Showing papers on "Affine transformation published in 2000"

Term Premia and Interest Rate Forecasts in Affine Models

Abstract: The standard class of affine models produces poor forecasts of future Treasury yields. Better forecasts are generated by assuming that yields follow random walks. The failure of these models is driven by one of their key features: Compensation for risk is a multiple of the variance of the risk. Thus risk compensation cannot vary independently of interest rate volatility. I also describe a broader class of models. These “essentially affine” models retain the tractability of standard models, but allow compensation for interest rate risk to vary independently of interest rate volatility. This additional f lexibility proves useful in forecasting future yields. CAN WE USE F INANCE THEORY to tell us something about the empirical behavior of Treasury yields that we do not already know? In particular, can we sharpen our ability to predict future yields? A long-established fact about Treasury yields is that the current term structure contains information about future term structures. For example, long-maturity bond yields tend to fall over time when the slope of the yield curve is steeper than usual. These predictive relations are based exclusively on the time-series behavior of yields. To rule out arbitrage, the cross-sectional and time-series characteristics of the term structure are linked in an internally consistent way. In principle, imposing these restrictions should allow us to exploit more of the information in the current term structure, and thus improve forecasts. But in practice, existing no-arbitrage models impose other restrictions for the sake of tractability; thus their value as forecasting tools is a priori unclear. I examine the forecasting ability of the affine class of term structure models. By “affine,” I refer to models where zero-coupon bond yields, their physical dynamics, and their equivalent martingale dynamics are all affine functions of an underlying state vector. A variety of nonaffine models have been developed, but the tractability and apparent richness of the affine class has led the finance profession to focus most of its attention on such models. Although forecasting future yields is important in its own right, a model that is consistent with finance theory and produces accurate forecasts can make a deeper contribution to finance. It should allow us to address a key

Specification Analysis of Affine Term Structure Models

Abstract: This paper explores the structural differences and relative goodness-of-fits of affine term structure models ~ATSMs!. Within the family of ATSMs there is a tradeoff between f lexibility in modeling the conditional correlations and volatilities of the risk factors. This trade-off is formalized by our classification of N-factor affine family into N 1 1 non-nested subfamilies of models. Specializing to three-factor ATSMs, our analysis suggests, based on theoretical considerations and empirical evidence, that some subfamilies of ATSMs are better suited than others to explaining historical interest rate behavior. IN SPECIFYING A DYNAMIC TERM STRUCTURE MODEL—one that describes the comovement over time of short- and long-term bond yields—researchers are inevitably confronted with trade-offs between the richness of econometric representations of the state variables and the computational burdens of pricing and estimation. It is perhaps not surprising then that virtually all of the empirical implementations of multifactor term structure models that use time series data on long- and short-term bond yields simultaneously have focused on special cases of “affine” term structure models ~ATSMs! .A n ATSM accommodates time-varying means and volatilities of the state variables through affine specifications of the risk-neutral drift and volatility coefficients. At the same time, ATSMs yield essentially closed-form expressions for zero-coupon-bond prices ~Duffie and Kan ~1996!!, which greatly facilitates pricing and econometric implementation. The focus on ATSMs extends back at least to the pathbreaking studies by Vasicek ~1977! and Cox, Ingersoll, and Ross ~1985!, who presumed that the instantaneous short rate r~t! was an affine function of an N-dimensional state vector Y~t!, r~t! 5 d 0 1 d y ’ Y~t!, and that Y~t! followed Gaussian and square-root diffusions, respectively. More recently, researchers have explored formulations of ATSMs that extend the one-factor Markov represen

Single View Metrology

Abstract: We describe how 3D affine measurements may be computed from a single perspective view of a scene given only minimal geometric information determined from the image This minimal information is typically the vanishing line of a reference plane, and a vanishing point for a direction not parallel to the plane It is shown that affine scene structure may then be determined from the image, without knowledge of the camera's internal calibration (eg focal length), nor of the explicit relation between camera and world (pose) In particular, we show how to (i) compute the distance between planes parallel to the reference plane (up to a common scale factor)s (ii) compute area and length ratios on any plane parallel to the reference planes (iii) determine the camera's location Simple geometric derivations are given for these results We also develop an algebraic representation which unifies the three types of measurement and, amongst other advantages, permits a first order error propagation analysis to be performed, associating an uncertainty with each measurement We demonstrate the technique for a variety of applications, including height measurements in forensic images and 3D graphical modelling from single images

Reliable feature matching across widely separated views

Abstract: We present a robust method for automatically matching features in images corresponding to the same physical point on an object seen from two arbitrary viewpoints. Unlike conventional stereo matching approaches we assume no prior knowledge about the relative camera positions and orientations. In fact in our application this is the information we wish to determine from the image feature matches. Features are detected in two or more images and characterised using affine texture invariants. The problem of window effects is explicitly addressed by our method-our feature characterisation is invariant to linear transformations of the image data including rotation, stretch and skew. The feature matching process is optimised for a structure-from-motion application where we wish to ignore unreliable matches at the expense of reducing the number of feature matches.

