The Wall Street Journal is helping Leonard Mlodinow tout his book The Drunkard’s Walk: How Randomness Rules Our Lives. Among other things, Mlodinow, like academics Tversky, Kahneman, and Gilovich before him, wants to show that streaks in games like basketball don’t exist. Or, rather, they do exist, but they can be “explained by randomness.”
Listen: randomness can’t explain anything.
Statisticians imagine—I choose this word carefully—a basketball player has an ineffable probability of making a free throw, and they try to guess the probability’s value through modeling. Suppose a guess is 80% for a particular player and then suppose our player has just made his last 10 shots. A fan might say our man has a hot hand. Mlodinow:
If a person tossing a coin weighted to land on heads 80% of the time produces a streak of 10 heads in a row, few people would see that as a sign of increased skill. Yet when an 80% free throw shooter in the NBA has that level of success people have a hard time accepting that it isn’t. [Tversky and others] showed that despite appearances, the “hot hand” is a mirage. Such hot and cold streaks are identical to those you would obtain from a properly weighted coin.
This statement is confused. Each time a “properly weighted coin” is tossed something makes it fall heads or tails, some physical cause. “Randomness” does not make the coin choose a side. Spin and momentum cause it to land on one side or the other. There is nothing “random” in a coin toss: there is only physics. If you knew the amount of force propelling the coin upwards, and the amount of spin imparted, you can predict with certainty the outcome of the flip. (Persi Diaconis and Ed Jaynes—both non-traditional statisticians—have written multiple papers on this subject.)
“Randomness” is not a physical property; it does not exist inside the coin. Mlodinow acknowledges this in the words “weighted coin” used to describe his thought experiment. He is aware implicitly that modifying a physical property of a coin like the weight changes whether it shows head or tail. But he fails to realize that there is no difference in philosophy between changing the weight or modifying the spin or the momentum. Like Nelson reading the signal flags, he has turned a blind eye to the physics and has taken refuge in “randomness” to explain how the coin behaves.
Similarly, something, some physical—and biological and mental—process is causing the basketball player to make his shot. Again, the spin, the momentum, the aim, and the mental pathways that give rise to those properties are what determines whether the shot falls through the hoop or misses.
Our man has made his last 10 and is setting up for the 11th. Now the fans behind the basket distract him, or maybe he starts thinking too much about the shot, or there is excess sweat on his finger, or whatever, but he misses his shot and his streak ends. Randomness does not explain why the streak ends: physics and biology do.
Random means unknown and nothing more. Before the player takes the shot, or Mlodinow flips his weighted coin, we do not know what the outcome will be because we do not know what the values of the physical properties that determine the outcome are: it is these properties that change from shot to shot and from flip to flip that cause the different endings. If we did know the physics—like we can if we practice with coins—we can predict the outcome. That is, the outcome becomes certain, or known, and is therefore not random.
Our knowledge any outcome depends on what information we condition on. What might be random to you might not be random to me if I have different information than you. For example, right now my cell phone is either in my left- or right-hand pocket. To you, the outcome (finding out which pocket), is random because your conditional information consists solely of the knowledge that it might be in one or other pocket. The information I condition on allows me to know with certainty the outcome.
Same thing in basketball: if we knew what amount of force a player uses etc. we can predict whether his shot will go in. But that sort of information is hard to measure, so we look for proxies, like statistical models of the player’s past performance. Conditional only on those models, we can say “There is an 80% chance the next shot will go in.”
If the 11th shot is sunk, our man’s streak continues. The mental state of the player certainly played a part in that shot and so did his “hot hands.” Because we cannot predict who will have a hot hand or when, does not mean that hot-hand streaks do not exist. We should not mistake our models for reality.
I haven’t finished the book, but it’s not my impression that Mlodinow is saying all streaks are random, only that streaks can occur randomly, and without a sufficiently large sample, you will not be able to assess with meaningful certainty whether a streak is one or the other. I expect that he’ll elaborate on this in his discussion of the law of large numbers.
