6,744 research outputs found
Prior specification for binary Markov mesh models
We propose prior distributions for all parts of the specification of a Markov
mesh model. In the formulation we define priors for the sequential
neighborhood, for the parametric form of the conditional distributions and for
the parameter values. By simulating from the resulting posterior distribution
when conditioning on an observed scene, we thereby obtain an automatic model
selection procedure for Markov mesh models. To sample from such a posterior
distribution, we construct a reversible jump Markov chain Monte Carlo algorithm
(RJMCMC). We demonstrate the usefulness of our prior formulation and the
limitations of our RJMCMC algorithm in two examples
Minkowski tensor density formulas for Boolean models
A stationary Boolean model is the union set of random compact particles which
are attached to the points of a stationary Poisson point process. For a
stationary Boolean model with convex grains we consider a recently developed
collection of shape descriptors, the so called Minkowski tensors. By combining
spatial and probabilistic averaging we define Minkowski tensor densities of a
Boolean model. These densities are global characteristics of the union set
which can be estimated from observations. In contrast local characteristics
like the mean Minkowski tensor of a single random particle cannot be observed
directly, since the particles overlap. We relate the global to the local
properties by density formulas for the Minkowski tensors. These density
formulas generalize the well known formulas for intrinsic volume densities and
are obtained by applying results from translative integral geometry. For an
isotropic Boolean model we observe that the Minkowski tensor densities are
proportional to the intrinsic volume densities, whereas for a non-isotropic
Boolean model this is usually not the case. Our results support the idea that
the degree of anisotropy of a Boolean model may be expressed in terms of the
Minkowski tensor densities. Furthermore we observe that for smooth grains the
mean curvature radius function of a particle can be reconstructed from the
Minkowski tensor densities. In a simulation study we determine numerically
Minkowski tensor densities for non-isotropic Boolean models based on ellipses
and on rectangles in two dimensions and find excellent agreement with the
derived analytic density formulas. The tensor densities can be used to
characterize the orientational distribution of the grains and to estimate model
parameters for non-isotropic distributions.Comment: 36 pages, 6 figure
Jumping VaR: Order Statistics Volatility Estimator for Jumps Classification and Market Risk Modeling
This paper proposes a new integrated variance estimator based on order
statistics within the framework of jump-diffusion models. Its ability to
disentangle the integrated variance from the total process quadratic variation
is confirmed by both simulated and empirical tests. For practical purposes, we
introduce an iterative algorithm to estimate the time-varying volatility and
the occurred jumps of log-return time series. Such estimates enable the
definition of a new market risk model for the Value at Risk forecasting. We
show empirically that this procedure outperforms the standard historical
simulation method applying standard back-testing approach.Comment: 31 pages, 29 figures, source code available at
https://github.com/sigmaquadro/VolatilityEstimato
Statistical Mechanics of Online Learning of Drifting Concepts : A Variational Approach
We review the application of Statistical Mechanics methods to the study of
online learning of a drifting concept in the limit of large systems. The model
where a feed-forward network learns from examples generated by a time dependent
teacher of the same architecture is analyzed. The best possible generalization
ability is determined exactly, through the use of a variational method. The
constructive variational method also suggests a learning algorithm. It depends,
however, on some unavailable quantities, such as the present performance of the
student. The construction of estimators for these quantities permits the
implementation of a very effective, highly adaptive algorithm. Several other
algorithms are also studied for comparison with the optimal bound and the
adaptive algorithm, for different types of time evolution of the rule.Comment: 24 pages, 8 figures, to appear in Machine Learning Journa
Quantum Mechanics from Symmetry and Statistical Modelling
A version of quantum theory is derived from a set of plausible assumptions
related to the following general setting: For a given system there is a set of
experiments that can be performed, and for each such experiment an ordinary
statistical model is defined. The parameters of the single experiments are
functions of a hyperparameter, which defines the state of the system. There is
a symmetry group acting on the hyperparameters, and for the induced action on
the parameters of the single experiment a simple consistency property is
assumed, called permissibility of the parametric function. The other
assumptions needed are rather weak. The derivation relies partly on quantum
logic, partly on a group representation of the hyperparameter group, where the
invariant spaces are shown to be in 1-1 correspondence with the equivalence
classes of permissible parametric functions. Planck's constant only plays a
role connected to generators of unitary group representations.Comment: The paper has been withdrawn because it is outdate
Graph Sampling for Covariance Estimation
In this paper the focus is on subsampling as well as reconstructing the
second-order statistics of signals residing on nodes of arbitrary undirected
graphs. Second-order stationary graph signals may be obtained by graph
filtering zero-mean white noise and they admit a well-defined power spectrum
whose shape is determined by the frequency response of the graph filter.