Robust template matching for affine resistant image watermarks

Abstract: Digital watermarks have been proposed as a method for discouraging illicit copying and distribution of copyrighted material. This paper describes a method for the secure and robust copyright protection of digital images. We present an approach for embedding a digital watermark into an image using the Fourier transform. To this watermark is added a template in the Fourier transform domain to render the method robust against general linear transformations. We detail a new algorithm based on polar maps for the accurate and efficient recovery of the template in an image which has undergone a general affine transformation. We also present results which demonstrate the robustness of the method against some common image processing operations such as compression, rotation, scaling, and aspect ratio changes.

As-rigid-as-possible shape interpolation

Abstract: We present an object-space morphing technique that blends the interiors of given two- or three-dimensional shapes rather than their boundaries. The morph is rigid in the sense that local volumes are least-distorting as they vary from their source to target configurations. Given a boundary vertex correspondence, the source and target shapes are decomposed into isomorphic simplicial complexes. For the simplicial complexes, we find a closed-form expression allocating the paths of both boundary and interior vertices from source to target locations as a function of time. Key points are the identification of the optimal simplex morphing and the appropriate definition of an error functional whose minimization defines the paths of the vertices. Each pair of corresponding simplices defines an affine transformation, which is factored into a rotation and a stretching transformation. These local transformations are naturally interpolated over time and serve as the basis for composing a global coherent least-distorting transformation.

Lp Affine Isoperimetric Inequalities

Abstract: The Lp analogues of the Petty projection inequality and the BusemannPetty centroid inequality are established. An affine isoperimetric inequality compares two functionals associated with convex (or more general) bodies, where the ratio of the functionals is invariant under non-degenerate linear transformations. These isoperimetric inequalities are more powerful than their better-known Euclidean relatives.

On the interpretation and identification of dynamic Takagi-Sugeno fuzzy models

Abstract: Dynamic Takagi-Sugeno fuzzy models are not always easy to interpret, in particular when they are identified from experimental data. It is shown that there exists a close relationship between dynamic Takagi-Sugeno fuzzy models and dynamic linearization when using affine local model structures, which suggests that a solution to the multiobjective identification problem exists. However, it is also shown that the affine local model structure is a highly sensitive parametrization when applied in transient operating regimes. Due to the multiobjective nature of the identification problem studied here, special considerations must be made during model structure selection, experiment design, and identification in order to meet both objectives. Some guidelines for experiment design are suggested and some robust nonlinear identification algorithms are studied. These include constrained and regularized identification and locally weighted identification. Their usefulness in the present context is illustrated by examples.

An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals

Abstract: Variational approaches for nonlinear elasticity show that Hill’s incremental formulation for the prediction of the overall behaviour of heterogeneous materials yields estimates which are too stiff and may even violate rigorous bounds. This paper aims at proposing an alternative ‘affine’ formulation, based on a linear thermoelastic comparison medium, which could yield softer estimates. It is first described for nonlinear elasticity and specified by making use of Hashin–Shtrikman estimates for the linear comparison composite; the associated affine self-consistent predictions are satisfactorily compared with incremental and tangent ones for power-law creeping polycrystals. Comparison is then made with the second-order procedure (Ponte Castaneda, P., 1996. Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids, 44 (6), 827–862) and some limitations of the affine method are pointed out; explicit comparisons between different procedures are performed for isotropic, two-phase materials. Finally, the affine formulation is extended to history-dependent behaviours; application to the self-consistent modelling of the elastoplastic behaviour of polycrystals shows that it offers an improved alternative to Hill’s incremental formulation.

Robust image registration using log-polar transform

Abstract: This paper describes a hierarchical image registration algorithm for affine motion recovery. The algorithm estimates the affine transformation parameters necessary to register any two digital images misaligned due to rotation, scale, shear, and translation. The parameters are computed iteratively in a coarse-to-fine hierarchical framework using a variation of the Levenberg-Marquadt nonlinear least squares optimization method. This approach yields a robust solution that precisely registers images with subpixel accuracy. A log-polar registration module is introduced to accommodate arbitrary rotation angles and a wide range of scale changes. This serves to furnish a good initial estimate for the optimization-based affine registration stage. We demonstrate the hybrid algorithm on pairs of digital images subjected to large affine motion.

