Mathematics posts
- September 13, 2010
- 07:13 PM
- 823 views
Finding Truth in a Messy World
by jebyrnes in I'm a chordata, urochordata!
*-note, this was derived from a combination of emails between myself and my former phd advisor. See if you can pick out who is arguing what and where. It’s fun – well, for some of you, anyway. How do we know the world? This is a seemingly simple and vast question – one with no [...]... Read more »
Paine, R. (2010) Macroecology: Does It Ignore or Can It Encourage Further Ecological Syntheses Based on Spatially Local Experimental Manipulations?. The American Naturalist, 176(4), 385-393. DOI: 10.1086/656273
Ellison, A., & Dennis, B. (2010) Paths to statistical fluency for ecologists. Frontiers in Ecology and the Environment, 8(7), 362-370. DOI: 10.1890/080209
- September 13, 2010
- 04:04 PM
- 1,362 views
Scientific hubris, or: Everything you thought you knew about straight line fits is wrong
by sarah in One Small Step
Think you’ve got your least squares down to a tee? Think again. In a paper posted to the Arxiv in late August, David Hogg of NYU and his collaborators take us to task on our sloppy data fitting habits. And he’s not in the mood to mince his words. It is conventional to begin any [...]... Read more »
David W. Hogg, Jo Bovy, & Dustin Lang. (2010) Data analysis recipes: Fitting a model to data. Arxiv . arXiv: 1008.4686v1
- August 29, 2010
- 04:45 PM
- 697 views
A Mathematical Description of Cell Aggregate Mechanical Deformation
by Michael Long in Phased
Luigi Preziosi (Politecnico di Torino, Italy) and coworkers have developed a mathematical model for the mechanical stress experienced by cell aggregates, relevant to cellular function in normal health (blood flow) and disease (cancer). This news feature was written on August 29, 2010.... Read more »
Preziosi, L., Ambrosi, D., & Verdier, C. (2010) An elasto-visco-plastic model of cell aggregates. Journal of Theoretical Biology, 262(1), 35-47. DOI: 10.1016/j.jtbi.2009.08.023
- August 27, 2010
- 10:28 AM
- 1,132 views
Please explain the end of kin selection
by Zen Faulkes in NeuroDojo
As an evolutionary biologist, I’m very familiar with the idea of kin selection. When I saw a paper titled “The evolution of eusociality” in the table of contents of Nature, and read the abstract saying, “Kin selection? Don’t need it,” I thought to myself, “Ooooh, this is big.”
I’ve read blog posts about it on Plektix and Wired. I listened to first author Martin Novak being interviewed on the Nature podcast.
Novak does a good job of explaining why kin selection is invoked to invoke the evolution of sterile castes. I’ll also buy his argument that kin selection needs special conditions to work. But I have yet to read or hear a decent summary for how natural selection can pull off this feat. Novak seems to be saying that mathematically, they are the same.
The Wired article suggests that they are resorting to revived form of group selection. Third author, E.O. Wilson, has certainly been suggesting that group selection should be revived for some time (his co-author on that piece, David Sloan Wilson, is quoted in the Wired article.)
I understand that it can be hard to convey mathematical propositions verbally. But I am currently very unsatisfied with the explanations I’ve heard so far. I may not be along in this.
I am not going to have a chance to read the full paper for a while yet. The first day of our fall semester is Monday, and our library only has a subscription to the print edition of Nature.
So here is a challenge to you, fellow science bloggers! Can anyone explain the gist of this paper and how it shows natural selection explains eusociality – and do it without resorting to equations?
