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 A dichotomy is a partition of a whole (or a set) into two parts (subsets). In other words, this couple of parts must be * jointly exhaustive: everything must belong to one part or the other, and * mutually exclusive: nothing can belong simulta ...

for modeling purposes, as in binary classification). Discretization is also related to

discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...

, and is an important component of granular computing. 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. Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level considered

negligible {{Short pages monitor (Specifically, the two terms share a semantic field.) The same is true of discretization error and

quantization error Quantization, in mathematics and digital signal processing, is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements. Rounding and ...

. Mathematical methods relating to discretization include the Euler–Maruyama method and the zero-order hold.

Discretization of linear state space models

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 :

\dot(t) = \mathbf A \mathbf(t) + \mathbf B \mathbf(t) + \mathbf(t)

\mathbf(t) = \mathbf C \mathbf(t) + \mathbf D \mathbf(t) + \mathbf(t)

where ''v'' and ''w'' are continuous zero-mean white noise sources with power spectral densities :

\mathbf(t) \sim N(0,\mathbf Q)

\mathbf(t) \sim N(0,\mathbf R)

can be discretized, assuming zero-order hold for the input ''u'' and continuous integration for the noise ''v'', to :

\mathbf +1 = \mathbf A_d \mathbf + \mathbf B_d \mathbf + \mathbf /math>
: \mathbf = \mathbf C_d \mathbf + \mathbf D_d \mathbf +  \mathbf /math>

with covariances

: \mathbf \sim N(0,\mathbf Q_d) : \mathbf \sim N(0,\mathbf R_d) where

: \mathbf A_d = e^ = \mathcal^\_: \mathbf B_d = \left( \int_^e^d\tau \right) \mathbf B = \mathbf A^(\mathbf A_d - I)\mathbf B, if \mathbf A is nonsingular : \mathbf C_d = \mathbf C : \mathbf D_d = \mathbf D : \mathbf Q_d = \int_^ e^ \mathbf Q e^  d\tau : \mathbf R_d = \mathbf R \frac and T is the sample time, although \mathbf A^\top is the transposed matrix of \mathbf A . The equation for the discretized measurement noise is a consequence of the continuous measurement noise being defined with a power spectral density.

A clever trick to compute ''A''

_''d'' and ''B''_''d'' in one step is by utilizing the following property: :

e^ = \begin \mathbf & \mathbf \\
                                                            \mathbf & \mathbf \end

Where

\mathbf A_d

and

\mathbf B_d

are the discretized state-space matrices.

Discretization of process noise

Numerical evaluation of

\mathbf_d

is a bit trickier due to the matrix exponential integral. It can, however, be computed by first constructing a matrix, and computing the exponential of itCharles Van Loan: ''Computing integrals involving the matrix exponential'', IEEE Transactions on Automatic Control. 23 (3): 395–404, 1978 :

\mathbf = 
\begin -\mathbf & \mathbf \\
                 \mathbf & \mathbf^\top \end T

\mathbf = e^\mathbf =
\begin \dots & \mathbf_d^\mathbf_d \\
           \mathbf & \mathbf_d^\top             \end.

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: :

\mathbf_d = (\mathbf_d^\top)^\top (\mathbf_d^\mathbf_d) = \mathbf_d (\mathbf_d^\mathbf_d).

Derivation

Starting with the continuous model :

\mathbf(t) = \mathbf A\mathbf x(t) + \mathbf B \mathbf u(t)

we know that the matrix exponential is :

\frace^ = \mathbf A e^ = e^ \mathbf A

and by premultiplying the model we get :

e^ \mathbf(t) = e^ \mathbf A\mathbf x(t) + e^ \mathbf B\mathbf u(t)

which we recognize as :

