Paul Alper points to this news report by Ian Sample, which goes:

Psychologists believe they can identify progressive changes in work of artists who went on to develop Alzheimer’s disease

The first subtle hints of cognitive decline may reveal themselves in an artist’s brush strokes many years before dementia is diagnosed, researchers believe. . . .

Forsythe found that paintings varied in their fractal dimensions over an artist’s career, but in the case of de Kooning and Brooks, the measure changed dramatically and fell sharply as the artists aged. “The information seems to be like a footprint that artists leave in their art,” Forsythe said. “They paint within a normal range, but when something is happening the brain, it starts to change quite radically.” . . .

The research provoked mixed reactions from other scientists. Richard Taylor, a physicist at the University of Oregon, described the work as a “magnificent demonstration of art and science coming together”. But Kate Brown, a physicist at Hamilton College in New York, was less enthusiastic and dismissed the research as “complete and utter nonsense”. . . .

“The whole premise of ‘fractal expressionism’ is completely false,” Brown said. “Since our work came out, claims of fractals in Pollock’s work have largely disappeared from peer-reviewed physics journals. But it seems that the fractal zealots have managed to exert some influence in psychology.”

Totally off topic, but here’s another example of a Sontag: i.e., where the top person in a field is unexpected from a diversity standpoint: e.g., most magazine film critics are men, but the most influential one of the 20th Century was a woman (assuming Roger Ebert was the top newspaper movie reviewer of the last century).

The most influential baseball stadium designer of the last three decades has been Janet Marie Smith, who invented the “retro look” at Baltimore’s Camden Yard in the 1990s:

http://www.sportsbusinessdaily.com/Journal/Issues/2017/03/20/Champions/Janet-Marie-Smith.aspx

Steve:

You’re talking about Objects of the class “Pauline Kael”, which also includes Alice Waters, Mata Hari, Agatha Christie, and Helen Keller. Also good cases can be made for Queen Elizabeth 1, Margaret Mead, Oprah Winfrey, Rachel Carson, Mary “Mother” Jones, Temple Grandin, Marie Curie, Margaret Thatcher, Mary Baker Eddy, and Ellen Willis. I wouldn’t put Susan Sontag in this class.

Do you know of anything useful that has come from the study of fractals per se? The ideas of critical slowing down, basins of attraction, complex dynamics, and generally being careful with sets in real analysis have been valuable in their own right, but I can’t think of any benefit to getting excited about fractals. Other than that they are pretty. Every time I hear about them in the popular science press, it’s attached to some study like this.

Hausdorff measure and dimension, Minkowski content, etc. are important within mathematics. Falconer’s ‘Fractal Geometry: Mathematical foundations and applications (2nd ed.)’, like the name suggests, talks about a range of applications.

I guess it depends on your definition of “fractals” and “useful”. But see for example http://www.medicographia.com/2013/01/fractals-and-their-contribution-to-biology-and-medicine or https://books.google.ch/books?id=eNM1OylCljIC

The biological applications in the first link are interesting, thanks. That’s exactly the kind of thing I was looking for. My problem is that I always seem to see language like this along with the science (from the first link, first paragraph after the abstract):

“Obviously, this [fractal geometry] is in opposition to the ancient, conventional vision based on Euclidean geometry and widely adopted concepts, such as homeostasis, linearity, smoothness, and thermodynamic reversibility, which stems from a more intuitive—but artificially ideal—view of reality.”

I don’t know what to call that. Given the recent discussion on this blog, I’m getting more careful with words like “disingenous” and “dishonest”. It certainly seems designed to play to the kind of postmodern attitude that dismisses classical techniques just because they’re classical, without discussing how well they work. After all, Hausdorff dimension is a generalization, not a rejection, of Lebesgue measure, which was developed in order to deal with the stuffy, linear topic of Fourier series. Those in turn were developed to deal with the (equally classical, linear, and smooth) heat equation, which is intimately bound up with Brownian motion, one of the central examples of roughness and self-similarity. I don’t have the expertise to comment on the quality of Forsythe’s work or Brown’s criticism, but I think the term “fractal zealot” gets at something real.

+1

Yeah, it does seem like there are some fringe uses of “nonlinear dynamical systems”. I was briefly interested in it and read up a bit, but honestly, I couldn’t make that much sense of it in the end. The general principle seems to be that it can predict things close in time, but the further ahead you look the more impossible it is to predict the outcome (e.g., like weather). I’ve never read anything that I would characterize as “useful” … but then, I honestly find it almost impossible to understand.

All this talk of “non-Euclidean geometry” makes me feel like I’m in a H.P. Lovecraft story though…

To be clear, nonlinear dynamical systems themselves are definitely useful, especially in physical science. The most common versions break down into two kinds: ordinary differential equations (ODEs), which have only one independent variable, usually time but sometimes distance in a particular direction; and partial differential equations (PDEs), which have multiple independent variables, chief among which are usually time and space. Nonlinear ODEs are necessary and useful for describing and predicting the behaviour of a whole whack of things, including neuron activation potentials, charge carrier concentrations in semiconductors and electrolytes, predatory-prey interactions in animal populations, and, from what I understand, some of the models in Stan. Nonlinear PDEs are even more essential to physical science: two of the most successful models in continuum physics (Einstein’s equations of general relativity and the Navier-Stokes equations of fluid flow) are nonlinear PDEs, and the techniques developed to get concrete results out of them are behind a good deal of modern numerical computing. Many of the concepts related to fractals have natural applications in this area. It’s just when people seem more excited about the fractals than the models and the real systems the models describe that I get worried.