Reference
Nowak M, Tarnita C, Wilson E. 2010. The evolution of eusociality Nature 466(7310): 1057-1062. DOI: 10.1038/nature09205... Read more »
Nowak, M., Tarnita, C., & Wilson, E. (2010) The evolution of eusociality. Nature, 466(7310), 1057-1062. DOI: 10.1038/nature09205
- August 26, 2010
- 03:35 AM
- 739 views
Global Temperature Proxy Reconstructions ~ now with CO2 forcing
by apeescape in mind of a Markov chain
Previously, I did a simple Bayesian projection of recent temperature using proxy data and the methods shown in McShane and Wyner (2010). I showed that when you take out the last 30 years of data (1969~1998), the projection does not track the recent uptick in temperatures well. The “projection” is a simple unparametric bootstrap which [...]... Read more »
BLAKELEY B. MCSHANE AND ABRAHAM J. WYNER. (2010) A STATISTICAL ANALYSIS OF MULTIPLE TEMPERATURE PROXIES: ARE RECONSTRUCTIONS OF SURFACE TEMPERATURES OVER THE LAST 1000 YEARS RELIABLE?. Annals of Applied Statistics, 4(3). info:/
- August 24, 2010
- 11:54 AM
- 870 views
Differentiating Skill and Luck in Financial Markets with Streaks
by Samuel Arbesman in arbesman.net
Speaking of luck, we just released a paper onto SSRN about luck and skill entitled Differentiating Skill and Luck in Financial Markets with Streaks. This paper, which I worked on with Andrew Mauboussin (a brilliant high school student who worked in our lab this summer), examines the relationship between skill and luck using mutual fund [...]... Read more »
Andrew Mauboussin, & Samuel Arbesman. (2010) Differentiating Skill and Luck in Financial Markets with Streaks. SSRN: http://ssrn.com/abstract. info:/
- August 22, 2010
- 04:10 AM
- 666 views
Global Temperature Proxy Reconstructions ~ Bayesian extrapolation of warming w/ rjags
by apeescape in mind of a Markov chain
There are a bunch of “hockey sticks” that calculate past global temps. through the use of proxies when instrumental data is absent. There is a new one out there by McShane and Wyner (2010) that’s creating quite a stir in the blogosphere (here, here, here, here). The main take out being, that the uncertainty is [...]... Read more »
BLAKELEY B. MCSHANE AND ABRAHAM J. WYNER. (2010) A STATISTICAL ANALYSIS OF MULTIPLE TEMPERATURE PROXIES: ARE RECONSTRUCTIONS OF SURFACE TEMPERATURES OVER THE LAST 1000 YEARS RELIABLE?. Annals of Applied Statistics, 4(3). info:/
Mann, M., Zhang, Z., Hughes, M., Bradley, R., Miller, S., Rutherford, S., & Ni, F. (2008) Proxy-based reconstructions of hemispheric and global surface temperature variations over the past two millennia. Proceedings of the National Academy of Sciences, 105(36), 13252-13257. DOI: 10.1073/pnas.0805721105
- August 20, 2010
- 04:10 PM
- 727 views
Genome-Scale Epigenetic Marker Detection Across Populations
by Michael Long in Phased
Lior Pachter (University of California at Berkeley, United States) and coworkers have developed MetMap software for uncovering epigenetic data hidden by standard MethylSeq analysis, which will advance our understanding of the role of epigenetics in human health and medicine. This news feature was written on August 20, 2010.... Read more »
Singer, M., Boffelli, D., Dhahbi, J., Schoenhuth, A., Schroth, G. P., Martin, D. I. K., & Pachter, L. (2010) MetMap Enables Genome-Scale Methyltyping for Determining Methylation States in Populations. PLoS Computational Biology, 6(8). DOI: 10.1371/journal.pcbi.1000888
- August 19, 2010
- 07:30 AM
- 585 views
Numbers on the Brain: Neurobiology of Mathematics
by Jason Goldman in Child's Play
Nearly everyone has heard of developmental dyslexia – a learning disorder characterized by poor reading skills despite otherwise sufficient schooling – but have you heard of developmental dyscalculia? Many people have not. Here is part 4 in a week-long series on this lesser-known learning disorder. Case-studies of patients with various brain lesions have demonstrated the [...]... Read more »
Ardila A, & Rosselli M. (2002) Acalculia and dyscalculia. Neuropsychology review, 12(4), 179-231. PMID: 12539968
Dehaene, S. (2004) Arithmetic and the brain. Current Opinion in Neurobiology, 14(2), 218-224. DOI: 10.1016/j.conb.2004.03.008
Isaacs EB, Edmonds CJ, Lucas A, & Gadian DG. (2001) Calculation difficulties in children of very low birthweight: a neural correlate. Brain : a journal of neurology, 124(Pt 9), 1701-7. PMID: 11522573
Molko N, Cachia A, Rivière D, Mangin JF, Bruandet M, Le Bihan D, Cohen L, & Dehaene S. (2003) Functional and structural alterations of the intraparietal sulcus in a developmental dyscalculia of genetic origin. Neuron, 40(4), 847-58. PMID: 14622587
Dehaene, S, Piazza, M, Pinel, P, & Cohen, L. (2003) Three Parietal Circuits for Number Processing. Cognitive Neuropsychology, 487-506. info:/
- August 19, 2010
- 06:00 AM
- 1,335 views
Evolution of Colour Terms: 3 Perceptual Constraints
by Sean Roberts in A Replicated Typo 2.0
Continuing my series on the Evolution of Colour terms, this post reviews evidence for perceptual constraints on colour terms. For the full dissertation and for references, go here.
