Shimazaki and Shinomoto, Journal of Computational Neuroscience, 2010, 29 (1-2) 171-182

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* The Gauss kernel density was used. Its standard deviation is optmized.

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ver 0.4 (101231) accelarating computation by excluding >5 std samples.

I. Let be your data. Let the kernel with bandwidth be .

II. Compute a formula,

III. Find that minimizes .

From the observed data only, the method estimates a bandwidth that minimizes expected L2 loss between the kernel estimate and an unknown underlying density function. An assumption made here is merely that samples are drawn from the density independently each other.

In the referenced paper, we developed a locally adaptive bandwidth optimization method.

The locally adaptive bandwidth is obtained by iteratively computing optimal bandwidths wihtihn local intervals using the algorithm above. In this approach, the local interval length is a critical parameter that determines the smoothness of bandwidth, and therefore goodness-of-fit of the density estimate. We suggested, at every points of estimation, to select the interval length such that the locally optimal bandwidth covers \gamma *100 % of the interval length (0<\gamma<=1). In other words, the selected bandwidth, w, is optimal as evaluated within the local interval of the length w / \gamma . The parameter, \gamma, serves as a smoothness (or stiffness) parameter of the variable bandwidth just like the bandwidth is the smoothness parameter of the estimated density. The stiffness constant, \gamma, is optimized by minimizing the L2 risk estimate.

Example: An underlying densityi is a mixture of two normal densities and an exponential density.

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A Matlab code for locally adaptive kernel method is available below.

Shimazaki H. and Shinomoto S., Kernel Bandwidth Optimization in Spike Rate Estimation. Journal of Computational Neuroscience (2010) Vol. 29 (1-2) 171-182 doi:10.1007/s10827-009-0180-4 [Open Access: PDF, LINK]

Presentation Slides

Kernel density estimation by a fixed bandwidth

Matlab code: sskernel.m

Function `sskernel' returns optimal bandwidth (standard deviation) of the Gauss kernel function used in kernel density estimation.

Locally adaptive kernel density estimation

Matlab code: ssvkernel.m

Function `ssvkernel' returns an optimized kernel density estimate using a Gauss kernel function with bandwidths locally adapted to the data.

tutorial.m

The codes are available at GitHub. http://github.com/shimazaki/density_estimation

Experimental Julia code: here.

Q. Is the method applicable to probability density estimation?

A. Yes.

Q. You provide a histogram optimization method, too. Which of the methods, kernel and histogram, do you recommend for density estimation.

A. Kernel density estimation is generally recommended. See our original paper for comparison of the methods.

Q. Is it different from the least squares cross-validation method?

A. Yes. The above formula was derrived under a Poisson point process assumption [see our original paper], not by the cross-validation. As a result, chosen bandwidths by the two methods are not identical.

Q. Can I use the method for the 2-dimensional density estimation?

A. Yes. The same optimization formula can be used in 2-d density estimation. Specifically, the formula for two dimensional vector data is given by

Here the kernel function is defined in 2-dimension. For example, the 2-d (symmetric) Gauss kernel is defined as

where .

An example of the optimized 2d kernel density estimate is displayed below.

Here are the example matlab code sskernel2d_demo.m and data.

Note: if you have two dimensional variables with different dynamic ranges, it should be careful to use the one-parameter 2-d kernel. It would be recommended to either i) optimize two-parameters kernel, or ii) optimize the one-parameter kernel using the standardized data, for example, by dividing the data by standard deviations of each component.

Q. Can I compare the goodness-of-fit of kernel density estimate with that of a histogram?

The definition of the cost function is different in the two papers (2007 and 2010). To compare the cost function of a histogram with the cost function of a kernel, please adjust the histogram cost function as T*C(D)+K^2/T, where K is the total number of samples, and T is the observation length.

The method was developed in collaboration with Prof. Shinomoto in Kyoto University.

Other applications for analyzing spike data: SULAB ( Prof. Shinomoto )