Tribute to Howell Tong by Professor Peter Whittle FRS
19th December 2009
It gives us all great pleasure to celebrate Howell’s 65th birthday at this meeting and, more than that, to celebrate his remarkable record of achievement: the steady perseverance and probing insight with which he has led the development of a whole theory for the extension of time series analysis into the nonlinear domain, coping with phenomena which in some cases had scarcely been recognised. We celebrate also his great warmth and generosity, familiar to the many who have become first his followers and then his friends and colleagues, and familiar also to the institutions which have so benefited from his good sense and entirely altruistic initiative.
Howell began his research career about 1972 in control theory, progressing to stochastic control. By about 1976 he had picked up the time series theme, but soon became convinced of the inadequacy of the existing linear theory. As many had, but who felt themselves helpless, partly because the familiar linear theory seemed not to transfer at all, but also because the very richness of possibilities meant that they were lost for a guiding theme. However, Howell’s insight led him to such a theme in the threshold models and their various generalisations whose presentation he and Professor Lim began in their RSS paper of 1980. At the same time, Howell did what he knew he had to do; something at which statisticians, including myself, flinched. He mastered dynamic systems theory, the mathematical theory which seeks to discern a conceptual pattern in the nonlinear models suggested by physics and technology.
However, this is a deterministic theory, and on to it one has to graft the randomness of a stochastic model and the statistics of inference. The purely stochastic aspects of such a generalisation had been considered by authors such as Richard Tweedie, and Professor Cline has given an excellent account and extension of such work in his contribution to the Festschrift. Howell had his own insights, recognising, for example, that the Liapunov functions of a deterministic stability analysis often give the right tool for the study of ergodicity in the stochastic case. However, over and above this, he tackled the daunting inference question.
Characteristically, he produced the complete package: nonlinear stochastic models in full rigour, practicable statistical analyses, well-posed computer simulations and analyses of real data. However, what is also revealed is his intuition and instinct for the essential, which guides him so well on the path to conceptualisation.
For example, after model-fitting he homes on to the question of determination of dimension, as an essential characteristic. One would imagine that this is a difficult plum to pull out of a complicated pudding, but his instinct led him to a cross-validation approach which reveals a remarkable simplification, reassuringly confirming the instinct which had led Howell to this point.
One of the reasons for considering nonlinear models was to find an explanation for the robust limit cycles which are so often observed. These generalise the notion of a stable static equilibrium, and are associated with existence of an attractor in state space. However, even a limit cycle corresponds to a rather special case of such an attractor, and it is now realised that one does not have to look very far before one finds attractors whose equilibrium has the character now termed chaotic. A chaotic solution looks random, and a principal question has been: can one distinguish such a solution from the random sequences of probability theory?
Howell immediately saw this as an inference question. Suppose one had a model which, in its deterministic form, generates a chaotic sequence. Suppose that one stochasticises the model by introducing process noise into the dynamics; would it then be possible to distinguish the chaotic and the random effects in the time series generated by such a model? The natural reaction is to dismiss this as an impossibly delicate task. On the other hand, there could be quite gross effects; the sensitivity of the whole course of a chaotic variable to initial values presumably implies a similar sensitivity to process noise. Howell was of course quite undaunted; he opened up the problem and, jointly with Professor Chan, found a successful treatment. This required a strong generalisation of existing concepts, e.g by generalisation of the deterministic Liapunov indices and by insistence on identification of the embedding dimension, the dimension of the space within which the attractor lies.
I mentioned Professor Chan. In the course of his work Howell has attracted many students, who went on to become co-workers and co-authors, making their own independent contributions in the field which Howell had opened. Several are here. These would certainly acknowledge the great stimulus that Howell has been to them, but he has the lightest of touches in such matters. He is unfailingly kind, generous, good-tempered and with a very appealing sense of humour.
Professor Cutler refers to just these qualities when she writes in her contribution to the Festschrift that “Howell brings not only incredible talent and creativity to his work, but generosity, enthusiasm and an unmistakable uniqueness of style”. She also very appositely quotes Pascal: “When we see a natural style we are quite surprised and delighted, for we expected to see an author and we find a man”.
Howell’s quotations from Lao Tze in his 1990 book prompt one to ask whether there is anything distinctly Chinese in his work. I don’t regard this as a delicate question, but as a very proper and interesting one, whose answer is in fact positive. Howell completes, over several years, the construction of a large and complex theory, with all its technical detail and all its real-world latency. He seeks clarification, particularly of issues which, although simply expressed, are the subtlest and of the essence. His style is such as to imbue the whole work with a kind of imaginative vividness and even, at times, an impish humour. I believe that in this he maintains, with great distinction, a long and rich tradition from which we all plainly continue to benefit.
May I propose a hearty toast to this most remarkable and most human of colleagues: Howell Tong.
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