However… I think you may be on to something. The statement you quote is more expansive than it should be. His language does seem a bit too loose at times, which can be confusing for a neophyte like myself. I have several times thought he was overstating his case. Still, the book’s entertaining, and I’m being acquainted with a lot of basic, powerful ideas about randomness.
finally: maybe I should just get your book, but this was free….
You know the old saying , Clazy, about getting what you paid for…
Matt:
This is excellent and ties neatly into your earlier discussion of smoothing. The argument for randomness is essentially an argument about the nature of the “model” that accounts for a given outcome. It also raises the question as to what “error” actually refers.
You are so right. There is clearly some luck involved in shooting a basketball (a slight miss can hit the inside of the rim and bound away; a worse miss can bounce off the top of the rim, hit the backboard and go in. And of course, the unintentional bank shot is lucky.) But good shooters DO have stretches where their mechanics get grooved, their confidence soars, and they “get hot”. And they can have stretches where a flaw creeps into their mechanics and their consistency falters.
Someone did a study claiming to debunk the hot shooter myth where he asked shooters before they shot whether they were going to make the next shot and showed no increased ability to predict whether they were “hot”. My problem with the study is that the assumption is wrong. Being hot isn’t about making every shot or predicting same. Even when a shooter is “in the zone”, he doesn’t make every shot. And even poor mechanics will sometimes result in a made shot.
Part of the reason is that people assume coin flips are random while streaks in basketball may denote improved skill is related to the difficulty in improving ones skill.
We know that it is very, very difficult to increase one’s skill in a way that lets them be able to control which face shows when flipping coins. This is particularly true if you impart plenty of momentum to the coin. While, in principle, if we specify the momentum and spin precisely and flip in a vacuum, one can predict which face will show when the coin lands, we also know that very, very tiny differences in amount of momentum imparted will affect the outome. The difference is so small that almost no one can improve their skill; the result is we don’t even bother to host competitive coin flipping competitions.
In contrast, people can improve their hoop shooting ability with practice. Their mental state could, conceivably affect their ability to aim, apply the correct pressure to the ball etc. People even take lessons and practice to improve their skills, and this often works.
So, of course we can see runs even if athlete’s skill level does not improve. But, when we see long runs, we might suspect the athlete’s skill has changed. In contrast, if we see long runs in coin flips, we tend to suspect a biased coin. (Doesn’t this just mean that people’s opinions are implicitly Bayesian?)
In any event, either notion can, hypothetically, be tested by either continuing to watch the athlete or finding another person to flip the coin .
I remember reading in Scientific American an article about Persi Diaconis, a professional mathematician who also dabbled in magic (or was it the other way around?) – the article indicated that Diaconis was talented enough at controlling coin flips that he could flip 50 “heads” in a row.
So even the old standard of coin flips – a 50:50 proposition for most people – is (if the article referred to above is correct) decidedly not so for someone like Persi Diaconis.
In a similar vein, I remember one of the “Beating Vegas” programs on television describing a group of sharps who had developed their ability at throwing dice to the extent that they could make money off the craps tables just by optimizing their chances of rolling specific combinations (maybe sevens, I don’t fully recall). There again a process that was supposedly random was, with sufficient practice, something that in the correct hands was anything but random.
Lucia:
I was thinking along similar lines. My thought though is that if you suspect a biased coin then you do not ask for someone else to flip the coin, you subject the coin to another test to determine if it is fair or not. The problem is that the outcome of the coin toss (Y) is determined by a large number of factors (X1, X2, …,Xn) one of which is that the coin is fair (X1). To efficiently determine whether the outcome is due to the coin or some other factors then one should test the coin. It gets a bit complex if you assume that the outcome is due to the flippers skill (X2) because this may be more difficult to test independently of flipping a fair coin.
For me this comes back to the fact that many of our models are simply underspecified and we mask our ignorance by resorting to the equivalent of dust bowl empiricism.
Clazy’s first post has it (mostly) right.
The book is non-technical and has an entirely different purpose / audience than Matt’s narrow post suggests. In fact, given the current political climate, and Brigg’s conservative politics, I would have thought that he would have given the book at least some high marks.
