171,581 research outputs found
A theory of cross-validation error
This paper presents a theory of error in cross-validation testing of algorithms for predicting
real-valued attributes. The theory justifies the claim that predicting real-valued
attributes requires balancing the conflicting demands of simplicity and accuracy. Furthermore,
the theory indicates precisely how these conflicting demands must be balanced, in
order to minimize cross-validation error. A general theory is presented, then it is
developed in detail for linear regression and instance-based learning
Convex Optimization In Identification Of Stable Non-Linear State Space Models
A new framework for nonlinear system identification is presented in terms of
optimal fitting of stable nonlinear state space equations to input/output/state
data, with a performance objective defined as a measure of robustness of the
simulation error with respect to equation errors. Basic definitions and
analytical results are presented. The utility of the method is illustrated on a
simple simulation example as well as experimental recordings from a live
neuron.Comment: 9 pages, 2 figure, elaboration of same-title paper in 49th IEEE
Conference on Decision and Contro
Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations
Recent results in the literature provide computational evidence that
stabilized semi-implicit time-stepping method can efficiently simulate phase
field problems involving fourth-order nonlinear dif- fusion, with typical
examples like the Cahn-Hilliard equation and the thin film type equation. The
up-to-date theoretical explanation of the numerical stability relies on the
assumption that the deriva- tive of the nonlinear potential function satisfies
a Lipschitz type condition, which in a rigorous sense, implies the boundedness
of the numerical solution. In this work we remove the Lipschitz assumption on
the nonlinearity and prove unconditional energy stability for the stabilized
semi-implicit time-stepping methods. It is shown that the size of stabilization
term depends on the initial energy and the perturba- tion parameter but is
independent of the time step. The corresponding error analysis is also
established under minimal nonlinearity and regularity assumptions
Evaluation of stochastic effects on biomolecular networks using the generalised Nyquist stability criterion
Abstract—Stochastic differential equations are now commonly used to model biomolecular networks in systems biology, and much recent research has been devoted to the development of methods to analyse their stability properties. Stability analysis of such systems may be performed using the Laplace transform, which requires the calculation of the exponential
matrix involving time symbolically. However, the calculation of the symbolic exponential matrix is not feasible for problems of even moderate size, as the required computation time increases exponentially with the
matrix order. To address this issue, we present a novel method for approximating the Laplace transform which does not require the exponential matrix to be calculated explicitly. The calculation time associated with
the proposed method does not increase exponentially with the size of the system, and the approximation error is shown to be of the same order as existing methods. Using this approximation method, we show how a straightforward application of the generalized Nyquist stability criterion
provides necessary and sufficient conditions for the stability of stochastic biomolecular networks. The usefulness and computational efficiency of the proposed method is illustrated through its application to the problem of analysing a model for limit-cycle oscillations in cAMP during aggregation of Dictyostelium cells
A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation
We linearize the Einstein-scalar field equations, expressed relative to
constant mean curvature (CMC)-transported spatial coordinates gauge, around
members of the well-known family of Kasner solutions on . The Kasner solutions model a spatially uniform scalar field
evolving in a (typically) spatially anisotropic spacetime that expands towards
the future and that has a "Big Bang" singularity at . We
place initial data for the linearized system along and study the linear solution's behavior in the collapsing
direction . Our first main result is the proof of an
approximate monotonicity identity for the linear solutions. Using it, we
prove a linear stability result that holds when the background Kasner solution
is sufficiently close to the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW)
solution. In particular, we show that as , various
time-rescaled components of the linear solution converge to regular functions
defined along . In addition, we motivate the preferred
direction of the approximate monotonicity by showing that the CMC-transported
spatial coordinates gauge can be viewed as a limiting version of a family of
parabolic gauges for the lapse variable; an approximate monotonicity identity
and corresponding linear stability results also hold in the parabolic gauges,
but the corresponding parabolic PDEs are locally well-posed only in the
direction . Finally, based on the linear stability results, we
outline a proof of the following result, whose complete proof will appear
elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing
direction under small perturbations of its data at .Comment: 73 page
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