Kernel Density Estimation Toolbox for MATLAB (R13)

# MATLAB KDE Class Description

The KDE class is a general matlab class for k-dimensional kernel density estimation. It is written in a mix of matlab ".m" files and MEX/C++ code. Thus, to use it you will need to be able to compile C++ code for Matlab. Note that the default compiler for Windows does not support C++, so you will need GCC under Linux, or GCC or Visual C++ for Windows. Bloodshed supplies a nice development environment along with the MinGW compiler. See this page for help setting up MEX with MinGW.

[NOTE: Since several compiled mexglx and dll files are included, you may not need to re-compile the toolbox at all; however, I recommend it if possible for compatibility reasons.]

Kernels supported are:

• Gaussian

• Laplacian (Double-exponential)

For multivariate density estimates, the code supports product kernels -- kernels which are products of the kernel function in each dimension. For example, for Gaussian kernels this is equivalent to requiring a diagonal covariance. It can also support non-uniform kernel bandwidths -- i.e. bandwidths which vary over kernel centers.

The implementation uses "kd-trees", a heirarchical representation for point sets which caches sufficient statistics about point locations etc. in order to achieve potential speedups in computation. For the Epanetchnikov kernel this can translate into speedups with no loss of precision; but for kernels with infinite support it provides an approximation tolerance level, which allows tradeoffs between evaluation quality and computation speed. In particular, we implement Alex Gray's "Dual Tree" evaluation algorithm; see Gray and Moore, "Very Fast Multivariate Kernel Density Estimation using via Computational Geometry", in Proceedings, Joint Stat. Meeting 2003 for more details. This gives a tolerance parameter which is a percent error (from the exact, N^2 computation) on the value at any evaluated point. In general, "tolerance" parameters in the matlab code / notes refers to this percent tolerance. This percentage error translates to an absolute additive error on the mean log-likelihood, for example. An exception to this is the gradient calcuation functions, which calculate using an absolute tolerance value. This is due to the difficulty of finding a percentage bound when the function calculated is not strictly positive.

We have also recently implemented the so-called Improved Fast Gauss Transform, described in [Yang, Duraiswami, and Gumerov, "Improved Fast Gauss Transform", submitted to the Siam Journal of Scientific Computing]. This often performs MUCH faster than the dual tree algorithm mentioned above, but the error bounds which control the computation are often quite loose, and somewhat unwieldy (for example, it is difficult to obtain the fractional error bounds provided & used by the dual tree methods and other functions in the KDE toolbox). Thus for the moment we have left the IFGT separate, with alternate controls for computational complexity (see below, and the file "evalIFGT.m").

# Getting Started

(If desired) Compile the MEX functions. This can be done by going to the "@kde/mex" directory in Matlab, and copying and pasting the code from the "makemex.m" file into the Matlab window. If this fails, make sure that MEX / C++ compilation works. The KDE toolbox is tested in Matlab R13 and later; it may work in ealier versions as well. Re-compiling may not be required, depending on your platform and version; "mexglx" (Linux 32-bit), "mexa64" (Linux 64-bit), and "dll" (Windows 32-bit) files are included. If you have trouble, recompile.

Thanks to Ankur Datta for compiling Mac versions of the MEX files and making them available. (I do not even own a Mac and cannot vouch for their operation, etc.; use at your own risk.)

NOTE: MS Visual C++ has a bug in dealing with "static const" variables; I think there is a patch available, or you can change these to #defines.

Operate from the class' parent directory, or add it to your MATLAB path (e.g. if you unzip to "myhome/@kde", cd in matlab to the "myhome" dir, or add it to the path.)

Objects of type KDE may be created by e.g.

p = kde( rand(2,1000), [.05;.03] ); % Gaussian kernel, 2D % BW = .05 in dim 1, .03 in dim 2.

p = kde( rand(2,1000), .05, ones(1,1000) ); % Same as above, but uniform BW and % specifying weights

p = kde( rand(2,1000), .05, ones(1,1000), 'Epanetchnikov'); % Quadratic kernel; just 'E' or 'e' also works

p = kde( rand(2,1000), 'rot' ); % Gaussian kernel, 2D, BW chosen by "rule of thumb" (below)

To see the kernel shape types, you can use:

plot(-3:.01:3, evaluate(kde(0,1,1,T),-3:.01:3) ); % where T = 'G', 'E', or 'L'

Kernel sizes may be selected automatically using e.g.

p = ksize(p, 'lcv'); % 1D Likelihood-based search for BW

p = ksize(p, 'rot'); % "Rule of Thumb"; Silverman '86 / Scott '92

p = ksize(p, 'hall'); % Plug-in type estimator (estimates each dim. separately)

Density estimates may be visualized using e.g.

plot(p);

or

mesh(hist(p));

Also, the demonstration programs @kde/examples/demo_kde_#.m may be helpful.

# Usage Examples

The demonstration programs in `@kde/examples/demo_kde_#.m` (where `#` is one of `1,2,3`) may be helpful.

# KDE Matlab class definition

The following is a simple list of all accessible functions for the KDE class. Use "help functionname" in Matlab for more information.

