Kernel Density Estimation- Computation, Applications, and More

kernel density estimation

Introduction

 Kernel Density Estimation is a concept wherein data probability values are taken from non-parametric models. Now, what is your understanding on what non-parametric models are all about?

Well, a non-parametric model is a method of assessment or evaluation wherein data values are not taken from a known pool or resources of information. Here, one takes data values after deriving them through underlying formulae or derivation methods post which the segregated data values are taken up for assessment or evaluation. In other words, all non-parametric models derive their data values and not pick them from a pool or from tabulated rows of statistical figures.

Examples of Non-parametric data models

Here are certain examples of non-parametric data models you can have hands-on to:

1.   Ranked or Ordinal data

You have histograms or statistical data figures that do not represent known sources wherein the data is explicitly taken from. The data values are taken from cluster lists or ranked algorithms. These data values are then computed using Kernel Density Estimation techniques to arrive at statements that invoke decision-making.

2.   Non-parametric regression

Here, you have data that does not have a strong or powerful link to the known distribution of data lists. Say, for instance, you have non-parametric regression figures wherein data values are computed from derived formulae. Hence, you create data lists and do not pick values from known data sets. K-nearest data algorithms or other techniques are devised to compute data values.

3.   Data values with anomalies

Again, when you have manufacturing companies that produce goods and services, set machine values have to be derived for data configuration purposes. Say, for instance, outliers, shifts, or heavy tails, and these are Kernel Density techniques that are computed for Support Vector machines or other complicated algorithms to function.

Using an illustrative example to study how the Kernel density formula works

Let us discover how Kernel Density Estimation can work using an illustrative example. Hence, let us get started here:

Suppose, we are talking about the marks that have been obtained by 5 students in a particular subject. Here a kernel estimation has to be done for every data value. Let us have an overview of how the computations are being carried out here:

The values are inputted this way:

xi = {65, 75, 67, 79, 81, 91} and x1 = 65, x2 = 75 … x6 = 91. Typically, you would be requiring three types of data for a kernel curve to be estimated. And, these are:

  1. You have the point of observation which is the xi.
  2. Computation of the value for h
  3. A K series integration algorithm. Here, we take into account the nearest data points from where observations are being done. In other words, you have Xj = {50,51,52 …. 99}
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Here goes the table pertaining to the same:

Kernel density data

And, the K values are calculated for each data point which stays at the data prototype xj for the given set of values that is xi and h. Here, xi = 65 and h= 5.5.

The Kernel density curve is done this way:

kernel density curve

Similarly, the kernel values are plotted according to the table as given below:

kernel density estimation

When you have a look at it, the kernel value is nearly 0 for those data points that

are away from Xi. For instance, the kernel value is 0 when Xi= 99 as against the data point wherein Xi= 65.

Here is a table wherein kernel curves are drawn up at different data points

kernel density estimation kde

Kernel Density Estimation or KDE

So, far, the Kernel estimation values were done for individual data values. Now, comes the time to compute composite values and these are density values that are computed for the whole data set. The process behind computing composite values covering whole-length data sets is what is clearly known as Kernel Density Estimation or KDE as what it is called using acronym.

How do you think KDE is arrived at? Well, this is simple and straightforward. We just add up all values of K or what is there under Xj. In other words, the KDE is estimated once we add all the rows of the given data set. The sum is then normalized by dividing the said number by the number of data points that are taken up for calculation. In this example, the number of data points used for the computation of Kernel Density is 6.

There is also a specific formula by which you can compute KDE values every single time. Here goes the same:

formula to calculate kde values

Here n refers to the number of data points and KDE is obtained according to the plotted values as depicted here on this chart:

kernel density estimation example chart

Here is one more example that depicts KDE values and graphs representing data computation

kernel function example

Here, using actual data sets and values let us arrive at the figures:

kernel functions example

Here x1 = 30 and this is how the data table looks like:

kernal density example

Here kernel values take the fulcrum computation around xi which equals 30. Here is the plotted graph pertaining to the KDE values:

kernel values graph example

And, when we traverse through all data points, here are individual kernel estimation values that we get through this statistical chart. Here it goes for you to have a reference into:

kde example

In this example too, we sum up individual kernel functions at each data point to arrive at Kernel Density Estimation or KDE.

Bandwidth Optimization

When you have a look at it, the bandwidth that is denoted by the letter ‘h’ gets an important part to play with respect to data computation via kernel density techniques indeed. This value helps data fit appropriately. The lower the value of ‘h’, the higher the density of variance while the higher value of ‘h’ denotes that the given data values are subject to a lot of bias. Therefore, the Kernel Density estimation techniques involve evaluation of the value ‘h’ in order to derive intensive and meaningful computation of data sets.

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Using a statistical plot, let us move on to discover how the funda works

kernel density statistical plot

Here, the value of ‘h’ represented by the curve ensures that the data sets are accurate and insightful. The density of variance represented by the black curve is at its best. On the other hand, when you have a look at the purple curve that has the highest value of ‘h’ measuring 10 on the graph, you find that the data sets are inaccurate and misleading. Therefore, the purple curve hides itself from the more relevant curves. In other words, the purple curve fails to represent data with accurate density variance as it hides information.

