E1: Optimal information extraction and classification of multi-sensor time-series
data from large-scale physical systems using compressive sampling
E2 AIMS AND BACKGROUND
Introduction
The closure between model based calculations and experimental observations
in large, multi-scale
[1], space science [2],
complex systems, such as plasma physics and nuclear
fusion
experiments
3
4
meteorological systems such as the atmosphere [ ] and oceans [ ] and many others areas, is one of
the continuing challenges across the sciences, engineering and mathematics. A typical example of
such a multi-scale complex
system is the heliac plasma experiment at the H-1 Major National
5
Research Facility, ANU[ ].
Typical data acquisition in all these systems produces time-series from multiple arrays of sensors.
In spite of copious data, all the above systems suffer from two major limitations:

Firstly, they are notoriously under-sampled in the spatial domain due to physical constraints,
aperture limitations and instrument cost.

Secondly, the measurements are often taken with high or varying noise levels and inherent
uncertainties.
The reconstruction of spatial and temporal quantities, such as magnetic fluctuations in a fusion
plasma, is therefore limited by strong assumptions about the nature of the physical model. The
intermediate but equally important task of signal classification is hampered in much the same way
by the abovementioned constraints. Typically, it also requires either strong model assumptions
about spatial structures or reduction to classification using a single time-series, a problem that has
been studied for a long time [6].
Exciting developments in signal processing, statistics and applied mathematics have gone beyond
the limits of classical information theory; that to avoid loss of information,
a signal has to be
7
sampled at least at twice its bandwidth. Compressive sampling [ ] results have recently
demonstrated that, for certain classes of signals and with a modified measurement scheme, signals
can be recovered from samples taken at a rate significantly below the classical limit.
E2.1 Aims
The heart of this proposal is information recovery from observations - scarcely sampled in space but
highly sampled in time - from large-scale, complex physical systems. We will exploit recent
breakthroughs in compressive sampling in order to enhance the information recovery and to
streamline classification. We propose to develop the framework and application of approximation
theory and compressive sampling for this class of distributed signals in a particular context: the
study of plasma confinement and fluctuation phenomena, research which is leading to the
production of electricity from fusion reactions like those that power the Sun.
Specifically we aim to

develop the mathematical theory underlying optimal recovery of spatial information from
multiple sensor input,
 investigate the relative merits of adaptive approximation vs. compressive sampling for
classification of the experimental data, and
 implement an automated classification scheme on the H-1NF data acquisition system.
Details are given in E4. Initially, we will use as our reference data the evolving repository (now
~300 gigabytes) from the H-1NF Heliac Major National Research Facility located at the ANU.
1
After development of suitable algorithms and bases, we will apply our tools on the estimated 20
terabytes (20,000 gigabytes) of fusion plasma experimental data available in a standard format in
major plasma laboratories worldwide via Internet links. To validate our methods, we will draw on
the results of the previous data mining project [ARC Discovery Grant DP0451960] which has
already enabled us to classify certain types of events (“Alfvén” waves) with respect to their basic
spatial and temporal structure.
E2.2 Background:
Signal Processing and Approximation Theory in Data Analysis
Key to our approach is the observation that the "essential" information content of most signals is
actually quite low. In other words, if it is represented in a suitable set of basis functions, the signal
is "sparse" in this basis. In Fourier space, the continuously played notes of a recorder can be
represented by fewer than 20 frequencies. A familiar standard from photography is JPEG image
compression[Pennebaker, W. B. and Mitchell, J. L. JPEG - Still Image Data Compression
Standards, Van Nostrand Reinhold, 1993.], which exploits sparsity in order to significantly reduce
storage requirements with little perceptible loss of information. Approximation theory, which
provides a formal framework, is also applicable to typical observations of H-1 plasmas where the
spatial domain is a three dimensional twisted torus. Of major interest in this work are the detection
of irregularities in the magnetic field and the density as well as the onset of
instability of the
8
[
]
plasma. Many types of such events have been identified in the literature .
