2,809 research outputs found
Estimation of Markov Chain via Rank-Constrained Likelihood
This paper studies the estimation of low-rank Markov chains from empirical
trajectories. We propose a non-convex estimator based on rank-constrained
likelihood maximization. Statistical upper bounds are provided for the
Kullback-Leiber divergence and the risk between the estimator and the
true transition matrix. The estimator reveals a compressed state space of the
Markov chain. We also develop a novel DC (difference of convex function)
programming algorithm to tackle the rank-constrained non-smooth optimization
problem. Convergence results are established. Experiments show that the
proposed estimator achieves better empirical performance than other popular
approaches.Comment: Accepted at ICML 201
Factored expectation propagation for input-output FHMM models in systems biology
We consider the problem of joint modelling of metabolic signals and gene
expression in systems biology applications. We propose an approach based on
input-output factorial hidden Markov models and propose a structured
variational inference approach to infer the structure and states of the model.
We start from the classical free form structured variational mean field
approach and use a expectation propagation to approximate the expectations
needed in the variational loop. We show that this corresponds to a factored
expectation constrained approximate inference. We validate our model through
extensive simulations and demonstrate its applicability on a real world
bacterial data set
Hybrid methodology for Markovian epidemic models
In this thesis, we introduce a hybrid discrete-continuous approach suitable
for analysing a wide range of epidemiological models, and an approach
for improving parameter estimation from data describing the early stages
of an outbreak. We restrict our attention to epidemiological models with
continuous-time Markov chain (CTMC) dynamics, a ubiquitous framework
also commonly used for modelling telecommunication networks, chemical
reactions and evolutionary genetics. We introduce our methodology in the
framework of the well-known Susceptible–Infectious–Removed (SIR) model,
one of the simplest approaches for describing the spread of an infectious
disease. We later extend it to a variant of the Susceptible–Exposed–Infectious–
Removed (SEIR) model, a generalisation of the SIR CTMC that is more
realistic for modelling the initial stage of many outbreaks.
Compartmental CTMC models are attractive due to their stochastic
individual-to-individual representation of disease transmission. This feature is
particularly important when only a small number of infectious individuals are
present, during which stage the probability of epidemic fade out is considerable.
Unfortunately, the simple SIR CTMC has a state space of order N², where
N is the size of the population being modelled, and hence computational
limits are quickly reached as N increases. There are a number of approaches
towards dealing with this issue, most of which are founded on the principal of restricting one’s attention to the dynamics of the CTMC on a subset of its
state space. However, two highly-efficient approaches published in 1970 and
1971 provide a promising alternative to these approaches.
The fluid limit [Kurtz, 1970] and diffusion limit [Kurtz, 1971] are large-population
approximations of a particular class of CTMC models which
approximate the evolution of the underlying CTMC by a deterministic trajectory
and a Gaussian diffusion process, respectively. These large-population
approximations are governed by a compact system of ordinary differential
equations and are suitably accurate so long as the underlying population is
sufficiently large. Unfortunately, they become inaccurate if the population of
at least one compartment of the underlying CTMC is close to an absorbing
boundary, such as during the initial stages of an outbreak. It follows that a
natural approach to approximating a CTMC model of a large population is to
adopt a hybrid framework, whereby CTMC dynamics are utilised during the
initial stages of the outbreak and a suitable large-population approximation
is utilised otherwise.
In the framework of the SIR CTMC, we present a hybrid fluid model and
a hybrid diffusion model which utilise CTMC dynamics while the number of
infectious individuals is low and otherwise utilises the fluid limit and the diffusion
limit, respectively. We illustrate the utility of our hybrid methodology in
computing two key quantities, the distribution of the duration of the outbreak
and the distribution of the final size of the outbreak. We demonstrate that
the hybrid fluid model provides a suitable approximation of the distribution
of the duration of the outbreak and the hybrid diffusion model provides a
suitable approximation of the distribution of the final size of the outbreak. In
addition, we demonstrate that our hybrid methodology provides a substantial
advantage in computational-efficiency over the original SIR CTMC and is superior in accuracy to similar hybrid large-population approaches when
considering mid-sized populations.
During the initial stages of an outbreak, calibrating a model describing the
spread of the disease to the observed data is fundamental to understanding and
potentially controlling the disease. A key factor considered by public health
officials in planning their response to an outbreak is the transmission potential
of the disease, a factor which is informed by estimates of the basic reproductive
number, Râ‚€, defined as the average number of secondary cases resulting from
a single infectious case in a naive population. However, it is often the case
that estimates of Râ‚€ based on data from the initial stages of an outbreak
are positively biased. This bias may be the result of various features such as
the geography and demography of the outbreak. However, a consideration
which is often overlooked is that the outbreak was not detected until such
a time as it had established a considerable chain of transmissions, therefore
effectively overcoming initial fade out. This is an important feature because
the probability of initial fade out is often considerable, making the event that
the outbreak becomes established somewhat unlikely. A straightforward way
of accounting for this is to condition the model on a particular event, which
models the disease overcoming initial fade out.
In the framework of both the SIR CTMC and the SEIR CTMC we present
a conditioned approach to estimating Râ‚€ from data on the initial stages of an
outbreak. For the SIR CTMC, we demonstrate that in certain circumstances,
conditioning the model on effectively overcoming initial fade out reduces bias
in estimates of Râ‚€ by 0.3 on average, compared to the original CTMC model.
Noting that the conditioned model utilises CTMC dynamics throughout,
we demonstrate the flexibility of our hybrid methodology by presenting a
conditioned hybrid diffusion approach for estimating Râ‚€. We demonstrate that our conditioned hybrid diffusion approach still provides estimates of Râ‚€
which exhibit less bias than under an unconditioned hybrid diffusion model,
and that the diffusion methodology enables us to consider larger outbreaks
then would have been computationally-feasible in the original conditioned
CTMC framework. We demonstrate the flexibility of our conditioned hybrid
approach by applying it to a variant of the SEIR CTMC and using it to
estimate Râ‚€ from a range of real outbreaks. In so doing, we utilise a truncation
rule to ensure the initial CTMC dynamics are computationally-feasible.Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 201
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