Therefore, via this plot, you clearly understand the proportion between a desired density variance as denoted by the letter ‘h’ versus incompetent data curves denoted by a higher value ‘h’.

Using a few more set of graphs, let us discover how the bandwidth optimization chronology works:

Here, the previous data values have been used for this illustration as well. We will just study the trends of curves as represented by changing bandwidths:

 kernel density curves represented by changing bandwidth

kernel density estimation graph

Through this graph, these are the observations you can possibly make:

  1. When Xj< or = 25 and Xj > or = 35, the density value almost goes down to zero which means data sets have steeper and infertile density values.
  2. Again, as the density variance widens, do we find a smoother flow of data values?

Discovering and using the ‘Old Faithful’ method of data computation

Here, we are going to help you compute data sets using yet another method. This is the ‘Old Faithful Data set’ method we are going to use while arriving at computational values. Through this statistical figure, you can typically figure out how the empirical data values have been distributed within the given graph:

kde old faithdul eruption

Again, the data is actually loaded from the given URL:

old faithful method for kernel density values

As you see here, we are going to use the ‘Old Faithful Method’ to arrive at the Kernel Density values you are looking for. So, let us take a sneak peek into how this is being done:

a) Firstly, you require a kernel functional specification.

b) Then, you require the bandwidth specification

c) Finally, ‘Kernel Density structure’ for advanced settings and control.

Here is the formula we are going to use here:

kernel density structure formulla

Let us have an explanation of what each data protocol that is used here typically stands for. Do you want to get started over the same?

1.   Data sets

Here we talk about data values and the values can be numeric or alpha-numeric as a matter of fact. You can also give other acronyms to data sets. In other words, these values can be called data matrix or data file, or data frame.

2.   Kernel

The kernel is the manner in wherein you focus your argumentative scales. The default value can be set to zero. It can be a scalar or a vector kernel for you to estimate data variances, and their densities and then deduce the accuracy of data computation.

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3.   Bandwidth

The bandwidth can be an optional statement as stated by the programming guidelines. Or, it can be a scalar, vector, or matrix variable as the computation protocols demand. Say for instance, for a scalar co-efficient, you must maintain the same bandwidth values for every single row. If the bandwidth has a row vector, then the coefficient will be different for each column of the pertaining data set. If the component value starts at zero, then the necessary computations will be made to arrive at the bandwidth. The default figures must initially be set to zero so that you arrive at the exact data density variance.

4.   Ctl

This is the Kernel Density Estimation Control that is being talked about here. The structure controls the features of the KDE modules in general. You have plot customization, variable names, and other data components that are involved in the computation of information on the whole.

Default Estimation

Sometimes, you can evaluate Kernel Density using default settings too. Again, through default settings, you can evaluate data variance and plot the same on a density variance scale. Here, you just use a single input to evaluate Kernel Density.

Say, for example, this is the programming nomenclature for KDE using default settings. Here we go with the same:

programming nomenculture for kde

And here is the graph pertaining to the same:

kde graph

Supporting Kernel function Estimates

Actually, there are 13 different support functions that the Kernel density supports using a scalar or vector medium as the base. Here is the chart pertaining to the same:

kernel density functions

 

Applications associated with Kernel Density Estimation or KDE

Today, quite a lot of sophisticated functionalities take their anomalies via KDE techniques in an effective and hassle-free manner indeed. Let us have an overview of what these are:

1.  Machine learning

As the data sets are computed using unsupervised data models, the Kernel Density Estimation is integrated with Python to form Kernel Density Estimation Python-supported techniques and functionalities. Programming capabilities and machine learning derivations are already making great inroads using Kernel Density Estimation Python methodologies.

1.   Financial frauds detected

 Rare forms of financial fraud and their networks are detected using Python-enabled Kernel Density Estimation techniques. Here, banks or financial institutions benefit as major financial frauds are exposed to the limelight this way. Python Kernel Density Estimation is the savior for share markets wherein scams can be revealed easier and quicker.

In a nutshell, Python Kernel Density Estimation techniques help follow complicated URLs and compute graphs using unbiased data computation standards and financial companies are taking a lucrative advantage of the same.

2. Self-supervised learning made possible

As KDE depends on unclassified data sets, machine models pick self-supervised learning modules through these techniques. Hence, data accuracy is at a more precise level than data that is done using biased resources. So, companies are able to drive information using the highest standards of excellence and precision. Eventually, the figures help companies arrive at better decisions and elevate brands in a more robust way.

Concluding lines

We have seen varied forms of data components Kernel Estimation uses. Graphs and relevant examples give you a clear yardstick on how data sets are computed and the resultant figures arrived at. We have also seen how KDE techniques with Python modalities work hand in hand with financial banks, corporations, and machine learning companies in a robust and compatible manner.

 

 

 

 

 

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