Closed-form discrete fractional and affine Fourier transforms

Abstract: The discrete fractional Fourier transform (DFRFT) is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. We introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by the FFT. Since the continuous FRFT can be generalized into the continuous affine Fourier transform (AFT) (the so-called canonical transform), we also extend the DFRFT into the discrete affine Fourier transform (DAFT). We derive two types of the DFRFT and DAFT. Type 1 is similar to the continuous FRFT and AFT and can be used for computing the continuous FRFT and AFT. Type 2 is the improved form of type 1 and can be used for other applications of digital signal processing. Meanwhile, many important properties continuous FRFT and AFT are kept in the closed-form DFRFT and DAFT, and some applications, such as filter design and pattern recognition, are also discussed. The closed-form DFRFT we introduce has the lowest complexity among all current DFRFTs that is still similar to the continuous FRFT.

Time Series and Cross-section Information in Affine Term-Structure Models

Abstract: In this article I provide an empirical analysis of the term structure of interest rates using the affine class of term-structure models introduced by Duffie and Kan. I estimate these models by combining time series and cross-section information in a theoretically consistent way. In the estimation I use a Kalman filter based on a discretization of the continuous-time factor process and allow for a general measurement-error structure. I provide evidence that a three-factor affine model with correlated factors is able to provide an adequate fit of the cross-section and the dynamics of the term structure. The three factors can be given the usual interpretation of level, steepness, and curvature.

Affine mappings of translation surfaces: geometry and arithmetic

Abstract: 1. Introduction. Translation surfaces naturally arise in the study of billiards in rational polygons (see [ZKa]). To any such polygon P , there corresponds a unique translation surface, S = S(P), such that the billiard flow in P is equivalent to the geodesic flow on S (see, e.g., [Gu2], [Gu3]). There is also a classical relation between translation surfaces and quadratic differentials on a Riemann surface S. Namely, each quadratic differential induces a translation structure on a finite puncturing of S or on a canonical double covering of S. Quadratic differentials have a natural interpretation as cotangent vectors to Teich-müller space, and this connection has proven useful in the study of billiards (see, e.g., [Ma2], [V1]). With a translation surface, S, one associates various algebraic and geometric objects: the induced affine structure of S and the group of affine diffeomorphisms, Aff(S); the holonomy homomorphism, hol : π 1 (S) → R 2 and the holonomy group Hol(S) = hol(π 1 (S)); the flat structure on S and the natural cell decompositions of its metric completion S. In the present paper, we study the relations between these objects, as well as relations among different translation surfaces. Our main focus is the group Aff(S) and the associated group of differentials, (S) ⊂ SL(2, R). The study of these groups began as part of W. Thurston's classification of surface diffeomorphisms in [Th2]. This study continued with the work of W. Veech in [V1] and [V2]. Veech produced explicit examples of translation surfaces S for which (S) is a nonarithmetic lattice. He showed that if (S) is a lattice, then the geodesic flow on S exhibits remarkable dynamical properties. For these reasons, we call (S) the Veech group of S, and if this group is a lattice, then we call S a Veech surface. We now describe the structure of the paper and our main results. In §2, we establish the setting. In particular, we recall the notion of a G-manifold and associated objects: the developing map, the holonomy homomorphism, and the holonomy group. We introduce the notion of the differential of a G-map with respect to a normal subgroup H ⊂ G. We also introduce the spinal triangulation, one of several cell decompositions canonically associated to a flat surface with cone points. In §3, we study G-coverings of G-manifolds. Given such a covering, p : X → Y , we characterize the group …

A Library for doing polyhedral operations

Abstract: The design and implementation of a library of C-code procedures to perform operations on rational polyhedra is described. The library supports intersection, union, difference, simplification in context, convex hull, affine image, affine preimage, and computation of dual forms. Since not all of these functions are closed over polyhedra, the library is extended to operate on finite unions of polyhedra. The major design decisions made during the implementation of the library are discussed. The data structure used for representing finite unions of polyhedra is developed and validity rules for the representation of polyhedra are derived. And finally, the algorithms used to implement the various functions in the library are presented.

