Skip to content. Change Language. Related Articles. Table of Contents. Improve Article. Save Article. Like Article. Last Updated : 04 Jan, Previous Network Security. This is due to the binning where we effectively expand each line horizontally to the bin width, i. This influences the occupied area and we need some kind of compensation. Let's create a small example which we can also use later when we dive into the Parzen window approach.
The first question is: where do we start binning, i. This is actually application-specific since sometimes we have a natural range where it could be valid to start e. The following figure shows a graphical representation of the histogram. So, why do we need something else when we can already obtain a valid PDF via the histogram method? The problem is that histograms provide only a rough representation of the underlying distribution.
This is where the Parzen window estimator enters the field. Our goal is to improve the histogram method by finding a function which is smoother but still a valid PDF. The general idea of the Parzen window estimator is to use multiple so-called kernel functions and place them at the positions of the data points. We are superposing all of these kernels and scale the result to our needs.
The resulting function from this process is our PDF. Then, we can use the Parzen window estimator. A common choice is to use standard normal Gaussians which we are also going to use in the example later.
Two operations are performed in the argument of the kernel function. We don't have a clear bin-structure anymore since the bins are now implemented by the kernel functions. In the end, we have to compensate for this as well. We now could directly start and use this approach to retrieve a smooth PDF. This does not give us a smooth function, but hopefully some insights in the approach itself.
The uniform kernels have a length of 1 and are centred around the origin. How does the kernel change with this argument? Note also that this kernel is symmetric and hence has the same absolute value on both sides. Even though the two PDFs are not identical, they are very similar in the sense that both are step functions, i. So much for the detour to uniform kernels.
This kernel choice makes the Parzen estimator more complex but is also a requirement to achieve our smoothness constraint. The good thing is, though, that our recent findings remain valid. Even though the function is now composed of more terms, we can still see the individual Gaussian functions and how they are placed at the locations of the data points.
You can find out in the following animation. But compared to the uniform kernels before, we don't have the clear endings of the bins anymore. It is only noticeable that the Gaussians get wider. However, this is exactly what makes our resulting function smooth and this is, after all, precisely what we wanted to achieve in the first place. Before we move further to an example in the two-dimensional case, let us make a small detour and reach the Parzen window estimator from a different point of view: convolution.
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