A fortnightly seminar series is planned to be held at venues throughout New
Zealand between the first and second workshops, starting in late August. Presenters will
include New Zealanders and Visiting Fellows and Post-Doctoral Fellows of the Programme. Details
are given below.
Wednesday August 31st, 4 pm.
David Harte: "An Introduction to Discrete Time Hidden Markov Models"
370
KB and go to talk.pdf
This was originally intended as a first introduction for members of the
Statistical Seismology group, but
could be a tutorial for anyone else wanting a first introduction to HMMs. The
room may change.
Friday September 9th,
12 noon, CO249
Andrea Ridolfi: "Normal Mixture Models and HMM Parameter
Estimation"
Image segmentation is one of the many interesting engineering
applications of hidden Markov models. When the underlying Markov field
is a Picard one, N-dimensional images can be segmented using a uni-dimensional
approach. HMM parameters can be then efficiently estimated since the
problem is reduced to a Mixture model identification. In this talk, the
estimation of the parameters of a mixture of Gaussian densities is
considered. In this particular context, it is well known that the
maximum likelihood approach is statistically ill posed, i.e., the
likelihood function is not bounded above. We show that this difficulty
can be avoided by adopting a suitable penalized likelihood function.
Local maximization of the likelihood function can be performed by mean
of Green's modified EM algorithm. Provided that an inverse gamma is
chosen as penalty function, EM reestimation equations are still explicit
and automatically ensure that the estimates are not singular. We
consider both i.i.d. and dependent mixtures of Gaussian densities, with
particular reference to the important case of hidden Markov models.
Tuesday September
13th, 12 noon, CO431
Andrea Ridolfi: "Power Spectra of Point Processes and Related
Signals"
Point processes are commonly used by engineers, physicists and
biologists to describe events associated with time records, locations in
space, or more generally, space-time events. We are concerned with point
processes and the complex signals resulting from various operations on
the basic event stream, such as filtering, jittering, delaying,
thinning, clustering, sampling and modulating. We present a systematic
study of their second order properties, which are conveniently
represented by the spectrum of the signal and play an important role in
signal analysis. We develop a modular approach for the construction of
complex signals, and derive formulas for the computation of the spectrum
that preserve such modularity: each additional feature added to a basic
model appears as a separate and explicit contribution in the
corresponding basic spectrum. These formulae are very general and
provide useful tools for model analysis and parametric spectral
estimation.
Friday September 16th,
12 noon, CO249
Junko Murukami: "Introduction to particle filters for simple
HMMs"
Particle filters are sequential Monte Carlo methods, which are widely
used in various stochastic systems. This is a very basic introductory
presentation focused on an example where the particle filters are used
for parameter estimation of a simplest possible hidden Markov chain
system. In this example, the method approximates the least square error
estimate of the parameter set.
Friday
Sept 23rd, 12 noon, CO249
Marcus Frean: "Particle Filters and Inference for Stochastic Processes
on Graphs"
1.31 MB
Hidden Markov models apply probabilistic inference to a particularly regular
graphical structure. However with modest enhancements the forward-backward and
Baum-Welch algorithms of HMMs can be applied to inference in much more general
graphical structures, such as belief nets (inuence diagrams) and Markov random fields.
The seminar will be an introduction to this general algorithm. Time permitting,
I'll discuss how particle filters fit into this picture, and give an example of
their use in enhancing HMM predictions.
Friday
Sept 30, 12 noon, CO249
Pierre Ailliot (VUW and NIWA): "Markov-switching autoregressive models for wind time series"
270 KB
Hidden Markov Models (HMM) have successfully been used to describe different
kinds of meteorological time-series, the hidden Markov chain representing the meteorological
regime (or "weather type"). In the case of wind time series, HMM cannot
catch the strong relation which exists between successive observations. In this case,
Markov Switching AutoRegressive (MS-AR) models, which are simple extensions of
HMMs, suit the data better. In this talk, I will present several specific MS-AR models
which have been introduced for wind time-series and briefly discuss the statistical
inference in these models.
Friday Oct 7, 12 noon
Paul Malcolm (Canberra) "Parameter Estimation for Asset-Price Evolution
Dynamics via M-ary Detection"
This seminar reviews joint work with R.J. Elliott. In it we consider a dynamic M-ary
detection problem for Markov modulated partially observed systems. Here, "Markov
modulated" refers to dynamics with one or more parameters which change value according
to a known law. Such systems are sometimes referred to as jump stochastic
systems, or stochastic hybrid systems. The basic detection objective is to estimate the
so-called mode probabilities from an observation process. The mode probabilities are
the estimated conditional probabilities of a given model parameter set, (taken from a
finite list of candidate parameter sets), being in effect at the time of estimation, or best
explaining the data. The corresponding filtering problem usually concerns utilising
these estimated probabilities to estimate a hidden state process.
