Tuesday, 12 June 2018

Mechanistic models versus machine learning: a fight worth fighting for the biological community?

90% of the world’s data have been generated in the last five years. A small fraction of these data is collected with the aim of validating specific hypotheses. These studies are led by the development of mechanistic models focussed on the causality of input-output relationships. However, the vast majority of the data are aimed at supporting statistical or correlation studies that bypass the need for causality and focus exclusively on prediction.

Along these lines, there has been a vast increase in the use of machine learning models, in particular in the biomedical and clinical sciences, to try and keep pace with the rate of data generation. Recent successes now beg the question of whether mechanistic models are still relevant in this area. Why should we try to understand the mechanisms of disease progression when we can use machine learning tools to directly predict disease outcome?

Oxford Mathematician Ruth Baker and Antoine Jerusalem from Oxford's Department of Engineering argue that the research community should embrace the complementary strengths of mechanistic modelling and machine learning approaches to provide, for example, the missing link between patient outcome prediction and the mechanistic understanding of disease progression. The full details of their discussion can be found in Biology Letters.

Tuesday, 5 June 2018

Lifting the curse of ill-posedness by hybridization

At the beginning of the 20th century, Jacques Hadamard gave the definition of well-posed problems, with a view to classifying “correct” mathematical models of physical phenomena. Three criteria should be fulfilled: a solution exists, that solution is unique, and it should depend continuously on the parameters.

This point of view has led to the foundation of the functional analysis approach to the resolution of partial differential equations, which was developed during the last century and is commonly accepted nowadays. The continuous dependence on the parameters means that a small change will not affect the outcome noticeably. Its immediate consequence is that inferring cause from effect, that is, parameters from measurement, is an ill-posed problem – the opposite of a well-posed problem.

Elliptic boundary value problems, such as the conduction equation, are well-posed problems. The voltage potential depends continuously on the imposed boundary condition, and even analytically (the smoothest possible dependence) on the conductivity parameter of the material inside the object. Electrical Impedance Tomography is the corresponding ill-posed problem. A current is imposed on the surface of the object to be imaged, and the resulting voltage potential on the surface of the object is measured. The target parameter to be reconstructed is the internal impedance. It is an appealing imaging method, as it is non-invasive, non-destructive, and the equipment required is inexpensive. But this problem is severely ill-posed : a small error on the measurements can result in a very large error on the reconstructed impedance.

To address this difficulty, the so-called hybrid imaging methods have been developed. In Acousto-Electric Tomography, one creates internal perturbations of the impedance of the medium to be imaged by an acoustic pulse. Because these perturbations are controlled, they provide additional data that reduce very significantly the ill-posedness of the problem. In a recent book, Yves Capdeboscq and his former Oxford student Giovanni Alberti explore the mathematics of these hybrid problems and the corresponding reconstruction methods. They show that this problem has deep connections to the Rado-Kneser-Choquet Theorem for holomorphic functions of the complex variables, and how structures such as Antoine's necklace create additional difficulties in three dimensions. Quasi-static approximations prove unhelpful in such a case. But time is on our side.

Thursday, 31 May 2018

Sir Andrew Wiles appointed as the first Regius Professor of Mathematics at Oxford

Oxford mathematician Sir Andrew Wiles, renowned for his proof of Fermat’s Last Theorem, has been appointed by Her Majesty the Queen to be Oxford’s first Regius Professor of Mathematics.

The Regius Professorship – a rare, sovereign-granted title – was granted to Oxford’s Mathematical Institute as part of the Queen’s 90th birthday celebrations. It is the first Regius Professorship awarded to Oxford since 1842.

Sir Andrew is the world’s most celebrated mathematician. In 2016 he was awarded the highest honour in mathematics, the Abel Prize, for his stunning proof of Fermat’s Last Theorem, a conundrum that had stumped mankind for 350 years. In recognition of this transformative work, he was also awarded the Copley medal, the Royal Society’s oldest and most prestigious award.

Professor Louise Richardson, Vice-Chancellor of Oxford University, said: ‘I know my colleagues join me in offering our warmest congratulations to Sir Andrew on being named Oxford’s newest Regius Professor. It is a fitting recognition of his outstanding contribution to the field of mathematics.’

