Code covered by the BSD License  

Highlights from
kernel density estimation

4.1 | 30 ratings Rate this file 133 Downloads (last 30 days) File Size: 3.8 KB File ID: #17204
image thumbnail

kernel density estimation


Zdravko Botev


28 Oct 2007 (Updated )

fast and accurate state-of-the-art bivariate kernel density estimator

| Watch this File

File Information

% fast and accurate state-of-the-art
% bivariate kernel density estimator
% with diagonal bandwidth matrix.
% The kernel is assumed to be Gaussian.
% The two bandwidth parameters are
% chosen optimally without ever
% using/assuming a parametric model for the data or any "rules of thumb".
% Unlike many other procedures, this one
% is immune to accuracy failures in the estimation of
% multimodal densities with widely separated modes (see examples).
% INPUTS: data - an N by 2 array with continuous data
% n - size of the n by n grid over which the density is computed
% n has to be a power of 2, otherwise n=2^ceil(log2(n));
% the default value is 2^8;
% MIN_XY,MAX_XY- limits of the bounding box over which the density is computed;
% the format is:
% MIN_XY=[lower_Xlim,lower_Ylim]
% MAX_XY=[upper_Xlim,upper_Ylim].
% The dafault limits are computed as:
% MAX=max(data,[],1); MIN=min(data,[],1); Range=MAX-MIN;
% MAX_XY=MAX+Range/4; MIN_XY=MIN-Range/4;
% OUTPUT: bandwidth - a row vector with the two optimal
% bandwidths for a bivaroate Gaussian kernel;
% the format is:
% bandwidth=[bandwidth_X, bandwidth_Y];
% density - an n by n matrix containing the density values over the n by n grid;
% density is not computed unless the function is asked for such an output;
% X,Y - the meshgrid over which the variable "density" has been computed;
% the intended usage is as follows:
% surf(X,Y,density)
% Example (simple Gaussian mixture)
% clear all
% % generate a Gaussian mixture with distant modes
% data=[randn(500,2);
% randn(500,1)+3.5, randn(500,1);];
% % call the routine
% [bandwidth,density,X,Y]=kde2d(data);
% % plot the data and the density estimate
% contour3(X,Y,density,50), hold on
% plot(data(:,1),data(:,2),'r.','MarkerSize',5)
% Example (Gaussian mixture with distant modes):
% clear all
% % generate a Gaussian mixture with distant modes
% data=[randn(100,1), randn(100,1)/4;
% randn(100,1)+18, randn(100,1);
% randn(100,1)+15, randn(100,1)/2-18;];
% % call the routine
% [bandwidth,density,X,Y]=kde2d(data);
% % plot the data and the density estimate
% surf(X,Y,density,'LineStyle','none'), view([0,60])
% colormap hot, hold on, alpha(.8)
% set(gca, 'color', 'blue');
% plot(data(:,1),data(:,2),'w.','MarkerSize',5)
% Example (Sinusoidal density):
% clear all
% X=rand(1000,1); Y=sin(X*10*pi)+randn(size(X))/3; data=[X,Y];
% % apply routine
% [bandwidth,density,X,Y]=kde2d(data);
% % plot the data and the density estimate
% surf(X,Y,density,'LineStyle','none'), view([0,70])
% colormap hot, hold on, alpha(.8)
% set(gca, 'color', 'blue');
% plot(data(:,1),data(:,2),'w.','MarkerSize',5)
% Notes: If you have a more accurate density estimator
% (as measured by which routine attains the smallest
% L_2 distance between the estimate and the true density) or you have
% problems running this code, please email me at

% Reference: Z. I. Botev, J. F. Grotowski and D. P. Kroese
% Annals of Statistics, 2009

MATLAB release MATLAB 7.4 (R2007a)
Tags for This File   Please login to tag files.
Please login to add a comment or rating.
Comments and Ratings (44)
24 Dec 2014 yaq zhou

yaq zhou

I ran into the same error as described by Oliver,how can I fix it?

