Type I and type II errors | EPFL Graph Search

In statistical hypothesis testing, a type I error is the mistaken rejection of an actually true null hypothesis (also known as a "false positive" finding or conclusion; example: "an innocent person is convicted"), while a type II error is the failure to reject a null hypothesis that is actually false (also known as a "false negative" finding or conclusion; example: "a guilty person is not convicted"). Much of statistical theory revolves around the minimization of one or both of these errors, though the complete elimination of either is a statistical impossibility if the outcome is not determined by a known, observable causal process. By selecting a low threshold (cut-off) value and modifying the alpha (α) level, the quality of the hypothesis test can be increased. The knowledge of type I errors and type II errors is widely used in medical science, biometrics and computer science. Intuitively, type I errors can be thought of as errors of commission (i.e., the researcher unluckily conc

Deep(ly) Learning the Depth

Xun Victor Suo

We report on the use of deep learning algorithms to perform depth recovery in multiview imaging. We show that if enough training data are provided, a neural network such as multilayer perceptron can be trained to recover the depth in multiview imaging as a regression problem. Such a method can replace camera calibration since no knowledge on the camera configuration is required during training. Another advantage of deep learning for this problem, is the speed of testing; typically a few microseconds per point in the scene. This is a lot better than state-of-art algorithms that require to solve a full optimization problem. In a second part, we have studied a related problem: detecting changes in the camera setting. We have shown that deep learning classifiers can recognize amongst a few (4 or 5) camera settings based only on the projections of points on the cameras, with less than 1% classification error. This is a promising step towards the SLAM problem.

2016

The detection of two-component mixture alternatives

Daria Rukina

In this thesis, we deal with one of the facets of the statistical detection problem. We study a particular type of alternative, the mixture model. We consider testing where the null hypothesis corresponds to the absence of a signal, represented by some known distribution, e.g., Gaussian white noise, while in the alternative one assumes that among observations there might be a cluster of points carrying a signal, which is characterized by some distribution G. The main research objective is to determine a detectable set of alternatives in the parameter space combining the parameters of G and the mixture proportion p. We focus is on the finite sample sizes and wish to study the possibility of detecting alternatives for fixed n, given pre-specified error levels. The first part of the thesis covers theoretical results. Specifically, we introduce a parametrization which relates the parameter space of an alternative to the sample size, and present the regions of detectability and non-detectability. The regions of detectability are the subsets of the new parameter space (induced by the parametrization) where for a prespecified type I error rate, the type II error rate of the likelihood ratio test (LRT) is bounded from above by some constant. To move towards the real data applications, we also check the performance of some non-parametric testing procedures proposed for this problem and some widely used distributions. In the second part of the thesis, we use this argument to develop a framework for clinical trial designs aimed at detecting a sensitive-to-therapy subpopulation. The idea of modeling treatment response as a mixture of subpopulations originates from treatment effect heterogeneity. Methods studying the effects of heterogeneity in the clinical data are referred to as subgroup analyses. However, designs accounting for possible response heterogeneity are rarely discussed, though in some cases they might help to avoid trial failure due to the lack of efficacy. In our work, we consider two possible subgroups of patients, drug responders and drug non-responders. Given no preliminary information about patients' memberships, we propose a framework for designing randomized clinical trials that are able to detect a responders' subgroup of desired characteristics. We also propose strategies to minimize the number of enrolled patients whilst preserving the testing errors below given levels and suggest how the design along with all testing metrics can be generalized to the case of multiple centers. The last part of the thesis is not directly related to the preceding parts. We present two supervised classification algorithms for real-data applications.

EPFL2018

Modélisation et analyse numérique de problèmes de réaction-diffusion provenant de la solidification d'alliages binaires

In this work, we are interested in elaborating models and numerical methods in order to study some phenomena which arise in the solidification of binary alloys. We propose two distinct models based on the mass and energy conservation laws as well as on classical thermodynamics. The first model is based on the irreversible process theory and therefore requires the computation of the entropy of the system to complete the description. To obtain this quantity, we develop a formalism and a method which yields numerical results. This construction is made so that the entropy function inherits some interesting mathematical properties from physical requirements. These properties allow us to elaborate and analyze mathematically an original scheme using a time discretization depending on a real parameter to solve the solidification problem. In particular, we are able to prove that the scheme is stable for all values of the time step if the parameter is chosen correctly. Furthermore, we give some numerical results to support the theory. The second model, called phase-field model, is used to describe dendrites' formation during the solidification of binary alloys. The nature of the problem constrains us to use very fine meshes in some physical regions. To reduce the number of discrete unknowns, we develop an adaptive mesh strategy based on an ad-hoc error estimator. We make numerical tests showing that the method converges on regular meshes and that the estimator is in good agreement with the true error. We also show that the mesh refinement strategy gives good results in academic and physical cases.

EPFL1999

The detection of two-component mixture alternatives

Daria Rukina

EPFL2018