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Parzen Windows density estimation technique

Parzen Window is a non-parametric density estimation technique. Density estimation in Pattern Recognition can be achieved by using the approach of the Parzen Windows. Parzen window density estimation technique is a kind of generalization of the histogram technique. 

It is used to derive a density function, 
is used to implement a Bayes Classifier. When we have a new sample feature and when there is a need to compute the value of the class conditional densities, is used. 
takes sample input data value and returns the density estimate of the given data sample. 

An n-dimensional hypercube is considered which is assumed to possess k-data samples. 
The length of the edge of the hypercube is assumed to be hn

 



Hence the volume of the hypercube is: Vn = hnd 

We define a hypercube window function, φ(u) which is an indicator function of the unit hypercube which is centered at origin.: 
φ(u) = 1 if |ui| <= 0.5 
φ(u) = 0 otherwise 
Here, u is a vector, u = (u1, u2, …, ud)T
φ(u) should satisfy the following:


Let 

Since, φ(u) is centered at the origin, it is symmetric. 
φ(u) = φ(-u) 
 


Hence the estimated density function is : 

*** QuickLaTeX cannot compile formula:
 

*** Error message:
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Also Since, Vn = hnd, Density Function becomes : 



would satisfy the following conditions:  

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