Kernel Bandwidth Optimization
Web application for optimizing kernel density estimate 
Shimazaki and Shinomoto. J Comput Neurosci, 2010, 29 (12) 171182 
Web Application for Kernel Bandwidth
Optimization (Ver. 0.4)
2. 3. * The Gauss kernel density was used. Its standard deviation is optmized. * Feel safe. The data is processed on your own computer (NEVER transferred to our server). * This web application uses the HTML5 Canvas technology. ver 0.4 (101231) accelarating computation by excluding >5 std samples.
Web Application for Kernel Bandwidth
Optimization © 20092011Hideaki Shimazaki



Shimazaki H. and Shinomoto S., Kernel Bandwidth Optimization in Spike Rate Estimation. Journal of Computational Neuroscience (2010) Vol. 29 (12) 171182 doi:10.1007/s1082700901804 [Open Access: PDF, LINK] 
The method has been used in a variety of scientific fields. Example usages in Neuroscience are found at Hill, M.R., Fried, I. and Koch, C., 2014. Quantification and classification of neuronal responses in kernelsmoothed peristimulus time histograms. Journal of neurophysiology, 113(4), pp.12601274. J Neurophysiol2015. Ito, H.T., Zhang, S.J., Witter, M.P., Moser, E.I. and Moser, M.B., 2015. A prefrontal–thalamo–hippocampal circuit for goaldirected spatial navigation. Nature, 522(7554), p.50. Nature 2015. Manita, S., Suzuki, T., Homma, C., Matsumoto, T., Odagawa, M., Yamada, K., Ota, K., Matsubara, C., Inutsuka, A., Sato, M. and Ohkura, M., 2015. A topdown cortical circuit for accurate sensory perception. Neuron, 86(5), pp.13041316. Neuron 2015. 
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. The codes are available at GitHub. http://github.com/shimazaki/density_estimation Experimental Julia code: here. 
Lee Cooper created python program of kernel and histogram optimization. This is an extended version of the translation by Subhasis Ray.  2016/5/6 The adaptive kernel density estimation module is can be installed by typing pip adaptivekde See also 
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 crossvalidation method? A. Yes. The above formula was derrived under a Poisson point process assumption [see our original paper], not by the crossvalidation. As a result, chosen bandwidths by the two methods are not identical. Q. Can I use the method for the 2dimensional density estimation? A. Yes. The same optimization formula can be used in 2d density estimation. Specifically, the formula for two dimensional vector data is given by Here the kernel function is defined in 2dimension. For example, the 2d (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 oneparameter 2d kernel. It would be recommended to either i) optimize twoparameters kernel, or ii) optimize the oneparameter kernel using the standardized data, for example, by dividing the data by standard deviations of each component. Q. Can I compare the goodnessoffit 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 FAQ includes my (HS) opinion. They are not opinions neither by my collaborators nor institutions I belong to. 
The method was developed in collaboration with Prof. Shinomoto in Kyoto University. Other applications for analyzing spike data: SULAB ( Prof. Shinomoto ) 