The perceptual space that results from the processing of opponent colours is non-uniform (see Figure below), meaning that there are optimal ways to describe it (Jameson & D’Andrade, . . . → Read More: Evolution of Colour Terms: 3 Perceptual Constraints... Read more »
Jameson, K., & D'Andrade, R.G. (1997) It's not really Red, Green, Yellow, Blue: An Inquiry into cognitive color space. Color Categories in Thought and Language. DOI: 10.1017/CBO9780511519819.014
Regier, T., Kay, P., & Khetarpal, N. (2007) Color naming reflects optimal partitions of color space. Proceedings of the National Academy of Sciences, 104(4), 1436-1441. DOI: 10.1073/pnas.0610341104
Liljencrants, J., Lindblom, B., & Lindblom, B. (1972) Numerical Simulation of Vowel Quality Systems: The Role of Perceptual Contrast. Language, 48(4), 839. DOI: 10.2307/411991
DEBOER, B. (2000) Self-organization in vowel systems. Journal of Phonetics, 28(4), 441-465. DOI: 10.1006/jpho.2000.0125
Buchsbaum, G. (2002) Color categories revealed by non-negative matrix factorization of Munsell color spectra. Vision Research, 42(5), 559-563. DOI: 10.1016/S0042-6989(01)00303-0
- August 18, 2010
- 09:30 AM
- 489 views
Developmental Dyscalculia Explained: Strategy, Memory, Attention
by Jason Goldman in Child's Play
Nearly everyone has heard of developmental dyslexia – a learning disorder characterized by poor reading skills despite otherwise sufficient schooling – but have you heard of developmental dyscalculia? Many people have not. Here is part 3 in a week-long series on this lesser-known learning disorder. (See parts one, and two, and a companion post at [...]... Read more »
Gallistel CR, & Gelman R. (1992) Preverbal and verbal counting and computation. Cognition, 44(1-2), 43-74. PMID: 1511586
Siegler, R. (1994) Cognitive Variability: A Key to Understanding Cognitive Development. Current Directions in Psychological Science, 3(1), 1-5. DOI: 10.1111/1467-8721.ep10769817
Geary, D. (1993) Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114(2), 345-362. DOI: 10.1037/0033-2909.114.2.345
Butterworth, B. (2005) The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46(1), 3-18. DOI: 10.1111/j.1469-7610.2004.00374.x
Rosenberg PB. (1989) Perceptual-motor and attentional correlates of developmental dyscalculia. Annals of neurology, 26(2), 216-20. PMID: 2774508
- August 5, 2010
- 06:52 PM
- 1,242 views
R and Google Earth ~ comparing tuna tracks vs. Gulf of Mexico oil spill extent
by apeescape in mind of a Markov chain
There is a lot of interest in how the Gulf of Mexico oil gusher will affect the ecosystem and its marine species. One such species is the Western Atlantic bluefin tuna that holds the Gulf of Mexico as one of its major spawning grounds. Recent tag data show that the location of the gusher is [...]... Read more »
Teo SL, & Block BA. (2010) Comparative influence of ocean conditions on yellowfin and atlantic bluefin tuna catch from longlines in the gulf of Mexico. PloS one, 5(5). PMID: 20526356
- August 4, 2010
- 03:44 PM
- 732 views
Real Time fMRI
by Neuroskeptic in Neuroskeptic
Wouldn't it be cool if you could measure brain activation with fMRI... right as it happens?You could lie there in the scanner and watch your brain light up. Then you could watch your brain light up some more in response to seeing your brain light up, and watch it light up even more upon seeing your brain light up in response to seeing itself light up... like putting your brain between two mirrors and getting an infinite tunnel of activations.Ok, that would probably get boring, eventually. But there'd be some useful applications too. Apart from the obvious research interest, it would allow you to attempt fMRI neurofeedback: training yourself to be able to activate or deactivate parts of your brain. Neurofeedback has a long (and controversial) history, but so far it's only been feasible using EEG because that's the only neuroimaging method that gives real-time results. EEG is unfortunately not very good at localizing activity to specific areas.Now MIT neuroscientists Hinds et al present a new way of doing right-now fMRI: Computing moment to moment