Discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts.
This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers.
Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification).
In this context, discretization may also refer to modification of variable or category granularity, as when multiple discrete variables are aggregated or multiple discrete categories fused.
The goal is to reduce the amount to a level considered negligible for the modeling purposes at hand.
The terms discretization and quantization often have the same denotation but not always identical connotations.
(Specifically, the two terms share a semantic field.)
Discretization is also concerned with the transformation of continuous differential equations into discrete difference equations, suitable for numerical computing.
The following continuous-time state space model
where v and w are continuous zero-mean white noise sources with power spectral densities
can be discretized, assuming zero-order hold for the input u and continuous integration for the noise v, to
The equation for the discretized measurement noise is a consequence of the continuous measurement noise being defined with a power spectral density.
[1] A clever trick to compute Ad and Bd in one step is by utilizing the following property:[2]: p. 215
Where Ad and Bd are the discretized state-space matrices.
Numerical evaluation of Qd is a bit trickier due to the matrix exponential integral.
The discretized process noise is then evaluated by multiplying the transpose of the lower-right partition of G with the upper-right partition of G:
which is an analytical solution to the continuous model.
, and the second term can be simplified by substituting with the function
We also assume that u is constant during the integral, which in turn yields
which is an exact solution to the discretization problem.
Exact discretization may sometimes be intractable due to the heavy matrix exponential and integral operations involved.
It is much easier to calculate an approximate discrete model, based on that for small timesteps
Each of these approximations has different stability properties.
The bilinear transform preserves the instability of the continuous-time system.
In statistics and machine learning, discretization refers to the process of converting continuous features or variables to discretized or nominal features.
In generalized functions theory, discretization arises as a particular case of the Convolution Theorem on tempered distributions where
is a rapidly decreasing tempered distribution (e.g. a Dirac delta function
is the (unitary, ordinary frequency) Fourier transform.
which, interpreted as the coefficients of a linear combination of Dirac delta functions, forms a Dirac comb.
If additionally truncation is applied, one obtains finite sequences, e.g.
