# 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 = (u _{1}, u_{2}, …, u_{d})^{T}**.

**φ(u)**should satisfy the following:

Let

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

**φ(u) = φ(-u)**

- is a hypercube of size h cenetered at
**u**_{0} - Let D =
**{x**be the data samples._{1}, x_{2}, …, x_{n}} - 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 : **

Also Since, **V _{n} = h_{n}^{d}**, Density Function becomes :

would satisfy the following conditions:

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