The base signal (in the absence of "events") is very smooth, so it can be compressed using wavelets
with a high number of vanishing moments, for example Daubechies or spline-based wavelets. An
optimal compression method is obtained by thresholding the wavelet coefficients. Thresholding
is
9
an adaptive approximation procedure, and the coarsening approach suggested by Mallat [ ] requires
the determination of all wavelet coefficients first, which is computationally expensive. Good
compression is achieved for signals with coefficient sequences which are decreasing very fast to
zero or - in mathematical terms - are in lp for small p. This is the case for the base signal. However,
the occurrence of an event is localised in time so that the resulting coefficient sequences are in an lp
with a slightly larger p. The application of these sequence spaces which are only quasi-Banach10
spaces for p<1
is now standard in the theory of best approximation (see Lorentz and DeVore [ ] or
11
A. Cohen [ ] for more details on the application of wavelets).
Representation of our signals in this basis is sparse, forming the mathematical foundation for the
application of compressed sensing. Here, we will investigate the selection and development of such
a basis. In particular, curvelets [http://www.curvelet.org] in space seem to be good candidates as
they provide sparse representations of curved spatial singularities and of the propagators of the
wave equations. This is desirable as the evolution of plasma properties such as density and magnetic
field is described by wave equations.
The robustness of the compressive sampling scheme in the presence of noise and the treatment of
multi-channel signals, termed "distributive
compressive sampling", is the subject of current research
12
in leading signal processing institutions [ ]. Present application areas include techniques of image
compression, biomedical and geophysical image reconstruction, analogue-to-digital conversion and
communications.
The Challenge of Under-Sampled Data
With few exceptions, economic, practical and other considerations place significant constraints on
the deployment and sampling rates of sensors in large scale experiments. H-1NF observations are
time sampled spatial functionals of density and magnetic field. The main problem in this case is
that spatial resolution is extremely low in one dimension (around the major axis of the torus) due to
1
constraints on placement of sensor arrays. Thus one faces the problem of extracting additional
spatial information from (the finely sampled) information in the temporal and minor axis spatial
dimensions. In the simpler case of one-dimensional waves the problem is straight-forward as the
functions are of the form f(x-ct) and so sampling even at just one point over time recovers the full
information of f. We will investigate how much of this simple time/spatial tradeoff can be observed
for the case of complex three dimensional spatial geometry when the magnetohydrodynamic wave
operator replaces the simple wave function.
We intend to develop new (computational) detectors of instability based on a sparse featureset
obtained from the observed data. We will combine ideas from signal processing and optimal
reconstruction with ideas from machine learning.
Formally, if A : X -> Y is the observation operator mapping the function space X into the
observation space Y, and g : X -> R is a decision function, then we would want to find a function f
: Y -> R such that the composition of f and A approximates g. Of course, g is also only known for a
discrete set of points, the learning set. The main tool to achieve this classification will be the basis
functions developed in the first part and compressed sensing which will provide us with sparse and
efficient approximations. We will study the performance of g both theoretically and in practice on a
test data set. The main challenges to overcome are spatial under-sampling and the ill-posedness of
this inverse problem.
The results will be utilised to develop compressive signal classification of the H-1 NF data which
extends the previous, successful data mining project to a larger class of events and phenomena that
can be sufficiently resolved by the new methods. The composition of the project team (section E7)
enhances collaboration between experts in applied mathematics and the domain of the application,
which is crucial to such an endeavor.
Automated classification of multi-sensor timeseries
As stated in the third aim, the ultimate application will focus on efficient, automated analysis of
multi-sensor timeseries such as magnetic field, density etc., measured with large arrays of sensors.
This is an area in which ANU experimentalists have world-leading expertise, and one which is ripe
for the deployment of advanced signal processing ideas, an essential requirement for the
demonstration of compressed sensing in the physical sciences. Fluctuations seen in these timeseries
are due to plasma instabilities, which enhance the outward transport of energy and ultimately limit
the ability of confinement devices to reach fusion conditions. These topics are of special importance
to plasma physics and the design of fusion reactors because they determine the reactor size (→ cost)
and the heating power required. As large international facilities such as the International Tokamak
Experimental Reactor come into operation, experimental access will increasingly become more
difficult and spatial resolution poorer and such advanced data analysis techniques will become
essential research tools.