Professor Martin Bridson, Head of Oxford’s Mathematical Institute, said: ‘The award of the Regius Professorship to Oxford recognised both our pre-eminence in fundamental research and the enormous benefits that flow to society from mathematics.

‘It is entirely fitting that the first holder of this Professorship should be Sir Andrew Wiles. Nobody exemplifies the relentless pursuit of mathematical understanding in the service of mankind better than him. His dedication to solving problems that have defied mankind for centuries, and the stunning beauty of his solutions to these problems, provide a beacon to inspire and sustain everyone who wrestles with the fundamental challenges of mathematics and the world around us. We are immensely proud to have Andrew as a colleague at the Mathematical Institute in Oxford.’

Sir Andrew, who will remain the Royal Society Research Professor of Mathematics at Oxford and a Fellow of Merton College, dedicated much of his early career to solving Fermat’s Last Theorem. First formulated by the French mathematician Pierre de Fermat in 1637, the theorem states:

There are no whole number solutions to the equation $x^n + y^n = z^n$ when n is greater than 2, unless xyz=0

Fermat himself claimed to have found a proof for the theorem but said that the margin of the text he was making notes on was not wide enough to contain it. Sir Andrew first became fascinated with the problem as a boy, and after years of intense private study at Princeton University, he announced he had found a proof in 1993, combining three complex mathematical fields – modular forms, elliptic curves and Galois representations.

The Norwegian Academy of Science and Letters, which presents the Abel Prize, said in its citation that ‘few results have as rich a mathematical history and as dramatic a proof as Fermat’s Last Theorem’. The proof has subsequently opened up new fields of inquiry and approaches to mathematics, and Sir Andrew himself continues to pursue his fascination with the subject. In his current research he is developing new ideas in the context of the Langlands Program, a set of far-reaching and influential conjectures connecting number theory to algebraic geometry and the theory of automorphic forms.

The new Regius Professorship in mathematics was one of a dozen announced by the government to celebrate the increasingly important role of academic research in driving growth and improving productivity during Queen Elizabeth II’s reign. The creation of Regius Professorships falls under the Royal Prerogative, and each appointment is approved by the monarch on ministerial advice.

Sir Andrew’s father, Maurice Wiles, was Regius Professor of Divinity at Oxford from 1970 to 1991.

You can watch Sir Andrew's Oxford Mathematics London Public Lecture and interview with Hannah Fry here.

Tuesday, 29 May 2018

Wytham Woods Photography Exhibition in the Andrew Wiles Building - Celebrating 75 Years of Science

If you are ever in the centre of Oxford and are getting tired of the endless beautiful buildings, then make your way to Wytham Woods. Covering 1000 acres of ancient and beautiful woodland 3 miles NW of Oxford, Wytham is exceptionally rich in flora and fauna, with over 500 species of plants, a wealth of woodland habitats, and 800 species of butterflies and moths. And it is so wonderfully peaceful.

But if you don't make it down to the woods today, you'll notice that Wytham Woods has come to Dunsinane (aka the Andrew Wiles Building, home to Oxford Mathematics). Wytham are celebrating 75 years of scientific research with a photographic exhibition on the Mezzanine level. You are very welcome.

Tuesday, 22 May 2018

Alchemy on a Saturday night & Sunday morning - Oxford and UCL mathematicians go mad after midnight in a search for Newton's baldness cure

It is a little known (and entirely untrue) fact that Isaac Newton's alchemical investigations led him to a formula for a potion to cure baldness. Ten mathematicians from Oxford and UCL spent Saturday night (and Sunday morning) running around central London solving puzzles and gathering clues and ingredients to recreate this potion, before a pedalo race across the Serpentine to present a vial of the wonder cure to the President of the Royal Society.

This wasn't just for fun (although it was certainly enormously enjoyable): the event was raising funds for the charity Raise Your Hands. The other teams, comprising the City's finest, were vying for a trophy, and the Oxford and UCL team (aka 'Crackers') of academics was there to set the benchmark. Which they achieved coming in as honorary winners within minutes of the trophy winners from Cantab Capital.

