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International Workshop
on
Spatio-Temporal Statistics
Imperial College London
18–20 April 2016
Organisers:
Almut Veraart
Mikko Pakkanen
The workshop is funded by A. Veraart’s Marie Curie FP7 Integration Grant (within the
7th European Union Framework Programme).
Programme
All sessions take place in Lecture Theatre 139, 1st floor of Huxley Building, unless
stated otherwise.
Monday, 18 April 2016
Registration, 8:30–8:50.
Morning Session, 8:50–12:15. Chair: Almut Veraart
8:50–9:00 Opening Remarks
9:00–9:35 Emanuele Giorgi
9:40–10:15 Ying Sun
10:15–11:00 Coffee Break
11:00–11:35 David Bolin
11:40–12:15 Gregoire Mariethoz
Lunch Break, 12:15–14:30.
Afternoon Session, 14:30–17:05. Chair: Ragnhild Noven
14:30–15:05 Michele Nguyen
15:10–15:45 Mikkel Bennedsen
15:45–16:30 Coffee Break
16:30–17:05 Denis Allard
17:15–17:50 Poster Flash Talks
Poster Session, 18:00–20:00. Mathematics Common Room, 5th floor of Huxley
Building.
Tuesday, 19 April 2016
Morning Session, 9:00–12:15. Chair: Mikkel Bennedsen
9:00–9:35 Jesper Møller
9:40–10:15 Orimar Sauri
10:15–11:00 Coffee Break
11:00–11:35 Sofia Olhede
11:40–12:15 Shahin Tavakoli
Lunch Break, 12:15–14:30.
Afternoon Session, 14:30–17:45. Chair: Michele Nguyen
14:30–15:05 Sebastian Reich
15:10–15:45 Theresa Smith
15:45–16:30 Coffee Break
16:30–17:05 Hajo Holzmann
17:10–17:45 Peter Tankov
Conference Dinner at 19:00 (for invited speakers).
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Wednesday, 20 April 2016
Morning Session, 9:00–12:15. Chair: Mikko Pakkanen
9:00–9:35 Marc G. Genton
9:40–10:15 Krzysztof Podgorski
10:15–11:00 Coffee Break
11:00–11:35 Ed Cohen
11:40–12:15 Marie-Colette van Lieshout
Lunch Break, 12:15–14:30.
Afternoon Session, 14:30–17:15. Chair: Ed Cohen
14:30–15:05 Martin Schlather
15:10–15:45 Ragnhild Noven
15:45–16:30 Coffee Break
16:30–17:05 Franz Kiraly
17:05–17:15 Closing Remarks
End of the Workshop, 17:15.
3
Information
Scope and Aims of the Workshop
This workshop brings together international experts in statistics and probability
theory to discuss recent advances in spatio-temporal statistics. Topics of particular
interest are:
? Stochastic and statistical modelling of spatio-temporal phenomena (including
time series analysis and spatial statistics)
? Statistical inference for hierarchical models
? Stochastic volatility modelling and inference (including in particular extensions
to a spatio-temporal set-up)
? Extreme value theory
? Image analysis
Venue
The workshop (including coffee breaks) takes place in Lecture Theatre 139, 1st floor
of Huxley Building (180 Queen’s Gate, London SW7 2RH) on the South Kensington
Campus of Imperial College London, except for the Poster Session (on Monday, 18
April, 18:00–20:00), which is held in the Mathematics Common Room, 5th floor of
Huxley Building. The invited speakers will receive additional information regarding
the conference dinner and arrangements for lunch breaks.
AHOI Network
This workshop is part of the activities of the AHOI Network. AHOI (AarHus,
Oslo, Imperial) is a collaborative network between researchers in Stochastics at
Aarhus University, University of Oslo and Imperial College London. The purpose
of the network is to foster basic research in the theory and applications of Ambit
Stochastics, a new field of mathematical stochastics that has its origin in the study
of turbulence, but is in fact of broad applicability in science, technology and finance,
in relation to modelling of spatio-temporal processes. For further information, visit:
https://sites.google.com/site/ahoinet/
Contact Details of the Organisers
Almut Veraart, email: [email protected]
Mikko Pakkanen, email: [email protected]
Postal Address: Department of Mathematics
Imperial College London
South Kensington Campus
London SW7 2AZ, UK
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Abstracts of Talks
A Flexible Class of Non-separable Cross-Covariance Functions for Multivariate
Space-Time Data
Denis Allard L’Institut national de la recherche agronomique
Multivariate space-time data are increasingly available in various scientific disci-
plines. When analyzing these data, one of the key issues is to describe the multi-
variate space-time dependences. Under the Gaussian framework, one needs to pro-
pose relevant models for multivariate space-time covariance functions, i.e. matrix-
valued mappings with the additional requirement of non-negative definiteness. A
flexible parametric class of cross-covariance functions for multivariate space-time
Gaussian random fields is presented. Space-time components belong to the (uni-
variate) Gneiting class of space-time covariance functions, with Matern or Cauchy
covariance functions in the spatial margins. The smoothness and scale parameters
can be different for each variable. Sufficient conditions for positive definiteness are
shown. A simulation study shows that the parameters of this model can be effi-
ciently estimated using weighted pairwise likelihood, which belongs to the class of
composite likelihood methods. The model is then illustrated on a French dataset of
weather variables.
Bootstrapping the roughness index of Brownian semistationary and related
processes
Mikkel Bennedsen Aarhus University
In this talk we are concerned with newly developed bootstrap methods for estimators
of the roughness index of a class of continuous, conditionally Gaussian, processes.
In particular, we present the local fractional bootstrap of Bennedsen, Hounyo, Lunde
and Pakkanen (The local fractional bootstrap, working paper, 2016) and its appli-
cation to the change-of-frequency estimator of the roughness index of the Brownian
semistationary process. The same method is applied in a semiparametric setup to
a different estimator of the roughness index in the spirit of Bennedsen (Semipara-
metric estimation and inference of the fractal index of a time series using the local
fractional bootstrap, working paper, 2016). We compare the methods and consider
some empirical applications.
Geostatistical Modelling Using Non-Gaussian Matern Fields
David Bolin Chalmers University of Technology
We present a class of non-Gaussian spatial models useful for analysing geostatisti-
cal data. The models are constructed as solutions to stochastic partial differential
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equations driven by generalized hyperbolic noise and are incorporated in a standard
geostatistical setting with irregularly spaced observations, measurement errors and
covariates. We present a likelihood-based parameter estimation method and discuss
various model extensions. Finally, an application to precipitation data is presented
and the models are compared with Gaussian and trans-Gaussian models.
Spatio-temporal processes in super-resolution microscopy imaging
Ed Cohen Imperial College London
Super-resolution microscopy is a collection of imaging techniques allowing experi-
menters to delve beyond classical resolution limits to image cellular structures in the
nanometer scale. The key element to the success of super-resolution techniques is the
stochastic blinking of fluorophores (light emitting molecules) allowing sparse subsets
to be localised with very high precision and then localizations collected across time
to build a spatial point pattern of molecular positions. In this talk I will present a
brief overview of super-res microscopy and show that while these blinking properties
are key to the technique they produce spurious artefacts that can hinder rigorous
analysis of underlying spatial structures. We present a new model for the observed
blinking of molecules in an imaging experiment and demonstrate its effectiveness in
estimating key characteristics of the blinking fluorophores, followed by a look what
this model might unlock for the super-resolution community in the future. This is
joint work with Lekha Patel (Imperial), Ricardo Henriques (UCL) and Raimund
Ober (Texas A&M).
Tukey g-and-h Random Fields
Marc G. Genton King Abdullah University of Science and Technology
We propose a new class of trans-Gaussian random fields named Tukey g-and-h
(TGH) random fields to model non- Gaussian spatial data. The proposed TGH ran-
dom fields have extremely flexible marginal distributions, possibly skewed and/or
heavy-tailed, and, therefore, have a wide range of applications. The special for-
mulation of the TGH random field enables an automatic search for the most suit-
able transformation for the dataset of interest while estimating model parameters.
An efficient estimation procedure, based on maximum approximated likelihood, is
proposed and an extreme spatial outlier detection algorithm is formulated. The
probabilistic properties of the TGH random fields, such as second-order moments,
are investigated. Kriging and probabilistic prediction with TGH random fields are
developed long with prediction confidence intervals. The predictive performance of
TGH random fields is demonstrated through extensive simulation studies and an
application to a dataset of total precipitation in the south east of the United States.
The talk is based on joint work with Ganggang Xu.
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Geostatistical modelling of zero-inflated prevalence data
Emanuele Giorgi Lancaster University
When prevalence data are collected in areas where transmission of the disease in
question is highly restricted by environmental factors, the resulting spatial datasets
often exhibit an excessive number of sampled locations with no reported disease
cases by comparison with the standard binomial geostatistical model, even after
adjustment for covariate effects and both spatially structured and spatially inde-
pendent random effects. This behaviour, usually called zero-inflation, occurs when
some parts of the study-region are fundamentally unsuitable for transmission as
distinct from general extra-binomial variation. We propose two extensions of the
standard geostatistical model for prevalence data that exhibit zero-inflation. To
motivate the extensions, let q(x) and r(x) respectively denote the probability that
location x is suitable for the transmission of the disease and the probability of con-
tracting the disease at location x given that x is suitable. In the first extension, we
model the logit transformations of q(x) and p(x) as a pair of Gaussian processes,
possibly correlated, with the resulting prevalence given by p(x) = q(x)r(x). In the
second extension we model q(x) as a binary indicator equal to 0 or 1 if location x is
unsuitable or suitable, respectively. Using an Ising model, we then allow for discon-
tinuities in prevalence to occur at the boundaries between suitable and unsuitable
areas. Finally, we describe an application of the proposed models to river blindness
prevalence data from Mozambique, Malawi and Tanzania.
Scoring functions for forecast evaluation and the role of the information set
Hajo Holzmann Philipps-Universitat Marburg
Scoring functions are an essential tool to evaluate point forecasts, and scoring rules
to evaluate probabilistic forecasts. We start by reviewing some recent results on the
construction of scoring functions and scoring rules.
Point forecasts are issued on the basis of certain information. If the forecasting
mechanisms are correctly specified, a larger amount of available information should
lead to better forecasts. We show how the effect of increasing the information set
on the forecast can be quantified by using strictly consistent scoring functions, and
also discuss the role of the information set for evaluating probabilistic forecasts by
using strictly proper scoring rules (Holzmann, H. and Eulert, M.: The role of the
information set for forecasting — with applications to risk management. The Annals
of Applied Statistics 8, 595–621, 2014).
Further, a method is proposed to test whether an increase in a sequence of infor-
mation sets leads to distinct, improved h-step point forecasts. For the value at risk
(VaR), we show that increasing the information set will result in VaR forecasts which
lead to smaller expected shortfalls, unless an increase in the information set does not
7
change the VaR forecast. The effect is illustrated in simulations and applications
to stock returns for unconditional versus conditional risk management as well as
univariate modeling of portfolio returns versus multivariate modeling of individual
risk factors.
Kernels for sequentially ordered data
Franz Kiraly University College London
We present a novel framework for kernel learning with sequential data of any kind,
such as time series, sequences of graphs, or strings. Our approach is based on signa-
ture features which can be seen as an ordered variant of sample (cross-)moments; it
allows to obtain a “sequentialized” version of any static kernel. The sequential ker-
nels are efficiently computable for discrete sequences and are shown to approximate
a continuous moment form in a sampling sense.
A number of known kernels for sequences arise as “sequentializations” of suitable
static kernels: string kernels may be obtained as a special case, and alignment ker-
nels are closely related up to a modification that resolves their open non-definiteness
issue. Our experiments indicate that our signature-based sequential kernel frame-
work may be a promising approach to learning with sequential data, such as time
series, that allows to avoid extensive manual pre-processing. (Joint work with Harald
Oberhauser)
Non-parametric indices of dependence for inhomogeneous multivariate random
closed sets
Marie-Colette van Lieshout CWI & University of Twente
We propose new summary statistics for intensity-reweighted moment stationary mul-
tivariate random closed sets. The new statistics are based on the cumulant densities
and reduce to cross K- and D-functions when stationarity holds. We explore the
relationships between the various functions and discuss their explicit forms under
specific model assumptions. We derive ratio-unbiased minus sampling estimators
for our statistics and illustrate their use in practice.
Multiple-point geostatistics with spatio-temporal training images
Gregoire Mariethoz Universite de Lausanne
Multiple-point geostatistics (MPS) has received a lot of attention in the last decade
for modeling complex spatial patterns. The underlying principle consists in repre-
senting spatial variability using training images. A common conception is that a
training image can be seen as a prior for the desired spatial variability. As a result,
a variety of algorithmic tools have been developed to generate geostatistical realiza-
8
tions of spatial processes based on what can be seen broadly as texture generation
algorithms.
While the initial applications of MPS were dedicated to the characterization of 3D
subsurface structures and the study of geological/hydrogeological reservoirs, a new
trend is to use MPS for the modeling of earth surface processes. In this domain,
the availability of remote sensing data as a basis to construct training images offers
new possibilities for represent complexity with such non-parametric data-driven ap-
proaches. Repeated satellite observations or climate models outputs, available at a
daily frequency for periods of several years, provide the required patterns repetition
for having robust statistics on high-order patterns that vary in both space and time.
This presentation will delineate recent results in this direction, including MPS appli-
cations to the stochastic downscaling of climate models, the completion of partially
informed satellite images, the removal of noise in remote sensing data, and modeling
of complex spatio-temporal phenomena such as precipitations.
Second-order pseudo-stationary random fields and point processes on graphs
and their edges
Jesper Møller Aalborg University
Suppose we are given
(i) an undirected connected graph with vertex set V and a countable set E of
edges, where each edge apart from specifying a relation between two vertices
is viewed as a set e which is in bijective correspondence with some non-empty
open interval;
(ii) V and any edge e ∈ E are disjoint, and the edges are pairwise disjoint.
Then we call the triple of V , E , and the bijective mappings/edge coordinates for a
graph with Euclidean edges, and we denote this triple by G and the whole graph
set by L = V ∪⋃
e∈E e. In the special case where each edge e is just an open line
segments whose endpoints agree with the adjacent vertices associated to e, then
L is a linear network as considered in connection to for example road networks,
dendrite networks of neurons, and brick walls. Now, for any points u, v ∈ L, the
edge coordinates lead naturally to a geodesic distance dG(u, v) given by shortest
path distance in G. If the vertex set is contained in the Euclidean space Rk and the
edges are smooth subsets Rk, we may require that condition (i) and not necessarily
(ii) is satisfied: In fact there is then a natural one-to-one correspondence to a graph
with Euclidean edges, and this naturally induces a geodesic distance dG(u, v) but
taking into consideration whether u (or v) is a certain vertex or it belongs to a
certain edge. We notice that dG(u, v) may then be different from the usual geodesic
distance dL(u, v) on L which is given by shortest path-connected curve distance.
9
Our main goal is to establish sufficient conditions on the existence of positive def-
inite functions of the form K(dG(u, v)) for all u, v ∈ L. Then the Kolmogorov
Extension Theorem establishes the existence of a separable (Gaussian) random field
Z = {Z(u) : u ∈ S} on G with covariance function
cov(Z(u), Z(v)) = K (dG(u, v)) ∀u, v ∈ L.
(Since the covariance function depends on the graph with Euclidean edges, we prefer
using the terminology ”random filed on G” rather than ”random field on L”.) We
say then that the covariance function is pseudo-stationary and that the random
field Z is second-order pseudo-stationary, noting that we do not require that the
mean function EZ(u) is constant. Note that our setting is different from that in
research on random fields on directed trees such as in a network of rivers or streams
where water flows in one direction. Then special techniques are appropriate for
constructing covariance functions of the form above. However, our techniques will
be different, since we deal with undirected graphs.
One motivation for considering a second-order pseudo-stationary random field Z is
that for any geodesic path puv ⊆ L connecting two points u, v ∈ L, the restriction
of Z to puv has the same covariance structure as the random field Z(t) defined on
t ∈ [0, t0] ⊂ R where cov(Z(t), Z(s)) = K(|t − s|) and t0 = dG(t, s). In brief, Z
restricted to a geodesic path is indistinguishable from a corresponding Gaussian
random field on a closed interval.
Another motivation is that given a covariance function of the form above, we can
construct second-order intensity-reweighted pseudo-stationary (SOIRPS) point pro-
cesses on G, meaning that the point process has a pair correlation function of the
form g(u, v) = g0(dG(u, v)) for all u, v ∈ L. A Poisson process on L is SOIRPS but
to the best of our knowledge, apart from the Poisson process, models for SOIRPS
point processes on point processes with Euclidean edges have not yet been specified
in the literature. We show that for a log Gaussian Cox process (LGCP) X, i.e. when
X conditional on a Gaussian random field Z on L is a Poisson process with intensity
function exp(Z(u)), u ∈ L, second-order pseudo-stationarity of Z is equivalent to
SOIRPS of X. We also specify moment and Palm measure theoretical results for
LGCPs. Further examples of SOIRPS point processes on graphs with Euclidean
edges will be discussed in the talk. (Joint work with Ethan Anderes, University of
California at Davis, and Jakob G. Rasmussen, Aalborg University)
Modeling spatial heteroskedasticity by volatility modulated moving averages
Michele Nguyen Imperial College London
Spatial heteroskedasticity, which refers to changing variances and covariances in
space, is a feature that has been observed in environmental data. While promi-
nent models in the literature have accounted for this behaviour by multiplying the
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Gaussian error process with a stochastic volatility process, we propose a model that
intricately blends the effects of spatial volatility across space. This is related to the
way stochastic volatility is modelled in financial partial differential equations.
Let t ∈ Rd for some d ∈ N. Our model, which we call the volatility modulated
moving average (VMMA), is defined by:
Y (t) =
∫Rd
g(t− s)σ(s)W (ds),
where g is a deterministic (kernel) function, W is a homogeneous standard Gaussian
basis or white noise and σ is a stationary stochastic volatility field, independent ofW .
Without σ, this model reverts to the Gaussian moving average which is frequently
used in Geostatistics to design covariance structures.
In this project, we develop the theory of VMMAs and show how to simulate from
the models. We also discuss methods of inference.
Modelling complex distributions and dependence structures with trawl-type
processes
Ragnhild Noven Imperial College London
Trawl processes are a class of stationary, continuous-time stochastic processes driven
by an independently scattered random measure. They belong to the wider class of so-
called Ambit fields, and give rise to a flexible class of models that can accommodate
non-Gaussian distributions and a wide range of tempo-spatial covariance structures.
We develop the fundamentals of trawl processes with a view to statistical modelling,
and introduce a new representation that enables exact simulation and suggests novel
estimation methods. Then we use these processes to construct a general hierarchical
modelling framework, and present an application to modelling temporal dependence
in extreme values.
Characterising anisotropy in random fields
Sofia Olhede University College London
Detecting and analyzing directional structures in images is important in many appli-
cations since one-dimensional patterns often correspond to important features such
as object contours or trajectories. Classifying a structure as directional or nondirec-
tional requires a measure to quantify the degree of directionality and a threshold,
which needs to be chosen based on the statistics of the image. In order to do this,
we model the image as a random field. So far, little research has been performed on
analyzing directionality in random fields. In this paper, we propose a novel measure
to quantify the degree of anisotropy, and show how it can be applied to determine
novel and interesting features in the topography of Venus. This is joint work with
11
David Ramirez and Peter Schreier.
