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)
is a hypercube of size h centered at u0
- Let D = {x1, x2, …, xn} 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 :
*** QuickLaTeX cannot compile formula: *** Error message: Error: Nothing to show, formula is empty
Also Since, Vn = hnd, Density Function becomes :
would satisfy the following conditions: