Vector field reconstruction
Encyclopedia
Vector field reconstruction is a method of creating a vector field
from experimental or computer generated data, usually with the goal of finding a differential equation
model
of the system.
A differential equation
model
is one that describes the value of dependent variables as they evolve in time or space by giving equations involving those variables and their derivative
s with respect to some independent variables, usually time and/or space. An ordinary differential equation
is one in which the system's dependent variables are functions of only one independent variable. Many physical, chemical, biological and electrical systems are well described by ordinary differential equations. Frequently we assume a system is governed by differential equations, but we do not have exact knowledge of the influence of various factors on the state of the system. For instance, we may have an electrical circuit that in theory is described by a system of ordinary differential equations, but due to the tolerance of resistors, variations of the supply voltage
or interference from outside influences we do not know the exact parameters of the system. For some systems, especially those that support chaos
, a small change in parameter values can cause a large change in the behavior of the system, so an accurate model is extremely important. Therefore, it may be necessary to construct more exact differential equations by building them up based on the actual system performance rather than a theoretical model. Ideally, one would measure all the dynamical variables involved over an extended period of time, using many different initial conditions, then build or fine tune a differential equation model based on these measurements.
In some cases we may not even know enough about the processes involved in a system to even formulate a model. In other cases, we may have access to only one dynamical variable for our measurements, i.e., we have a scalar time series
. If we only have a scalar time series, we need to use the method of time delay embedding or derivative coordinates to get a large enough set of dynamical variables to describe the system.
In a nutshell, once we have a set of measurements of the system state over some period of time, we find the derivatives of these measurements, which gives us a local vector field, then determine a global vector field consistent with this local field. This is usually done by a least squares
fit to the derivative data.
for
beginning at several different initial conditions. Then the task of finding a vector field, and thus a differential equation model consists of fitting functions, for instance, a cubic spline, to the data to obtain a set of continuous time functions
computing time derivatives dx1/dt, dx2/dt,...,dxk/dt of the functions, then making a least squares
fit using some sort of orthogonal basis functions (orthogonal polynomials
, radial basis functions, etc.) to each component of the tangent vectors to find a global vector field. A differential equation then can be read off the global vector field.
There are various methods of creating the basis functions for the least squares fit. The most common method is the Gram–Schmidt process
. Which creates a set of orthogonal basis vectors, which can then easily be normalized. This method begins by first selecting any standard basis β={v1, v2,...,vn}. Next, set the first vector v1=u1. Then, we set u2=v2-proju1v2. This process is repeated to for k vectors, with the final vector being uk= vk-∑(j=1)(k-1)projukvk. This then creates a set of orthogonal standard basis vectors.
The reason for using a standard orthogonal basis rather than a standard basis arises from the creation of the least squares fitting done next. Creating a least-squares fit begins by assuming some function, in the case of the reconstruction an nth degree polynomial, and fitting the curve to the data using constants. The accuracy of the fit can be increased by increasing the degree of the polynomial being used to fit the data. If a set of non-orthogonal standard basis functions was used, it becomes necessary to recalculate the constant coefficients of the function describing the fit. However, by using the orthogonal set of basis functions, it is not necessary to recalculate the constant coefficients.
s and singular value decomposition
. Gouesbet and Letellier used a multivariate polynomial approximation and least squares to reconstruct their vector field. This method was applied to the Rössler system, and the Lorenz system, as well as thermal lens oscillations.
The Rossler system, Lorenz system and Thermal lens oscillation follows the differential equations in standard system as
where F(X,Y,Z) is known as the standard function.
Usually the system being modeled in this way is a chaotic dynamical system, because chaotic systems explore a large part of the phase space
and the estimate of the global dynamics based on the local dynamics will be better than with a system exploring only a small part of the space.
Frequently, one has only a single scalar time series measurement from a system known to have more than one degree of freedom
. The time series may not even be from a system variable, but may be instead of a function of all the variables, such as temperature in a stirred tank reactor using several chemical species. In this case, one must use the technique of delay coordinate embedding, where a state vector consisting of the data at time t and several delayed versions of the data is constructed.
A comprehensive review of the topic is available from
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
from experimental or computer generated data, usually with the goal of finding a differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
of the system.
A differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
is one that describes the value of dependent variables as they evolve in time or space by giving equations involving those variables and their derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s with respect to some independent variables, usually time and/or space. An ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
is one in which the system's dependent variables are functions of only one independent variable. Many physical, chemical, biological and electrical systems are well described by ordinary differential equations. Frequently we assume a system is governed by differential equations, but we do not have exact knowledge of the influence of various factors on the state of the system. For instance, we may have an electrical circuit that in theory is described by a system of ordinary differential equations, but due to the tolerance of resistors, variations of the supply voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
or interference from outside influences we do not know the exact parameters of the system. For some systems, especially those that support chaos
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
, a small change in parameter values can cause a large change in the behavior of the system, so an accurate model is extremely important. Therefore, it may be necessary to construct more exact differential equations by building them up based on the actual system performance rather than a theoretical model. Ideally, one would measure all the dynamical variables involved over an extended period of time, using many different initial conditions, then build or fine tune a differential equation model based on these measurements.
In some cases we may not even know enough about the processes involved in a system to even formulate a model. In other cases, we may have access to only one dynamical variable for our measurements, i.e., we have a scalar time series
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
. If we only have a scalar time series, we need to use the method of time delay embedding or derivative coordinates to get a large enough set of dynamical variables to describe the system.
In a nutshell, once we have a set of measurements of the system state over some period of time, we find the derivatives of these measurements, which gives us a local vector field, then determine a global vector field consistent with this local field. This is usually done by a least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
fit to the derivative data.
Formulation
In the best possible case, one has data streams of measurements of all the system variables, equally spaced in time, say- s1(t), s2(t), ... , sk(t)
for
- t = t1, t2,..., tn,
beginning at several different initial conditions. Then the task of finding a vector field, and thus a differential equation model consists of fitting functions, for instance, a cubic spline, to the data to obtain a set of continuous time functions
- x1(t), x2(t), ... , xk(t),
computing time derivatives dx1/dt, dx2/dt,...,dxk/dt of the functions, then making a least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
fit using some sort of orthogonal basis functions (orthogonal polynomials
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
, radial basis functions, etc.) to each component of the tangent vectors to find a global vector field. A differential equation then can be read off the global vector field.
There are various methods of creating the basis functions for the least squares fit. The most common method is the Gram–Schmidt process
Gram–Schmidt process
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space Rn...
. Which creates a set of orthogonal basis vectors, which can then easily be normalized. This method begins by first selecting any standard basis β={v1, v2,...,vn}. Next, set the first vector v1=u1. Then, we set u2=v2-proju1v2. This process is repeated to for k vectors, with the final vector being uk= vk-∑(j=1)(k-1)projukvk. This then creates a set of orthogonal standard basis vectors.
The reason for using a standard orthogonal basis rather than a standard basis arises from the creation of the least squares fitting done next. Creating a least-squares fit begins by assuming some function, in the case of the reconstruction an nth degree polynomial, and fitting the curve to the data using constants. The accuracy of the fit can be increased by increasing the degree of the polynomial being used to fit the data. If a set of non-orthogonal standard basis functions was used, it becomes necessary to recalculate the constant coefficients of the function describing the fit. However, by using the orthogonal set of basis functions, it is not necessary to recalculate the constant coefficients.
Applications
Vector field reconstruction has several applications, and many different approaches. Some mathematicians have not only used radial basis functions and polynomials to reconstruct a vector field, but they have used Lyapunov exponentLyapunov exponent
In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...
s and singular value decomposition
Singular value decomposition
In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
. Gouesbet and Letellier used a multivariate polynomial approximation and least squares to reconstruct their vector field. This method was applied to the Rössler system, and the Lorenz system, as well as thermal lens oscillations.
The Rossler system, Lorenz system and Thermal lens oscillation follows the differential equations in standard system as
- X'=Y, Y'=Z and Z'=F(X,Y,Z)
where F(X,Y,Z) is known as the standard function.
Implementation issues
In some situation the model is not very efficient and difficulties can arise if the model has a large number of coefficients and demonstrates a divergent solution. For example, nonautonomous differential equations give the previously described results. In this case the modification of the standard approach in application gives a better way of further development of global vector reconstruction.Usually the system being modeled in this way is a chaotic dynamical system, because chaotic systems explore a large part of the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
and the estimate of the global dynamics based on the local dynamics will be better than with a system exploring only a small part of the space.
Frequently, one has only a single scalar time series measurement from a system known to have more than one degree of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
. The time series may not even be from a system variable, but may be instead of a function of all the variables, such as temperature in a stirred tank reactor using several chemical species. In this case, one must use the technique of delay coordinate embedding, where a state vector consisting of the data at time t and several delayed versions of the data is constructed.
A comprehensive review of the topic is available from