Nach oben pdf Regularization of Least Squares Problems

Regularization of Least Squares Problems

Regularization of Least Squares Problems

Newton’s method converges globally to a solution of φ(µ) − ε 2 = 0. However, for large dimension the evaluation of φ(µ) and φ 0 (µ) for fixed µ is much too expensive (or even impossible due to the high condition number of A). Calvetti and Reichel (2003) took advantage of partial bidiagonalization of A to construct bounds to φ(µ) and thus determine a ˆ µ satisfying (61).

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Least squares residuals and minimal residual methods

Least squares residuals and minimal residual methods

for some tn, and for the residual norm we have the identities (3.5)–(3.6). At first sight the second approach seems more attractive because it gives a cleaner relation between the residual norm (which is minimized at every step) and the conditioning of the computed basis. Its implementation is also simpler. On the other hand, the fact that the approximate solution is in this approach determined via the basis vec- tors v 1 , w 1 , . . . , w n −1 which are not mutually orthogonal raises some suspicions about potential numerical problems. In this section we recall implementations of both ap- proaches resulting in different variants of the GMRES algorithm. In section 5 we will discuss their numerical properties.
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A matrix nullspace approach for solving equality-constrained multivariable polynomial least-squares problems

A matrix nullspace approach for solving equality-constrained multivariable polynomial least-squares problems

The method we introduce is based on computing the nullspace of a large sparse matrix, and computing the zeros of a scalar, uni- variate polynomial. We introduce a novel nullspace algorithm to accomplish this goal, although any nullspace method (QR, SVD, etc.) could easily be substituted in the main algorithm. In our view, the main contribution of this paper is the formulation of these multivariable optimization problems in a way for which stan- dard tools such as nullspace computation methods and eigenvalue solvers can be directly applied. In this way, advances in sparse nullspace techniques can be easily and directly applied to this large class of optimization problems. The method we present does not rely on an initial guess, and will yield the set of local and global minimizers to equality-constrained multivariable polynomial op- timization problems when there exist a finite number of local and
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The Application of Partial Least Squares Method in Hedonic Modelling

The Application of Partial Least Squares Method in Hedonic Modelling

The theory of hedonic models states that it is possible to precisely describe the price of a heterogeneous commodity by a set of its characteristics. The assump- tion that consumers derive utility from attributes of goods, rather than the good itself, led to the conception in which the price of a commodity is determined as an aggregate of values estimated for each significant characteristic of this commodity (so called hedonic prices). However, in cases of some goods the number of characteristics which significantly influence its price can be quite large. For smaller data sets this might cause a problem with the degrees of freedom, if standard methods of model estimation (such as OLS) are applied. Moreover, the bigger the number of prediction variables the more probable that they will be correlated to one another, which might cause problems induced by multicollinearity, especially when a large number of explanatory variables is binary, as it is sometimes the case in hedonic models, where variables often represent the existence of specific features in the given variant of a good, as well as its brand name or model.
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Adaptive spectral least-squares techniques for reaction-diffusion equations

Adaptive spectral least-squares techniques for reaction-diffusion equations

only the temporal discretisation with the Crank-Nicolson method and adaptive time step size provided reasonable results. But it is worth noting that we were able to solve the Turing mechanism for all investigated large reaction coefficients using our method. Regarding the examined refinement criteria we have shown that the criteria based on the spectral method are well-suited for reaction-diffusion equations. The results from chapter 7 prove that we obtain very good results with the refined grids from these criteria. Comparing the two spectral criteria we have seen that the output of both methods is nearly the same. Therefore we recommend to use the refinement criterion based on the coefficients along the spectrum’s tail as it has much less computational effort. The gradient based criterion caused some problems applying it to the Turing mechanisms. In section 7.4 we developed some ideas to explain these issues. We came to the conclusion that the gradient based criterion is simply unsuitable for Turing mechanisms as they represent kind of smooth problems.
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Sparsity Constraints and Regularization for Nonlinear Inverse Problems

Sparsity Constraints and Regularization for Nonlinear Inverse Problems

We have first investigated sparsity regularization for two parameter identification problems. For these problems, sparsity regularization incorporated with the energy functional approach was analyzed. In diffusion coefficient identification problem, the regularized problem was proven to be well-posed and convergence rates of the method was obtained under the simple source condition. An advantage of the new approach is to work with a convex and weakly lower semicontinuous functional. Therefore, the problem can be solved numerically by the fast algorithms and the proof of the well-posedness is obtained without further conditions. Another advantage is that the source condition of obtaining convergence rates is very simple. We want to emphasize that our source condition is the simplest when it is compared with the others in the least squares approach. We do not need the requirement of smallness (or generalizations of it) in the source condition.
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Regularization of linear inverse problems with total generalized variation

