Standard error

Summary

The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM). The sampling distribution of a mean is generated by repeated sampling from the same population and recording of the sample means obtained. This forms a distribution of different means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean. Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the

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Nondyadic and nonlinear multiresolution image approximations

Maria Arrate Munoz Barrutia

This thesis focuses on the development of novel multiresolution image approximations. Specifically, we present two kinds of generalization of multiresolution techniques: image reduction for arbitrary scales, and nonlinear approximations using other metrics than the standard Euclidean one. Traditional multiresolution decompositions are restricted to dyadic scales. As first contribution of this thesis, we develop a method that goes beyond this restriction and that is well suited to arbitrary scale-change computations. The key component is a new and numerically exact algorithm for computing inner products between a continuously defined signal and B-splines of any order and of arbitrary sizes. The technique can also be applied for non-uniform to uniform grid conversion, which is another approximation problem where our method excels. Main applications are resampling and signal reconstruction. Although simple to implement, least-squares approximations lead to artifacts that could be reduced if nonlinear methods would be used instead. The second contribution of the thesis is the development of nonlinear spline pyramids that are optimal for lp-norms. First, we introduce a Banach-space formulation of the problem and show that the solution is well defined. Second, we compute the lp-approximation thanks to an iterative optimization algorithm based on digital filtering. We conclude that l1-approximations reduce the artifacts that are inherent to least-squares methods; in particular, edge blurring and ringing. In addition, we observe that the error of l1-approximations is sparser. Finally, we derive an exact formula for the asymptotic Lp-error; this result justifies using the least-squares approximation as initial solution for the iterative optimization algorithm when the degree of the spline is even; otherwise, one has to include an appropriate correction term. The theoretical background of the thesis includes the modelisation of images in a continuous/discrete formalism and takes advantage of the approximation theory of linear shift-invariant operators. We have chosen B-splines as basis functions because of their nice properties. We also propose a new graphical formalism that links B-splines, finite differences, differential operators, and arbitrary scale changes.

EPFL2002

Tumor control and radiobiological fingerprint after Gamma Knife radiosurgery for posterior fossa meningiomas: A series of 46 consecutive cases

Constantin Tuleasca

Introduction: Gamma Knife radiosurgery (GKR) can be a valuable treatment option for posterior cranial fossa meningiomas (PCFM). We retrospectively analyzed outcomes of GKR for PCFM.& nbsp;Methods: Were included forty-six patients with 47 PCFM. Primary endpoint was tumor control. Secondary endpoint was clinical improvement. Biologically effective dose (BED) was evaluated in relationship to primary and secondary outcomes. Mean marginal dose was 12.4 Gy (median 12, 12-14). Mean BED was 63.6 Gy (median 65, 49.1-88.3). Mean target volume (TV) was 2.21 cc (range 0.3-8.9 cc).& nbsp;Results: Overall tumor control rate was 93.6% (44/47) after mean follow-up of 47.8 months & PLUSMN; 28.46 months (median 45.5, range 6-108). Radiological progression-free survival at 5 years was 94%. Higher pretherapeutic TVs were predictive for higher likelihood of tumor progression (Odds ratio, OR 1.448, 95% confidence interval CI 1.001-2.093, p = 0.049). At last clinical follow-up, 28 patients (71.8%) remained stable, 10 (25.6%) improved and 1 patient (2.6%) worsened. Using logistic regression, the relationship between BED and clinical improvement was assessed (OR 0.903, standard error 0.59, coefficient 0.79-1.027, CI-0.10; 0.01; p = 0.14). The highest probability of clinical improvement corresponded to a range of BED values between 56 and 61 Gy.& nbsp;Conclusion: Primary GKR for PCFM is safe and effective. Higher pretherapeutic TV was predictor of volumetric progression. Highest probability of clinical improvement might correspond to a range of BED values between 56 and 61 Gy, although this was not statistically significant. The importance of BED should be further validated in larger cohorts, other anatomical locations and other pathologies.

2022

, , ,

In this work, we present a PDE-aware deep learning model for the numerical solution to the inverse problem of electrocardiography. The model both leverages data availability and exploits the knowledge of a physically based mathematical model, expressed by means of partial differential equations (PDEs), to carry out the task at hand. The goal is to estimate the epicardial potential field from measurements of the electric potential at a discrete set of points on the body surface. The employment of deep learning techniques in this context is made difficult by the low amount of clinical data at disposal, as measuring cardiac potentials requires invasive procedures. Suitably exploiting the underlying physically based mathematical model allowed circumventing the data availability issue and led to the development of fast-training and low-complexity models. Physical awareness has been pursued by means of two elements: the projection of the epicardial potential onto a space-time reduced subspace, spanned by the numerical solutions of the governing PDEs, and the inclusion of a tensorial reduced basis solver of the forward problem in the network architecture. Numerical tests have been conducted only on synthetic data, obtained via a full order model approximation of the problem at hand, and two variants of the model have been addressed. Both proved to be accurate, up to an average

\ell^1

-norm relative error on epicardial activation maps of 3.5%, and both could be trained in

\approx$$15

min. Nevertheless, some improvements, mostly concerning data generation, are necessary in order to bridge the gap with clinical applications.

2022

Nondyadic and nonlinear multiresolution image approximations

Maria Arrate Munoz Barrutia

EPFL2002

, , ,

\ell^1

-norm relative error on epicardial activation maps of 3.5%, and both could be trained in

\approx$$15

min. Nevertheless, some improvements, mostly concerning data generation, are necessary in order to bridge the gap with clinical applications.

2022