Application to Plasma Fusion Research
Extensive research and investment have been undertaken worldwide towards the “grand challenge”
of power production through nuclear fusion. Using magnetically confined plasmas, fusion energy
output has improved by many orders of magnitude in recent decades, and the fundamental physics
of this ubiquitous fourth state of matter is much better understood. However, due to the dynamically
complex nature of magnetically confined plasmas, significant gaps remain in our understanding of
the physics of these systems.
The H-1 heliac (Figure 1) is unique among the magnetic confinement devices in its configurational
flexibility, providing a wide ranging experimental parameter space. Automation has been employed
in H-1 experimental campaigns to exploit the high resolution in magnetic geometry, specifically the
1
'rotational transform', or pitch of the magnetic field. Data mining techniques have successfully been
used for the analysis of data from these configuration scans, and have greatly assisted in the
interpretation of observed fluctuations in the plasma.
Figure 1: (a) Simplified top view of the H-1 flexible heliac at the ANU, which has a major radius of 1 m.
(b) Side view, with technician visible at lower left. (c) Poincaré plot in poloidal cut of magnetic field lines
within the plasma and the position of twenty magnetic field probe coils around the plasma. The toroidal
direction is poorly sampled in the toroidal direction, with coils at only three distinct toroidal angles.
Figure 2: (a) Time-frequency graph of magnetic fluctuations from one sensor with electron density
is overlaid, and (b) corresponding time-series data.
As an example of H-1 timeseries data, a time-frequency graph (“sonogram”) for a single magnetic
probe for one plasma pulse is shown in Figure 2. Depending on its physical nature, the phase
velocity of an instability may depend on plasma density; indeed, in this case we see such correlation
with the “chirps” in frequency in the range 40 to 60ms coinciding with sudden changes in density.
Typically, such signals are highly coherent between coils, suggesting a global mode structure whose
spatial structure may have a low-dimensional representation given an appropriate selection of bases.
13
Harris et. al[ ] developed one of the first applications of spectrum and mode analysis to magnetic
1
fluctuation data in plasma physics. With the limited computational power available at the time, they
identified the “second stability” phenomenon, in which finite plasma pressure effects distort an
unstable magnetic configuration so that it becomes linearly stable. This work was based on a largely
manual, ad-hoc analysis of time-frequency data, which identified particular frequencies with spatial
mode numbers, then correlated the onset of instability of those modes with gross plasma
parameters, such as plasma pressure. Since then, much progress has been made in the signal
processing of multi-channel time-series and efficient classification of the latter.
The existing and emerging data from the H-1 heliac is an ideal target for the work proposed here as
it provides relatively good spatial resolution and a wide spectrum of data characteristics. The
machine and fluctuation diagnostics are fully operational, and, as the heliac is under control of the
investigators, adjustments to the type of data taken, the recording sensors, and their signal
processing electronics can be made to best suit the goals of the signal processing research as it
progresses. H-1NF data is accessed using the open-source MDSPlus data system, a standard used in
laboratories around the world. This will allow technologies developed for the H-1 heliac to be
readily applied to experimental data repositories for other devices.
Compressed Sensing in other Applications
Research in the field of compressed sensing started with the realization that signals that are sparse
in some basis can be completely recovered
by a number of samples that is below the “traditional”
14 15 16
[
,
,
]
Shannon/Nyquist sampling limit
. The novelty of exploiting signal sparseness
at the
[17] and high-speed
sampling stage has sparked a number
of
applications
in
medical
imaging
systems
18
analogue-to-digital converters[ ], where an increase in sampling rate is very cost intensive.
19
Distributed compressive sampling research[ ] has sprung from the communication sector where
bandwidth and signal power are scarce resources and the reduction of direct communication in a
sensor network is a primary goal.
The application of compressive sampling20to21signal classification is currently in its infancy and
mostly limited to single channel signals[ , ]. Extensions to multi-channel, distributed compressive
sampling for signal classification is one of the goals of this proposal. The results can then be
evaluated against an adaptive approximation method which will be optimized for the H-1 data.