BOLD activation for real-time neurofeedback. It's not in fact the first such method, but they argue that it's the only one that provides reliable, truly real-time signals.Essentially the approach is closely related to standard fMRI analysis processes, except instead of waiting for all of the data to come in before starting to analyze it, it incrementally estimates neural activation every time a new scan of the brain arrives, while accounting for various forms of noise. They first show that it works well on some simulated data, and then discuss the results of a real experiment in which 16 people were asked to alternately increase or decrease their own neural response to hearing the noise of the MRI scanner (they are very noisy). Neurofeedback was given by showing them a "thermometer" representing activity in their auditory cortex.The real-time estimates of activation turned out to be highly correlated with the estimates given by conventional analysis after the experiment was over - though we're not told how well people were able to use the neurofeedback to regulate their own brains.Unfortunately, we're not given all of the technical details of the method, so you won't be able to jump into the nearest scanner and look into your brain quite yet, though they do promise that "this method will be made publicly available as part of a real-time functional imaging software package."Hinds, O., Ghosh, S., Thompson, T., Yoo, J., Whitfield-Gabrieli, S., Triantafyllou, C., & Gabrieli, J. (2010). Computing moment to moment BOLD activation for real-time neurofeedback NeuroImage DOI: 10.1016/j.neuroimage.2010.07.060... Read more »
Hinds, O., Ghosh, S., Thompson, T., Yoo, J., Whitfield-Gabrieli, S., Triantafyllou, C., & Gabrieli, J. (2010) Computing moment to moment BOLD activation for real-time neurofeedback. NeuroImage. DOI: 10.1016/j.neuroimage.2010.07.060
- July 30, 2010
- 11:00 AM
- 621 views
Measuring Synchrony - pt 2 of ??
by Brandon Goodell in Bored Lunatic
This is the second in a multi-part series analyzing the paper linked below. The paper uses several measures of synchrony and tests them against some real-world data to compare their performance. Today I will be talking about three different measures of nonlinear interdependence between two signals, all of which are based on nearest neighbors.
The main thing here is that the frequency coherence method of measuring synchrony primarily measures linear interdependency. In other words, if we have two clocks moving at the same rate, but are set to different times, the frequency coherence method measures the difference in those times. It does this spectacularly well. In fact, if we have one clock running at a fixed, faster rate than the other, the frequency coherence method should (hypothetically) measure that perfectly well too. However, if you have one clock that is running faster and faster as time goes on, the frequency coherence method fails miserably... the phase difference between the two clocks changes nonlinearly.
Hence, we need a definition of some nonlinear interdependence between two signals - after all, signals are very chaotic sometimes, very nonlinear. One way of doing this is to construct some "nearest-neighbor" definition. We first take a signal and consider a sliding window along the signals. That is, if we have a 100-second signal, I will break that signal up into 90, 10-second signals (now I have X1, X2, ..., X90, and Y1, Y2, ..., Y90). The first will start at the first second, the second will start at the second, and so on. I do this for both signals, and compute the first k nearest neighbors of each window. I will then compute the mean Euclidean distance between each Xi's nearest neighbors with the Yi's nearest neighbors... and by that, I mean that if the 10th, 13th, and 75th window are Y17's 3 nearest neighbors, I compute the mean distance between X17 and X10, X13, and X75. This is the Y-conditioned mean Euclidean distances. Finally, what I do is take the ratio of the mean Euclidean distance with the Y-conditioned mean Euclidean distance, and take the average of this value across all windows. Call this S.
