Probability Density Functions and Kernel Density Estimation

Abstract

Stochastic modeling loop in the stochastic optimization framework involves dealing with evaluation of a probabilistic objective function and constraints from the output data. Probability density functions (PDFs) are a fundamental tool used to characterize uncertain data. Equation 3.1 shows the definition of a PDF f of variable X.

Notations

d :

dimension

E :

expected value function

f :

probability density function

h :

bin width

$h_{opt}$ :

optimum bin width

K :

kernel density function

n :

number of observations

w :

weight function

Z :

output variable

Greek letters

σ:

standard deviation