The signal processing framework and data analysis techniques we develop for data that is poorly
sampled in the spatial dimension may be applied to other systems involving the costly or difficult
implementation of many sensors taking time-series data. These include, in addition to large
experiments like H-1, seismology[FJ Herrmann, P Moghaddam, CC Stolk, Sparsity- and continuitypromoting seismic image recovery with curvelet frames, Applied and Computational Harmonic
Analysis, 24,150-173.(2008)], meteorology and climatology, and the current space weather
observations which range from ground based measurements from radar and magnetometers[Chen, J
, Sharma, A , (2006), Title, Eos Trans. AGU,
87(36), Jt. Assem. Suppl., Abstract SM41C-04] to highly expensive and selective satellite
measurements[H Hasegawa, BU Sonnerup, MW Dunlop, A Balogh, E Georgescu, S E Haaland, B
Klecker, G Paschmann, B Lavraud, I Dandouras, H Reme, A Vaivads (2004), Title, Eos Trans.
AGU,
85(17), Jt. Assem. Suppl., Abstract SM52B-02]. We are in particularly looking into collaboration
with the Australian "space weather" research community[http://plasma.newcastle.edu.au/plasma/ et
al] which possesses a comprehensive data repository at the Australian Space Weather Agency
[http://www.ips.gov.au/].
1
E3 SIGNIFICANCE AND INNOVATION
This project is significant because it will utilize synergies between two complementary domains
(and skill sets of the investigators) to advance knowledge in both disciplines:
On the one hand, it will use insights of different areas of computational mathematics, harmonic
analysis and partial differential equations in order to advance the important field of plasma fusion
research.
Conversely, the application to the large-scale fusion experiment H-1 NF at the ANU will lead to
the derivation of new data analysis techniques which would have a much wider applicability to
other systems with a high time sampling and low spatial resolution.
We seek innovative solutions to the following questions, which are motivated by the analysis of
plasma fusion data and are of interest in a broad area of applications:
Recovery from under-sampling: Under which conditions (mathematical and physical) on the
observed field, e.g. magnetic field, is it possible to recover spatial information from incomplete
spatial and oversampled temporal information? Can techniques from approximation theory and
partial differential equations (wave equations) be used to generalize a simple one-dimensional
system?
Compression: How well can this type of data be compressed? What mathematical spaces and basis
functions are most suitable for the representation of (emerging) instabilities of such a necessarily
lossy compression?
Classification, Prediction and Control: How can the innovations, that result from the work on the
first two questions, be utilized to classify and predict (physical) events? If the prediction can be
implemented in real time, can we ultimately control and avoid instabilities?
The research will thus combine insights of different mathematical areas, computational, harmonic
analysis and partial differential equations to derive innovative data analysis techniques on which
one can then base new control algorithms. The resulting methods would have a wider applicability
to time-space data with a high time sampling and low spatial resolution, combining investigations
of irregularities in complex systems and of their detection with modern nonlinear approximation
theory. These goals are unique and rely on the availability of specific skill sets of the investigators
and their collaborators at the ANU and other Australian universities.
E4 APPROACH
The initial phase of the project will draw on the pre-processing work performed in the earlier data
mining project. Together with standard theoretical models about instabilities in magnetized
plasmas, these results have provided us with a good knowledge about the nature of some of the
underlying phenomena and the application of techniques such as singular value decomposition,
principal component analysis and independent component analysis has already yielded information
about frequent spatial structures.
In this phase, limits on signal to noise ratio for signal recovery of the experiment will be calculated
and the minimum requirements of the number of sensors in the spatial domain will be determined.
In the second phase we will develop distributed compressive sampling algorithms for space-time
data with high time resolution and poor spatial resolution. We will apply compressive sampling to
the H-1 magnetic fluctuation signals and explore the sparseness of the data within different basis
functions. These will include for example wavelets, local Fourier basis and more recent
1
22
developments such as curvelets , which are ideally suited for wave propagation problems.
The sparseness of the23respective representations will be compared with an adapted “dictionary”
based representation , and selected basis functions will be evaluated on a hybrid digital-analog
hardware implementation.
The final step is the development of an automatic classification scheme which will feed into the
developed data mining techniques of the previous proposal.