What do statisticians do? Explain randomness by using (or inventing) statistical tools and probability. So we, statisticians, try to explain something that cannot explain anything! *_^
All,
It’s physics all the way down. My number one son found this for me today.
Bad dice are killing off characters that deserve to live!
These are fascinating videos, incidentally. I urge you to take the twenty minutes and watch both.
Quite so.
Except on quantum mechanical scales, and not necessarily then, randomness as such does not exist in Nature, in this universe at least. Chaos does.
Only mechanical artifacts can produce nearly random results, the classic example is the old fashioned fruit machine: or indeed the roulette wheel. And by the way a useful check as to whether dice are loaded is to drop them into your drink.
If you want the classical analysis of casino and card games see Scarney.
Which does not mean we cannot assess with reasonable accuracy the chances of a particular outcome of a non mechanical process. In English horse racing the handicapper will impose a weight penalty on the various horses with the aim of producing a dead heat. This seldom happens but what is surprising is that analysis of the results over a hundred years shows just how good the handicapper’s judgement is overall: and consequently how closely the race usually runs. A good example is the last Grand National.
So don’t bet on it: unless the price in the market is heavily in your favour. And even then wonder why the price is what it is before hazarding your money. You know better than the market?
But what really bugs me is the arrogance and ignorance of climate modellers. Of course they say weather is chaotic in the short term but in the long term this all evens out so we can confidently predict what the climate will be like in thirty, forty, take your pick, years.
This is akin to the gambler’s fallacy. The martingale, the doubling up etc.
True if we take a machine designed to produce a random outcome, say a perfect roulette wheel and ignore the zeroes, then over time we would expect to see the numbers of reds and blacks turn up evenly irrespective of short runs of one or the other. Which is why old fashioned casinos still give you a preprinted card to record the numbers as they come up. And why they love a run because evrybody tries to bet against it: and lose.
But weather is not random, chaotic yes but NOT random. Unlike mechanical chance the summation of chaotic events over time does not necessarily even out however long the time period, so the idea of a long term average or a mean is meaningless.
Of course we can predict to some extent what tomorrow’s weather might be like based on yesterday, and that some seasons are likely to be hotter than others and so forth.
What we cannot do is predict that because the average weather of the last few years, centuries or whatever is such like then the next year, years or centuries will be like that.
The analogy that somehow weather is weather and varies and cannot be reliably foretold exept in the short term BUT that climate is the summation of weather over the long term and therefore an be predicted far into the future from the past weather record is utterly fallacious.
And finally if you must play games of chance remember this ancient advice which is found in Seneca: but the old fraud probably pinched it from some other ancient. It says the Gambler must ride his luck.
In modern Americanese this probably translates as ‘ When you’re hot you’re hot and when you’re not you’re not.’
If you are losing walk away, if you are winning drive it for all you’re worth. It could just be your lucky day.
Kindest Regards
I take issue with this. Maybe you get nice predictable qualities in your classical mechanics, but the in world models I am familiar with, this is only due to the Law of Large Numbers. It seems to me there is lots and lots of truly “random” embedded deeply in the system.
If you truly believe this to be true and have the physics to back it up, would you be willing to sign me up as your agent ? I am sure there is a boatload of cash available for skills like deterministically predicting nuclear decay.
If someone believes that ‘randomness’ is a real thing and not merely a place holder for something we do not yet understand, or for some process we do not have a practical method for measuring, then write out a computer program that generates ‘random’ numbers.
The algorithm must take a purely mathematical form, and if copied and run on a second machine, must be unable to anticipate what number the first machine will generate next.
Once it can be described mathematically, then I’ll accept people understand what it actually is.
Where do I send the paperwork, Artifex?
I’m betting on this book: Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference by Guido Bacciagaluppi, Antony Valentini, forthcoming.
Reviews say it contains a strong argument for de Broglie’s pilot-wave theory, i.e. non-randomness at the quantum level. Somewhere, Bohm is smiling.
Will,
Lots of computer code exists to create numbers called pseudo-random numbers. As you suggest, they are wholly deterministic. The hilarious thing about these is that the algorithms are subjected to a battery of tests to ensure the numbers “look random.” And again, a blind eye is turned to the evidence that shows the numbers are perfectly predictable.