Event based distributions for spatio-temporal random fields
Krzysztof Podgorski Lund University
The sea surface is a classical example of stochastic field that is evolving in time.
Extreme events that are occurring on such a surface are random and of interest for
practitioners - ocean engineers are interested in large waves and damage they may
cause to an oil platform or to a ship. Thus data on the ocean surface elevation are
constantly collected by system of buoys, ship- or air-borne devices, and satellites all
around the globe. These vast data require statistical analysis to answer important
questions about random events of interest. For example, one can ask about statistical
distribution of wave sizes, in particular, how distributed large waves are or how steep
they are. Waves often travel in groups and a group of waves typically causes more
damage to a structure or a ship than an individual wave even if the latter is bigger
than each one in the group. So one can be interested in how many waves there is
per group or how fast groups are traveling in comparison to individual waves.
In the talk, a methodology that analyze statistical distributions at random events
defined on random process is presented. It is based on a classical result of Rice and
allows for computation of statistical distributions of events sampled from the sea
surface. The methodology initially was applied to Gaussian models but in fact, it is
also valid for quite general dynamically evolving stochastic surfaces.
In particular, it is discussed how sampling distributions for non-Gaussian processes
can be obtained through Slepian models that describe the distributional form of
a stochastic process observed at level crossings of a random process. This is used
for efficient simulations of the behavior of a random processes sampled at crossings
of a non-Gaussian moving average process. It is observed that the behavior of the
process at high level crossings is fundamentally different from that in the Gaussian
case, which is in line with some recent theoretical results on the subject.
Non-Gaussian data assimilation via a hybrid ensemble transform filter
Sebastian Reich Universitat Potsdam
Most current data assimilation (DA) algorithms for numerical weather prediction
(NWP) are based on variational and/or ensemble-based approaches, which rely on
a Gaussian approximation to forecast uncertainties. Such Gaussian representations
are less likely to be appropriate for characterizing uncertainties arising from fully
three dimensional and multi-phase convection-driven atmospheric circulation pat-
terns. Thus high-resolution NWP has triggered the exploration of non-Gaussian
DA methods. This talk will contribute to this development by presenting a hybrid
ensemble transform filter which bridges the ensemble Kalman filter with sequential
12
Monte Carlo methods. The proposed hybrid filter also allows for localization and
inflation as necessary for filtering spatio-temporal processes under model errors.
On the class of distributions of subordinated Levy processes and bases
Orimar Sauri Aarhus University
In this talk we study the class of innitely divisible distributions obtained by sub-
ordinating a Levy basis via an independent meta-time. We show that, for a xed
Levy basis, the law of the subordinated Levy basis is uniquely determined by the
law of the associated meta-time. In particular, we use our results to solve the so-
called recovery problem for Levy bases as well as moving average processes driven
by subordinated Levy processes. This talk is based on a joint work with Almut
Veraart.
Exact and Fast Simulation of Max-Stable Processes
Martin Schlather Universitat Mannheim
The efficiency of simulation algorithms for max-stable processes relies on the choice
of the spectral representation: different choices result in different sequences of fi-
nite approximations to the process. We modify the general optimization problem so
that a relatively simple solution can be obtained, which is essentially de Haan’s nor-
malized spectral representation. Compared to other simulation algorithms hitherto,
our approach has at least two advantages. First, it allows the exact simulation of a
comprising class of max-stable processes. Second, the algorithm has a stopping time
with finite expectation. In practice, our approach has the potential of considerably
reducing the simulation time of max-stable processes.
Spatio-temporal log-Gaussian Cox processes for public health data
Theresa Smith Lancaster University
Health data with high spatial and temporal resolution are becoming more common,
but there are several practical and computational challenges to using such data to
study the relationships between disease risk and possible predictors. These diffi-
culties include lack of measurements on individual-level covariates/exposures, inte-
grating data measured on difference spatial and temporal units, and computational
complexity.
In this talk, I outline strategies for jointly estimating systematic (i.e., parametric)
trends in disease risk and assessing residual risk with spatio-temporal log-Gaussian
Cox processes (LGCPs). In particular, I will present a Bayesian methods and MCMC
tools for using spatio-temporal LGCPs to investigate the roles of environmental and
socio-economic risk-factors in the incidence of Campylobacter in England.
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Approximating Likelihoods for Large Environmental Datasets
Ying Sun King Abdullah University of Science and Technology
For Gaussian process models, likelihood based methods are often difficult to use with
large irregularly spaced spatial datasets due to the prohibitive computational bur-
den and substantial storage requirements. Although various approximation methods
have been developed to address the computational difficulties, retaining the statisti-
cal efficiency remains an issue. This talk focuses on statistical methods for approx-
imating likelihoods and score equations. The proposed new unbiased estimating
equations are both computationally and statistically efficient, where the covariance
matrix inverse is approximated by a sparse inverse Cholesky approach. A unified
framework based on composite likelihood methods is also introduced, which allows
for constructing different types of hierarchical low rank approximations. The perfor-
mance of the proposed methods is investigated by numerical and simulation studies,
and parallel computing techniques are explored for very large datasets. Our meth-
ods are applied to nearly 90,000 satellite-based measurements of water vapor levels
over a region in the Southeast Pacific Ocean, and nearly 1 million numerical model
generated soil moisture data in the area of Mississippi River basin. The fitted models
facilitate a better understanding of the spatial variability of the climate variables.
