3,239 research outputs found
Bias Reduction of Long Memory Parameter Estimators via the Pre-filtered Sieve Bootstrap
This paper investigates the use of bootstrap-based bias correction of
semi-parametric estimators of the long memory parameter in fractionally
integrated processes. The re-sampling method involves the application of the
sieve bootstrap to data pre-filtered by a preliminary semi-parametric estimate
of the long memory parameter. Theoretical justification for using the bootstrap
techniques to bias adjust log-periodogram and semi-parametric local Whittle
estimators of the memory parameter is provided. Simulation evidence comparing
the performance of the bootstrap bias correction with analytical bias
correction techniques is also presented. The bootstrap method is shown to
produce notable bias reductions, in particular when applied to an estimator for
which analytical adjustments have already been used. The empirical coverage of
confidence intervals based on the bias-adjusted estimators is very close to the
nominal, for a reasonably large sample size, more so than for the comparable
analytically adjusted estimators. The precision of inferences (as measured by
interval length) is also greater when the bootstrap is used to bias correct
rather than analytical adjustments.Comment: 38 page
Higher-Order Improvements of the Sieve Bootstrap for Fractionally Integrated Processes
This paper investigates the accuracy of bootstrap-based inference in the case
of long memory fractionally integrated processes. The re-sampling method is
based on the semi-parametric sieve approach, whereby the dynamics in the
process used to produce the bootstrap draws are captured by an autoregressive
approximation. Application of the sieve method to data pre-filtered by a
semi-parametric estimate of the long memory parameter is also explored.
Higher-order improvements yielded by both forms of re-sampling are demonstrated
using Edgeworth expansions for a broad class of statistics that includes first-
and second-order moments, the discrete Fourier transform and regression
coefficients. The methods are then applied to the problem of estimating the
sampling distributions of the sample mean and of selected sample
autocorrelation coefficients, in experimental settings. In the case of the
sample mean, the pre-filtered version of the bootstrap is shown to avoid the
distinct underestimation of the sampling variance of the mean which the raw
sieve method demonstrates in finite samples, higher order accuracy of the
latter notwithstanding. Pre-filtering also produces gains in terms of the
accuracy with which the sampling distributions of the sample autocorrelations
are reproduced, most notably in the part of the parameter space in which
asymptotic normality does not obtain. Most importantly, the sieve bootstrap is
shown to reproduce the (empirically infeasible) Edgeworth expansion of the
sampling distribution of the autocorrelation coefficients, in the part of the
parameter space in which the expansion is valid
Filtering Random Graph Processes Over Random Time-Varying Graphs
Graph filters play a key role in processing the graph spectra of signals
supported on the vertices of a graph. However, despite their widespread use,
graph filters have been analyzed only in the deterministic setting, ignoring
the impact of stochastic- ity in both the graph topology as well as the signal
itself. To bridge this gap, we examine the statistical behavior of the two key
filter types, finite impulse response (FIR) and autoregressive moving average
(ARMA) graph filters, when operating on random time- varying graph signals (or
random graph processes) over random time-varying graphs. Our analysis shows
that (i) in expectation, the filters behave as the same deterministic filters
operating on a deterministic graph, being the expected graph, having as input
signal a deterministic signal, being the expected signal, and (ii) there are
meaningful upper bounds for the variance of the filter output. We conclude the
paper by proposing two novel ways of exploiting randomness to improve (joint
graph-time) noise cancellation, as well as to reduce the computational
complexity of graph filtering. As demonstrated by numerical results, these
methods outperform the disjoint average and denoise algorithm, and yield a (up
to) four times complexity redution, with very little difference from the
optimal solution
CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters
The rise of graph-structured data such as social networks, regulatory
networks, citation graphs, and functional brain networks, in combination with
resounding success of deep learning in various applications, has brought the
interest in generalizing deep learning models to non-Euclidean domains. In this
paper, we introduce a new spectral domain convolutional architecture for deep
learning on graphs. The core ingredient of our model is a new class of
parametric rational complex functions (Cayley polynomials) allowing to
efficiently compute spectral filters on graphs that specialize on frequency
bands of interest. Our model generates rich spectral filters that are localized
in space, scales linearly with the size of the input data for
sparsely-connected graphs, and can handle different constructions of Laplacian
operators. Extensive experimental results show the superior performance of our
approach, in comparison to other spectral domain convolutional architectures,
on spectral image classification, community detection, vertex classification
and matrix completion tasks
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