Drift Vector Research Articles

BorderShift: toward optimal MeanShift vector for cluster boundary detection in high-dimensional data

We present a cluster boundary detection scheme that exploits MeanShift and Parzen window in high-dimensional space. To reduce the noises interference in Parzen window density estimation process, the kNN window is introduced to replace the sliding window with fixed size firstly. Then, we take the density of sample as the weight of its drift vector to further improve the stability of MeanShift vector which can be utilized to separate boundary points from core points, noise points, isolated points according to the vector models in multi-density data sets. Under such circumstance, our proposed BorderShift algorithm doesn’t need multi-iteration to get the optimal detection result. Instead, the developed Shift value of each data point helps to obtain it in a liner way. Experimental results on both synthetic and real data sets demonstrate that the F-measure evaluation of BorderShift is higher than that of other algorithms.

Impact of Sea Ice Drift Retrieval Errors, Discretization and Grid Type on Calculations of Ice Deformation

We studied two issues to be considered in the calculation of parameters characterizing sea ice deformation: the effect of uncertainties in an automatically retrieved sea ice drift field, and the influence of the type of drift vector grid. Sea ice deformation changes the local ice mass balance and the interaction between atmosphere, ice, and ocean, and constitutes a hazard to marine traffic and operations. Due to numerical effects, the results of deformation retrievals may predict, e.g., openings and closings of the ice cover that do not exist in reality. We focus specifically on fields of ice drift obtained from synthetic aperture radar (SAR) imagery and analyze the Propagated Drift Retrieval Error (PDRE) and the Boundary Definition Error (BDE). From the theory of error propagation, the PDRE for the calculated deformation parameters can be estimated. To quantify the BDE, we devise five different grid types and compare theoretical expectation and numerical results for different deformation parameters assuming three scenarios: pure divergence, pure shear, and a mixture of both. Our findings for both sources of error help to set up optimal deformation retrieval schemes and are also useful for other applications working with vector fields and scalar parameters derived therefrom.

Empirical evaluated SDE modelling for dimensionality-reduced systems and its predictability estimates

This paper develops and validates a method of empirical modelling for a dimensionality-reduced system of a nonlinear dynamical system based on the framework of the stochastic differential equation (SDE). Following the mathematical theorem corresponding to some inverse problem of the probability theory, we derive the empirically evaluating formulae for the drift vector and diffusion matrix. Focusing on a low-dimensional dynamical system of the Lorenz system, we empirically reconstruct an SDE that approximates the original time-series on the projected 2-dimensional plane. The distribution of the ensemble variance of solutions generated by the numerical SDE well agrees with that of the trajectories of the projected time-series, which indicates the ability of the SDE modelling to represent local predictability. Moreover, we also compare our SDE constructing method with the conventional Mori–Zwanzig projected operator method, which is used to derive a generalised Langevin equation for dimensionality-reduced systems, to assess the applicability of the obtained SDE model derived by the presented method.

Dealing with Drift Uncertainty: A Bayesian Learning Approach

One of the main challenges investors have to face is model uncertainty. Typically, the dynamic of the assets is modeled using two parameters: the drift vector and the covariance matrix, which are both uncertain. Since the variance/covariance parameter is assumed to be estimated with a certain level of confidence, we focus on drift uncertainty in this paper. Building on filtering techniques and learning methods, we use a Bayesian learning approach to solve the Markowitz problem and provide a simple and practical procedure to implement optimal strategy. To illustrate the value added of using the optimal Bayesian learning strategy, we compare it with an optimal non-learning strategy that keeps the drift constant at all times. In order to emphasize the prevalence of the Bayesian learning strategy above the non-learning one in different situations, we experiment three different investment universes: indices of various asset classes, currencies and smart beta strategies.