Matt:
Didn’t Einstein make the same point about randomness?
a jones, define the difference between ‘random’ and ‘chaotic’. Don’t both words really mean ‘unpredictable’? Of course, there are long term averages which can be used for prediction. It isn’t fallacious to make those predictions either. So weather is chaotic? I think I can say safely that in the next 500 years, the average global temperature of the Earth will not be remotely close to either 0K or 5000K as a result of ‘chaotic’ weather. I think I can also safely say that ‘weather’ will not likely produce a snowflake the size of Manhattan. How can anyone apply ranges without resorting to averages?
The fallacious reasoning in climate predictions is that enough is known about the workings of weather to safely predict a far-term average within a few degrees precision when the evidence indicates otherwise.
Well William don’t count much on the 101th attempt to revive De Broglie Bohm theories !
They have been killed 100 times and so will be killed for the 101st time too (and for the same reasons) . They are actually already dead .
Most important reason is that this and similar theories give and must give exactly the same predictions as the standard QM already used with unparalleled success for 80 years .
So the only difference is postulating weird entities (pilot waves) that are both real and not observable , material but not made out of matter .
A theory saying that real and non observable little green men did it would have as many virtues as the pilote wave theory .
A more technical but deadly flaw is that the pilot wave theory is non relativist , you need to pile on the heap of hypothesis another one – interactions can propagate with infinite speed (equivalent to saying that the theory is non local . Argh !) .
I suppose that this guy is just somebody who is deeply unhappy (misunderstanding ?) with the modern physics so he found the only way to release his frustrations which is to tell in a popular book to the large and uneducated public that BOTH quantum mechanics AND relativity are wrong and that he is the genius who will show where Einstein , Bohr , Dirac , Heisenberg and everybody else went all wrong .
Let us only say that the probability for that to be true is fairly low .
On top when a physicist reads that Lee Smolin supports some “theory” , it is an almost certain sign that the theory in question is either crazy or wrong 🙂
To come back on topic of this thread , even the pilot wave theory doesn’t (can’t) do away with the Heisenberg uncertainty principle and the probabilistic character of QM (the square of the wave function is still giving the probability of a measure) .
It will make its difference to “classical” QM by only saying that a mysterious an unobservable process “simulates” the randomness that you see . That this process violates the relativity principle by passing is a detail .
A bit like the randomness generating computer program you mentionned .
But as , per definition , you cannot access the program , it is in any case a time waste to speculate about its existence or non existence .
DAV
The difference is that random is not deterministic and chaotic is deterministic .
As far as their deterministic character goes , they are exactly the opposite of each other .
There are many other ways to distinguish them but they all base on this fundamental difference .
For example in random anything can happen albeit sometimes with a very low probability , in chaotic only some things may happen (regions outside of the chaotic attractor in the phase space have a zero probability) .
Etc
You’re probably right about Valentini’s book, Tom. Things that have failed usually continue failing. From what I can tell from the two-page Science write up, Valentini’s is a better attempt than Smolin’s, reproducing the entire Solvay conference papers and then commenting on them (including the idea that quantum entanglements are non-local and require infinite speed, too). And there is also the element of fun to be had.
DAV,
Chaotic does not mean unpredictable, but randomness does. Sensitivity to initial conditions is a better portrayal of the former. If you knew the initial conditions (and the operators) then you could predict any outcome exactly.
Vonk,
So when did weather become deterministic except in the philosophical sense that all natural phenomena are ultimately predictable?
Seems to me then, that the practical difference between ‘random’ and ‘chaotic’ is that ‘chaotic’ systems have a large number of states with unequal probability. I maintain that such a system still has a predictable average range. For example, the turbulence in a flowing pipe is chaotic but I can still calculate an average flow rate.
‘Chaotic’ is a euphemism for ‘random’ with a nod toward the topic of this blog post.
Another fallacy the climate modelers are making: they are averaging their models then claiming the average will be within a few degrees of the average of yet-to-be experienced weather. Especially fallacious when they haven’t even shown that the models are reasonably close to reality.
