Archive for October, 2010

The Ensemble Kalman Filter

October 27, 2010

The standard Kalman filter and even the Extended Kalman filter (for nonlinear problems) proved inadequate. I’ve now placed my hope in what’s known as the Ensemble Kalman Filter:

Another sequential data assimilation method which has received a lot of attention is named the Ensemble Kalman Filter (EnKF). The method was originally proposed as a stochastic or Monte Carlo alternative to the deterministic [Extended Kalman filter] by Evensen (1994a).  The EnKF was designed to resolve the two major problems related to the use of the [Extended Kalman filter] with nonlinear dynamics in large state spaces, i.e. the use of an approximate closure scheme and the huge computational requirements associated with the storage and forward integration of the error covariance matrix.

The EnKF gained popularity because of its simple conceptual formulation and relative ease of implementation, e.g. it requires no derivation of a tangent linear operator or adjoint equations and no integrations backward in time. Furthermore, the computational requirements are affordable and comparable to other popular sophisticated assimilation methods […].*

* Excerpt from Geir Evensen’s Data Assimilation: The Ensemble Kalman Filter, 2007, p. 38.

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B squared + RTF = 0

October 25, 2010

I read Sylvia Nasar’s biography of John Nash, A Beautiful Mind (1998), which also has been made a movie. John Nash won the Nobel in economics in 1994, but is equally famous for his serious mental disorders which he recovered from in the early nineteen ninties. ‘B squared + RTF = 0’ is a ‘very personal’ equation Nash used, while he was ill, to describe his life when he was at MIT (see chapter 21, A Beautiful Mind). Of course, he used an equation to describe his life. ‘The equation represents a three-dimensional hyperspace, which has a singularity at the origin, in four-dimensional space. Nash is the singularity, the special point, and the other variables are people who affected him’ Sylvia Nasar writes about the equation. Sometimes, I recognize something of myself in the Nash Nasar describes, bar the genius, of course. I have not come to the point where I feel like describing my life with an equation, though, and if I do, it probably won’t be four-dimensional.

Nash did marvelous mathematical research at MIT, but he also lead a chaotic life. My life is quite marvelous most of the time; my research, however…

Kuhn vs. Popper by Steve Fuller, Part 2

October 19, 2010

Part 2? Take Two, rather (This is Take One). It surprises me how difficult it is to get to grips with this book, particularly given its apparent brevity (the main body of the book runs through page 215 in a relative small format). Of course, I’m not even an amateur philosopher of science, but still.

A part of the difficult lies in the chaotic or at least hidden structure of the book. Fuller announces his motives in the introduction (‘to recapture the full range of issues that separate [Kuhn and Popper],’ see p. 3*). The ‘full range’ is presumably a lot of material; the already mentioned brevity is thus surprising. But Fuller do not list nor declear the ‘issues’ he wants to address. There seem to be no plan or structure. Rather, he seems to move from issue to issue in a haphazard fasion, and the motive or aim of the discussion is often out of sight and elusive. The conclusion of the book is also something of an anti-climax. The last chapter seemingly only discusses one of the issues separating Kuhn and Popper; there are no final remarks, no conclusion, or anything that resembles a closure.

Kuhn vs. Popper did increase my understanding and knowledge of the ideas of both Kuhn and Popper, and also how their ideas connect to the ideas of other important thinkers. Perhaps more importantly, Fuller has helped me see the important differences between Kuhn and Popper. Throughout the book, for one thing, Fuller comes up with comparative statements.

Kuhn and Popper represent two radically different ways of specifying the ends of inquiry: What drives our understanding of reality? Where is the truth to be found? [p. 56].

Kuhn was indeed authoritarian and Popper liertarian in their attitudes to science. This point has been largely lost, if not inverted, by those who regard ‘Kuhn vs Popper’ as a landmark in 20th-century philosophy of science [p. 13]

Popper was a democrat concerned with science as a form of dynamic inquiry and Kuhn an élitist focused on science as a stabilising social practice. Nevertheless, they normally appear with these qualities in reverese. How can this be? [p. 68].

To dig deeper into these differences, one has to dig into the actual ideas. Kuhn first:

For Kuhn, science begins in earnest with the adoption of a ‘paradigm’, which means both an exemplary piece of research and the blueprint it provides for future research […] Kuhn deliberately selects the phrase ‘puzzle-solving’ (as in crossword puzzles) over ‘problem-solving’ to underscore the constrained nature of normal science […] A ‘revolution’ occurs [upon a ‘crisis’] when a viable alternative paradigm has been found. The revolution is relatively quick and irreversible. In practice, this means that an intergenerational shift occurs [pp. 19-20].

An important aspect of Kuhn’s philosophy of science is how history is rewritten after a scientific revolution, such that the scientific development appears streamlined and meaningful. In Kuhn’s view, Fuller writes,

[…] the secret of science’s success – its principled pursuit of paradigmatic puzzles – would be underminded if scientists had the professional historian’s demythologised sense of their history. After all, in the great scheme of things, most actual scientific work turns out to be inconsequential or indeterminately consequential [p. 20].

Another important feature of Kuhn’s ideas regards how people become scientists. One becomes a scientist through a “conversion experience or ‘Gestalt switch,’ whereby one comes to see the world in a systematically different way” (p. 21).  These features, combined with the conservative flavor of Kuhn’s The Structure of Scientific Revolutions, led Popperians to liken Kuhn to ‘religious and politcal indoctrinators’ (p. 21).

But of course, this was not how Structure was read by most of its admirers – if they actually did read the book. For while Kuhn’s examples are drawn almost exclusively from the physical sciences, these are the disciplines that have probably paid the ‘least’ attention to Structure, even though Kuhn himself was qualified only in physics. Kuhn’s admirers are to be found instead in the humanities and the social and biological sciences [p. 21].

Kuhn’s admirers persisted in wrenching Structure from its original context and treating it as an all-purpose manual for converting one’s lowly discipline into a full-fledged science. These wishful readings of Structure have been helped by its readers’ innocence of any alternative accounts of the history of science – often including their own – with which to compare Kuhn’s [p. 22].

When Fuller turns to discuss Popper, his sympathies with Popper become obvious:

[Popper] was always a ‘philosopher’ in the grand sense, for whom science happened to be an apt vehicle for articulating his general world-view [pp. 22-23].

For the ‘grand philosopher,’ philosophy of science is only a reflection of more fundamental attitudes:

Once Popper’s philosophy of science is read alongside his political philosophy, it becomes clear that scientific inquiry and democratic politics are meant to be alternative expressions of what Popper called ‘the open society’ [p. 26].

Popper grew up intellectually among the positivists in the Vienna Circle, but disagreed with them on their attitude towards the role of logical deduction.

For the positivists, deduciton demonstrates the coherence of a body of thought, specifically by showing how more general knowledge claims explain less general ones, each of which provide some degree of confirmation for the more general ones. For Popperians, deduction is mainly a tool for compelling scientists to thest th econesequences fo their general knowledge claims in particular cases by issuing predictions that can be contradicted by the findings of empirical research. This is the falsifiability principle in a nutshell [p. 25].

Fuller neatly sums up the difference between the 20th century’s giants in the philosophy of science:

Whereas actual scientific communities existed for Popper only as more or less corrupt versions of the scientific ideal, for Kuhn the scientific ideal is whatever has historically emerged as the dominant scientific communities [p. 6].

* Page numbers refer to the Icon Books 2006 paperback edition.

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