Reinforced concrete seismically-excited frame design with a new mixed $$\varvec{H}_2/\varvec{H}_{\infty }$$ H 2 / H ∞ optimization approach

A new approach is proposed for the optimal design of seismically excited reinforced concrete frames. Unlike most existing methods that do not exploit the input/output relationship between the base acceleration and the structural response, the main feature of the presented approach is to directly shape the input/output transfer function so as to reduce the dynamic amplification factors that govern the structural response. The approach is general with respect to distinctive issues: on the one side the structural output may encompass different quantities of engineering interest such as the overall compliance, the lateral displacement of a representative point (typically the averaged top-storey displacement) and the interstorey drift vector, on the other two system norms may be considered (or a combination thereof) as to the transfer-function amplitude to be minimized, namely the $$H_{\infty }$$ -norm and the $$H_2$$ -norm. The former allows to reduce the peak-gain response whereas the latter the squared power energy of the response. By cleverly combining the two, typically by means of a convex combination, one may end up with significant peak gain as well as power response reduction. A numerical investigation on a 2D frame is conducted to validate the theoretical framework that is modeled following Eurocode 8 (CEN in Eurocode 8: design of structures for earthquake resistance. Part 1: general rules, seismic actions and rules for buildings. European Standard EN 1998-1, Brussels (2004)) in medium ductility class (DCM) but the method applies to any regulations including modern displacement based codes such as the fib Model Code 2010 (Bulletins Nos. 65/66, Federation Internationale du Beton, Lausanne 2012).

ON THE FUNCTIONAL ESTIMATION OF MULTIVARIATE DIFFUSION PROCESSES

We propose a nonparametric estimation theory for the occupation density, the drift vector, and the diffusion matrix of multivariate diffusion processes. The estimators are sample analogues to infinitesimal conditional expectations constructed as Nadaraya-Watson kernel averages. Mild assumptions are imposed on the statistical properties of the multivariate system to obtain limiting results. Harris recurrence is all that we require to show consistency and asymptotic (mixed) normality of the proposed functional estimators. The identification method and asymptotic theory apply to both stationary and nonstationary multivariate diffusion processes of the recurrent type.

Asymptotics for the normalized error of the Ninomiya–Victoir scheme

In Gerbi et al. (2016) we proved strong convergence with order 1∕2 of the Ninomiya–Victoir scheme XNV,η with time step T∕N to the solution X of the limiting SDE. In this paper we check that the normalized error defined by NX−XNV,η converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields. The limit does not depend on the Rademacher random variables η. This result can be seen as a first step to adapt to the Ninomiya–Victoir scheme the central limit theorem of Lindeberg Feller type, derived in Ben Alaya and Kebaier (2015) for the multilevel Monte Carlo estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes. This suggests that the rate of convergence is greater than 1∕2 in this case and we actually prove strong convergence with order 1 and study the limit of the normalized error NX−XNV,η. The limiting SDE involves the Lie brackets between the Brownian vector fields and the Stratonovich drift vector field. When all the vector fields commute, the limit vanishes, which is consistent with the fact that the Ninomiya–Victoir scheme coincides with the solution to the SDE on the discretization grid.

Open-source sea ice drift algorithm for Sentinel-1 SAR imagery using a combination of feature tracking and pattern matching

Abstract. An open-source sea ice drift algorithm for Sentinel-1 SAR imagery is introduced based on the combination of feature tracking and pattern matching. Feature tracking produces an initial drift estimate and limits the search area for the consecutive pattern matching, which provides small- to medium-scale drift adjustments and normalised cross-correlation values. The algorithm is designed to combine the two approaches in order to benefit from the respective advantages. The considered feature-tracking method allows for an efficient computation of the drift field and the resulting vectors show a high degree of independence in terms of position, length, direction and rotation. The considered pattern-matching method, on the other hand, allows better control over vector positioning and resolution. The preprocessing of the Sentinel-1 data has been adjusted to retrieve a feature distribution that depends less on SAR backscatter peak values. Applying the algorithm with the recommended parameter setting, sea ice drift retrieval with a vector spacing of 4 km on Sentinel-1 images covering 400 km × 400 km, takes about 4 min on a standard 2.7 GHz processor with 8 GB memory. The corresponding recommended patch size for the pattern-matching step that defines the final resolution of each drift vector is 34 × 34 pixels (2.7 × 2.7 km). To assess the potential performance after finding suitable search restrictions, calculated drift results from 246 Sentinel-1 image pairs have been compared to buoy GPS data, collected in 2015 between 15 January and 22 April and covering an area from 80.5 to 83.5° N and 12 to 27° E. We found a logarithmic normal distribution of the displacement difference with a median at 352.9 m using HV polarisation and 535.7 m using HH polarisation. All software requirements necessary for applying the presented sea ice drift algorithm are open-source to ensure free implementation and easy distribution.

