Of course, KDE requires a lot of expensive operations. Making the above happen in a few seconds required a few efficiency tricks at the cost of perfectly accurate clusters. So, I put together this mini-tutorial to explain my approach.

#### What is Kernel Density Estimation?

KDE helps identify the density of a distribution of data. That is, it can be useful in finding where a lot of data is grouped together and where it isn't. Naturally, it can be used for 1D clustering by creating clusters about the points of highest density (local maxima), separated by the points of lowest density (local minima). There are many great online tutorials about KDE, and I recommend familiarizing yourself before moving on.1D clustering with KDE can be done in roughly 4 steps:

- Normalize data (0 to 1) and sort
- Compute densities
- Find local maxima
- Find minima and cluster

####
__1.) Normalize data and sort__

With standard KDE you don't need to sort, because density is calculated from every other point in your dataset. Since the following section makes use of a limited neighborhood, sorting is necessary.####
__2) Compute densities__

This is the only step that requires a little bit of work and some knowledge of KDE. It's *also*where we can start taking shortcuts for faster clustering.

The general idea is for a given point along the data, we compare the distance of that point to its neighbors. Neighbors that are really close add more to the density and neighbors that are far away add much less. How much they add is dependent on the choice of a smoothing kernel with radius

*h*. A good choice for

*h*is the Silverman's rule of thumb:

*h*=std(

*x*)*(4/3/n

*)*^(1/5). For example, the density at point

*i*, data

*x*, number of elements

*n,*and

*often-used Gaussian/normal kernel would be:*

sum = 0;

for j = 1:n

v = ( x(i) - x(j) )/h;

sum = sum + exp(-0.5*v*v) / sqrt(2*pi);

end

density = sum/(n*h);

For full KDE, that's an unseemly

*n*^2 checks. To reduce the workload, we can make two simplifications:

- Compute density in bins instead of every point
- Check only nearby neighbors that have a greater impact on density

*improve run time but may give you noisier results, especially when an ignored neighbor would add a not-so-insignificant amount to the density. Sometimes is sufficient to add a round of smoothing, where the density of a bin is recomputed as the average of itself and left and right bins. The following code snippet applies these simplifications:*

h = std(x) * (4/(3*n))^(1/5);

nn = 10; % number of neighbors to check on either side

nb = 10; % number of bins

bh = 1/10; % bin step size

bins = zeros(nb,1); % nb bins for computing density

for i = 1:n

bin = ceil(x(i)/bh); % bin index

sum = 0;

for j = max(1,i-nn):min(n,i+nn)

v = (x(i)-x(j))/h;

sum = sum + exp( -0.5*v*v ) / sqrt(2*pi);

end

dens = sum/(2*nn*h);

bins(bin) = bins(bin) + dens;

end

for i = 2:nb-1 % Smoothing

bins(i) = (bins(i-1)+bins(i)+bins(i+1))/3;

end

Of course, the bins aren't exactly centered at the points of highest density. For a lot of applications this is perfectly fine. Plotting the resulting density and data gives us:

####
__3) Find local maxima__

Local maxima are the peaks of the curves in the above plot. They can easily be discovered by checking the bins on either side. If it's the largest, it's a local maximum! If your curve is jaggy with too many maxima, you'll need to increase the number of neighbors sampled when computing density. Play with the tunable variables until you have something that looks nice:####
__4) Find local minima__

The minima are simply the bins of lowest density between two maximums:Local minima define where the clusters split. In the above example, we can see the data is split into three clusters.

And that's it. The script I used to create the plots above can be found right here.

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