Regularization of linear inverse problems with total generalized variation

The framework for TGV-regularization with arbitrary order established in this paper may be applied to recover functions or symmetric tensor fields which can be modeled as piecewise polynomials of a prescribed order. For instance, for the regularization of height maps, it might be favorable to utilize order 3 or higher, since then, solutions tend to be piecewise polynomials of at least order 2 which appear smooth in three-dimensional visualizations. In contrast to that, TGV 2 fa- vors piecewise linear solutions which typically exhibit “polygonalization artifacts” in 3D-plots (or “second-order staircasing”, see [10] for a discussion in dimen- sion one). The results in this paper may moreover be useful for the develop- ment of numerical solution algorithms: A discrete version of (1.2) can directly be plugged into L 1 -type minimization methods. For instance, based on ideas in [5], one can easily compute, given a Tikhonov-functional with TGV k;l ˛ regularization, an equivalent convex-concave saddle-point formulation and derive a convergent primal-dual algorithm from general saddle-point solvers such as, e.g., the one in- troduced in [12].
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Multivariate factorisable sparse asymmetric least squares regression

Multivariate factorisable sparse asymmetric least squares regression

Abstract More and more data are observed in form of curves. Numerous applications in finance, neuroeconomics, demographics and also weather and climate analysis make it necessary to extract common patterns and prompt joint modelling of individual curve variation. Focus of such joint variation analysis has been on fluctuations around a mean curve, a statistical task that can be solved via functional PCA. In a variety of questions concerning the above applications one is more interested in the tail asking therefore for tail event curves (TEC) studies. With increasing dimension of curves and complexity of the covariates though one faces numerical problems and has to look into sparsity related issues.
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Least Squares Importance Sampling for Monte Carlo Security Pricing

Least Squares Importance Sampling for Monte Carlo Security Pricing

proaches proposed in the literature, essentially differ in the way in which such change of drift is found, and can be roughly divided into two families depending on the strategy adopted. The first strategy, common to the so- called adaptive Monte Carlo methods [5, 8–10], aims to determine the optimal drift through stochastic optimiza- tion techniques that typically involve an iterative algo- rithm. On the other hand, the second strategy, proposed in a remarkable paper by Glasserman, Heidelberger, and Shahabuddin (GHS) [6], relies on a deterministic opti- mization procedure that can be applied for a specific class of payouts. This approach, although approximate, turns out to be very effective for several pricing problems, in- cluding the simulation of a single factor Heath-Jarrow- Morton model [7], and portfolio credit scenarios [12].
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Model-Checks Based on Least Squares Residual Partial Sums Processes

Model-Checks Based on Least Squares Residual Partial Sums Processes

Chapter 6 Discussions and conclusions By virtue of being a new approach, model-check methods based on the residual partial sums process applied to spatial data analysis present open problems, both from a pure mathematical viewpoint as well as from the perspective of applications. Throughout this work, even under the simplest model, several difficulties and challenging mathe- matical problems are encountered. Since only few results concerning this subject are available, we preserve these problems as challenging open problems and directions for future research.
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Least-squares finite element methods with applications in fluid and solid mechanics

Least-squares finite element methods with applications in fluid and solid mechanics

[2003b] can be related to the thesis of Codd [2002]. Moreover, Lee et al. [2000] show an approach based on a least-squares formulation for the fluid regime and a Galerkin-FEM formulation for the solid part. Here, the coupling is solved based on a staggered scheme. The publications Heys et al. [2004a], Heys et al. [2004b], and Heys et al. [2006a] come to a LSFEM-FSI approach from a mathematical perspective, as they consider the influence of different coupling strategies on solver performance and efficiency. Heys et al. [2004a] provide a method for solving the fluid and solid problems in a single stage, and additionally consider mesh motion for the fluid phase resulting from the movement of the solid domain. The movement of the mesh is solved in an intermediate stage. The authors demonstrate their methodology for three-dimensional problems of blood flow through elastic blood vessels. Likewise, the further publication from 2004 discusses the coupling of fluiddynamical and elastic problems in one method and investigates the difference of a fully coupled, partly coupled (mapping decoupled) and fully decoupled algorithm, see Heys et al. [2004b] . Hereby, the influence of an elliptic grid generation (EGG) is used to map the deformed fluid region back to the original computational region, which is determined by the “undeformed” fluid domain. The advantages of the EGG method can be found in Codd et al. [2003b] in companion with Codd et al. [2003a]. Heys et al. [2004b] deal with the incompressible Navier-Stokes equations and formulate a first-order system in terms of velocity-velocity gradient-pressure to describe the fluid phase. The solid phase is considered to be a linear elastic material, but enhancements for a nonlinear material behavior of Neo-Hookean type are announced. The solid problem is rewritten in terms of displacement and displacement gradient in order to obtain a first-order system. The numerical testings are two-dimensional problems. Moreover, in Heys et al. [2006a] the deliberations of previously mentioned articles are expanded to three dimensional investigations with applications to blood flow through vessels. The methods under investigation are tested in conjunction with an algebraic multigrid/conjugate gradient solver, compare to Heys et al. [2006a].
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On the relationship between the Method of Least Squares and Gram-Schmidt orthogonalization

