87 research outputs found
New statistical goodness of fit techniques in noisy inhomogeneous inverse problems
No full textThe assumption that a parametric class of functions fits the
data structure sufficiently well is common in
fitting curves and surfaces to regression data. One then derives
a parameter estimate resulting from a least squares fit, say, and
in a second step various kinds of goodness of fit measures,
to assess whether the deviation between data and estimated surface is due to
random noise and not to systematic departures from the model.
In this paper we show that commonly-used -measures are invalid
in regression models, particularly when inhomogeneous noise is present.
Instead we present a bootstrap algorithm which is applicable in problems
described by noisy versions of Fredholm integral equations of the first kind.
We apply the suggested method to the problem of recovering
the luminosity density in the Milky Way from
data of the DIRBE experiment on board the
COBE satellite
Convergence rates of general regularization methods for statistical inverse problems and applications
No full textDuring the past the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut-off and Tikhonov type estimators. Spectral cut-off estimators achieve minimax rates for a broad range of smoothness classes and operators, but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator. Tikhonov estimators are simpler to compute, but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes. In this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods (spectral methods) including the aforementioned estimators as well as many iterative methods, such as í-methods and the Landweber iteration. The latter estimators converge at the same rate as spectral cut-off, but only require matrixvector products. Our results are applied to various problems, in particular we obtain precise convergence rates for satellite gradiometry, L2-boosting, and errors in variable problems
Gas dynamics in the Milky Way: second pattern speed and large-scale morphology
No full textWe present new gas flow models for the Milky Way inside the solar circle. We use smoothed particles hydrodynamics (SPH) simulations in gravitational potentials determined from the near-infrared (NIR) luminosity distribution of the bulge and disc, assuming a constant NIR mass-to-light ratio, with an outer halo added in some cases. The luminosity models are based on the COBE/DIRBE maps and on clump giant star counts in several bulge fields and include a spiral arm model for the disc. Gas flows in models that include massive spiral arms clearly match the observed (CO)-C-12 (l, v) diagram better than if the potential does not include spiral structure. Furthermore, models in which the luminous mass distribution and the gravitational potential of the Milky Way have four spiral arms are better fits to the observed (l , v) diagram than two-armed models. Besides single-pattern speed models we investigate models with separate pattern speeds for the bar and spiral arms. The most important difference is that in the latter case the gas spiral arms go through the bar corotation region, keeping the gas aligned with the arms there. In the (l, v) plot this results in characteristic regions that appear to be nearly devoid of gas. In single-pattern speed models these regions are filled with gas because the spiral arms dissolve in the bar corotation region. Comparing with the (CO)-C-12 data we find evidence for separate pattern speeds in the Milky Way. From a series of models the preferred range for the bar pattern speed is Omega(p) = 60 +/- 5 Gyr(-1), corresponding to corotation at 3.4 +/- 0.3 kpc. The spiral pattern speed is less well constrained, but our preferred value is Omega(sp) approximate to 20 Gyr(-1). A further series of gas models is computed for different bar angles, using separately determined luminosity models and gravitational potentials in each case. We find acceptable gas models for 20degreesless than or similar tophi(bar) less than or similar to 25degrees. The model with (phi(bar) = 20degrees, Omega(p) = 60 Gyr(-1), Omega(sp) = 20 Gyr(-1)) gives an excellent fit to the spiral arm ridges in the observed (l , v) plot
Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise
No full textWe consider nonlinear inverse problems described by operator equations F(a) = u. Here a is an element of a Hilbert space H which we want to estimate, and u is an L-2-function. The given data consist of measurements of u at n points, perturbed by random noise. We construct an estimator (a) over cap (n) for a by a combination of a local polynomial estimator and a nonlinear Tikhonov regularization and establish consistency in the sense that the mean integrated square error Eparallel to(a) over cap (n) - aparallel to(H)(2) (MISE) tends to 0 as n --> infinity under reasonable assumptions. Moreover, if a satisfies a source condition, we prove convergence rates for the MISE of (a) over cap (n) as well as almost surely. Further, it is shown that a cross-validated parameter selection yields a fully data-driven consistent method for the reconstruction of a. Finally, the feasibility of our algorithm is investigated in a numerical study for a groundwater filtration problem and an inverse obstacle scattering problem, respectively
Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators
No full text--
Shape constrained estimators in inverse regression models with convolution-type operator
No full textIn this paper we are concerned with shape restricted estimation in inverse regression problems with convolution-type operator. We use increasing rearrangements to compute increasingand convex estimates from an (in principle arbitrary) unconstrained estimate of the unknown regression function. An advantage of our approach is that it is not necessary that prior shape information is known to be valid on the complete domain of the regression function. Instead, it is sufficient if it holds on some compact interval. A simulation study shows that the shape restricted estimate on the respective interval is significantly less sensitive to moderate undersmoothing than the unconstrained estimate, which substantially improves applicability of estimates based on data-driven bandwidth estimators. Finally, we demonstrate the application of the increasing estimator by the estimation of the luminosity profile of an elliptical galaxy. Here, a major interest is in reconstructing the central peak of the profile, which, due to its small size, requires to select the bandwidth as small as possible. --convexity,increasing rearrangements,image reconstruction,inverse problems,monotonicity,order restricted inference,regression estimation,shape restrictions
Testing for lack of fit in inverse regression - with applications to photonic imaging.
No full textRegression; Problems; Lack-of-fit; Applications;
Statistical inference for inverse problems
No full textIn this paper we study statistical inference for certain inverse problems. We go beyond mere estimation purposes and review and develop the construction of confidence intervals and confidence bands in some inverse problems, including deconvolution and the backward heat equation. Further, we discuss the construction of certain hypothesis tests, in particular concerning the number of local maxima of the unknown function. The methods are illustrated in a case study, where we analyze the distribution of heliocentric escape velocities of galaxies in the Centaurus galaxy cluster, and provide statistical evidence for its bimodality. --Asymptotic normality,confidence interval,deconvolution,heat equation,modality,statistical inference,statistical inverse problem
Confidence bands for inverse regression models with application to gel electrophoresis
No full textWe construct uniform confidence bands for the regression function in inverse, homoscedastic regression models with convolution-type operators. Here, the convolution is between two non-periodic functions on the whole real line rather than between two period functions on a compact interval, since the former situation arguably arises more often in applications. First, following Bickel and Rosenblatt [Ann. Statist. 1, 10711095] we construct asymptotic confidence bands which are based on strong approximations and on a limit theorem for the supremum of a stationary Gaussian process. Further, we propose bootstrap confidence bands based on the residual bootstrap. A simulation study shows that the bootstrap confidence bands perform reasonably well for moderate sample sizes. Finally, we apply our method to data from a gel electrophoresis experiment with genetically engineered neuronal receptor subunits incubated with rat brain extract. --Confidencebands,Inverseproblems,Deconvolution,Rates of convergences,Nonparametric Regression
Analysing observed star cluster SEDs with evolutionary synthesis models: systematic uncertainties
No full textWe discuss the systematic uncertainties inherent to analyses of observed (broad-band) spectral energy distributions (SEDs) of star clusters with evolutionary synthesis models. We investigate the effects caused by restricting oneself to a limited number of available passbands, choices of various passband combinations, finite observational errors, non-continuous model input parameter values, and restrictions in parameter space allowed during analysis. Starting from a complete set of UBVRIJH passbands (respectively, their Hubble Space Telescope/WFPC2 equivalents) we investigate to what extent clusters with different combinations of age, metallicity, internal extinction and mass can or cannot be disentangled in the various evolutionary stages throughout their lifetimes and what are the most useful passbands required to resolve the ambiguities. We find the U and B bands to be of the highest significance, while the V band and near-infrared data provide additional constraints. A code is presented that makes use of luminosities of a star cluster system in all of the possibly available passbands, and tries to find ranges of allowed age-metallicity-extinction-mass combinations for individual members of star cluster systems. Numerous tests and examples are presented. We show the importance of good photometric accuracies and of determining the cluster parameters independently without any prior assumptions
- …