Briggs says:
Given that you may want a bunch of points in [0, 1] that don’t obviously look biased, you might very well cook up a bunch of (necessarily) deterministic processes and pick one that fails as many tests for “predictibility” as possible. It isn’t SO ridiculous as long as you remember what’s going on, is it? 🙂
Briggs says:
8 July 2009 at 9:20 am
Tom Vonk says:
8 July 2009 at 8:58 am
Although they differ at a fundamental level, is it not often impossible to distinguish them in practice (e.g., through any reasonable observational program)?
I seem to recall Tom Vonk (?) discussing a MHD model working on a (very) small domain to study the R-T instability (perhaps at another blog?) and comparing this to traditional continuum model which requires a small amount of “noise” in the interface to trigger the instability and therefore is in some sense “unfaithful.”
Although the behavior is theoretically different, isn’t also essentially true that a small amount of “noise” will necessarily appear in your (measured) initial conditions?
Interesting post and comments.
Here is a slow film of lightning:
http://www.youtube.com/watch?v=8waV48897o4&NR=1
And here one of a bullet, making a mess:
http://www.youtube.com/watch?v=4zSm-J4oG58&feature=related
Tom: “A theory saying that real and non observable little green men did it would have as many virtues as the pilote wave theory.”
Sounds like modern cosmology 🙂 “22% [of the universe] is thought to be composed of dark matter and The remaining 74% is thought to consist of dark energy” where “dark” means unobservable.
Tom: “For example in random anything can happen albeit sometimes with a very low probability”
Like one of Hawking’s black holes ejecting a complete Elvis Presley 😉
Briggs: “Listen: randomness can’t explain anything.”
A tautology.
The good thing about tautologies, PG, is that they are always true.
Briggs: “The good thing about tautologies, PG, is that they are always true.”
Ain’t that the truth? 🙂
Where would logic be without them?
DAV
You do the classical error that do all those who are not familiar with the non linear dynamics .
You confuse deterministic and predictable .
When did weather become deterministic ?
Well about at the same time when the fluids began to obey the Navier Stokes equations what was some 15 billions years ago .
The N-S equations are perfectly deterministic but at the same time untractable in practice because to solve them , you need the initial and boundary conditions where you run in the problem characteristic for chaotic systems – the sensibility to initial conditions .
Thare is absolutely NOTHING “random” in a chaotic system , and it is impossible both in practice and in theory to attach a “probability” to a point in the phase space .
The same old tired example for turbulence and flow rate has everything to do with mass conservation , constant section and incompressibility and nothing with the dynamics of the system .
If you took the R-T flow (OMS you indeed correctly remember the subject :)) , you would see that you can’t calculate anything (at least for longer periods of time) and a flow rate are not more than something else .
.
You may consider a geometrical equivalent of a chaotic system – the Mandelbrot set .
Like in every fractal set , every point of the plane is either black or white (e.g a point of a phase space belongs or doesn’t belong to a chaotic attractor) .
It is impossible to define even in principle a “probability” that a given point of the plane belongs to the Mandelbrot’s set .
So the function f(x,y) of the plane giving a value of 1 if the point of coordinates (x,y) belongs to the Mandelbrot’s set and 0 if it doesn’t is perfectly deterministic but there is no general rule allowing to predict the value of f for every point (x,y) .
Of course f doesn’t define any probability either and is trivially not random .
.
Btw QM is also perfectly deterministic Рthe wave function obeys the deterministic Schr̦dinger equation and is defined at every time and in every point .
The complexity (no pun intended) comes from the fact that it is complex valued and that the MEASURE results are eigenvalues of an operator (f.ex the Hamiltonian for energy)
But as there is generally an infinity of eigenvalues yet the measure can yield only one , there is a “selection” process .
This process is random and the probability density function is given by Psi.Q.Psi* where Psi is the wave function and Q the operator and this result has been verified over and over these last 80 years .
Now the whole “big” discussion (de Broglie, Bohm , Bohr etc) is whether this process is really random (the “Shut up and calculate” attributed to Feynman) or if it is a “simulated” randomness where there is something mysterious , non random and per definition non observable behind the scene making us believe that we observe randomness .
