1,831 research outputs found
Performance analysis of the Least-Squares estimator in Astrometry
We characterize the performance of the widely-used least-squares estimator in
astrometry in terms of a comparison with the Cramer-Rao lower variance bound.
In this inference context the performance of the least-squares estimator does
not offer a closed-form expression, but a new result is presented (Theorem 1)
where both the bias and the mean-square-error of the least-squares estimator
are bounded and approximated analytically, in the latter case in terms of a
nominal value and an interval around it. From the predicted nominal value we
analyze how efficient is the least-squares estimator in comparison with the
minimum variance Cramer-Rao bound. Based on our results, we show that, for the
high signal-to-noise ratio regime, the performance of the least-squares
estimator is significantly poorer than the Cramer-Rao bound, and we
characterize this gap analytically. On the positive side, we show that for the
challenging low signal-to-noise regime (attributed to either a weak
astronomical signal or a noise-dominated condition) the least-squares estimator
is near optimal, as its performance asymptotically approaches the Cramer-Rao
bound. However, we also demonstrate that, in general, there is no unbiased
estimator for the astrometric position that can precisely reach the Cramer-Rao
bound. We validate our theoretical analysis through simulated digital-detector
observations under typical observing conditions. We show that the nominal value
for the mean-square-error of the least-squares estimator (obtained from our
theorem) can be used as a benchmark indicator of the expected statistical
performance of the least-squares method under a wide range of conditions. Our
results are valid for an idealized linear (one-dimensional) array detector
where intra-pixel response changes are neglected, and where flat-fielding is
achieved with very high accuracy.Comment: 35 pages, 8 figures. Accepted for publication by PAS
Analysis of the Bayesian Cramer-Rao lower bound in astrometry: Studying the impact of prior information in the location of an object
Context. The best precision that can be achieved to estimate the location of
a stellar-like object is a topic of permanent interest in the astrometric
community.
Aims. We analyse bounds for the best position estimation of a stellar-like
object on a CCD detector array in a Bayesian setting where the position is
unknown, but where we have access to a prior distribution. In contrast to a
parametric setting where we estimate a parameter from observations, the
Bayesian approach estimates a random object (i.e., the position is a random
variable) from observations that are statistically dependent on the position.
Methods. We characterize the Bayesian Cramer-Rao (CR) that bounds the minimum
mean square error (MMSE) of the best estimator of the position of a point
source on a linear CCD-like detector, as a function of the properties of
detector, the source, and the background.
Results. We quantify and analyse the increase in astrometric performance from
the use of a prior distribution of the object position, which is not available
in the classical parametric setting. This gain is shown to be significant for
various observational regimes, in particular in the case of faint objects or
when the observations are taken under poor conditions. Furthermore, we present
numerical evidence that the MMSE estimator of this problem tightly achieves the
Bayesian CR bound. This is a remarkable result, demonstrating that all the
performance gains presented in our analysis can be achieved with the MMSE
estimator.
Conclusions The Bayesian CR bound can be used as a benchmark indicator of the
expected maximum positional precision of a set of astrometric measurements in
which prior information can be incorporated. This bound can be achieved through
the conditional mean estimator, in contrast to the parametric case where no
unbiased estimator precisely reaches the CR bound.Comment: 17 pages, 12 figures. Accepted for publication on Astronomy &
Astrophysic
How to Reuse and Compose Knowledge for a Lifetime of Tasks: A Survey on Continual Learning and Functional Composition
A major goal of artificial intelligence (AI) is to create an agent capable of
acquiring a general understanding of the world. Such an agent would require the
ability to continually accumulate and build upon its knowledge as it encounters
new experiences. Lifelong or continual learning addresses this setting, whereby
an agent faces a continual stream of problems and must strive to capture the
knowledge necessary for solving each new task it encounters. If the agent is
capable of accumulating knowledge in some form of compositional representation,
it could then selectively reuse and combine relevant pieces of knowledge to
construct novel solutions. Despite the intuitive appeal of this simple idea,
the literatures on lifelong learning and compositional learning have proceeded
largely separately. In an effort to promote developments that bridge between
the two fields, this article surveys their respective research landscapes and
discusses existing and future connections between them
Orbits for eighteen visual binaries and two double-line spectroscopic binaries observed with HRCAM on the CTIO SOAR 4m telescope, using a new Bayesian orbit code based on Markov Chain Monte Carlo
We present orbital elements and mass sums for eighteen visual binary stars of
spectral types B to K (five of which are new orbits) with periods ranging from
20 to more than 500 yr. For two double-line spectroscopic binaries with no
previous orbits, the individual component masses, using combined astrometric
and radial velocity data, have a formal uncertainty of ~0.1 MSun. Adopting
published photometry, and trigonometric parallaxes, plus our own measurements,
we place these objects on an H-R diagram, and discuss their evolutionary
status. These objects are part of a survey to characterize the binary
population of stars in the Southern Hemisphere, using the SOAR 4m
telescope+HRCAM at CTIO. Orbital elements are computed using a newly developed
Markov Chain Monte Carlo algorithm that delivers maximum likelihood estimates
of the parameters, as well as posterior probability density functions that
allow us to evaluate the uncertainty of our derived parameters in a robust way.
