014 KP

KP (view profile)

Hi, very useful one.How can I plot in Log scale, something like semilogy . I need to show the y axis in log without converting the y data. Thanks in advance

Comment only
18 Oct 2012 rito

rito (view profile)

Very Cool. I'd recommend making the functional form the standard. We can plot it using contourf etc on the user's end.

03 Sep 2012 Mark Matusevich  
14 Dec 2009 Yang Liu

Really glad to find this helpful file! Thanks!

Comment only
12 Nov 2009 Hubsi Hubert

Glad I finally found the script, but have some difficulties...
Don't know wheather this is the right place to post questions reagarding this skript, but I'll try:
How can change the axis: I would like to have the point of origin in the lower left corner and not at the upper right one.
Thanks for any suggestions!

Comment only
24 Aug 2009 Scott Beaver

Indeed, excellent file and simple to use.

17 Feb 2009 Emanuele Ronchi

Hi, I modified the function so that you con get the histogram and axes out and so that you can enter the edges as well (instead of only the bin numbers)

let me know if there are problems. From my tests it looks ok but didn't try on non uniform grids yet


function [F,ctrs1,ctrs2]=smoothhist2D(X,lambda,nbins,outliercutoff,plottype)
% SMOOTHHIST2D Plot a smoothed histogram of bivariate data.
% [H,X,Y]=SMOOTHHIST2D(X,LAMBDA,NBINS) plots a smoothed histogram of the bivariate
% data in the N-by-2 matrix X. Rows of X correspond to observations. The
% first column of X corresponds to the horizontal axis of the figure, the
% second to the vertical. LAMBDA is a positive scalar smoothing parameter;
% higher values lead to more smoothing, values close to zero lead to a plot
% that is essentially just the raw data. NBINS is a two-element vector
% that determines the number of histogram bins in the horizontal and
% vertical directions.
% SMOOTHHIST2D(X,LAMBDA,NBINS,CUTOFF) plots outliers in the data as points
% overlaid on the smoothed histogram. Outliers are defined as points in
% regions where the smoothed density is less than (100*CUTOFF)% of the
% maximum density.
% SMOOTHHIST2D(X,LAMBDA,NBINS,[],'surf') plots a smoothed histogram as a
% surface plot. SMOOTHHIST2D ignores the CUTOFF input in this case, and
% the surface plot does not include outliers.
% SMOOTHHIST2D(X,LAMBDA,NBINS,CUTOFF,'image') plots the histogram as an
% image plot, the default.
% 1. you can also enter the histogram edges instead of the bin numbers
% by making NBINS a CELL array. Example (using X defined below)
% 2. Added outputs (histogram and edges)
% [h,xg,yg]=smoothhist2D(X,5,{[-5:0.1:10],[0:0.1:15]},.05);
% Example:
% X = [mvnrnd([0 5], [3 0; 0 3], 2000);
% mvnrnd([0 8], [1 0; 0 5], 2000);
% mvnrnd([3 5], [5 0; 0 1], 2000)];
% smoothhist2D(X,5,[100, 100],.05);
% smoothhist2D(X,5,[100, 100],[],'surf');
% Reference:
% Eilers, P.H.C. and Goeman, J.J (2004) "Enhancing scaterplots with
% smoothed densities", Bioinformatics 20(5):623-628.

% Written by Peter Perkins, The MathWorks, Inc.
% Revision: 1.0 Date: 2006/12/12
% This function is not supported by The MathWorks, Inc.
% Requires MATLAB R14.

if nargin < 4 || isempty(outliercutoff), outliercutoff = .05; end
if nargin < 5, plottype = 'image'; end

minx = min(X,[],1);
maxx = max(X,[],1);
if ~iscell(nbins) %mode: bins
edges1 = linspace(minx(1), maxx(1), nbins(1)+1);
edges2 = linspace(minx(2), maxx(2), nbins(2)+1);
ctrs1 = edges1(1:end-1) + .5*diff(edges1);
ctrs2 = edges2(1:end-1) + .5*diff(edges2);
else %mode: edges

edges1 = [-Inf edges1(2:end-1) Inf];
edges2 = [-Inf edges2(2:end-1) Inf];

[n,p] = size(X);
bin = zeros(n,2);
% Reverse the columns of H to put the first column of X along the
% horizontal axis, the second along the vertical.
[dum,bin(:,2)] = histc(X(:,1),edges1);
[dum,bin(:,1)] = histc(X(:,2),edges2);

H = accumarray(bin,1,[nbins2,nbins1]) ./ n;
%H = accumarray(bin,1,nbins([2 1])) ./ n;

% Eiler's 1D smooth, twice
G = smooth1D(H,lambda);
F = smooth1D(G',lambda)';
% % An alternative, using filter2. However, lambda means totally different
% % things in this case: for smooth1D, it is a smoothness penalty parameter,
% % while for filter2D, it is a window halfwidth
% F = filter2D(H,lambda);

relF = F./max(F(:));
if outliercutoff > 0
outliers = (relF(nbins2*(bin(:,2)-1)+bin(:,1)) < outliercutoff);

nc = 256;
switch plottype
case 'surf'
case 'image'
image(ctrs1,ctrs2,floor(nc.*relF) + 1);
hold on
% plot the outliers
if outliercutoff > 0
plot(X(outliers,1),X(outliers,2),'.','MarkerEdgeColor',[.8 .8 .8]);
% % plot a subsample of the data
% Xsample = X(randsample(n,n/10),:);
% plot(Xsample(:,1),Xsample(:,2),'bo');
hold off

function Z = smooth1D(Y,lambda)
[m,n] = size(Y);
E = eye(m);
D1 = diff(E,1);
D2 = diff(D1,1);
P = lambda.^2 .* D2'*D2 + 2.*lambda .* D1'*D1;
Z = (E + P) \ Y;
% This is a better solution, but takes a bit longer for n and m large
% opts.RECT = true;
% D1 = [diff(E,1); zeros(1,n)];
% D2 = [diff(D1,1); zeros(1,n)];
% Z = linsolve([E; 2.*sqrt(lambda).*D1; lambda.*D2],[Y; zeros(2*m,n)],opts);

function Z = filter2D(Y,bw)
z = -1:(1/bw):1;
k = .75 * (1 - z.^2); % epanechnikov-like weights
k = k ./ sum(k);
Z = filter2(k'*k,Y);

06 Oct 2008 Jimmy Shen

The author added 2D filter to make it look unique, although there were many version of MATLAB codes for the cited paper. e.g.: http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=6037&objectType=file

15 Sep 2008 krishna mahale

helpful... but i believe such tools should be a requisite /mandatory for a product like matlab

17 Jan 2007 Adam A

Really excellent file.

Should have default parameters for lamda and nbins tho.

12 Jun 2009 1.2

Cleaned up file header information.

18 Jun 2009 1.3

Added copyright.

18 Jun 2009 1.5

Remove author.

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