Making decisions with probabilistic forecasts
Peter Tankov Universite Paris-Diderot
We consider a sequential decision making process (trading, investment, production
scheduling etc.) whose outcome depends on the realization of a random factor,
such as a meteorological variable. We assume that the decision maker disposes of a
probabilistic forecast (predictive distribution) of the random factor, which is regu-
larly updated. We propose a stochastic model for the evolution of the probabilistic
forecast inspired by the Kushner’s equation of nonlinear filtering, and show how
this model may be estimated from the historical forecast data. We then show how
this stochastic model can be used to determine optimal decision making strategies
depending on the forecast updates. Applications to wind energy trading are given.
Tests for separability in nonparametric covariance operators of random surfaces
Shahin Tavakoli University of Cambridge
The assumption of separability of the covariance operator for a random image or
hypersurface can be of substantial use in applications, especially in situations where
the accurate estimation of the full covariance structure is unfeasible, either for com-
putational reasons or due to a small sample size. However, inferential tools to verify
this assumption are somewhat lacking in high-dimensional or functional settings
where this assumption is most relevant. We propose here to test separability by
14
focusing on K-dimensional projections of the difference between the covariance op-
erator and its nonparametric separable approximation (Aston, J. A. D., Pigoli, D.
and Tavakoli, S.: Tests for separability in nonparametric covariance operators of
random surfaces, working paper, http://arxiv.org/abs/1505.02023, 2015). The
subspace we project onto is one generated by the eigenfunctions estimated under the
separability hypothesis, negating the need to ever estimate the full non-separable
covariance. We show that the rescaled difference of the sample covariance operator
with its separable approximation is asymptotically Gaussian. As a by-product of this
result, we derive asymptotically pivotal tests under Gaussian assumptions, and pro-
pose bootstrap methods for approximating the distribution of the test statistics when
multiple eigendirections are taken into account. We probe the finite sample perfor-
mance through simulations studies, and present an application to log-spectrogram
images from a phonetic linguistics dataset.
15
Abstracts of Posters
Asymptotic high frequency theory for the multivariate Brownian semistationary
process
Andrea Granelli Imperial College London
In our work we formally prove limit theorems for the asymptotic high frequency
theory of the multivariate Brownian semistationary (BSS) process, that can be
defined as: ∫ t
−∞g(t− s)σsdWs.
This process offers very flexible modelling possibilities and has recently been used
in the context of energy prices. Depending on the behaviour of the function g, it
may lie outside the semimartingale class, and this is typically the commonest use in
practice. The literature has only dealt with the univariate case, so far. Our work
develops the fundamental high frequency asymptotic theory needed for this process,
using Malliavin calculus, to significantly extend the array of results available in the
literature, covering the multivariate case for the first time.
We look at the realised covariation of two correlated BSS processes, defined as the
limit of the sum of the product of the increments of two processes along an equally
spaced partition, as the size of the partition shrinks to zero. We give conditions to
ensure convergence of this quantity in an appropriate sense, since it does not follow
from the usual general theorems concerning semimartingales. We give very general
conditions needed for a law of large numbers to hold, and the main result of the
paper is a central limit theorem, showing convergence in law to the Gaussian, under
some more restrictive assumptions.
This result is important on its own, but also in the applications, allowing to un-
derstand the asymptotic theory for quantities of great interest in finance, like the
correlation coefficient between two assets, and the realised beta coefficient.
This poster is based on joint work with Almut Veraart.
Limit theory for Levy semistationary processes
Claudio Heinrich Aarhus University
Levy semistationary (LSS) processes are processes of the form
Xt =
∫ t
−∞g(t− s)σsdLs,
where g is a deterministic kernel, σ is a predictable process, and L is a Levy process
on the real line. These processes form an important subclass of ambit fields, a
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flexible class of spatio-temporal processes that have found manifold applications in
various sciences such as biology, finance, and physics. We investigate the limiting
behavior for n→∞ of the realised power variation
V (p)n =n∑
i=1
|Xi/n −X(i−1)/n|p,
when X is an LSS process driven by a pure jump Levy process L.
Substitute CT generation using Markov random field mixture models
Anders Hildeman Chalmers University of Technology
Computed tomography (CT) equivalent information is needed for attenuation cor-
rection in PET imaging and for dose planning in radiotherapy. Prior work has shown
that Gaussian mixture models can be used to generate a substitute CT image from
a specific set of MRI modalities. This is highly attractive since MR images can
be acquired without exposing the subject to hazardous ionizing radiation and MRI
information is often of interest in its own right.