The puzzle hunt was inspired by a similar charity fundraising event in New York. Teams started at Banking Hall at 20.00 hours on Saturday night, and the first teams reached the finish line at the Serpentine Sackler Gallery in Hyde Park just after 08.30 on Sunday morning. Each team had to collect three sets of alchemical information: from the criminal leader of the Chain Gang on the 7th floor of an NCP car park, from a celebrity businessman on level 32 of the Gherkin, and from a chemistry professor standing next to a statue of a goat in Spitalfields. 

All of which led the team to a street corner a few minutes from Tower Bridge where they had to assemble a marble run to decode the alchemical information into a list of ingredients. Three team members went off for a boat trip from Tower Pier, which led to a sequence of puzzles on the south side of the river. A quick call to criminal mastermind Ray then led the team to Newspaper Charlie on Tower Bridge, who was persuaded to hand over a newspaper containing clues to several more locations where the team found ingredients as diverse as apple pip oil and slumber dust. 

To get the butterfly tears, team members had to locate a group of mime artists on the South Bank near Waterloo Bridge and learn to mime, then race to Waterloo Vaults to add Newton's family crest to the walls of graffiti. Meanwhile other members of the team were getting apples in Borough Market, and solving a murder in a boxing club using a code written in blood on towels. No ordinary Saturday night.

From their various locations round London, the team gathered at the Institute of Engineering and Technology, to help police solve the mystery of the theft of Newton's flask. Having solved many further puzzles and pieced together the clues, the team identified the culprit and found the flask in her locker, along with the fourth and final piece of a puzzle cube, which revealed that a lab and chemist would be found at the Institute of Contemporary Arts. 

Cue a dash along to Carlton House Terrace, where the chemist combined the ingredients to produce a spectacular reaction and a long snaking coil of baldness-curing foam.  The team thought they'd finished, but instead had to take a vial of the potion along to the President of the Royal Society, who was fishing on the Serpentine.  Other teams were reaching the lake at a similar time, and there was a nail-biting pedalo race to the President to collect a certificate.  Then a final dash on a Santander bike to the Sackler Gallery, where the team found one last challenge: conduct an orchestra in a rendition of Bizet's Habanera, complete with violin solo.

Time for breakfast, while the other teams found their way to the finish line.  As though that wasn't exhausting enough, even after leaving to head home, the group on the train back to Oxford were working on filling in the details of one of the puzzles. The ultimate 'completer-finishers'.

This was an amazing experience, seeing London from a whole new perspective, with a diverse collection of ingenious puzzles, immersive theatre, and stamina and fitness elements - and all in a good cause. The Raise Your Hands fundraising page is still open. 

The Crackers team was captained by Oxford Prof Jon Chapman, and also featured Head of IT Waldemar Schlackow (Oxford), faculty members Ian Hewitt (Oxford), Vicky Neale (Oxford) and Karen Page (UCL), and graduate students James Aaronson (Oxford), Ed Goldsmith (UCL), Momchil Konstantinov (UCL), Johnny Nicholson (UCL) and Spike Smith (Oxford). 

Some of the team members are disappointed to confirm that Newton's potion does not in fact cure baldness.

Friday, 18 May 2018

Using mathematical modelling to identify future diagnoses of Alzheimer's disease

Oxford Mathematician Paul Moore talks about his application of mathematical tools to identify who will be affected with Alzheimer's.

"Alzheimer's disease is a brain disorder which progressively affects cognition and results in an impairment in the ability to perform daily activities.  It is the most common form of dementia in older people affecting about 6% of the population aged over 65 and it increases in incidence with age. The initial stage of Alzheimer's disease is characterised by memory loss, and this is the usual presenting symptom. 

Psychiatrists would like to predict which individuals will develop the condition both for selecting participants for clinical trials and for finding which variables are used in prediction because this gives insights into the disease process. These variables might be individual characteristics such as age and genetic status or the results of brain scans and cognitive tests.  The graph shows some time plots of scaled brain volumes from successive MRI scans of an individual who has Alzheimer's disease.  The whole brain volume is shown as the blue markers, the hippocampus is marked in red and the entorhinus in yellow. The diamonds at the foot of the graph represent diagnosis points where the red diamonds are a diagnosis of Alzheimer's disease.  The trend in time seems to be downwards, but this feature might also be found in many healthy people as they age.  So our research question is: can we distinguish the changes of relative brain volumes in people who are healthy from those who will subsequently be diagnosed with Alzheimer's disease?