In our seminar we suppose that one of M candidate volatility models best explains
a given asset price process. Sequential estimators are computed for each of the M
candidate models. These schemes compute an estimate for the relative likelihood of a
given model explaining an observation process. Two classes of model are considered.
In the first model, volatility states are determined by a continuous-time Markov chain.
An important practical feature of the detection schemes we compute for this model,
is that they do not include stochastic integration. Here we develop a version of the
J. M. C. Clark Transformation based on a Hadamard product, resulting in detector
dynamics where the observation process appears as a parameter, rather than an integrator.
Our main objective is to illustrate how M-ary detection ideas and techniques,
developed largely in Electrical Engineering, can be applied to solve common problems
in mathematical finance and to present a new transformation technique to eliminate
certain stochastic integrations.
Friday Oct 14, 12 noon
Xiaogu Zheng (NIWA) "A Mixture Model for Simulation of Precipitation in the
Upper Waitaki Catchment, New Zealand, and its Relation with Interdecadal Pacific
Oscillation"
1.64
MB
We aim to simulate time series of daily precipitation amounts within a season over many years. The simulated intra-seasonal variability, such as distributions of
dry and wet-day durations, and the means and tails of the distribution of daily precipitation, should be close to that observed. Simulated inter-annual variability, such as the
mean and variance of seasonal precipitation totals, should also be close to the observed.
If the observed precipitation is related to a climate variable that varies on yearly time-scales, then the simulated precipitation should also show this relation. Such simulations
are highly desirable in hydroclimatic research, particularly, in forecasting the capacity
of future hydroelectricity generation. In this study, we proposed a rainfall generator
based on a mixture model for both precipitation and a climate variable, Interdecadal
Pacific Oscillation index. An EM algorithm is used to estimate the parameters of the
generator. Its application in simulating precipitation in the upper Waitaiki catchment,
New Zealand, over 1950-2000 shows that specified the requirements are achieved to
acceptable levels.
Friday Oct 21, 12 noon
Paul Mullowney (Christchurch)
"The role of variance in capped-rate stochastic growth models"
The role of environmental variability in the growth of larval fish and their subsequent
recruitment into the adult population is poorly understood. In this talk, a capped-rate
stochastic growth model is considered where the underlying feeding mechanism of the
fish is based on an M/G/1 or G/D/1 queue. In the first scenario, larval fish (typically
cod or herring) encounter and consume prey (plankton) according to a Poisson process.
The service time of the consumed prey depends on its size and linear (capped-rate)
growth occurs during the ”busy periods” of the queue. Distributions for the time to
maturity and recruitment (those fish not consumed by a whale) are analyzed as a function of the moments of the prey spectra. These results are compared to the limiting
case where all prey have unit size (no variance). In the second situation (G/D/1),
the consumed prey are assumed to have unit size. Here however, the predator-prey
encounter rate is no longer Poisson, with variance independent from the mean. Distributions for the time to maturity and recruitment are studied (numerically) as a
function of the variance.
Friday
Oct 28, 12 noon,
Mike Paulin (Department of Zoology and Centre for Neuroscience, University of Otago):
"The Neural Particle Filter: A model of neural computations for dynamical state estimation in the brain"
Recent experimental work in collaboration with Larry Hoffman at UCLA has
shown that, as a consequence of fractional order dynamical characteristics of vestibular
sensory transduction mechanisms, single spikes generated by vestibular motion-sensing
neurons can be regarded as measurements of the dynamical state of the head.
We hypothesize that this measurement is translated into an explicit Monte Carlo representation
in the brainstem vestibular nucleus, which forms a central map of head
state. In this representation, neural spikes are regarded as particles and their spatial
distribution over the map at any instant represents the brain’s knowledge of head state.
Particles are constrained to move along axons, corresponding to pre-defined state trajectories.
A network can be constructed so that the distribution of spikes in the map
approximates the Bayesian posterior distribution of states given the sense data. The
neural particle filter model generates the circuit topology and response properties of
real neurons in the brain, from purely statistical principles.
Friday
Nov 4, 12 noon,
David Bryant (Auckland)