Now, S is sensitive to nonlinearities. In particular, it is sensitive to small dependencies in the data. When S is small, the signals are not synchronized, and as S approaches 1, the signals are synchronous. On the other hand, instead of taking the average of the ratios between the X-conditioned and the Y-conditioned mean Euclidean distances, I can take the average of the logarithm of the ratios. Call that H. Finally, I can take the average of the relative ratio between X and Y-conditioned Euclidean distances. Call that N.
These are 3 different measures of synchrony, all based on the idea of nearest neighbors. We are basically trying to compare the contribution of our second signal to each of the nearest neighbors of each window from the first signal. It's pretty nifty, and surprisingly, not symmetric: the synchrony between X and Y is different than the synchrony between Y and X. This is a good idea for a lot of reasons - for one, a signal such as your metabolic rate will be strongly synchronized with the oscillation of the sun in the sky. However, the sun in the sky is not strongly synchronized with your metabolic rate. This method gives researchers a way to tease out such a relationship in the data, allowing us to make some causative statements we are normally not capable of making via correlation arguments.
On the other hand, the abstraction at this level of synchrony measure is pretty heavy-handed. This prevents an easy visualization of what's going on - very similar to taking the Fourier transform of something that is not time-dependent. Interpreting frequencies in a non-temporal, non-spatial context is a pain.
Quian Quiroga, R., Kraskov, A., Kreuz, T., & Grassberger, P. (2002). Performance of different synchronization measures in real data: A case study on electroencephalographic signals Physical Review E, 65 (4) DOI: 10.1103/PhysRevE.65.041903
... Read more »
Quian Quiroga, R., Kraskov, A., Kreuz, T., & Grassberger, P. (2002) Performance of different synchronization measures in real data: A case study on electroencephalographic signals. Physical Review E, 65(4). DOI: 10.1103/PhysRevE.65.041903
- July 20, 2010
- 07:09 AM
- 932 views
Why you REALLY can’t trust small studies: the small study effect
by Michael Slezak in Good, Bad, and Bogus
You’ll often see loony zealots refer you to a study showing how effective their preferred treatment is — there usually is some small study supporting the use of almost any treatment.
You’ll also often hear people reply that the study was only small, so shouldn’t be trusted. But why shouldn’t you trust small studies? Sure, they [...]... Read more »
Nüesch E, Trelle S, Reichenbach S, Rutjes AW, Tschannen B, Altman DG, Egger M, & Jüni P. (2010) Small study effects in meta-analyses of osteoarthritis trials: meta-epidemiological study. BMJ. PMID: 20639294
- July 19, 2010
- 09:32 AM
- 559 views
Measuring Synchrony - pt 1 of ??
by Brandon Goodell in Bored Lunatic
This is the first in a multi-part series analyzing the paper linked below. The paper uses several measures of synchrony and tests them against some real-world data to compare their performance. Today I will be talking about the frequency-coherence measure of synchrony.
I really love being a scientist. The coolest bit about research, as far as I can tell, is that it's hard. You are exploring the unknown. This paper is exploring the unknown - in the past two decade, there was this big surge in popularity in the idea that synchrony was the mechanism exploited by the brain to do big and complex tasks. The popularity behind that idea eventually dwindled, as more papers came out suggesting that synchrony is not a mechanism to be exploited, but either an emergent property or a fluke coincidence more likely to cause a seizure than perform computations. Now, it seems as if synchrony falls somewhere in between these extremes - it may be exploited to perform computations, but is not the primary computational mechanism. Unsurprisingly, over the past two decades, there has been a lot of interest in developing some method of measuring synchrony.
