237,845 research outputs found
Sequential Quantiles via Hermite Series Density Estimation
Sequential quantile estimation refers to incorporating observations into
quantile estimates in an incremental fashion thus furnishing an online estimate
of one or more quantiles at any given point in time. Sequential quantile
estimation is also known as online quantile estimation. This area is relevant
to the analysis of data streams and to the one-pass analysis of massive data
sets. Applications include network traffic and latency analysis, real time
fraud detection and high frequency trading. We introduce new techniques for
online quantile estimation based on Hermite series estimators in the settings
of static quantile estimation and dynamic quantile estimation. In the static
quantile estimation setting we apply the existing Gauss-Hermite expansion in a
novel manner. In particular, we exploit the fact that Gauss-Hermite
coefficients can be updated in a sequential manner. To treat dynamic quantile
estimation we introduce a novel expansion with an exponentially weighted
estimator for the Gauss-Hermite coefficients which we term the Exponentially
Weighted Gauss-Hermite (EWGH) expansion. These algorithms go beyond existing
sequential quantile estimation algorithms in that they allow arbitrary
quantiles (as opposed to pre-specified quantiles) to be estimated at any point
in time. In doing so we provide a solution to online distribution function and
online quantile function estimation on data streams. In particular we derive an
analytical expression for the CDF and prove consistency results for the CDF
under certain conditions. In addition we analyse the associated quantile
estimator. Simulation studies and tests on real data reveal the Gauss-Hermite
based algorithms to be competitive with a leading existing algorithm.Comment: 43 pages, 9 figures. Improved version incorporating referee comments,
as appears in Electronic Journal of Statistic
Distinct counting with a self-learning bitmap
Counting the number of distinct elements (cardinality) in a dataset is a
fundamental problem in database management. In recent years, due to many of its
modern applications, there has been significant interest to address the
distinct counting problem in a data stream setting, where each incoming data
can be seen only once and cannot be stored for long periods of time. Many
probabilistic approaches based on either sampling or sketching have been
proposed in the computer science literature, that only require limited
computing and memory resources. However, the performances of these methods are
not scale-invariant, in the sense that their relative root mean square
estimation errors (RRMSE) depend on the unknown cardinalities. This is not
desirable in many applications where cardinalities can be very dynamic or
inhomogeneous and many cardinalities need to be estimated. In this paper, we
develop a novel approach, called self-learning bitmap (S-bitmap) that is
scale-invariant for cardinalities in a specified range. S-bitmap uses a binary
vector whose entries are updated from 0 to 1 by an adaptive sampling process
for inferring the unknown cardinality, where the sampling rates are reduced
sequentially as more and more entries change from 0 to 1. We prove rigorously
that the S-bitmap estimate is not only unbiased but scale-invariant. We
demonstrate that to achieve a small RRMSE value of or less, our
approach requires significantly less memory and consumes similar or less
operations than state-of-the-art methods for many common practice cardinality
scales. Both simulation and experimental studies are reported.Comment: Journal of the American Statistical Association (accepted
Particle approximations of the score and observed information matrix for parameter estimation in state space models with linear computational cost
Poyiadjis et al. (2011) show how particle methods can be used to estimate
both the score and the observed information matrix for state space models.
These methods either suffer from a computational cost that is quadratic in the
number of particles, or produce estimates whose variance increases
quadratically with the amount of data. This paper introduces an alternative
approach for estimating these terms at a computational cost that is linear in
the number of particles. The method is derived using a combination of kernel
density estimation, to avoid the particle degeneracy that causes the
quadratically increasing variance, and Rao-Blackwellisation. Crucially, we show
the method is robust to the choice of bandwidth within the kernel density
estimation, as it has good asymptotic properties regardless of this choice. Our
estimates of the score and observed information matrix can be used within both
online and batch procedures for estimating parameters for state space models.
Empirical results show improved parameter estimates compared to existing
methods at a significantly reduced computational cost. Supplementary materials
including code are available.Comment: Accepted to Journal of Computational and Graphical Statistic
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