2,672 research outputs found
Role of Bootstrap Averaging in Generalized Approximate Message Passing
Generalized approximate message passing (GAMP) is a computationally efficient
algorithm for estimating an unknown signal from a random
linear measurement , where
is a known measurement matrix and is
the noise vector. The salient feature of GAMP is that it can provide an
unbiased estimator , which
can be used for various hypothesis-testing methods. In this study, we consider
the bootstrap average of an unbiased estimator of GAMP for the elastic net. By
numerically analyzing the state evolution of \emph{approximate message passing
with resampling}, which has been proposed for computing bootstrap statistics of
the elastic net estimator, we investigate when the bootstrap averaging reduces
the variance of the unbiased estimator and the effect of optimizing the size of
each bootstrap sample and hyperparameter of the elastic net regularization in
the asymptotic setting . The results
indicate that bootstrap averaging effectively reduces the variance of the
unbiased estimator when the actual data generation process is inconsistent with
the sparsity assumption of the regularization and the sample size is small.
Furthermore, we find that when is less sparse, and the data size is
small, the system undergoes a phase transition. The phase transition indicates
the existence of the region where the ensemble average of unbiased estimators
of GAMP for the elastic net norm minimization problem yields the unbiased
estimator with the minimum variance.Comment: 6 pages, 5 figure
Fitting a function to time-dependent ensemble averaged data
Time-dependent ensemble averages, i.e., trajectory-based averages of some
observable, are of importance in many fields of science. A crucial objective
when interpreting such data is to fit these averages (for instance, squared
displacements) with a function and extract parameters (such as diffusion
constants). A commonly overlooked challenge in such function fitting procedures
is that fluctuations around mean values, by construction, exhibit temporal
correlations. We show that the only available general purpose function fitting
methods, correlated chi-square method and the weighted least squares method
(which neglects correlation), fail at either robust parameter estimation or
accurate error estimation. We remedy this by deriving a new closed-form error
estimation formula for weighted least square fitting. The new formula uses the
full covariance matrix, i.e., rigorously includes temporal correlations, but is
free of the robustness issues, inherent to the correlated chi-square method. We
demonstrate its accuracy in four examples of importance in many fields:
Brownian motion, damped harmonic oscillation, fractional Brownian motion and
continuous time random walks. We also successfully apply our method, weighted
least squares including correlation in error estimation (WLS-ICE), to particle
tracking data. The WLS-ICE method is applicable to arbitrary fit functions, and
we provide a publically available WLS-ICE software.Comment: 47 pages (main text: 15 pages, supplementary: 32 pages
Recommended from our members
A Modern Look at Freedman's Box Model
This paper revisits the box model, a metaphor developed by David Freedman to explain sampling distributions and statistical inference to introductory statistics students. The basic idea is to represent all random phenomena in terms of drawing tickets at random from a box. In this way, random sampling from a population can be described in the same way as everyday phenomena, like coin tossing and card dealing. For Freedman, box models were merely a thought experiment; calculations were still done using normal approximations. In this paper, we propose a more modern view that treats the box model as a practical simulation framework for conducting inference. We show how concepts in introductory statistics and probability classes can be motivated by simulating from a box model. To facilitate this simulation-based approach to teaching box models, we developed an online, open-source "box model simulator"
Self-Referential Noise and the Synthesis of Three-Dimensional Space
Generalising results from Godel and Chaitin in mathematics suggests that
self-referential systems contain intrinsic randomness. We argue that this is
relevant to modelling the universe and show how three-dimensional space may
arise from a non-geometric order-disorder model driven by self-referential
noise.Comment: Figure labels correcte
Force-dependent unbinding rate of molecular motors from stationary optical trap data
Molecular motors walk along filaments until they detach stochastically with a
force-dependent unbinding rate. Here, we show that this unbinding rate can be
obtained from the analysis of experimental data of molecular motors moving in
stationary optical traps. Two complementary methods are presented, based on the
analysis of the distribution for the unbinding forces and of the motor's force
traces. In the first method, analytically derived force distributions for slip
bonds, slip-ideal bonds, and catch bonds are used to fit the cumulative
distributions of the unbinding forces. The second method is based on the
statistical analysis of the observed force traces. We validate both methods
with stochastic simulations and apply them to experimental data for kinesin-1
Recommended from our members
Bootstrap confidence intervals for the contributions of individual variables to a Mahalanobis distance
Hotelling's T 2 and Mahalanobis distance are widely used in the statistical analysis of multivariate data. When either of these quantities is large, a natural question is: How do individual variables contribute to its size? The GarthwaiteâKoch partition has been proposed as a means of assessing the contribution of each variable. This yields point estimates of each variable's contribution and here we consider bootstrap methods for forming interval estimates of these contributions. New bootstrap methods are proposed and compared with the percentile, bias-corrected percentile, non-studentized pivotal and studentized pivotal methods via a large simulation study. The new methods enable use of a broader range of pivotal quantities than with standard pivotal methods, including vector pivotal quantities. In the context considered here, this obviates the need for transformations and leads to intervals that have higher coverage, and yet are narrower, than intervals given by the standard pivotal methods. These results held both when the population distributions were multivariate normal and when they were skew with heavy tails. Both equal-tailed intervals and shortest intervals are constructed; the latter are particularly attractive when (as here) squared quantities are of interest
- âŠ