Ion Velocity Measurements for the Ionospheric Connections Explorer.

The Ionospheric Connections Explorer (ICON) payload includes an Ion Velocity Meter (IVM) to provide measurements of the ion drift motions, density, temperature and major ion composition at the satellite altitude near 575 km. The primary measurement goal for the IVM is to provide the meridional ion drift perpendicular to the magnetic meridian with an accuracy of 7.5 ms-1 for all daytime conditions encountered by the spacecraft within 15° of the magnetic equator. The IVM will derive this parameter utilizing two sensors, a retarding potential analyzer (RPA) and an ion drift meter (IDM) that have a robust and successful flight heritage. The IVM described here incorporates improvements in the design and operation to produce the most sensitive device that has been fielded to date. It will specify the ion drift vector, from which the component perpendicular to the magnetic field will be derived. In addition it will specify the total ion density, the ion temperature and the fractional ion composition. These data will be used in conjunction with measurements from the other ICON instruments to uncover the important connections between the dynamics of the neutral atmosphere and the ionosphere through the generation of dynamo currents perpendicular to the magnetic field and collisional forces parallel to the magnetic field. Here the configuration and operation of the IVM instrument are described as well as the procedures by which the ion drift velocity is determined. A description of the subsystem characteristics, which allow a determination of the expected uncertainties in the derived parameters, is also given.

A Method to Improve High-Resolution Sea Ice Drift Retrievals in the Presence of Deformation Zones

Retrievals of sea ice drift from synthetic aperture radar (SAR) images at high spatial resolution are valuable for understanding kinematic behavior and deformation processes of the ice at different spatial scales. Ice deformation causes temporal changes in patterns observed in sequences of SAR images; which makes it difficult to retrieve ice displacement with algorithms based on correlation and feature identification techniques. Here, we propose two extensions to a pattern matching algorithm, with the objective to improve the reliability of the retrieved sea ice drift field at spatial resolutions of a few hundred meters. Firstly, we extended a reliability assessment proposed in an earlier study, which is based on analyzing texture and correlation parameters of SAR image pairs, with the aim to reject unreliable pattern matches. The second step is specifically adapted to the presence of deformation features to avoid the erasing of discontinuities in the drift field. We suggest an adapted detection scheme that identifies linear deformation features (LDFs) in the drift vector field, and detects and replaces outliers after considering the presence of such LDFs in their neighborhood. We validate the improvement of our pattern matching algorithm by comparing the automatically retrieved drift to manually derived reference data for three SAR scenes acquired over different sea ice covered regions.

Small-time fluctuations for sub-Riemannian diffusion loops

We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions, under the assumptions that the diffusivity has a sub-Riemannian structure and that the drift vector field lies in the span of the sub-Riemannian structure. In the case where the endpoints agree and the generator of the diffusion process is non-elliptic at that point, the deterministic Malliavin covariance matrix is always degenerate. We identify, after a suitable rescaling, another limiting Malliavin covariance matrix which is non-degenerate, and we show that, with the same scaling, the diffusion Malliavin covariance matrices are uniformly non-degenerate. We further show that the suitably rescaled fluctuations of the diffusion loop converge to a limiting diffusion loop, which is equal in law to the loop we obtain by taking the limiting process of the unconditioned rescaled diffusion processes and condition it to return to its starting point. The generator of the unconditioned limiting rescaled diffusion process can be described in terms of the original generator.