Unwarping of unidirectionally distorted EPI images

Abstract: Echo-planar imaging (EPI) is a fast nuclear magnetic resonance imaging (MRI) method. Unfortunately, local magnetic field inhomogeneities induced mainly by the subject's presence cause significant geometrical distortion, predominantly along the phase-encoding direction, which must be undone to allow for meaningful further processing. So far, this aspect has been too often neglected. In this paper, the authors suggest a new approach using an algorithm specifically developed for the automatic registration of distorted EPI images with corresponding anatomically correct MRI images. They model the deformation field with splines, which gives us a great deal of flexibility, while comprising the affine transform as a special case. The registration criterion is least squares. Interestingly, the complexity of its evaluation does not depend on the resolution of the control grid. The spline model gives the authors good accuracy thanks to its high approximation order. The short support of splines leads to a fast algorithm. A multiresolution approach yields robustness and additional speedup. The algorithm was tested on real as well as synthetic data, and the results were compared with a manual method. A wavelet-based Sobolev-type random deformation generator was developed for testing purposes. A blind test indicates that the proposed automatic method is faster, more reliable, and more precise than the manual one.

Convergence behavior of affine projection algorithms

Abstract: A class of equivalent algorithms that accelerate the convergence of the normalized LMS (NLMS) algorithm, especially for colored inputs, has previously been discovered independently. The affine projection algorithm (APA) is the earliest and most popular algorithm in this class that inherits its name. The usual APA algorithms update weight estimates on the basis of multiple, unit delayed, input signal vectors. We analyze the convergence behavior of the generalized APA class of algorithms (allowing for arbitrary delay between input vectors) using a simple model for the input signal vectors. Conditions for convergence of the APA class are derived. It is shown that the convergence rate is exponential and that it improves as the number of input signal vectors used for adaptation is increased. However, the rate of improvement in performance (time-to-steady-state) diminishes as the number of input signal vectors increases. For a given convergence rate, APA algorithms are shown to exhibit less misadjustment (steady-state error) than NLMS. Simulation results are provided to corroborate the analytical results.

Weyl Modules for Classical and Quantum Affine algebras

Abstract: We define a family of universal finite-dimensional highest weight modules for affine Lie algebras, we call these Weyl modules. We conjecture that these are the classical limits of the irreducible finite--dimensional representations of the quantum affine algebras. We prove this conjecture in the case of affine sl_2. We establish a criterion for these modules to be irreducible and prove a factorization theorem for them in the general case.

On Level Zero Representations of Quantized Affine Algebras

Abstract: We study the properties of level zero modules over quantized affine algebras. The proof of the conjecture on the cyclicity of tensor products by Akasaka and the present author is given. Several properties of modules generated by extremal vectors are proved. The weights of a module generated by an extremal vector are contained in the convex hull of the Weyl group orbit of the extremalweight. The universal extremal weight module with level zero fundamental weight as an extremal weight is irreducible, and isomorphic to the affinization of an irreducible finite-dimensional module.

Stability of perpetuities

Abstract: For a series of randomly discounted terms we give an integral criterion to distinguishbetween almost-sure absolute convergence and divergence in probability to $\infty$, these being the only possible forms of asymptotic behavior. This solves the existence problem for a one-dimensional perpetuity that remains from a 1979 study by Vervaat, and yields a complete characterization of the existence of distributional fixed points of a random affine map in dimension one.

Design and Use of Linear Models for Image Motion Analysis

Abstract: Linear parameterized models of optical flow, particularly affine models, have become widespread in image motion analysis. The linear model coefficients are straightforward to estimate, and they provide reliable estimates of the optical flow of smooth surfaces. Here we explore the use of parameterized motion models that represent much more varied and complex motions. Our goals are threefold: to construct linear bases for complex motion phenomenas to estimate the coefficients of these linear modelss and to recognize or classify image motions from the estimated coefficients. We consider two broad classes of motions: i) generic “motion features” such as motion discontinuities and moving barss and ii) non-rigid, object-specific, motions such as the motion of human mouths. For motion features we construct a basis of steerable flow fields that approximate the motion features. For object-specific motions we construct basis flow fields from example motions using principal component analysis. In both cases, the model coefficients can be estimated directly from spatiotemporal image derivatives with a robust, multi-resolution scheme. Finally, we show how these model coefficients can be use to detect and recognize specific motions such as occlusion boundaries and facial expressions.

