Discretization Explained

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).

Discretization is also related to discrete mathematics, 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 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.) The same is true of discretization error and quantization error.

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

$\begin \dot(t) &= \mathbf(t) + \mathbf(t) + \mathbf(t) \\[2pt] \mathbf(t) &= \mathbf(t) + \mathbf(t) + \mathbf(t)\end$

where and are continuous zero-mean white noise sources with power spectral densities

$\begin \mathbf(t) &\sim N(0,\mathbf Q) \\[2pt] \mathbf(t) &\sim N(0,\mathbf R)\end$

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

$\begin \mathbf[k+1] &= \mathbf[k] + \mathbf[k] + \mathbf[k] \\[2pt] \mathbf[k] &= \mathbf[k] + \mathbf[k] + \mathbf[k]\end$

with covariances

$\begin \mathbf[k] &\sim N(0,\mathbf) \\[2pt] \mathbf[k] &\sim N(0,\mathbf)\end$

where

$\begin\mathbf &= e^ = \mathcal^ \Bigl\_ \\[4pt] \mathbf &= \left(\int_^e^d\tau \right) \mathbf B \\[4pt] \mathbf &= \mathbf C \\[8pt] \mathbf &= \mathbf D \\[2pt] \mathbf &= \int_^ e^ \mathbf Q e^ d\tau \\[2pt] \mathbf &= \mathbf R \frac \end$

and is the sample time. If is nonsingular,

B_d

=A^-1

(A_d

-I)B.

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 and in one step is by utilizing the following property:^[2]

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

Where and are the discretized state-space matrices.

Discretization of process noise

Numerical evaluation of 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 it^[3] $\begin \mathbf &= \begin -\mathbf & \mathbf \\ \mathbf & \mathbf^\top \end T \\[2pt] \mathbf &= e^\mathbf = \begin \dots & \mathbf^\mathbf \\ \mathbf & \mathbf^\top \end\end$ The discretized process noise is then evaluated by multiplying the transpose of the lower-right partition of with the upper-right partition of : $\mathbf = (\mathbf^\top)^\top (\mathbf^\mathbf) = \mathbf (\mathbf^\mathbf).$

Derivation

Starting with the continuous model $\mathbf(t) = \mathbf(t) + \mathbf(t)$ we know that the matrix exponential is $\frace^ = \mathbfe^ = e^ \mathbf A$ and by premultiplying the model we get $e^ \mathbf(t) = e^ \mathbf(t) + e^ \mathbf(t)$ which we recognize as $\frac\Bigl[e^{-\mathbf{A}t}\mathbf x(t) \Bigr] = e^ \mathbf(t)$ and by integrating, $\begin e^\mathbf(t) - e^0\mathbf(0) &= \int_0^t e^ \mathbf(\tau) d\tau \\[2pt] \mathbf(t) &= e^\mathbf(0) + \int_0^t e^ \mathbf(\tau) d\tau \end$ which is an analytical solution to the continuous model.

Now we want to discretise the above expression. We assume that is constant during each timestep. $\begin \mathbf x[k] &\, \stackrel\ \mathbf x(kT) \\[6pt] \mathbf x[k] &= e^\mathbf x(0) + \int_0^ e^ \mathbf(\tau) d\tau \\[4pt] \mathbf x[k+1] &= e^\mathbf x(0) + \int_0^ e^ \mathbf(\tau) d \tau \\[2pt] \mathbf x[k+1] &= e^ \left[e^{\mathbf{A}kT}\mathbf x(0) + \int_0^{kT} e^{\mathbf A(kT-\tau)} \mathbf{Bu}(\tau) d \tau \right]+ \int_^ e^ \mathbf B\mathbf u(\tau) d\tau\end$ We recognize the bracketed expression as

x[k]

, 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 is constant during the integral, which in turn yields

$\begin \mathbf x[k+1] &= e^\mathbf x[k] - \left(\int_^ e^ dv \right) \mathbf[k] \\[2pt] &= e^\mathbf x[k] - \left(\int_T^0 e^ dv \right) \mathbf[k] \\[2pt] &= e^\mathbf x[k] + \left(\int_0^T e^ dv \right) \mathbf[k] \\[4pt] &= e^\mathbf x[k] + \mathbf A^\left(e^ - \mathbf I \right) \mathbf[k]\end$

which is an exact solution to the discretization problem.

When is singular, the latter expression can still be used by replacing

e^AT

by its Taylor expansion,

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

This yields

\begin \mathbf x[k+1] &= e^\mathbf x[k] + \left(\int_0^T e^ dv \right) \mathbf[k] \\[2pt] &= \left(\sum_^ \frac (\mathbfT)^k\right) \mathbf x[k] + \left(\sum_^ \frac \mathbf^ T^k\right) \mathbf[k], \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^AT ≈ I+AT

. The approximate solution then becomes:

\mathbf x[k+1] \approx (\mathbf I + \mathbfT) \mathbf x[k] + T \mathbf[k]

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

e^AT ≈ (I-AT)^-1

, otherwise known as the backward Euler method and

e^AT ≈ (I+\tfrac{1}{2}AT)(I-\tfrac{1}{2}AT)^-1

, which is known as the bilinear transform, 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

See main article: 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

In generalized functions theory, discretizationarises as a particular case of the Convolution Theoremon tempered distributions

l{F}\{f*\operatorname{III}\}=l{F}\{f\} ⋅ \operatorname{III}

l{F}\{\alpha ⋅ \operatorname{III}\}=l{F}\{\alpha\}*\operatorname{III}

where

\operatorname{III}

is the Dirac comb,

⋅ \operatorname{III}

is discretization,

*\operatorname{III}

isperiodization,

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

\delta

or any othercompactly supported function),

\alpha

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

or any other band-limited function)and

l{F}

is the (unitary, ordinary frequency) Fourier transform.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

yields the sequence

[..,1,1,1,..]

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.

[1,1,1,1]

. They are discrete in both, time and frequency.

External links

Discretization in Geometry and Dynamics: research on the discretization of differential geometry and dynamics

Notes and References

Book: Analytic Sciences Corporation. Technical Staff.. Applied optimal estimation.. limited. 1974. M.I.T. Press. Gelb, Arthur, 1937-. 0-262-20027-9. Cambridge, Mass.. 121. 960061.
Raymond DeCarlo: Linear Systems: A State Variable Approach with Numerical Implementation, Prentice Hall, NJ, 1989
Charles Van Loan: Computing integrals involving the matrix exponential, IEEE Transactions on Automatic Control. 23 (3): 395–404, 1978