applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemat ...

, discretization is the process of transferring continuous functions, models, variables, and equations into

discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...

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 Binary data is data whose unit can take on only two possible states. These are often labelled as 0 and 1 in accordance with the binary numeral system and Boolean algebra. Binary data occurs in many different technical and scientific fields, wher ...

(creating a dichotomy for modeling purposes, as in

binary classification Binary classification is the task of classifying the elements of a set into two groups (each called ''class'') on the basis of a classification rule. Typical binary classification problems include: * Medical testing to determine if a patient ha ...

). 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 continu ...

, 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 In linguistics, a semantic field is a lexical set of words grouped semantically (by meaning) that refers to a specific subject.Howard Jackson, Etienne Zé Amvela, ''Words, Meaning, and Vocabulary'', Continuum, 2000, p14. The term is also used in ...

.) 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 equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

s into discrete

difference equations In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

, 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 In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...

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 In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential giv ...

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 In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

, which in turn yields :

\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 In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...

, :

e^ = \sum_^ \frac (T)^k  .

This yields :

\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

In statistics 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

generalized function In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functio ...

s theory, discretization arises as a particular case of the Convolution Theorem on tempered distributions :

\mathcal\ = \mathcal\ \cdot \operatorname

\mathcal\= \mathcal\*\operatorname

where

\operatorname

is the Dirac comb,

\cdot \operatorname

is discretization,

* \operatorname

periodization In historiography, periodization is the process or study of categorizing the past into discrete, quantified, and named blocks of time for the purpose of study or analysis.Adam Rabinowitz. It's about time: historical periodization and Linked Ancie ...

f

is a rapidly decreasing tempered distribution (e.g. a Dirac delta function

\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 Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency. A band-limited signal is one whose Fourier transform or spectral density has bounded support (mathematics) ...

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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

.,1,1,1,.. /math> which, interpreted as the coefficients of a linear combination of Dirac delta function s, forms a Dirac comb . If additionally truncation 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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