E4.1 Previous ARC Data Mining project
Frank and Dave to do
E4.2 Detailed Implementation
Year 1: Develop solutions for the recovery of spatial information from multiple sensor input
A meta-study will be conducted to synthesize the current understanding of the spatial and temporal
structures from the theoretical models and the experimental results. The latter include in particular
the findings from the previous data mining proposal. For the application to H-1 data, the
experimental signal to noise ratio will be systematically quantified and included in the above
synthesis. Because of the limited data the analysis of spatial variation is much more difficult than
the analysis of temporal variation. Here, we plan to represent the data by curvelets, and in particular
consider the compression of various wave propagation operators P(t). The curvelet solution to the
linearised "Alfvén wave" propagator will be the subject of an Honours thesis project. Propagators of
other wave types which can appear in a single plasma discharge will be added to the compression
procedure.
These operators can be used to fit an initially sparse wavelet signal f to the data at time t=0 and P(t)
f to the data at larger t. The sparse initial representation will have to be chosen adaptively. A
challenge might also be the selection of the appropriate propagation operator P(t). For high noise
and nonlinear behaviour one may also need to consider how variations in the signal evolve over
time. For the appropriate nonlinear filtering approach, we would use very computing intensive
Fokker-Planck solvers to resolve the evolution of the probability distributions of the errors.
Year 2: Develop and evaluate the use of adaptive approximation vs. compressive sampling of
the data
In this part of the project we will use the results from the first year in order to develop an adaptive
approximation approach to the description of the data, optimized using the prior knowledge of H-1
data.
To this end, we will select or develop appropriate wavelet functions. Ideally, such wavelets would
be able to help in the detection of the events and even be able to distinguish between different types
of events. We will then compare this adaptive approximation with a compressed sensing solution
which is based on random sampling. The compressed sensing approach is linear and thus fast, and,
moreover, it does not require the wavelet transform of the full signal. However, we expect the best
approach based on thresholding to have a slightly higher quality. Using our data, however, we
would consider linear approaches as well, which would be effective for particular types of events.
The compressive sampling algorithm will be applicable to different basis functions, which is one of
its key advantages. We
will evaluate the performance of wavelets, local Fourier basis, curvelets and
24
adapted dictionaries [ ] with respect to maximally sparse signal representation.
Year 3: Implementation of an automatic classification scheme on the H-1 data acquisition
1
system
In this phase the previous results concerning spatial information and optimally sparse representation
of the data will be integrated into a classification scheme which will be developed. The trade-off
between time and quality between the fully adaptive approximation approach and the compressed
sensing approach using fixed bases or an adapted dictionary will determine which implementation
will be used on a large part of the data set. For a large number of experimental realisations, we will
apply the algorithm in order to produce a largely reduced data base of the sparse feature sets. A
classification program will be developed that is based on the sparse feature set.
In the first step of the machine learning process, we will
use the successful clustering and
25
[
]
classification from the previous data mining project
as the learning set for the classification of
different modes of Alfvén type waves. Other instabilities and events that are deemed important will
be added to the learning set in order to continuously increase the dictionary represented in the
classification scheme. Once the reliability of the classification system has been validated, we intend
to implement a real-time classification at the data acquisition stage.
In parallel, a prototype hardware signal compressor will be implemented by modification of the
signal processing chain on the 20 coil Mirnov array. This will be a hybrid design, using high speed
analog multiplication of the raw signal with a programmable repeating digital bit sequence. The
latter will be chosen from the basis sets found most efficient in the computational investigation
described above.
Finally, capitalising on both the control facility of H-1 NF and the proposed algorithms, we plan to
implement machine guided (agent-driven) data exploration at the data acquisition stage.
E5 NATIONAL BENEFIT
The project will bolster Australia’s presence in two fields: signal processing and data mining
research, and research into magnetically confined plasma and fusion energy. Its interdisciplinary
nature combines resources and strengthens both fields of research.
Novel algorithms and techniques will be developed which have considerable research and
commercial potential in space science, oceanography and medical and defence science:



-dimensional data sets,
automatic classification of space-time phenomena in large scale multi-dimensional complex
data sets and
agent-driven exploration and experimentation.
This novel signal processing research will ensure Australia's participation in the smart information
use ( Frontier Technologies ). Technology developed here will educate key personnel for
Australia’s role as major developer and producer of information technology. For the mathematical
community in Australia with its particular strengths in harmonic analysis, partial differential
equations and complex systems this research will provide new applications of their ideas.