Given two signals, how can one quantify the degree to which the signals are synchronized? Just like every other area of science and mathematics, about a billion different answers to this question popped up. The first answer was to compute frequency-coherence of two signals. In this method, you start by computing the cross correlation of the signals. This results in a function of time - the amount of time you input corresponds to a shift in the signals. That is, if you evaluate the function at 17 seconds, you get the correlation between the signals when one of the signals starts 17 seconds after it normally would. If this correlation is high, then the phase difference between the signals will be close to 17 seconds. This isn't the end of the story - while this method allows you to compute the maximum likelihood estimate of the phase angle between two signals (which is useful, don't get me wrong), a great many processes in the universe that are fairly unrelated to one another have a high correlation. For example, there is a very strong negative correlation between HIV patients and subscribers to Field and Stream magazine. This is not because HIV causes you to dislike hunting, and it is not because Field and Stream magazine convinces you to use a condom. There is an intervening factor that causes these two traits to be anticorrelated, namely, if you subscribe to Field and Stream magazine, you are less likely to be in an urban area, and hence less likely to get HIV.
To obtain a better idea of the synchrony between two signals, you can then take the Fourier transform of the cross-correlation. This is where the physical interpretation of these processes starts to suck a little. The Fourier transform of a time signal tells you how much of a signal is stored in a specific frequency; however, the Fourier transform of the cross correlation of two signals is harder to interpret. I suspect that, since the cross-correlation is a function of time, the interpretation is as straightforward as it seems - the Fourier transform of the cross correlation tells you how much of the correlation is packed into each frequency.
If you then normalize this according to the auto-correlations of the signals, then you get an array of complex numbers, each one corresponding to the degree to which two signals are synchronized in a specific frequency channel. The average of this across all possible frequencies ends up being a number (called the frequency coherence) between 0 and 1, which ends up being a particularly good measure of the synchrony between two signals. Theoretically, if you take a sine wave and a cosine wave, they will have synchrony 0. If you take a sine wave and a sine wave, they will have synchrony 1. Hell, if you have a sine wave f(x) and another signal -f(x), they will also have synchrony 1.
Pretty nifty. However, there are a couple of assumptions built into this approach. Primarily, it assumes that you have a pair of stationary signals - in other words, the same process is generating the signals regardless of the time you start measuring from. If you shift the time at all, you will still get the same statistical process underlying the signal. This is patently absurd, in particular for the brain itself, which is a highly nonstationary process - shift the time by -4 hours, and I'm asleep. A different statistical process has overtaken my brain*.
However, this particular drawback does not hamper my work at all, because I am purposely modeling a couple of stationary neurons. Hence, I'll be using this measure to kick off my thesis.
*Dreaming about your mom.
Quian Quiroga, R., Kraskov, A., Kreuz, T., & Grassberger, P. (2002). Performance of different synchronization measures in real data: A case study on electroencephalographic signals Physical Review E, 65 (4) DOI: 10.1103/PhysRevE.65.041903
... Read more »
Quian Quiroga, R., Kraskov, A., Kreuz, T., & Grassberger, P. (2002) Performance of different synchronization measures in real data: A case study on electroencephalographic signals. Physical Review E, 65(4). DOI: 10.1103/PhysRevE.65.041903
- July 18, 2010
- 12:07 AM
- 1,013 views
Cherry Picking to Generalize ~ retrospective meta-power analysis using Cohen’s f^2 of NASA temp visualization
by apeescape in mind of a Markov chain
Previously, I plotted a grid of NASA GISS global temps in ggplot2 to show general trends by the brute force method. Here, I will again use the brute force method to do a simple power analysis on a portion of the data (data here). The general aim is to figure out what the minimum sample [...]... Read more »
Thomas, L. (1997) Retrospective Power Analysis. Conservation Biology, 11(1), 276-280. DOI: 10.1046/j.1523-1739.1997.96102.x
- July 16, 2010
- 11:53 AM
- 536 views
Phase synchronization
by Brandon Goodell in Bored Lunatic
Consider two pendula (penduli? pendulums?) that are on a bit of a wobbly table, oscillating back and forth.
Eventually, they synchronize. Once a huge mystery to physics, the fact that placing two pendulums on the same surface causes minute vibrations to transmit through that surface, causing the devices to become weakly coupled. The result is that, after a sufficiently long period of time swinging back and forth, the two pendulums become fully synchronous.
The Scholarpedia article, "Phase Model" discusses this phenomenon, how it is modeled, and "what it all means." It lists the different methods of forming coupled oscillating systems, and describes a few coupled oscillating systems that we simply do not have the mathematics to analyze... yet.