On the relationship between the Method of Least Squares and Gram-Schmidt orthogonalization

Summary The method of Least Squares is due to Carl Friedrich Gauss. The Gram-Schmidt orthogonalization method is of much younger date. A method for solving Least Squares Problems is developed which automatically results in the appearance of the Gram-Schmidt orthogonalizers. Given these orthogonalizers an induction-proof is available for solving Least Squares Problems.

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Numerical solutions of BSDEs : a-posteriori estimates and enhanced least-squares Monte Carlo

Numerical solutions of BSDEs : a-posteriori estimates and enhanced least-squares Monte Carlo

One branch was engaged with the relaxation of the Lipschitz condition on the driver. For instance, see Lepeltier and San Mart´ın (1997), who examined BSDEs with continuous driver of linear growth, or Kobylanski (2000) on BSDEs with drivers of quadratic growth. An overview is given in El Karoui and Mazliak (1997). Another important aspect was the analysis of the connection between solutions of BSDEs and viscosity solutions for quasilinear parabolic partial differential equations by Pardoux and Peng (1992). Based on this, the notion of forward backward stochastic differential equations (FBSDEs) was developed and a generalization of the Feynman- Kac formula was obtained. A detailed introduction on this topic is also available in Ma and Yong (1999). Particularly, FBSDEs became a useful tool in the field of financial mathematics. Amongst these are pricing and hedging of European options in cases with constraints or utility optimization problems. An extension to American options by BSDEs with reflection was shown in El Karoui et al. (1997a). A comprehensive survey on the application of BSDEs in finance is given by El Karoui et al. (1997).
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Weighted-Average Least Squares Estimation of Generalized Linear Models

Weighted-Average Least Squares Estimation of Generalized Linear Models

Specifically, we show that many of the theoretical and computational advantages of the WALS approach to Gaussian linear models continue to hold in the wider class of GLMs by a simple linearization of the constrained maximum likelihood (ML) estimators. To establish the asymp- totic theory for WALS, some improvements had to be made to the semiorthogonal transformation procedure. These improvements address potential discontinuity problems in the eigenvalue decom- position used in earlier papers on WALS. In addition to developing the asymptotic theory for the WALS estimator of GLMs, we also investigate the finite-sample properties of our model averaging estimator by a Monte Carlo experiment whose design is based on a real empirical example, namely the analysis of attrition in the first two waves of the Survey of Health, Ageing and Retirement in Europe (SHARE). Here, we compare the performance of WALS with that of other popular estima- tion methods such as standard ML, strict BMA with conjugate priors for GLM (Chen and Ibrahim 2003; Chen et al. 2008), and strict FMA with weighting systems based on smooth information criteria (Buckland et al. 1997; Hjort and Claeskens 2003a).
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On tracking the motion of rigid bodies through edge detection
and least-squares fitting

On tracking the motion of rigid bodies through edge detection and least-squares fitting

Imaging techniques are an attractive option for study- ing a variety of fluid flow problems, as they are inher- ently non-intrusive. The use of optical imaging to track the boundaries of non-rigid bodies such as bubbles and droplets is well established (Thoroddsen et al, 2008); imaging techniques have also been employed for the tracking of rigid bodies moving within a fluid, in ap- plications as diverse as calculating the velocities of par- ticles colliding with one another or with walls (Yang and Hunt, 2006; Marston et al, 2010), and measuring the forces on free-flying models in a shock tunnel (War- ren et al, 1961; Bernstein, 1975). In such problems, the requirements are often somewhat different from those of typical computer-vision or medical-imaging applica- tions for which many of the relevant image-processing techniques have been developed: the body geometries are usually well-defined, if not known a priori, and the emphasis is on precise displacement measurements so that velocities or accelerations can be derived with minimal uncertainty. While earlier image-based mea- surements using photographic film were limited in ac- curacy (Canning et al, 1970), recent work has shown that, with appropriate image-processing techniques ap- plied to digital images, the motion of wind-tunnel-scale bodies can be tracked to the micron level (Laurence and Hornung, 2009; Laurence and Karl, 2010). Such accu- racy allows precise time-resolved velocity and accelera- tion measurements despite the requisite differentiation of the signals.
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Regularization of Total Least Squares Problems

Regularization of Total Least Squares Problems

The suggested eigensolver in line 7 of algorithm for finding the smallest eigenpair of B(θ k +1 ) is the Rayleigh quotient iteration. Due to the required LU factorizations in each step this method is very costly. An approach like this does not turn to account the fact that the matrices B(θ k ) converge as θ k approaches the root ˆ θ of g. We suggest a method which takes advantage of information acquired in previous iteration steps by thick starts.