A Combination of Feature Tracking and Pattern Matching with Optimal Parametrization for Sea Ice Drift Retrieval from SAR Data

Sea ice drift strongly influences sea ice thickness distribution and indirectly controls air-sea ice-ocean interactions. Estimating sea ice drift over a large range of spatial and temporal scales is therefore needed to characterize the properties of sea ice dynamics and to better understand the ongoing changes of the climate in the polar regions. An efficient algorithm is developed for processing SAR data based on the combination of feature tracking (FT) and pattern matching (PM) techniques. The main advantage of the combination is that the FT rapidly provides the first guess estimate of ice drift in a few unevenly distributed keypoints, and PM accurately provides drift vectors on a regular or irregular grid. Thorough sensitivity analysis of the algorithm is performed, and optimal sets of parameters are suggested for retrieval of sea ice drift on various spatial and temporal scales. The algorithm has rather high accuracy (error is below 300 m) and high speed (the time for one image pair is 1 min), which opens new opportunities for studying sea ice kinematic processes. The ice drift can now be efficiently observed in the Lagrangian coordinate system on an irregular grid and, therefore, used for pointwise evaluation of the models running on unstructured meshes or for assimilation into Lagrangian models.

Global optimization of fuel consumption in J2 rendezvous using interval analysis

This paper addresses an open-time Lambert problem under first-order gravitational perturbations with unfixed parking time and transfer time. The perturbations are compensated by introducing its analytical solutions derived from Lagrange's planetary equations into Lambert problem. A drift vector of aim position correction is defined to reduce the aim position bias caused by the perturbations. The first purpose of optimization is to find sufficiently small intervals involving the global optimal parking time, transfer time, drift vector and velocity increment. The second is to determine the global solution or the solution close to it in these intervals. Interval analysis and a double-deck gradient-based method with GA estimating the initial range of drift vector are utilized to obtain the sufficiently small intervals including the global minimum velocity increment and the global minimum solution or one sufficiently close to it in these intervals.

Nonlinear controllability of an underactuated two-link manipulator

In this contribution, we show the global controllability of an underactuated two-link manipulator model using the characteristic of the system's drift vector field and weak Poisson stability. The main idea is to proof the general possibility to let the system return to its starting equilibrium point even though the state space is not compact. After this analysis, a control method based on the controllability property of the system is suggested. More precisely, we concatenate flows of vector field of the system to generate appropriate constant input values. This method is especially useful for underactuated systems which fail to meet the so-called Brockett condition.

Optimal bounded noisy feedback control for damping random vibrations

We consider a stochastic optimal feedback control problem for a single-degree-of-freedom vibrational system, where uncertainty is described by two independent noises. The first of them is induced by the control actions and called internal, whereas the second one acts externally. The drift vector also depends on the control function. The set of pointwise control constraints is assumed to be bounded. The minimization functional is taken as the mean system response energy. The Cauchy problem for the corresponding Hamilton–Jacobi–Bellman (HJB) equation without the control constraints is first investigated. This allows us to find the sought-for feedback control strategy in a specific domain of the space of state and time variables. Then a proper extension to the remaining parts of the space is constructed, and the optimality of the resulting global feedback control strategy is proved. The obtained control law is compared with the dry friction and saturated viscous friction control laws.

Local uncontrollability for affine control systems with jumps

ABSTRACTThis paper investigates affine control systems with jumps for which the ideal If(g1, …, gm) generated by the drift vector field f in the Lie algebra L(f, g1, …, gm) can be imbedded as a kernel of a linear first-order partial differential equation. It will lead us to uncontrollable affine control systems with jumps for which the corresponding reachable sets are included in explicitly described differentiable manifolds.

On analytic characteristic functions and processes governed by SDEs

We study the analyticity of a characteristic function of a process defined by means of SDEs. Namely, starting with the simple case of a scalar Ito SDE we show that the corresponding characteristic function is entire. The proof is based on Grönwall's inequality technique and the classic analyticity criterion in terms of moments. Further, we extend this criterion and derive a handy sufficient condition of analyticity in the multidimensional case, which is used to prove the corresponding general result. We assume that the drift vector obeys the linear growth condition and the diffusion matrix is time-only-dependent but possibly degenerate. The approach used in this article can be extended to more general types of SDEs.