In this work we improve these models by incorporating spatial information through
assuming the mixture class probabilities to be distributed according to a discrete
Markov random field. Furthermore, the mixtures are extended from Gaussian to
normal inverse Gaussian distributions, allowing heavier tails and skewness.
Model parameters are estimated from training data using a maximum likelihood
approach. Due to the spatial model there is no closed form expression for the likeli-
hood function and a standard EM algorithm would not be possible. Instead, an EM
gradient algorithm utilizing MCMC approximations is developed. This procedure
yields acceptable convergence properties also when the large quantity of data makes
other common modifications of the EM-algorithm infeasible.
The estimation procedure is not only applicable to this specific problem but can
be used to a more general family of problems where the M-step is not possible to
perform but where the gradient of the likelihood can, at least approximately, be
evaluated.
The advantages of the spatial model and normal inverse Gaussian distributions are
evaluated with a cross-validation study based on data from 14 patients.
A statistical analysis of Tropical Cyclone Genesis
Thomas Patrick Leahy Imperial College London
According to the Fifth IPCC assessment report, in the 21st “the frequency of the
most intense storms will increase substantially in some ocean basins”. This poses a
significant risk to the vulnerable regions. Whilst studies suggest that there will be
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an increase in intensity and decrease in frequency, it is still uncertain by how much.
Quantifying this change in intensity and distribution of tropical cyclones is difficult.
As a natural first step, we examine the starting points or genesis of tropical cyclones.
This poster will give an insight into attributing and quantifying the influence of
physical covariates on tropical cyclone genesis. Generalised Linear Modelling pro-
vides a statistical framework to understand the physical variables that contribute to
the generation of tropical cyclones. An in depth understanding of the contributing
factors to genesis is particularly vital in a potential future climate.
Bayesian Inference for High Dimensional Dynamic Spatio-Temporal Models
Sofia Maria Karadimitriou University of Sheffield
The first reduced dimension Dynamic Spatio Temporal Model (DSTM) was intro-
duced by Wikle and Cressie (A dimension-reduced approach to space-time kalman
filtering. Biometrika, 86(4), 815–829, 1999) to jointly describe the spatial and tem-
poral evolution of a function observed subject to noise. A basic state space model is
adopted for the discrete temporal variation, while a continuous autoregressive struc-
ture describes the continuous spatial evolution. Application of Wikle and Cressies
DTSM relies upon the pre-selection of a suitable reduced set of basis functions
and this can present a challenge in practice. In this paper we propose an off-line
estimation method for high dimensional spatio-temporal data based upon DTSM
which attempts to resolve this issue allowing the basis to adapt to the observed
data. Specifically, we present a wavelet decomposition for the spatial evolution but
where one would typically expect parsimony. This believed parsimony can be in-
corporated by placing a Laplace prior distribution on the wavelet coefficients. The
aim of using the Laplace prior, is to filter wavelet coefficients with low contribution,
and thus achieve the dimension reduction with signicant computation savings. We
then propose a Hierarchical Bayesian State Space model, for the estimation of which
we offer an appropriate Forward Filtering Backward Sampling algorithm which in-
cludes Metropolis-Hastings steps and a Bayesian Graphical Lasso scheme (Wang,
H.: Bayesian graphical lasso models and efficient posterior computation. Bayesian
Analysis, 7(4), 867–886, 2012) for the covariance inference. (Joint work with Kostas
Triantafyllopoulos and Timothy Heaton)
Seasonality of mortality in the USA: identifying patterns and trends with
Bayesian spatiotemporal modelling
Robbie Parks Imperial College London
All-cause mortality is known to exhibit seasonal variation. In this study, we will
use all-cause mortality records of the entire USA from 1982-2010 for forecasting of
seasonal age-specific mortality on a state level, analysing differences in trends. The
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novel approach of this research comes from the Bayesian hierarchical model, which
borrows strength by neighbouring location, age group, and time. Our study will be
the first systematic analysis of seasonality of the entire USA throughout this time
period stratified by gender, age group, and state.
Initial results from the in-sample model indicate a distinct difference between sea-
sonal mortality profiles of younger and older age groups. Younger age groups relative
mortality peaks in the summer months, while older age groups peak in the winter
months. Further analysis over geography will determine if this variation is constant
across locations.
We expect the results to improve understanding on how distinct age groups and
locations are affected by season, as previous studies have looked at all age groups
combined. We also expect the model framework to enable coherent forecasts of
patterns and trends of seasonal mortality.
Rough Path Theory, Fractional Brownian Motion and connections to Statistical
Inference
Riccardo Passeggeri Imperial College London
Rough Path theory was originally developed in the late nineties by Terry Lyons.