28 Nov 2014 Lee


22 Oct 2014 Felipe Uribe

Felipe Uribe

16 Aug 2014 Jakob Nikolas

Jakob Nikolas

If you need to use a fixed bandwidth, just overwrite t_x and t_y in the main function. Thanks to Zdravko for this hint.

16 Aug 2014 Jakob Nikolas

Jakob Nikolas

I ran into the same error as described by Oliver and fixed it by replacing

t_star=fzero( @(t)(t-evolve(t)),[0,0.1]);


options = optimset('FunValCheck','off')
t_star=fzero( @(t)(t-evolve(t)),[0,0.1],options);
defaultError('override error, be cautious!');

30 Jan 2014 Oliver


I am running the code perfectly on sets of 45 points (each has a X and Y coordinate). But when I try to only use a subset of those 45 points I get:
Error using fzero (line 274)
The function values at the interval endpoints must differ in sign.

Error in kde2d (line 101)

Is code not able to handle the low sample size? Not sure what is happening. It seems to not be able to find a bandwidth.

Comment only
26 Jan 2014 sepideh


22 May 2013 Asha


Hi, I am using kde2d package for whale distribution data. I would like to plot the 50% and 95% kernels to look at where the core of the sightings are being made. Is this possible with this package? I am also not sure what the current legend means. I like the colour gradation that shows highly dense areas vs. lower density areas but it would be great if I could figure out why the numbers are as high as they are. I would appreciate any help on this matter please.

Comment only
05 Apr 2013 Simon


hey there
I have a problem: Im having a vector of log returns (1189 rows) whose distribution i want to estimate with a kernel distribution.
So i just type in: "ksdensity(returnvector)". I get a curve, which looks ok, but on the y axis, I get values up to 60 (nothing normalised). And the mean (by clicking on the curve, choosing tools and data statistics) is completely different from the normal mean calculated as "mean(returnvector)". I dont know what is wrong here, which mean do i need to chose and which std??


13 Feb 2013 Arnault


A quick question about the bandwidth parameter. Does it have a unit? Does it correspond the the larger of the kernel related to the size of the grid cell?

Comment only
16 Nov 2012 Harish S

Harish S

21 Sep 2012 Syed


I keep getting negative probability densities, which is surprising as KDE is an addition of strictly positive numbers ...

31 Aug 2012 Tobias


I need to input an own bandwidth for the density estimation.
Is it possible to make changes to the code that allow either the input of another bandwidth or to change the data over which the density is computed after bandwidth estimation?


Comment only
09 May 2012 Luca Baglivo

Luca Baglivo

20 Mar 2012 Rachel


No need to answer, I managed to do it by calculating the bin width from the [X, Y] (meshgrid) output.

Comment only
19 Mar 2012 Rachel


First of all, thank you for the code, it works great.

I want to get the marginal PDFs from the joint distribution, and am therefore doing:

m1 = sum(pdf_joint, 2); %marginal for signal 1 (column vector)
m2 = sum(pdf_joint, 1); %marginal for signal 2 (row vector)

How can I normalize m1 and m2 so that their total area is equal to unity?

Thank you.

Comment only
08 Jan 2012 John


I just wanted to verify that the volume under the output curve is 1. Right?

Comment only
09 Dec 2011 name



The 1d version of the kernel estimator also provides cdf values at the representative points. I would like to use these for 2d too. I don't have a strong background on this and I was not able to compute it.
Does anyone know how to compute cdf for 2d kernel?

Comment only
29 Nov 2011 El-ad David Amir

El-ad David Amir

25 Nov 2011 Alexander


Hi. For the 1-dimensional estimation you've fixed the "fzero at 293:The function values at the interval endpoints must" error with a try-catch block using :


What is the correct expression for kde2d?

I'm guessing the power is (-1/3), but what is the coefficient (any where is it coming from)?

11 Oct 2011 JERRY


I am having the same problem as Gil Tidhar. When using the WAFO kde function, the minimum is 0 (which is what I expected), but your function returns negative values.

test code:
>> [temp, pab] = kde2d(rand(10000, 2),...
>> 2^4, [-3, -3], [5, 5]);
>> min(min(pab))
returns -0.0026

Comment only
15 Sep 2011 Arthur Allen

Arthur Allen

I have exactly the same question as gwideman (21 Aug 2010) "Is it possible to use this procedure when the input data are weighted? " The 1-D ksdensity.m function has this ability. Thanks.