Grassmann phase space methods for fermions. I. Mode theory

In both quantum optics and cold atom physics, the behaviour of bosonic photons and atoms is often treated using phase space methods, where mode annihilation and creation operators are represented by c-number phase space variables, with the density operator equivalent to a distribution function of these variables. The anti-commutation rules for fermion annihilation, creation operators suggest the possibility of using anti-commuting Grassmann variables to represent these operators. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of Grassmann phase space methods in quantum–atom optics to treat fermionic systems is rather rare, though fermion coherent states using Grassmann variables are widely used in particle physics.The theory of Grassmann phase space methods for fermions based on separate modes is developed, showing how the distribution function is defined and used to determine quantum correlation functions, Fock state populations and coherences via Grassmann phase space integrals, how the Fokker–Planck equations are obtained and then converted into equivalent Ito equations for stochastic Grassmann variables. The fermion distribution function is an even Grassmann function, and is unique. The number of c-number Wiener increments involved is 2n2, if there are n modes. The situation is somewhat different to the bosonic c-number case where only 2n Wiener increments are involved, the sign of the drift term in the Ito equation is reversed and the diffusion matrix in the Fokker–Planck equation is anti-symmetric rather than symmetric. The un-normalised B distribution is of particular importance for determining Fock state populations and coherences, and as pointed out by Plimak, Collett and Olsen, the drift vector in its Fokker–Planck equation only depends linearly on the Grassmann variables. Using this key feature we show how the Ito stochastic equations can be solved numerically for finite times in terms of c-number stochastic quantities. Averages of products of Grassmann stochastic variables at the initial time are also involved, but these are determined from the initial conditions for the quantum state. The detailed approach to the numerics is outlined, showing that (apart from standard issues in such numerics) numerical calculations for Grassmann phase space theories of fermion systems could be carried out without needing to represent Grassmann phase space variables on the computer, and only involving processes using c-numbers. We compare our approach to that of Plimak, Collett and Olsen and show that the two approaches differ.As a simple test case we apply the B distribution theory and solve the Ito stochastic equations to demonstrate coupling between degenerate Cooper pairs in a four mode fermionic system involving spin conserving interactions between the spin 1/2 fermions, where modes with momenta −k,+k—each associated with spin up, spin down states, are involved.

Time-Asymptotic Evolution of Spatially Uniform Gaussian Vlasov Fluctuation Fields

We consider Vlasov fluctuations from a spatially uniform, infinitely extended plasma equilibrium and derive the appropriate Braun–Hepp evolution equations (with regularized potential). Assuming a Gaussian initial fluctuation field (of the form associated with the relevant central limit theorem), we show that at long times the fluctuating force field converges in law to a stationary Gaussian process. We also illustrate how the time-asymptotic fluctuating force field is formally associated with the diffusion matrix and drift vector for the Balescu–Guernsey–Lenard kinetic equation.

Dispersion and Limit Theorems for Random Walks Associated with Hypergeometric Functions of Type BC

The spherical functions of the non-compact Grassmann manifolds $$G_{p,q}({\mathbb {F}})=G/K$$ over the (skew-)fields $${\mathbb {F}}={\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}$$ with rank $$q\ge 1$$ and dimension parameter $$p>q$$ can be described as Heckman–Opdam hypergeometric functions of type BC, where the double coset space G / / K is identified with the Weyl chamber $$ C_q^B\subset {\mathbb {R}}^q$$ of type B. The corresponding product formulas and Harish-Chandra integral representations were recently written down by M. Rosler and the author in an explicit way such that both formulas can be extended analytically to all real parameters $$p\in [2q-1,\infty [$$ , and that associated commutative convolution structures $$*_p$$ on $$C_q^B$$ exist. In this paper, we study the associated moment functions and the dispersion of probability measures on $$C_q^B$$ with the aid of this generalized integral representation. This leads to strong laws of large numbers and central limit theorems for associated time-homogeneous random walks on $$(C_q^B, *_p)$$ where the moment functions and the dispersion appear in order to determine drift vectors and covariance matrices of these limit laws explicitly. For integers p, all results have interpretations for G-invariant random walks on the Grassmannians G / K. Besides the BC-cases, we also study the spaces $$GL(q,{\mathbb {F}})/U(q,{\mathbb {F}})$$ , which are related to Weyl chambers of type A, and for which corresponding results hold. For the rank-one-case $$q=1$$ , the results of this paper are well known in the context of Jacobi-type hypergroups on $$[0,\infty [$$ .