Optimization-Based Verification and Stability Characterization of Piecewise Affine and Hybrid Systems

Abstract: In this paper, we formulate the problem of characterizing the stability of a piecewise affine (PWA) system as a verification problem. The basic idea is to take the whole IRn as the set of initial conditions, and check that all the trajectories go to the origin. More precisely, we test for semi-global stability by restricting the set of initial conditions to an (arbitrarily large) bounded set X(0), and label as "asymptotically stable in T steps" the trajectories that enter an invariant set around the origin within a finite time T, or as "unstable in T steps" the trajectories which enter a set Xinst of (very large) states. Subsets of X(0) leading to none of the two previous cases are labeled as "non-classifiable in T steps". The domain of asymptotical stability in T steps is a subset of the domain of attraction of an equilibrium point, and has the practical meaning of collecting the initial conditions from which the settling time to a specified set around the origin is smaller than T. In addition, it can be computed algorithmically in finite time. Such an algorithm requires the computation of reach sets, in a similar fashion as what has been proposed for verification of hybrid systems. In this paper we present a substantial extension of the verification algorithm presented in [6] for stability characterization of PWA systems, based on linear and mixedinteger linear programming. As a result, given a set of initial conditions we are able to determine its partition into subsets of trajectories which are asymptotically stable, or unstable, or non-classifiable in T steps.

Numerically Invariant Signature Curves

Abstract: Corrected versions of the numerically invariant expressions for the affine and Euclidean signature of a planar curve introduced by Calabi et al. in (Int. J. Comput. Vision, 26: 107–135, 1998) are presented. The new formulas are valid for fine but otherwise arbitrary partitions of the curve. We also give numerically invariant expressions for the four differential invariants parameterizing the three dimensional version of the Euclidean signature curve, namely the curvature, the torsion and their derivatives with respect to arc length.

PWP-078 Linear Supply Function Equilibrium: Generalizations, Application, and Limitations

Abstract: We consider a supply function equilibrium (SFE) model of interaction in an electricity market. We assume a linear demand function and consider a competitive fringe and several strategic players all having capacity limits and affine marginal costs. The choice of SFE over Cournot equilibrium and the choice of affine marginal costs is reviewed in the context of the existing literature. We assume that bid rules allow affine or piecewise affine non-decreasing supply functions and extend results of Green and Rudkevitch concerning the linear SFE solution. An incentive compatibility result is proved. We also find that a piecewise affine SFE can be found easily for the case where there are non-negativity limits on generation. Upper capacity limits, however, pose problems and we propose an ad hoc approach. We apply the analysis to the England and Wales electricity market, considering the 1996 and 1999 divestitures. The piecewise affine SFE solutions generally provide better matches to the empirical data than previous analysis.

Robust watermarking of polygonal meshes

Abstract: This paper presents two variations of a robust watermarking method for general polygonal meshes of arbitrary topology which can be used for copyright protection, tamper proofing or content annotation purposes. The proposed watermark is immune to translation, rotation, scaling or affine transformation of the mesh and is hard to detect unless the exact encoding parameters are disclosed. Several examples demonstrate the effectiveness of the algorithm.

Querying time series data based on similarity

Abstract: We study similarity queries for time series data where similarity is defined, in a fairly general way, in terms of a distance function and a set of affine transformations on the Fourier series representation of a sequence. We identify a safe set of transformations supporting a wide variety of comparisons and show that this set is rich enough to formulate operations such as moving average and time scaling. We also show that queries expressed using safe transformations can efficiently be computed without prior knowledge of the transformations. We present a query processing algorithm that uses the underlying multidimensional index built over the data set to efficiently answer similarity queries. Our experiments show that the performance of this algorithm is competitive to that of processing ordinary (exact match) queries using the index, and much faster than sequential scanning. We propose a generalization of this algorithm for simultaneously handling multiple transformations at a time, and give experimental results on the performance of the generalized algorithm.

Practical stability and stabilization

Abstract: Presents a practical stability result for dynamical systems depending on a small parameter. This result is applied to a practical stability analysis of fast time-varying systems studied in averaging theory, and of highly oscillatory systems studied by Sussmann and Liu (1999). Furthermore, the problem of practically stabilizing control affine systems with drift is discussed.

Apparatus and method for context-based indexing and retrieval of image sequences

Abstract: An apparatus (150) and method for implementing object motion segmentation and object trajectory segmentation for an image sequence. Specifically, block-based motion vectors for a pair of adjacent frames are used to derive optical flow, e.g., affine, motion parameters. Such optical flow motion parameters are employed to determine key objects where their motion and trajectory within a sequence of frames are calculated and stored. Such object motion information is used to improve or offer image processing functions such as context-based indexing of the input image sequence by using motion-based information.