\frac(e^\mathbf x(t)) = e^ \mathbf B\mathbf u(t)

and by integrating.. :

e^\mathbf x(t) - e^0\mathbf x(0) = \int_0^t e^\mathbf B\mathbf u(\tau) d\tau

\mathbf x(t) = e^\mathbf x(0) + \int_0^t e^ \mathbf B\mathbf u(\tau) d \tau

which is an analytical solution to the continuous model. Now we want to discretise the above expression. We assume that u is constant during each timestep. :

\ \stackrel\ \mathbf x(kT)

= e^\mathbf x(0) + \int_0^ e^ \mathbf B\mathbf u(\tau) d \tau

= e^\mathbf x(0) + \int_0^ e^ \mathbf B\mathbf u(\tau) d \tau

\int_^ e^ \mathbf B\mathbf u(\tau) d \tau

We recognize the bracketed expression as

\mathbf x /math>, and the second term can be simplified by substituting with the function v(\tau) = kT + T - \tau . Note that d\tau=-dv . We also assume that \mathbf u is constant during the integral, which in turn yields
: \begin \mathbf x +1 =& e^\mathbf x - \left( \int_^ e^ dv \right) \mathbf B\mathbf u \\
&=& e^\mathbf x - \left( \int_T^0 e^ dv \right) \mathbf B\mathbf u \\
&=& e^\mathbf x + \left( \int_0^T e^ dv \right) \mathbf B\mathbf u \\
&=&e^\mathbf x + \mathbf A^\left(e^-\mathbf I \right) \mathbf B\mathbf u \end which is an exact solution to the discretization problem.

When \mathbf is singular, the latter expression can still be used by replacing e^by its Taylor expansion,
: e^ = \sum_^ \frac (T)^k  . This yields
: \begin \mathbf x +1 =& e^\mathbf x + \left( \int_0^T e^ dv \right) \mathbf B\mathbf u \\
&=&\left(\sum_^ \frac (T)^k\right) \mathbf x + \left(\sum_^ \frac ^ T^k\right) \mathbf B\mathbf u \end which is the form used in practice.

Approximations

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

e^ \approx \mathbf I + \mathbf A T

. The approximate solution then becomes: :

\mathbf x +1 \approx (\mathbf I + \mathbf AT) \mathbf x + T\mathbf B \mathbf u

This is also known as the Euler method, which is also known as the forward Euler method. Other possible approximations are

e^ \approx \left( \mathbf I - \mathbf A T \right)^

, otherwise known as the backward Euler method and

e^ \approx \left( \mathbf I +\frac  \mathbf A T \right) \left( \mathbf I - \frac \mathbf A T \right)^

, which is known as the

bilinear transform The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa. The bilinear t ...

, or Tustin transform. Each of these approximations has different stability properties. The bilinear transform preserves the instability of the continuous-time system.

Discretization of continuous features

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

and machine learning, discretization refers to the process of converting continuous features or variables to discretized or nominal features. This can be useful when creating probability mass functions.

Discretization of smooth functions

In generalized functions theory, discretization arises as a particular case of the Convolution Theorem on

tempered distributions Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose de ...

\mathcal\ = \mathcal\ \cdot \operatorname

\mathcal\= \mathcal\*\operatorname

where

\operatorname

is the Dirac comb,

\cdot \operatorname

is discretization,

* \operatorname

is periodization,

f

is a rapidly decreasing tempered distribution (e.g. a

Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...

\delta

or any other compactly supported function),

\alpha

is a smooth, slowly growing ordinary function (e.g. the function that is constantly

1

or any other band-limited function) and

\mathcal

is the (unitary, ordinary frequency)

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

. Functions

\alpha

which are not smooth can be made smooth using a mollifier prior to discretization. As an example, discretization of the function that is constantly

1

yields the sequence

.,1,1,1,.. /math> which, interpreted as the coefficients of a linear combination of

s, forms a Dirac comb. If additionally

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

is applied, one obtains finite sequences, e.g.

,1,1,1 /math>. They are discrete in both, time and frequency.

References

External links

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