One possibility is to put the data points directly into a deep learning method like a neural network.  This approach might give accurate predictions, but it would not be easy to see which variables are important and how they change with respect to each other.  The method we use is to think of the way the variables change against each other over time as a path in Euclidean space and to characterise that path as a vector which uniquely identifies it.  This path signature was introduced by K.T.Chen in 1958 and has recently proved to be highly successful in machine learning applications.  It generates interpretable features and it can distinguish the time ordering of events: whether variable a or variable b changes value first.  Our results show that the hippocampus is shrinking abnormally fast in people who are subsequently diagnosed with Alzheimer’s disease - a finding that is already known from clinical research.  We are now expanding the number of brain regions that we investigate to improve the accuracy of our models and to learn more about the underlying process of this deadly disease."

Tuesday, 15 May 2018

What do mathematicians do on Saturday nights?

Doing anything Saturday night? Well, if you are an Oxford Mathematician you might just be rushing around London learning to ballroom dance or trying to get your head around the sound wave patterns of a theremin or perhaps cracking a safe or two.

Why? The answer is Midnight Madness, a series of challenges which lead participants on an intellectual treasure hunt around London. Starting at 8pm on Saturday (May 19th), the madness lasts until high noon on Sunday. The Oxford team, together with colleagues from University College London will be competing against the brightest minds in the City. Will the academics prove superior or will years of mathematics have left them soft and contemplative against the sharp intellectual elbows of their opponents? The Oxford and UCL team has been selected via rigorous mathematical assessment (sort of) and features ageing Professors, puzzles gurus, as well as nimble (we hope) graduate students and our brilliant Head of IT. Hours have been spent coming up with a team name and logo (see image). But will it be enough? 

Midnight Madness is in aid of Raise Your Hands, which supports small, effective charities that improve the lives of children across the UK.

Monday, 14 May 2018

Following up Turing - how reaction-diffusion models generate complex patterns

In a seminal 1952 paper, Alan Turing mathematically demonstrated that two reacting chemicals in a spatially uniform mixture could give rise to patterns due to molecular movement, or diffusion. This is a particularly striking result, as diffusion is considered to be a stabilizing mechanism, driving systems towards uniformity (think of a drop of dye spreading in water). Turing's idea was that this mechanism may underlie how patterning occurs in organisms, as it provides a way for the spontaneous formation of spatial patterns even in systems without any heterogeneity at all. This is reminiscent of Darwin's closing remarks in On the Origin of Species: "[F]rom so simple a beginning endless forms most beautiful and most wonderful[...]''

Since this pioneering work on morphogenesis (how organisms develop), a considerable amount of research has explored the tremendous power of reaction-diffusion models to generate patterns. Despite the success of the theory at capturing the attention of mathematicians, biologists, physicists, and chemists for many decades, there are still things we do not know which are necessary to apply these models to processes in developmental biology. An important example of this, presaged by Turing himself, is that most patterning processes of interest do not emerge from spatial homogeneity, but instead evolve in complex environments, and especially from previous patterns. To quote Turing, "Most of an organism, most of the time is developing from one pattern into another, rather than from homogeneity into a pattern."

Recently, researchers Andrew Krause and Eamonn Gaffney from Oxford Mathematics together with colleagues from the University of Cardiff, and the Czech Technical University in Prague have been considering Turing's theory to try and explain certain aspects of the patterning of whiskers on mice. They ran simulations of a reaction-diffusion system known to generate spot patterns, but varied some of the model parameters in space, in order to capture some of the observed variation in mice whiskers (in terms of their size and spacing). The figure above shows an example of these spots with variation in space from this model, as well as an example of the arrangement of whiskers in a typical mouse. This work is still in progress, but it has already led to several fundamental insights into these kinds of systems.

In particular, the researchers found novel patterns which occur in space and time that can be attributed purely to spatial heterogeneity, rather than things which mathematicians are more familiar with that lead to oscillations. Moreover, these oscillating patterns appear robustly in a wide range of different kinds of chemical systems, leading the researchers to think they might be ubiquitous in reaction-diffusion systems. An example of these spatiotemporal oscillations is in the other figure above, where spikes in one spatial dimension are created, move across the spatial domain, and are destroyed, with this pattern repeating periodically in time. The way in which the oscillation period changes as parameters are varied depends crucially on the entire complicated state of the system, contrary to many important models in thermodynamics which display 'universality'.