Estimating the graph power spectrum forms an important component of stationary
graph signal processing and related inference tasks such as Wiener prediction
or inpainting on graphs. The central result of this paper is that by sampling a
significantly smaller subset of vertices and using simple least squares, we can
reconstruct the second-order statistics of the graph signal from the subsampled
observations, and more importantly, without any spectral priors. To this end,
both a nonparametric approach as well as parametric approaches including moving
average and autoregressive models for the graph power spectrum are considered.
The results specialize for undirected circulant graphs in that the graph nodes
leading to the best compression rates are given by the so-called minimal sparse
rulers. A near-optimal greedy algorithm is developed to design the subsampling
scheme for the non-parametric and the moving average models, whereas a
particular subsampling scheme that allows linear estimation for the
autoregressive model is proposed. Numerical experiments on synthetic as well as
real datasets related to climatology and processing handwritten digits are
provided to demonstrate the developed theory.Comment: Under peer review for Jour. of Sel. Topics in Signal Proc. (special
issue on graph signal processing), Nov. 201
Assessing the Performance of Question-and-Answer Communities Using Survival Analysis
Question-&-Answer (QA) websites have emerged as efficient platforms for
knowledge sharing and problem solving. In particular, the Stack Exchange
platform includes some of the most popular QA communities to date, such as
Stack Overflow. Initial metrics used to assess the performance of these
communities include summative statistics like the percentage of resolved
questions or the average time to receive and validate correct answers. However,
more advanced methods for longitudinal data analysis can provide further
insights on the QA process, by enabling identification of key predictive
factors and systematic comparison of performance across different QA
communities. In this paper, we apply survival analysis to a selection of
communities from the Stack Exchange platform. We illustrate the advantages of
using the proposed methodology to characterize and evaluate the performance of
QA communities, and then point to some implications for the design and
management of QA platforms.Comment: 10 pages, 3 figures, example cod
On Spatial Transition Probabilities as Continuity Measures in Categorical Fields
Models of spatial transition probabilities, or equivalently, transiogram
models have been recently proposed as spatial continuity measures in
categorical fields. In this paper, properties of transiogram models are
examined analytically, and three important findings are reported. Firstly,
connections between the behaviors of auto-transiogram models near the origin
and the spatial distribution of the corresponding category are carefully
investigated. Secondly, it is demonstrated that for the indicators of excursion
sets of Gaussian random fields, most of the commonly used basic mathematical
forms of covariogram models are not eligible for transiograms in most cases; an
exception is the exponential distance-decay function and models that are
constructed from it. Finally, a kernel regression method is proposed for
efficient, non-parametric joint modeling of auto- and cross-transiograms, which
is particularly useful for situations where the number of categories is large
Tropical Geometry of Statistical Models
This paper presents a unified mathematical framework for inference in
graphical models, building on the observation that graphical models are
algebraic varieties.
From this geometric viewpoint, observations generated from a model are
coordinates of a point in the variety, and the sum-product algorithm is an
efficient tool for evaluating specific coordinates. The question addressed here
is how the solutions to various inference problems depend on the model
parameters. The proposed answer is expressed in terms of tropical algebraic
geometry. A key role is played by the Newton polytope of a statistical model.
Our results are applied to the hidden Markov model and to the general Markov
model on a binary tree.Comment: 14 pages, 3 figures. Major revision. Applications now in companion
paper, "Parametric Inference for Biological Sequence Analysis
Sensor Selection for Estimation with Correlated Measurement Noise
In this paper, we consider the problem of sensor selection for parameter
estimation with correlated measurement noise. We seek optimal sensor
activations by formulating an optimization problem, in which the estimation
error, given by the trace of the inverse of the Bayesian Fisher information
matrix, is minimized subject to energy constraints. Fisher information has been
widely used as an effective sensor selection criterion. However, existing
information-based sensor selection methods are limited to the case of
uncorrelated noise or weakly correlated noise due to the use of approximate
metrics. By contrast, here we derive the closed form of the Fisher information
matrix with respect to sensor selection variables that is valid for any
arbitrary noise correlation regime, and develop both a convex relaxation
approach and a greedy algorithm to find near-optimal solutions. We further
extend our framework of sensor selection to solve the problem of sensor
scheduling, where a greedy algorithm is proposed to determine non-myopic
(multi-time step ahead) sensor schedules. Lastly, numerical results are
provided to illustrate the effectiveness of our approach, and to reveal the
effect of noise correlation on estimation performance.Comment: IEEE Transactions on Signal Processing (accepted
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