For spectroscopic binaries, using our approach, it is possible to derive a
self-consistent parallax for the system from the combined astrometric plus
radial velocity data ("orbital parallax"), which compares well with the
trigonometric parallaxes. We also present a mathematical formalism that allows
a dimensionality reduction of the feature space from seven to three search
parameters (or from ten to seven dimensions - including parallax - in the case
of spectroscopic binaries with astrometric data), which makes it possible to
explore a smaller number of parameters in each case, improving the
computational efficiency of our Markov Chain Monte Carlo code.Comment: 32 pages, 9 figures, 6 tables. Detailed Appendix with methodology.
Accepted by The Astronomical Journa
Optimality of the Maximum Likelihood estimator in Astrometry
The problem of astrometry is revisited from the perspective of analyzing the
attainability of well-known performance limits (the Cramer-Rao bound) for the
estimation of the relative position of light-emitting (usually point-like)
sources on a CCD-like detector using commonly adopted estimators such as the
weighted least squares and the maximum likelihood. Novel technical results are
presented to determine the performance of an estimator that corresponds to the
solution of an optimization problem in the context of astrometry. Using these
results we are able to place stringent bounds on the bias and the variance of
the estimators in close form as a function of the data. We confirm these
results through comparisons to numerical simulations under a broad range of
realistic observing conditions. The maximum likelihood and the weighted least
square estimators are analyzed. We confirm the sub-optimality of the weighted
least squares scheme from medium to high signal-to-noise found in an earlier
study for the (unweighted) least squares method. We find that the maximum
likelihood estimator achieves optimal performance limits across a wide range of
relevant observational conditions. Furthermore, from our results, we provide
concrete insights for adopting an adaptive weighted least square estimator that
can be regarded as a computationally efficient alternative to the optimal
maximum likelihood solution. We provide, for the first time, close-form
analytical expressions that bound the bias and the variance of the weighted
least square and maximum likelihood implicit estimators for astrometry using a
Poisson-driven detector. These expressions can be used to formally assess the
precision attainable by these estimators in comparison with the minimum
variance bound.Comment: 24 pages, 7 figures, 2 tables, 3 appendices. Accepted by Astronomy &
Astrophysic
Embodied Lifelong Learning for Task and Motion Planning
A robot deployed in a home over long stretches of time faces a true lifelong
learning problem. As it seeks to provide assistance to its users, the robot
should leverage any accumulated experience to improve its own knowledge to
become a more proficient assistant. We formalize this setting with a novel
lifelong learning problem formulation in the context of learning for task and
motion planning (TAMP). Exploiting the modularity of TAMP systems, we develop a
generative mixture model that produces candidate continuous parameters for a
planner. Whereas most existing lifelong learning approaches determine a priori
how data is shared across task models, our approach learns shared and
non-shared models and determines which to use online during planning based on
auxiliary tasks that serve as a proxy for each model's understanding of a
state. Our method exhibits substantial improvements in planning success on
simulated 2D domains and on several problems from the BEHAVIOR benchmark
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