Comment only
23 Aug 2011 Paulo


I'm having the same problem reported above by Sue C (16 Feb 2010). A similar problem was found for the 1-d version of this code and, apparently, was only fixed there:

16 Aug 2011 Alexander


Great code. One small improvements that I came across:
When data across one of the dimensions is constant, auto scaling is zero which leads to an error. Moreover, in that case it's better to increase/decrease max/min value, so that resulting meshgrid would not be 1-dimensional (which can cause wrong behavior in interpolation, for example)
Smth like this:
for ind=1:2
if MAX_XY(ind)==MIN_XY(ind)

14 Jul 2011 Gil Tidhar

Gil Tidhar

The minimum density values returned are NEGATIVE. This cannot be true for a density estimator and does not happen with other kernel type estimators that use positive definite Kernel functions.

22 Apr 2011 Nick


14 Mar 2011 Mi Matthew

Mi Matthew

My god,you are so excellent.Thanks a lot!

21 Aug 2010 gwideman


Is it possible to use this procedure when the input data are weighted? That is to say, input data have an X and Y value, and should not simply count as "1" at that location, but as some variable amount?

Comment only
14 Jun 2010 Anton Haug

Anton Haug

The author of this subroutine contacted me and when I exchanged my code with him he pointed out where I had made an error and helped me to correct my code. His subroutine is not biased. The bias was createrd by my lack of understanding of how the subroutine worked. I thank the author for his help!!

08 Jun 2010 Anton Haug

Anton Haug

I have tried this subroutine and found that it gives me a biased result. For samples from a mixture distribution of two Gaussians with known means, the kernel density is offset from the true mean in one dimension.

16 Feb 2010 Sue C

Sue C

It's easy to use .

but i got a problem;I am trying to do density estimation.
here is the m-file

%--------------(density estimation)------------------------------
clear all
data=[118,50;151,13;152,17;156,202;208,43;256,402;260,204;263,415;] ;
contour3(X,Y,density,50), hold on
%------------(density estimation)------------------------------(end)

and an error happened

??? Error using ==> fzero at 293
The function values at the interval endpoints must
differ in sign

Does anyone got the same error?

07 Jan 2010 David


i have the same question, and posted it on the other kernel estimator.

Comment only
29 Dec 2009 Dazhi Jiang

Dazhi Jiang

Hi, Jiangguang,

I don't think the sum of the density values is necessary to be 1. In fact, the volume is.

By the way, even a single value of density function can be over than 1.


Comment only
28 Dec 2009 Jianguang


I sum the 'density' and find sum(sum(denstiy)) > 1. This is an obivous bug and i don't know why. Should the density be normalizated to unit ?

12 Nov 2009 Matt


Should this work when there are multiple points at one location?
If I input a vector or points, with s number of identical points, for example:
>> y=rand(400, 2); y(371:400, 1)=y(370,1); y(370:400,2)=y(370,1);
>> [bandwidth, density, X, Y]=kde2d(y);
I find that the minimum density value is negative, although the bandwidth is still real and positive.

Comment only
13 Jun 2009 Nikola Toljic

Nikola Toljic

Just what I needed.

25 Jan 2009 Michael Jordan

Michael Jordan

16 Dec 2008 Nathanael Yoder

Nathanael Yoder

05 Oct 2008 Ying Lin

Maximum recursion limit of 500 reached. Use set(0,'RecursionLimit',N)
to change the limit.

17 Jun 2008 Alexander Sepúlveda  
17 Apr 2008 jagabandhu paul


01 Jan 2008 AMin Gheibi

It's easy to use and good organized code. I suggest it.

07 Nov 2007 dan p

This is some really nice code. It runs very quickly and is easy to get up and running. The author also provides plenty of comments and examples.

29 Oct 2007 Thomas T.

Easy to use bivariate kernel density estimation procedure. Thanks for sharing the code.

26 May 2009

updated reference and added new license as requested by Matlab

Contact us