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Digital Filtering Algorithms for Decorrelation within Large Least Squares Problems

Digital Filtering Algorithms for Decorrelation within Large Least Squares Problems

In practice, one special item of concern is the handling of data gaps, which is problematic, because the filtering relies on an uninterrupted observation time series. There are several strategies to solve this problem. One strategy is to simply stop the filtering at each data gap and to restart the filtering after the data gap. However, due to an effect, which is called the filter warm-up (cf. Schuh 2003), the first observations at the beginning of the filtering cannot be used within the least squares adjustment. If there are many data gaps and if the filter warm-up is not short, one loses many observations in this way. Another strategy is therefore not to stop the filtering at each data gap, which then requires to fill-in the data gaps. However, to compute suitable fill-in values for the data gaps is problematic, as shown by Klees and Ditmar (2004). Therefore, Klees and Ditmar (2004) favor another approach. Here, the problem of integrating the covariance matrix into the least squares adjustment is reduced to the solution of a linear equation system with the covariance matrix as equation matrix. The linear equation system is solved using the preconditioned conjugate gradient algorithm, whereas the digital filters are used for preconditioning. However, this strategy introduces much higher computational costs, which is confirmed by Klees and Ditmar (2004).
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Two-Stage Least Squares Random Forests with a Replication of Angrist and Evans (1998)

Two-Stage Least Squares Random Forests with a Replication of Angrist and Evans (1998)

This paper has two goals. The first one is to develop a conditional instrumental variable estimator based on multiple instruments in the general framework introduced by Athey et al. (2019), and to work out the expressions for estimation, sample splitting and variance estimation needed for implementation in software. This contributes to completing the toolbox of machine learning techniques for classical econometric problems and to verifying the validity of Athey et al. (2019)’s general framework. We also address the problem of tuning an instrumental variables forest which, to our best knowledge, has not been considered in the literature before. Finally, we provide an implementation in R and C++, extending previous codes contributed by Athey et al. (2019). Our second goal is to use this estimator to revisit a classic application of instrumental variables by Angrist and Evans (1998), who used sibling-sex composition instruments in order to investigate the effect of family size on parental labor supply. Including coarse group categories in their 2SLS regressions, they also provided a basic analysis of heterogeneity of these effects across characteristics such as mother’s education or husband’s earnings. We revisit this question using instrumental variables random forests, which allow one to plot detailed maps of heterogenous effects across multiple dimensions which is not possible using standard regression techniques. This yields deeper insights into the nature of heterogeneity in these effects, going beyond the analysis in Angrist and Evans (1998).
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Discrete regularization for parameter identification problems

Discrete regularization for parameter identification problems

Investigation into convergence behavior of reconstructions using multi-bang regularization ( 1.4 ) faces considerable challenges. Although giving convergence rates of Bregman distances is at hand by the help of the so-called source conditions or variational inequalities, rates of error norms are still attractive due to explicit form of multi-bang functional. It motivates studying a particular condition and adapting it accordingly to the multi-bang penalty in corporation with a source condition to derive error norms for reconstructions, at least in L 2 -space. In [ 59 , 57 ], a regularity assumption on the active sets is introduced and used as a substitution of the source condition to obtain error estimates for the solutions of optimal control problems. This condition is also applied in [ 26 , 30 ]. Following these works, the active set is the subset of the domain, where the inequality constraints for the true parameter are active. In the spirit of this approach, [ 18 ] investigates error estimations for solutions of multi-bang control problems using the regularity assumption on the active sets. This motivates an idea of how to treat convergence rates for reconstructions given by multi-bang regularization in this thesis which is shown thereafter.
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Strong Consistency of the Least-Squares Estimator in Simple Regression Models with Stochastic Regressors

Strong Consistency of the Least-Squares Estimator in Simple Regression Models with Stochastic Regressors

Consequently, under the assumption of rational expecations, only α is identified, not, however, δ and β separately. More recently, however, economic agents are frequently assumed to be boundedly ra- tional and to form their expectations via adaptive learning. The basic idea underlying all adaptive learning procedures is that agents employ an auxiliary model, or so-called perceived law of motion (PLM), to form their expectations y t|t−1 e . One way to specify the PLM is to assume that its functional form corresponds to that of the REE in (5.3). Generally, the agents will not know the parameter α and therefore replace it by some estimate a t−1 , based on the information F t−1 . Typically, the parameter α will be esti-
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