In pursuing these goals, we will be developing and exercising recently developed signal processing
and machine learning on H-1 NF data and from a range of other magnetic fusion experiments.
The advances to plasma and fusion energy will contribute to Australia's effort in reducing and
capturing emissions in transport and energy generation in it's national research priority for an
environmentally sustainable Australia. Not only will the data available from the H-1 NF heliac be
better understood, but Australian scientists will be able to apply their know-how to international
data repositories. This will provide strong foundations for future remote collaborations.
Application of this to fusion plasma databases gives Australian scientists a home-grown “tool”
1
which will help guarantee Australian access and input to the world-wide technology development
effort of this sustainable energy source, which is free of greenhouse emissions.
The proposed studies will contribute fundamental results to the worldwide effort to develop fusion
energy, complementing experiments on the larger, more rigidly programmed experiments in the US,
Japan, and Europe. Australia has long been an important training ground for plasma physicists who
advance the field, and this project will continue that effort.
The scientific and technical challenges of the proposed research will also provide unique
educational opportunities for students—the array of research projects available in CI Hegland’s
Computational Mathematics Program and on the H-1 NF heliac attracts honours project students in
mathematics, engineering and physics as well as post-graduate and post-doctoral researchers.
E6 COMMUNICATION OF RESULTS
We plan to present results at international computational and data mining conferences such as
CTAC, ICIAM, SIAM or IEEE Signal Processing as listed in Section C2 in the first instance, with
possible alternatives of NIPS, KDD/PaKDD/PKDD or HPC conferences. One plasma database
specialist conference (MDSPlus IAEA Workshop) and one physics meeting with a fluctuation
emphasis have been selected.
The same criteria apply to the journal publication plans: we plan to publish the results in data
mining and computational journals such as the IEEE series in Computational Science.
E7 DESCRIPTION OF PERSONNEL
The project team will include expertise in applied mathematics, signal processing and data mining,
and in both theoretical and experimental plasma physics. This ensures a high degree of
collaboration between experts in applied mathematics and the domain of the application, which is
crucial to the success of this endeavour.
Dr. Blackwell is the architect of the H-1NF data system, including the acquisition, raw, summary
and electronic log databases. Dr. Blackwell will act as primary contact for the international
experimental collaborators, and will be responsible for data interface and exchange.
Dr. Hegland will have responsibility for the direction of the research in sparse compression methods
and theory, and the analysis of strategies. He will share responsibility for pre-processing strategies
with Dr. Blackwell, and will be the joint supervisor of the Research Fellow requested. Dr. Hegland
will contribute 30 percent of his time to the project. He has been teaching sparse representation
techniques for several years, he has also led the data mining group in the Advanced Computational
CRC and an APAC data mining expertise program.
The Research Fellow will implement and test the various techniques described above, and
contribute to the theory, analysis and publication. It is anticipated that the Research Fellow will
spend 40% time with the plasma group organising, processing and analysing plasma data under the
immediate supervision of Dr. Blackwell.
Other personnel: Dr. H. Gardner, FEIT, ANU will be the advisor and main contact point for
Computer Science graduate student projects in support of this proposal. Prof. R.L. Dewar,
RSPhysSE, ANU will apply his domain expert knowledge in plasma instabilities and plasma wave
modeling.
This proposal provides ideal opportunities for student projects from all levels from Honours to PhD.
Dr. Hegland will suggest several applied mathematics honours projects. In a first project the studens
would compare nonlinear adaptive compression with linear compressed sensing for the H-1 data.
1
Another project would consider fitting data using various wave propagation operators and possibly
a Fokker-Planck based nonlinear filtering approach. Dr. Blackwell will propose several physics and
hardware based honours and PhD project. One project would involve the student in the
implementation of the classification results into the already used data mining work performed. In a
hardware project, the student would design and build the proposed compressed sampling based
signal compressor.
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1
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24 Holger Rauhut, Karin Schass, and Pierre Vandergheynst, Compressed sensing and redundant dictionaries.
(Preprint, 2006)
25 David Pretty, PhD Thesis, 2007
D.G. Pretty and B.D. Blackwell. A data mining algorithm for automated characterisation of fluctuations in
multichannel timeseries. Submitted to Com. Phys. Comm. 2007
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