The researchers hypothesized that it is the interaction of nonlinear reaction and diffusion in a spatially varying medium with an 'open' system that allows for this behaviour. It also calls into question much of the work that has been done using homogeneous models - as Turing himself has said, we anticipate real chemical systems to be in complicated spatial environments. Importantly, very small gradients can lead to moving patterns. Does this occur biologically? Well there are animals who change patterns over time e.g. tapir (slowly changing in time) and flamboyant cuttlefish (rapidly changing in time), but most organisms with spots or stripes, such as zebras or tigers, do not present any kind of oscillations in their coats after birth. The recent work asks many more questions than it answers, both biologically and mathematically. While Turing's original ideas are over 60 years old, he knew that many of his ideas would keep scientists and mathematicians busy for a long time to come, saying "We can only see a short distance ahead, but we can see plenty there that needs to be done.'' The findings appear in the journal Physical Review E.

Figures in rotation:

Reaction-Diffusion model in two spatial dimensions which predicts varying sizes and wavelength between spots.

Typical mouse whisker arrangement (Source: Arrangement of whiskers on the rat's face. Credit: Yan S. W. Yu, Matthew M. Graff, Chris S. Bresee, Yan B. Man, Mitra J. Z. Hartmann (2016) Whiskers aid anemotaxis in rats, Science Advances.)

The chemical concentration over time and space.

Monday, 14 May 2018

Flagging corruption in Government contracting in Africa

Public procurement – or government contracting – is critical to development, accounting for as much as 50% of government spending in developing countries. The procurement process is known to be highly prone to corruption, however corruption is difficult to detect or measure. A recent project led by the University of Oxford in collaboration with Sussex University and Government Transparency Institute has been using and implementing new methodologies for analysing large open public procurement datasets to detect ‘red flags’ that could indicate risks of corruption. Now, researchers from Oxford Mathematics are supporting the delivery of workshops in Africa to share these new methodologies and software tools with anti-corruption groups and researchers to enable them to analyse corruption risks in public procurement data.

Danny Parsons from the African Maths Initiative and Postdoctoral Research Assistant with Prof Balazs Szendroi at the Mathematical Institute in Oxford and Dr Elizabeth David-Barrett (Sussex University) delivered a 2-day workshop at the African Institute of Mathematical Sciences (AIMS), Ghana on Analysing Public Procurement Data for Corruption Risks. This workshop came out of an earlier collaboration between Dr David-Barrett, Dr Mihaly Fazekas (Government Transparency Institute), Prof Szendroi and Danny Parsons on data driven approaches to measuring corruption risks in government contracting. During that project Danny Parsons worked on implementing new methodologies for detecting corruption risks into an open source front end to the R statistics language, to make it easier for researchers in political science, civil society organisations and anti-corruption agencies to detect patterns of corruption risk in public procurement data. In this latest workshop in Ghana, which brought together students and researchers in mathematical sciences and political science as well as civil society groups, Danny showed participants how they could use these recently developed software tools to investigate "red flag" indicators of corruption risk in large open public procurement data. The event highlighted the potential impact this could have on the fight against corruption in Africa - freely available software tools tailored to public procurement data and a growing movement towards governments opening up their data. Interestingly the workshop was picked up by local media (the Ghana News Agency and the Ghana Times) which stressed its relevance to ongoing discussions in Ghana around open government data and in particular the Right to Information Bill.

Wednesday, 9 May 2018

Andreas Sojmark awarded the Bar-Ilan Young Researcher Prize in Financial Mathematics

Oxford Mathematician Andreas Sojmark, a DPhil student in the EPSRC Centre for Doctoral Training in Partial Differential Equations has been awarded the Bar-Ilan Young Researcher Prize in Financial Mathematics. The prize is awarded to a PhD student or early career postdoctoral researcher for an outstanding paper in financial mathematics submitted for the Third Bar-Ilan Conference in Financial Mathematics.

Andreas' paper `An SPDE model for systemic risk with endogenous contagion' will be presented at the conference at the end of May.