251 research outputs found
Estimation of interventional effects of features on prediction
The interpretability of prediction mechanisms with respect to the underlying
prediction problem is often unclear. While several studies have focused on
developing prediction models with meaningful parameters, the causal
relationships between the predictors and the actual prediction have not been
considered. Here, we connect the underlying causal structure of a data
generation process and the causal structure of a prediction mechanism. To
achieve this, we propose a framework that identifies the feature with the
greatest causal influence on the prediction and estimates the necessary causal
intervention of a feature such that a desired prediction is obtained. The
general concept of the framework has no restrictions regarding data linearity;
however, we focus on an implementation for linear data here. The framework
applicability is evaluated using artificial data and demonstrated using
real-world data.Comment: To appear in Proc. IEEE International Workshop on Machine Learning
for Signal Processing (MLSP2017
A Fractional Variational Approach for Modelling Dissipative Mechanical Systems: Continuous and Discrete Settings
Employing a phase space which includes the (Riemann-Liouville) fractional
derivative of curves evolving on real space, we develop a restricted
variational principle for Lagrangian systems yielding the so-called restricted
fractional Euler-Lagrange equations (both in the continuous and discrete
settings), which, as we show, are invariant under linear change of variables.
This principle relies on a particular restriction upon the admissible variation
of the curves. In the case of the half-derivative and mechanical Lagrangians,
i.e. kinetic minus potential energy, the restricted fractional Euler-Lagrange
equations model a dissipative system in both directions of time, summing up to
a set of equations that is invariant under time reversal. Finally, we show that
the discrete equations are a meaningful discretisation of the continuous ones.Comment: Key words: Variational analysis, Mechanical systems, Lagrangian
mechanics, Damping, Fractional derivatives, Discretisation, Variational
integrators. 13 pages, no figures. Contributed paper to 6th IFAC Workshop on
Lagrangian and Hamiltonian Methods for Nonlinear Contro
Discrete mechanics and optimal control: An analysis
The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated
Optimal Reconfiguration of Formation Flying Spacecraft--a Decentralized Approach
This paper introduces a hierarchical, decentralized,
and parallelizable method for dealing with optimization
problems with many agents. It is theoretically based on a hierarchical
optimization theorem that establishes the equivalence
of two forms of the problem, and this idea is implemented using
DMOC (Discrete Mechanics and Optimal Control). The result
is a method that is scalable to certain optimization problems
for large numbers of agents, whereas the usual “monolithic”
approach can only deal with systems with a rather small
number of degrees of freedom. The method is illustrated with
the example of deployment of spacecraft, motivated by the
Darwin (ESA) and Terrestrial Planet Finder (NASA) missions
High order variational integrators in the optimal control of mechanical systems
In recent years, much effort in designing numerical methods for the
simulation and optimization of mechanical systems has been put into schemes
which are structure preserving. One particular class are variational
integrators which are momentum preserving and symplectic. In this article, we
develop two high order variational integrators which distinguish themselves in
the dimension of the underling space of approximation and we investigate their
application to finite-dimensional optimal control problems posed with
mechanical systems. The convergence of state and control variables of the
approximated problem is shown. Furthermore, by analyzing the adjoint systems of
the optimal control problem and its discretized counterpart, we prove that, for
these particular integrators, dualization and discretization commute.Comment: 25 pages, 9 figures, 1 table, submitted to DCDS-
C1-continuous space-time discretization based on Hamilton's law of varying action
We develop a class of C1-continuous time integration methods that are
applicable to conservative problems in elastodynamics. These methods are based
on Hamilton's law of varying action. From the action of the continuous system
we derive a spatially and temporally weak form of the governing equilibrium
equations. This expression is first discretized in space, considering standard
finite elements. The resulting system is then discretized in time,
approximating the displacement by piecewise cubic Hermite shape functions.
Within the time domain we thus achieve C1-continuity for the displacement field
and C0-continuity for the velocity field. From the discrete virtual action we
finally construct a class of one-step schemes. These methods are examined both
analytically and numerically. Here, we study both linear and nonlinear systems
as well as inherently continuous and discrete structures. In the numerical
examples we focus on one-dimensional applications. The provided theory,
however, is general and valid also for problems in 2D or 3D. We show that the
most favorable candidate -- denoted as p2-scheme -- converges with order four.
Thus, especially if high accuracy of the numerical solution is required, this
scheme can be more efficient than methods of lower order. It further exhibits,
for linear simple problems, properties similar to variational integrators, such
as symplecticity. While it remains to be investigated whether symplecticity
holds for arbitrary systems, all our numerical results show an excellent
long-term energy behavior.Comment: slightly condensed the manuscript, added references, numerical
results unchange
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