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Sunday, 24 May 2020

COMPOSING MUSIC BY SIMULATION FROM MULTIPLE POINT PROCESSES



      They are lots of articles in the literature about using stochastic processes e,g.Markov Chains to compose music. I don't know whether the following suggestion is original.


                                                           
                 A log-Gaussian Cox Process driven by an Ornstein-Uhlenbeck Process.

   This process was first proposed by Tom Leonard (Journal of the Royal Society, Series B, 1978) . It is a special case of the Doubly Stochastic Poisson Process described by my undergraduate mentor David Cox in the same journal in 1955 

          Suppose you wish  to compose a piece of music on the time interval (0,T) and assume that each note may assume one of M different characteristics, Then the times of occurrences of any particular characteristic in the tune may be represented by points in the interval (0,T), These could be specified by the composer  (see above diagram) . If repeated for each of the N characteristics, then the N sets of points (when superimposed with different labels on the same interval) defines the entire composition.

         For any particular characteristic, it would instead be possible to simulate the points from a random point process on the interval (0,T). If a different point process is used to generate the points for each characteristic, then the entire composition can be simulated.

         A log-Gaussian Cox Process can  be parameterised using a mean value function M(t) for t assuming values in the interval (0,T),  See, for example, the smoother curve in the above diagram.

         If the Cox Process is driven by an Ornstein-Uhlenbeck process. then a variance V and a correlation parameter R also need to be specified.

         It is left to the composer to come up with an imaginative choice of the continuous function M. He may then use a standard computer package to simulate the points of occurrence of that particular characteristic, for specified V and R (which could of course be fiddled with to improve the quality of the tune).

         If the composer can devise a different choice of continuous function M for each of the N characteristics, then his entire composition can be simulated from the N thus defined Cox Processes, e.g.by keeping V and R fixed and specified across all simulations.

          This suggests to me to suggest a whole neball park, where the ingenuity of the composer is still an essential ingredient.

                                                             

Friday, 1 May 2020

SAMPLE SPACE--A NEW PUBLICATION BY THE DEPT OF STATISTICAL SCIENCE AT UCL




                                                               SAMPLE SPACE


                                                                       

FROM  PROFESSOR RICHARD CHANDLER


Welcome to our first edition of Sample Space, a newsletter that aims to keep our community of past and present students and staff in touch with what’s happening in Statistical Science at UCL. I hope you find the contents interesting: we plan to do more of this kind of thing in the future, to keep you all updated with the latest news from the department. UCL Statistical Science is currently undergoing a period of significant expansion: this is part of a strategic investment in mathematics, statistics and operational research that will lead to the establishment of a UCL Institute for Mathematical and Statistical Sciences, or IMSS for short. We plan for the Statistical Science and Mathematics departments to be re-housed next to each other on a single site, in a purposedesigned facility that will put UCL on the map as one of the ‘go-to’ places for the mathematical sciences in London – and that will also provide our students with a much greater sense of community than is currently possible. Bringing this idea to fruition has already taken considerable effort, and will continue to do so in the coming months and years, although I strongly believe that the rewards will be well worth it. Apart from anything else, being bigger gives us additional resources to do things like producing Sample Space! The department has also recently established a new master’s programme in Data Science, which has proven very popular with students. A recent graduate, Mete Veyisoglu, has kindly taken the time to share his experiences of the programme in this issue. Further curriculum developments are being undertaken at the moment to ensure that we continue to equip our students with the tools to be statistically literate, perhaps a more desirable skill than ever in today's world. On the research side, the recent announcement of a large increase in UK government funding for the mathematical sciences is very welcome, and particularly timely in view of our IMSS expansion plans. Final preparations are underway for REF 2021, the latest iteration of the UK’s ‘Research Excellence Framework’. The department’s performance in this exercise will also contribute to the level of government funding we receive for the next few years. We have excellent staff doing exciting and incredibly diverse research, some of which is making a real impact outside academia: this issue highlights work being done by Serge Guillas and his team on understanding tsunami risk around the world. As always, there are external challenges: as I write, UK universities are grappling with the uncertainties posed by Brexit and, more urgently, by the worldwide Covid-19 outbreak. UCL Statistical Science is such a thoroughly international community that these issues will certainly affect us to some extent. I very much hope, however, that the effects will be minimal for all of us. Finally, I would like to extend my very sincere thanks to everyone who contributed to this first issue of Sample Space, and in particular to Samuel Livingstone for coordinating and editing it. If any readers have feedback, or potential contributions to future issues, or suggestions for material that they’d like to see, please contact him at samuel.livingstone@ucl.ac.uk. I think he and all the contributors have done a fantastic job: I hope you do too.