核密度估计matlab程序

============================================================================== MATLAB KDE Class Description & Specification ============================================================================== 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 (http://www.bloodshed.net) supplies a nice development environment along with the MinGW (http://www.mingw.org) compiler. See the page http://gnumex.sourceforge.net/ for help setting up MEX with MinGW. Kernels supported are: Gaussian, Epanetchnikov (truncated quadratic), and 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 ============================================================================== Unzip the KDE class to a directory called @kde. Compile the MEX functions. This can be done by running "makemex" from inside matlab, in the "@kde/mex" directory. If this fails, make sure that MEX / C++ compilation works. The KDE toolbox is tested in Matlab R13, but apparently has problems in R12; I'm planning to investigate this. 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 Density estimates may be visualized using e.g. plot(p); or mesh(hist(p)); See help kde/plot and help kde/hist for more information. Also, the demonstration programs @kde/examples/demo_kde_#.m may be helpful. ============================================================================== KDE Matlab class definition ============================================================================== The following is a simple list of all accessible functions for the KDE class. 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) getBW(kde,index) : return the bandwidth assoc. with x_i (Ndim x length(index)) adjustBW : 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 : 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)) getPoints(kde) : Ndim x Npoints array of kernel locations adjustPoints(p,delta) : shift points of P by delta (by reference!) getWeights : [1 x Npts] array of kernel weights adjustWeights : set kernel weights (by reference!) rescale(kde,alpha) : rescale a KDE by the (vector) alpha getDim : get the dimension of the data getNpts : get the # of kernel locations getNeff : "effective" # of kernels (accounts for non-uniform weights) 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) ===================================================== covar : find the (weighted) covariance of the kernel centers mean : find the (weighted) mean of the kernel centers modes