870 research outputs found
Familywise Error Rate Control via Knockoffs
We present a novel method for controlling the -familywise error rate
(-FWER) in the linear regression setting using the knockoffs framework first
introduced by Barber and Cand\`es. Our procedure, which we also refer to as
knockoffs, can be applied with any design matrix with at least as many
observations as variables, and does not require knowing the noise variance.
Unlike other multiple testing procedures which act directly on -values,
knockoffs is specifically tailored to linear regression and implicitly accounts
for the statistical relationships between hypothesis tests of different
coefficients. We prove that knockoffs controls the -FWER exactly in finite
samples and show in simulations that it provides superior power to alternative
procedures over a range of linear regression problems. We also discuss
extensions to controlling other Type I error rates such as the false exceedance
rate, and use it to identify candidates for mutations conferring
drug-resistance in HIV.Comment: 15 pages, 3 figures. Updated reference
A Methodology for Robust Multiproxy Paleoclimate Reconstructions and Modeling of Temperature Conditional Quantiles
Great strides have been made in the field of reconstructing past temperatures
based on models relating temperature to temperature-sensitive paleoclimate
proxies. One of the goals of such reconstructions is to assess if current
climate is anomalous in a millennial context. These regression based approaches
model the conditional mean of the temperature distribution as a function of
paleoclimate proxies (or vice versa). Some of the recent focus in the area has
considered methods which help reduce the uncertainty inherent in such
statistical paleoclimate reconstructions, with the ultimate goal of improving
the confidence that can be attached to such endeavors. A second important
scientific focus in the subject area is the area of forward models for proxies,
the goal of which is to understand the way paleoclimate proxies are driven by
temperature and other environmental variables. In this paper we introduce novel
statistical methodology for (1) quantile regression with autoregressive
residual structure, (2) estimation of corresponding model parameters, (3)
development of a rigorous framework for specifying uncertainty estimates of
quantities of interest, yielding (4) statistical byproducts that address the
two scientific foci discussed above. Our statistical methodology demonstrably
produces a more robust reconstruction than is possible by using
conditional-mean-fitting methods. Our reconstruction shares some of the common
features of past reconstructions, but also gains useful insights. More
importantly, we are able to demonstrate a significantly smaller uncertainty
than that from previous regression methods. In addition, the quantile
regression component allows us to model, in a more complete and flexible way
than least squares, the conditional distribution of temperature given proxies.
This relationship can be used to inform forward models relating how proxies are
driven by temperature
Optimal Sampling-Based Motion Planning under Differential Constraints: the Driftless Case
Motion planning under differential constraints is a classic problem in
robotics. To date, the state of the art is represented by sampling-based
techniques, with the Rapidly-exploring Random Tree algorithm as a leading
example. Yet, the problem is still open in many aspects, including guarantees
on the quality of the obtained solution. In this paper we provide a thorough
theoretical framework to assess optimality guarantees of sampling-based
algorithms for planning under differential constraints. We exploit this
framework to design and analyze two novel sampling-based algorithms that are
guaranteed to converge, as the number of samples increases, to an optimal
solution (namely, the Differential Probabilistic RoadMap algorithm and the
Differential Fast Marching Tree algorithm). Our focus is on driftless
control-affine dynamical models, which accurately model a large class of
robotic systems. In this paper we use the notion of convergence in probability
(as opposed to convergence almost surely): the extra mathematical flexibility
of this approach yields convergence rate bounds - a first in the field of
optimal sampling-based motion planning under differential constraints.
Numerical experiments corroborating our theoretical results are presented and
discussed
Optimal Sampling-Based Motion Planning under Differential Constraints: the Drift Case with Linear Affine Dynamics
In this paper we provide a thorough, rigorous theoretical framework to assess
optimality guarantees of sampling-based algorithms for drift control systems:
systems that, loosely speaking, can not stop instantaneously due to momentum.
We exploit this framework to design and analyze a sampling-based algorithm (the
Differential Fast Marching Tree algorithm) that is asymptotically optimal, that
is, it is guaranteed to converge, as the number of samples increases, to an
optimal solution. In addition, our approach allows us to provide concrete
bounds on the rate of this convergence. The focus of this paper is on mixed
time/control energy cost functions and on linear affine dynamical systems,
which encompass a range of models of interest to applications (e.g.,
double-integrators) and represent a necessary step to design, via successive
linearization, sampling-based and provably-correct algorithms for non-linear
drift control systems. Our analysis relies on an original perturbation analysis
for two-point boundary value problems, which could be of independent interest
Relaxing the Assumptions of Knockoffs by Conditioning
The recent paper Cand\`es et al. (2018) introduced model-X knockoffs, a
method for variable selection that provably and non-asymptotically controls the
false discovery rate with no restrictions or assumptions on the dimensionality
of the data or the conditional distribution of the response given the
covariates. The one requirement for the procedure is that the covariate samples
are drawn independently and identically from a precisely-known (but arbitrary)
distribution. The present paper shows that the exact same guarantees can be
made without knowing the covariate distribution fully, but instead knowing it
only up to a parametric model with as many as parameters,
where is the dimension and is the number of covariate samples
(which may exceed the usual sample size of labeled samples when unlabeled
samples are also available). The key is to treat the covariates as if they are
drawn conditionally on their observed value for a sufficient statistic of the
model. Although this idea is simple, even in Gaussian models conditioning on a
sufficient statistic leads to a distribution supported on a set of zero
Lebesgue measure, requiring techniques from topological measure theory to
establish valid algorithms. We demonstrate how to do this for three models of
interest, with simulations showing the new approach remains powerful under the
weaker assumptions
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