Yes, I gathered that these kinds of models are useful when I read up on them a bit! I never really buckled down to fully understand them well-enough though, because it was really challenging stuff and not really done at all in my area (so no incentives). It’s really complex stuff!

Just think of all that complex diff eq solving as just giving you an estimate of the expectation based on parameters just like in a linear regression or a glm (like a Poisson regression). Instead of linear effects on predictors (like the effect of height on income), we have things like transfer rates of a molecule between different tissues (say fat, blood, and bones in a very simple case). Then there’s some unexplained variance and perhaps some measurement error, so the data is modeled as being drawn from a distribution with the computed location/mean/expectation and some variance estimated from data. The Stan manual has a simple example and Charles Margossian wrote an excellent tutorial on pharmacology applications (on the web site under users >> documentation >> case studies >> StanCon 2017).

Bob: my impression wasn’t necessarily that for people like Sean, the hard part was figuring out what the ODE does in the stats model, it’s more figuring out how to write ODEs in a meaningful way. People often don’t have much experience with models built on ODEs unless they’ve got a physics, or chemistry background. PDEs even less.

> Just think of all that complex diff eq solving as just giving you an estimate of the expectation based on parameters just like in a linear regression

That’s a pretty unappealing way to think about dynamical systems imo!

More constructively, here is some material I use for teaching dynamical systems:

https://github.com/omaclaren/open-learning-material/tree/master/qualitative-analysis-dynamical-systems/qualitative-analysis-lectures-2017

Which might be helpful to some.

if i remember correctly from a nonlinear dynamics class many years ago, this exact sort of fractal dimension measurement can tell you useful things about dynamic systems. i think the same professor has some work on using those sorts of measurements to characterize the behavior & state of microprocessors and other extremely complicated systems.

Perhaps this is the sort of thing you have in mind:

Sapoval et al., “Self-stabilised fractality of sea-coasts through damped erosion,” arXiv:cond-mat/0311509

Leaving the flaws of this effort to the side, it is amazing to me as I age that a fair number of artists seem to not only maintain their skills but even innovate and improve into old age. Monet, Chagall, Cezanne, Wyeth, etc. At an age where a Newton got into alchemy and an Einstein became irrelevant, at least some painters continued to produce great works of beauty. A young Georgia O’keefe was a very beautiful woman, but we don’t remember her as a rather astonishing photographic model; rather we see her as an old woman in a long skirt who does great things.

The artist is trying to get their head into Heaven, while the scientist is trying to get Heaven into their head. It is the latter who go bonkers.

Said Chesterton, approximately.

This is the postmodern re-staging of Jesus Christ Superstar we’ve all been crying out for.

Thanks. I needed a good laugh.

Quantum physics, nonlinear dynamics, and the mathematics of fractals are hard and counterintuitive, but often beautiful in their own way.

Working out what is happening is someone’s head is hard, but sometimes those counterintuitive processes produce beautiful things.

Therefore, our psychological processes must be determined by quantum physics, nonlinear dynamics, and the mathematics of fractals. QED.

Counterintuitive can be a pretty subjective term.

“The first subtle hints of cognitive decline may reveal themselves in an artist’s brush strokes many years before dementia is diagnosed, researchers believe. . . .”

Subtle is the weasel word for swamped by other, more powerful, effects.

+1

I don’t know. I mean from a neurological/bio basis of behavior perspective, I might be willing to buy that (with a longitudinal single subjects model based on high frequency accelerometers) you’d probably be able to detect changes in how the individual’s brain is functioning. And that with sufficiently rich data, it would be interesting to apply nonlinear dynamical systems theory to the movement data.

That being said, I’m not sure how to parse this:

Mitra, S., Amazeen, P. G., & Turvey, M. T. (1998). Intermediate motor learning as decreasing active (dynamical) degrees of freedom. Human Movement Science, 17(1), 17-65.

“A recent view is that motor learning is the acquisition of an abstract equation of motion that specifies the time evolution of a pattern of coordination. The pattern is expressed by a collective variable that enslaves or orders component subsystems that, in turn, act on and generate the collective variable. In these latter terms, early learning resolves the collective variable and its motion equation, intermediate learning stabilizes and standardizes the subsystems or active degrees of freedom (DFs) producing the collective variable’s dynamics.”

This sure sounds like gobbledygook (or word salad?) to me.

Yeah I got nothing but neither am I an expert in this area so I don’t know. I mean I know in my areas of expertise that I’ve come across plenty of impenetrable journal abstracts.

I’m encouraged by your optimism, but I’m less optimistic.

I have some experience with chaos/fractals in ecology and epidemiology (measles epidemics, lynx/hare cycles, etc.) and more recently with neurological (EEG) data (an undergraduate student did a review of linear and nonlinear metrics for characterizing EEGs).

While it’s reasonable to hope that nonlinear metrics (such as fractal dimension, Lyapunov exponent, various entropies) would be a powerful way for capturing differences in time series driven by strongly nonlinear processes, in practice this is (1) harder than you think (a lot of the estimation methods do badly in very noisy systems, or where the noise is anything other than Gaussian/white) and (2) IMO, typically a dead end relative to pursuing mechanistic models. The *combination* of plausible mechanistic models with nonlinear metrics as probes is great, but trying to use nonlinear metrics by themselves, or nonlinear metrics accompanied by armwaving about the dynamics, doesn’t seem to have been very fruitful.

+1