The model I'm particularly interested in is the phase model. See, most of these models are described by considering two (or more) little beads on a circular wire hoop. All of the beads are marching along clockwise at a constant speed, but when two beads are near each other, they are attracted to one another; the bead in the lead (ha!) slows down a little bit and the trailing bead speeds up a little bit so that they can catch up with each other. The result is that, after a sufficiently long period of time has passed, the beads stay as close to each other as can be for the rest of eternity. At least, this is the case with two beads.
If we have an arbitrary number of n beads, far more complex behavior occurs - we can even use the phase model to describe chaotic systems, which is pretty neat. Keep in mind here we aren't talking about actual beads on an actual wire hoop, we are talking about points that can collide and pass each other as if they were not there - ghost beads marching on a hoop.
Okay, so the real-life analogy starts to break down. Whatever.
The point is that emergent behavior pops up left and right - while two beads behave in an entirely predictable way (converging to one another), three beads behave in a less-than-predictable fashion. The whole is greater than the sum of it's parts - take one bead, and it marches along boringly. Take two beads, and they converge to one another. Take three beads, and crazy shit happens, so the complexity of 3 does not equal the complexity of 2 plus the complexity of 1.
Get it?
So why the hell am I interested in this? Well, I'm currently testing some software I have written that estimates the average phase angle between two signals that vary in time. In the context of the phase model, I can interpret the two signals as the position of two separate beads on the same hoop. Consequentially, I'm going to use the phase model to test my software to make sure it is working properly. I am going to generate several simulations of these beads, then run those through my software to see if the output makes sense in the context of arbitrary phase oscillators.
In the meantime, I'm brushing up on my synchrony reading - to say that it can be a bit mind-bending is likely an understatement. For example, the quick and dirty rundown I just gave you is for weakly coupled phase oscillators - but they are continuously coupled. We could develop a pulse-coupled model, like neurons, which give each other little discontinuous nudges every time they get sufficiently close to one another, and that would be equally interesting, and there is a mathematical framework to analyze that as well.
I find it fascinating that there are no mainstream methods of analyzing non-continuous yet non-pulse coupled oscillators. I imagine something like the stock market could be described by millions of coupled oscillators, but that system has sharp discontinuities in momentum oftentimes, because the stock market can change at the speed of our brains. That is, I imagine the stock market is actually an amalgam of the continuously-coupled and pulse-coupled oscillator models.
Izhikevich, E., & Ermentrout, B. (2008). Phase model Scholarpedia, 3 (10) DOI: 10.4249/scholarpedia.1487
... Read more »
Izhikevich, E., & Ermentrout, B. (2008) Phase model. Scholarpedia, 3(10), 1487. DOI: 10.4249/scholarpedia.1487
- July 16, 2010
- 04:23 AM
- 1,072 views
‘Gravity doesn’t exist’, says philosophically naive scientist/journalist
by Michael Slezak in Good, Bad, and Bogus
Reports of a physicist “taking on gravity” have recieved a bit of attention recently, with a New York Times article outlining Erik Verlinde’s idea that gravity is an emergent property of thermodynamics.
I think it’s great that the piece was written — even though apparently it hasn’t excited any physicists since the start of the year. Regardless [...]... Read more »
Erik P. Verlinde. (2010) On the Origin of Gravity and the Laws of Newton. arxiv.org. arXiv: 1001.0785v1
Bertrand Russell. (1912) On the notion of cause. Proceedings of the Aristotelian Society. info:other/
- July 9, 2010
- 09:35 PM
- 1,061 views
Log Normal Distributions in Ecology ~ multiplications complications
by apeescape in mind of a Markov chain
The normal distribution is the “norm” when applying statistics to data. It is simple to interpret, simple to predict, simple to optimize, convenient software-wise and analytically elegant. But in many applications, this modeling assumption may not be optimal. The first is that the normal distribution doesn’t have a zero bound. In ecology, the data is [...]... Read more »
LIMPERT, E., STAHEL, W., & ABBT, M. (2001) Log-normal Distributions across the Sciences: Keys and Clues. BioScience, 51(5), 341. DOI: 10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2
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