Behavioural Approximation of Stochastic Processes by Rank Reduced Spectra

Behaviours provide an elegant, parameter free characterization of deterministic systems. We discuss a possible application of behaviours in the approximation of stochastic systems. This can be seen as an extension to the dynamic case of the well-known static factor analysis model. An essential difference is that we see modelling primarily as a matter of process approximation, not as a method to recover the true data generating process. In particular we see "noise properties" as a kind of prior model assumption that can be compared with the resulting quality of the process approximation.factor analysis;behaviours;least squares;lineair systems;stationary processes

Consistency of System Identification by Global Total Least Squares

Global total least squares (GTLS) is a method for the identification of linear systems where no distinction between input and output variables is required. This method has been developed within the deterministic behavioural approach to systems. In this paper we analyse statistical properties of this method when the observations are generated by a multivariable stationary stochastic process. In particular, sufficient conditions for the consistency of GTLS are derived. This means that, when the number of observations tends to infinity, the identified deterministic system converges to the system that provides an optimal appoximation of the data generating process. The two main results are the following. GTLS is consistent if a guaranteed stability bound can be given a priori. If this information is not available, then consistency is obtained (at some loss of finite sample efficiency) if GTLS is applied to the observed data extended with zero values in past and future

Behavioural Approximation of Stochastic Processes by Rank Reduced Spectra

Behaviours provide an elegant, parameter free characterization of deterministic systems. We discuss a possible application of behaviours in the approximation of stochastic systems. This can be seen as an extension to the dynamic case of the well-known static factor analysis model. An essential difference is that we see modelling primarily as a matter of process approximation, not as a method to recover the true data generating process. In particular we see "noise properties" as a kind of prior model assumption that can be compared with the resulting quality of the process approximation

Dirac Fields in Loop Quantum Gravity and Big Bang Nucleosynthesis

Big Bang nucleosynthesis requires a fine balance between equations of state for photons and relativistic fermions. Several corrections to equation of state parameters arise from classical and quantum physics, which are derived here from a canonical perspective. In particular, loop quantum gravity allows one to compute quantum gravity corrections for Maxwell and Dirac fields. Although the classical actions are very different, quantum corrections to the equation of state are remarkably similar. To lowest order, these corrections take the form of an overall expansion-dependent multiplicative factor in the total density. We use these results, along with the predictions of Big Bang nucleosynthesis, to place bounds on these corrections.Comment: 15 pages, 2 figures; v2: new discussion of relevance of quantum gravity corrections (Sec. II) and new estimates (Sec. V

Consistency of global total least squares in stochastic system identification

Global total least squares has been introduced as a method for the identification of deterministic system behaviours. We analyse this method within a stochastic framework, where the observed data are generated by a stationary stochastic process. Conditions are formulated so that the method is consistent in the sense that, when the number of observations tends to infinity, the identified deterministic behaviour converges to the behaviour that provides an optimal appoximation of the data generating process

Identification of System Behaviours by Approximation of Time Series Data

The behavioural framework has several attractions to offer for the identification of multivariable systems. Some of the variables may be left unexplained without the need for a distinction between inputs and outputs; criteria for model quality are independent of the chosen parametrization; and behaviours allow for a global (i.e., non-local) approximation of the system dynamics. This is illustrated with a behavioural least squares method with an application in dynamic factor analysis

Radiative transfer effects on Doppler measurements as sources of surface effects in sunspot seismology

We show that the use of Doppler shifts of Zeeman sensitive spectral lines to observe wavesn in sunspots is subject to measurement specific phase shifts arising from, (i) altered height range of spectral line formation and the propagating character of p mode waves in penumbrae, and (ii) Zeeman broadening and splitting. We also show that these phase shifts depend on wave frequencies, strengths and line of sight inclination of magnetic field, and the polarization state used for Doppler measurements. We discuss how these phase shifts could contribute to local helioseismic measurements of 'surface effects' in sunspot seismology.Comment: 12 pages, 4 figures, Accepted for publication in the Astrophysical Journal Letter

Cosmic String Formation from Correlated Fields

We simulate the formation of cosmic strings at the zeros of a complex Gaussian field with a power spectrum $P(k) \propto k^n$, specifically addressing the issue of the fraction of length in infinite strings. We make two improvements over previous simulations: we include a non-zero random background field in our box to simulate the effect of long-wavelength modes, and we examine the effects of smoothing the field on small scales. The inclusion of the background field significantly reduces the fraction of length in infinite strings for $n < -2$. Our results are consistent with the possibility that infinite strings disappear at some $n = n_c$ in the range $-3 \le n_c < -2.2$, although we cannot rule out $n_c = -3$, in which case infinite strings would disappear only at the point where the mean string density goes to zero. We present an analytic argument which suggests the latter case. Smoothing on small scales eliminates closed loops on the order of the lattice cell size and leads to a ``lattice-free" estimate of the infinite string fraction. As expected, this fraction depends on the type of window function used for smoothing.Comment: 24 pages, latex, 10 figures, submitted to Phys Rev

System Identification by Dynamic Factor Models

This paper concerns the modelling of stochastic processes by means of dynamic factor models. In such models the observed process is decomposed into a structured part called the latent process, and a remainder that is called noise. The observed variables are treated in a symmetric way, so that no distinction between inputs and outputs is required. This motivates the condition that also the prior assumptions on the noise are symmetric in nature. One of the central questions in this paper is how uncertainty about the noise structure translates into non-uniqueness of the possible underlying latent processes. We investigate several possible noise specifications and analyse properties of the resulting class of observationally equivalent factor models. This concerns in particular the characterization of optimal models and properties of continuity and consistency

Pair distribution function and structure factor of spherical particles

The availability of neutron spallation-source instruments that provide total scattering powder diffraction has led to an increased application of real-space structure analysis using the pair distribution function. Currently, the analytical treatment of finite size effects within pair distribution refinement procedures is limited. To that end, an envelope function is derived which transforms the pair distribution function of an infinite solid into that of a spherical particle with the same crystal structure. Distributions of particle sizes are then considered, and the associated envelope function is used to predict the particle size distribution of an experimental sample of gold nanoparticles from its pair distribution function alone. Finally, complementing the wealth of existing diffraction analysis, the peak broadening for the structure factor of spherical particles, expressed as a convolution derived from the envelope functions, is calculated exactly for all particle size distributions considered, and peak maxima, offsets, and asymmetries are discussed.Comment: 7 pages, 6 figure