Constructors:

 kde( ) empty kde kde( kde ) re-construct kde from points, weights, bw, etc. `kde( points, bw )` construct Gauss kde with weights 1/N `kde( points, bw, weights)` construct Gaussian kde `kde( points, bw, weights,type)` potentially non-Gaussian `marginal( kde, dim)` marginalize to the given dimensions `condition( kde, dim, A)` marginalize to ~dim and weight by K(x_i(dim),a(dim)) `resample( kde, [kstype] )` draw N samples from kde & use to construct a new kde `reduce( kde, ...)` construct a "reduced" density estimate (fewer points) `joinTrees( t1, t2 )` make a new tree with t1 and t2 as the children of a new root node

Accessors: (data access, extremely limited or no processing req'd)

 `getType(kde)` return the kernel type of the KDE ('Gaussian', etc) `getDim` get the dimension of the data `getNpts` get the # of kernel locations `getNeff` "effective" # of kernels (accounts for non-uniform weights) `getPoints(kde)` Ndim x Npoints array of kernel locations `adjustPoints(p,delta)` shift points of P by delta (by reference!) `rescale(kde,alpha)` rescale a KDE by the (vector) alpha `getBW(kde,index)` return the bandwidth assoc. with x_i (Ndim x length(index)) `adjustBW(kde,newBW)` set the bandwidth(s) of the KDE (by reference!) Note: cannot change from a uniform -> non-uniform bandwidth `ksize` automatic bandwidth selection via a number of methods `LCV` 1D search using max leave-one-out likelihood criterion `HALL, HJSM` Plug-in estimator with good asymptotics; MISE criterion `ROT, MSP` Fast standard-deviaion based methods; AMISE criterion `LOCAL` Like LCV, but makes BW propto k-th NN distance (k=sqrt(N)) `getWeights` [1 x Npts] array of kernel weights `adjustWeights` set kernel weights (by reference!) `sample(P,Np,KSType)` draw Np new samples from P and set BW according to KSType

Display: (visualization / description)

 `plot(kde...)` plot the specified dimensions of the KDE locations `hist(kde...)` discretize the kde at uniform bin lengths display : text output describing the KDE `double` boolean evaluation of the KDE (non-empty)

Statistics: (useful stats & operations on a kde)

 `mean` find the (weighted) mean of the kernel centers `covar` find the (weighted) covariance of the kernel centers `knn(kde, points, k)` find the k nearest neighbors of each of points in kde `entropy` estimate the entropy of the KDE `kld` estimate divergence between two KDEs `evaluate(kde, x[,tol])` evaluate KDE at a set of points x `evaluate(p, p2 [,tol])` same as above, x = p2.pts (if we've already built a tree) `evalIFGT(kde, x, N)` evaluate using the N-term IFGT (requires uniform BW Gaussian kernels) `evalIFGT(p, p2, N)` ` evalAvgLogL(kde, x)` compute Mean( log( evaluate(kde, x) )) `evalAvgLogL(kde, kde2)` same as above, but use the weights of kde2 `evalAvgLogL(kde)` self-eval; leave-one-out option `llGrad(kde,kde2)` estimate the gradient of log-likelihood for kde evaluated at the points of kde2 `entropyGrad(p)` estimate gradient of entropy (uses llGrad) `miGrad(p,dim)` estimate gradient for mutual information between p(dim), p(~dim) `klGrad(p1,p2)` estimate gradient direction of KL-divergence

Mixture products: (NBP stuff) (GAUSSIAN KERNELS ONLY)

 `productExact` exact computation (N^d kernel centers) `productApprox` accessor for other product sampling methods `prodSampleExact` sample N points exactly (N^d computation) `prodSampleEpsilon` kd-tree epsilon-exact sampler `prodSampleGibbs1` seq. index gibbs sampler `prodSampleGibbs2` product of experts gibbs sampler `prodSampleGibbsMS1` multiresolution version of GS1 `prodSampleGibbsMS2` multiresolution version of GS2 `prodSampleImportance` "mixture" importance sampling `prodSampleImportGauss` gaussian importance sampling

The kde package and all code were written by Alex Ihler and Mike Mandel, and are copyrighted under the (lesser) GPL:

This program is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; version 2.1 or later. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

The authors may be contacted via email at: ihler (at) alum (.) mit (.) edu

# CHANGELOG

```CHANGE LOG FOR KERNEL DENSITY ESTIMATION CLASS
==============================================

08/14/07   ATI   Fixed 64-bit support for productApprox functions + bugfix
07/16/07   ATI   Fixed 64-bit support for most functions
of norm. constant for LOO version of evaluate
11/22/04   ATI   Added (original) Fast Gauss Transform (Greengard & Strain '91),
using newer (correct) error bound of Baxter & Roussos '02
11/09/04   ATI   Fixed bug in IFGT eval (incorrect scale factor)
10/02/04   ATI   Added support for Yang, Duraiswami, and Gumerov's Improved Fast
Gauss Transform.  Extremely fast; loose bounds on absolute error.
09/17/04   ATI   Fixed permutation bug in adjustBW, and bug in llGrad for evals
between two distributions.  Added explicit discrete resampling.
08/04/04   ATI   Updated productApprox:import, reduceKD; added ISE type to reduceKD
07/09/04   ATI   Fixed small bugs: condition.m, jointrees.m, productExact.m
Improved speed of sample.m
02/19/04   ATI   Added "llHess" (Hessian) and "modes" (mode-finding) functions
01/29/04   ATI   fixed 0 vs 1-base err in "index" ret'n values of productApprox
01/22/04   ATI   Added "ise" method and epsilon-exact MEX implementation
Fixed bug in epsilon and exact products of variable-BW densities
Improved implementation of "condition" for fixed-BW densities
12/28/03   ATI   Removed "abs" KL method, replaced with "ise" estimate method
("abs" was not a good est. of KL but served as an est. of ISE)
12/13/03   ATI   Fixed bug in KNN function and some bugs with the "reduce" f'n & "kld"