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 h_n. 

Hence the volume of the hypercube is: V_n = h_n^d

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 |u_i| <= 0.5 
φ(u) = 0 otherwise 
Here, u is a vector, u = (u₁, u₂, …, u_d)^T. 
φ(u) should satisfy the following:

1. 
2. 

Let 

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

• is a hypercube of size h centered at u₀
• Let D = {x₁, x₂, …, x_n} be the data samples.
• For any would be 1 only if falls in a hypercube of side centered at .
• Hence the number of data points falling in a hypercube of side h centered at x is 

Hence the estimated density function is : 

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Also Since, V_n = h_n^d, Density Function becomes : 

would satisfy the following conditions:  

1. 
2.