A rough path is an analytical and algebraic object associated to an irregular path
allowing one to define and study solutions to differential equations controlled by
such irregular paths, for example a Brownian motion. In particular, given a path
we can consider its signature, which is the set of all iterated integrals of the path,
hence it is a map from the path to a tensor algebra. The signature give us additional
information about the path so that we can use them, for example, to find a solution
of a certain ODE driven by the path or to obtain additional properties of the same
path. The first objective of my research is to compute the rate of convergence of the
expected signature of a piecewise linear approximation of the fractional Brownian
motion to the expected signature of the fractional Brownian motion. This is an
open problem for any value in the interval (0, 1) of the Hurst parameter H. The
second objective is to compute the expected signature of a fractional Brownian
motion with the Hurst parameter strictly less than a half (H < 12). This research
is directly relevant to parameter estimation for rough differential equations. In
particular, Anastasia Papavasiliou and Christophe Ladroue (Parameter estimation
for rough differential equations. The Annals of Statistics, 39(4), 2047–2073, 2011)
constructed the Expected Signature Matching Estimator (ESME). The goal is to
estimate the parameter θ of the vector field f of a differential equations of the form
dYt = f(Yt; θ) · dXt, Y0 = y0
and in order to do this the authors use the ESME, which is based on the matching
between the theoretical and the empirical expected signature of the response {Yt, 0 <
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t < T}, for T > 0. Two of the main assumptions are that the expected signature
of the path X is computable and the theoretical expected signature of the path Y ,
if not computable, can be approximated. Thus my project is strongly linked with
their work and in particular, by achieving the objectives mentioned above, it will be
possible to eliminate these restrictive assumptions in the case of fractional Brownian
motion.
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Participants
Denis Allard (INRA, France)
Saoirse Amarteifio (Imperial College London, UK)
Mikkel Bennedsen (Aarhus University, Denmark)
Nick Bingham (Imperial College London, UK)
David Bolin (Chalmers University of Technology, Sweden)
Ignacio Bordeu Weldt (Imperial College London, UK)
Ricardo Carrizo (MINES ParisTech, France)
Ed Cohen (Imperial College London, UK)
Alice Corbella (University of Cambridge, UK)
Gabriela Czanner (University of Liverpool, UK)
Petros Dellaportas (University College London, UK)
Stefano De Marco (Ecole Polytechnique, France)
Guler Ergun (Imperial College London, UK)
Nicola Fitz-Simon (Imperial College London, UK)
Marc G. Genton (KAUST, Saudi Arabia)
Paul Ginzberg (Imperial College London, UK)
Emanuele Giorgi (Lancaster University, UK)
Andrea Granelli (Imperial College London, UK)
Zorana Grbac (Universite Paris-Diderot, France)
Claudio Heinrich (Aarhus University, Denmark)
Hessam Hessami (Universite Joseph Fourier, France)
Anders Hildeman (Chalmers University of Technology, Sweden)
Marcel Hirt (University College London, UK)
Till Hoffmann (Imperial College London, UK)
Hajo Holzmann (Philipps-Universitat Marburg, Germany)
Blanka Horvath (Imperial College London, UK)
Jack Jacquier (Imperial College London, UK)
Christopher Jarvis (London School of Hygiene and Tropical Medicine, UK)
Sofia Maria Karadimitriou (University of Sheffield, UK)
Yannis Karmpadakis (Imperial College London, UK)
Dimitrios Kiagias (University of Sheffield, UK)
Franz Kiraly (University College London, UK)
Thomas Patrick Leahy (Imperial College London, UK)
Fekadu Lemessa (Umea University, Sweden)
Marie-Colette van Lieshout (CWI & University of Twente, The Netherlands)
Gregoire Mariethoz (Universite de Lausanne, Switzerland)
Maciej Marowka (Imperial College London, UK)
Maxime Morariu-Patrichi (Imperial College London, UK)
Jesper Møller (Aalborg University, Denmark)
Michele Nguyen (Imperial College London, UK)
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Ragnhild Noven (Imperial College London, UK)
Sofia Olhede (University College London, UK)
Thomas Opitz (INRA, France)
Aidan O’Sullivan (University College London, UK)
Mikko Pakkanen (Imperial College London, UK)
Aristeidis Panos (University College London, UK)
Robbie Parks (Imperial College London, UK)
Riccardo Passeggeri (Imperial College London, UK)
Lekha Patel (Imperial College London, UK)
Roozbeh H. Pazuki (Imperial College London, UK)
Erika Pellegrino (Imperial College London, UK)
Krzysztof Podgorski (Lund University, Sweden)
Tuomas Rajala (University College London, UK)
Nicola Reeve (Coventry University, UK)
Sebastian Reich (Universitat Potsdam, Germany)
Orimar Sauri (Aarhus University, Denmark)
Martin Schlather (Universitat Mannheim, Germany)
Theresa Smith (Lancaster University, UK)
Ying Sun (KAUST, Saudi Arabia)
James Sweeney (University College Dublin, Ireland)
Peter Tankov (Universite Paris-Diderot, France)
Shahin Tavakoli (University of Cambridge, UK)
Kostas Triantafyllopoulos (University of Sheffield, UK)
Almut Veraart (Imperial College London, UK)
Jianfeng Wang (Umea University, Sweden)
Hanna Zdanowicz (University of Oslo)
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