This kind of discussion is not really interesting because the result is the same in both cases – we see randomness with well defined and computable probability density in this particular specific case (quantum measure) .
PG
“Like one of Hawking’s black holes ejecting a complete Elvis Presley ”
It is more trivial and you don’t even need black holes for that .
If a relatively small number of molecules randomly conspired to synchronise their movement in the “up” direction for a small period of time at the right place , then you would see a pig fly .
Given the great number of pigs having existed in the space time since the beginning of the Universe , I think that a pig already flew somewhere sometime .
And when they’re at it , they could change the pig in Elvis Presley too because it doesn’t need such a big effort 🙂
Vonk,
“perfectly deterministic but at the same time intractable in practice ” ==> unpredictable ==> random. One sense of ‘non-deterministic’ is ‘unpredictable’. One sense of ‘intractable’ is ‘too hard to calculate’. Defining ‘chaotic’ as ‘deterministic but intractable (therefore ‘unpredictable’)’ to distinguish it from ‘random’ (therefore ‘unpredictable’) is a lesson in tautology.
‘Chaotic’ is a euphemism for ‘random’ in the same sense that ‘challenged’ is a euphemism for ‘disabled’. Insisting on one word over another when describing the same thing is PC thought control.
—
“You may consider a geometrical equivalent of a chaotic system – the Mandelbrot set”
Just because you can find a mathematical relation that resembles a phenomenon you doesn’t mean you have a workable model. Claiming some phenomena are intractable because you can find intractable models is silly and pointless. Claiming all phenomena are ultimately intractable (a la QM) is claiming that the ultimate goal of science is pointless.
Mlodinow’s book is worth reading for the discussion of the Monty Hall Problem.
DAV
The fundamental intuition underlying randomness is the absence of order or pattern. I do not see an absence of pattern in chaotic systems (think of the Mandelbrot & Julia sets). Identical chaotic events produce identical outcomes. That is IFF all the initial conditions were met to an infinite degree of precision. We can say that such events are *effectively* random just because we cannot make measurements to infinite precision.
Quantum events are random in that otherwise *identical* events have non-identical outcomes. Electron A is not just immeasurably similar to electron B; it is believed to be identical to infinite precision. It may of course turn out that electrons and other fundamental particles have little tiny mechanisms inside that determine their individual behaviour, but I’m not holding my breath while I await their discovery.
Hope that helps 🙂
Nettles,
We had a discussion of Old Monty here, too.
DAV,
There are many chaotic systems (like the logistic function) which is perfectly predictable, but which might not naively appear so. Again, if you knew the starting values and the operators then you will always know the future perfectly. This is why chaotic systems are not consider random.
But they are random in our sense if you know, for example, we have the logistic function, but you do not know the value of the constant. Just looking at the series won’t help you much in estimating the constant, either.
I don’t think we are all disagreeing here. It’s just important to be careful with definitions.
P. Git,
“The fundamental intuition underlying randomness is the absence of order or pattern. I do not see an absence of pattern in chaotic systems (think of the Mandelbrot & Julia sets). ”
The problem with using patterns is that you’ve now shifted the definition to what constitutes a pattern. Hardly helpful.
Mersenne twister generators have pattern (as do ALL mathematical equations — or they wouldn’t BE equations). It’s just that the pattern is not easily discernible. I guess that means in reality Mersenne twisters are chaotic although most consider them more random-like.
So, let’s summarize. Random events are unpredictable with NO patterns. Chaotic events are unpredictable but with SOME pattern. But doesn’t pattern imply predictable? Are you saying there is a continuum of patterness with random and chaotic systems having different rankings within that continuum?
—
“Identical chaotic events produce identical outcomes. That is IFF all the initial conditions were met to an infinite degree of precision.” … “Quantum events are random in that otherwise *identical* events have non-identical outcomes”
How then could anyone say the market or weather are chaotic systems when identical events are impossible to create? How could you ever prove a model of a chaotic phenomenon?
I find the touchiness of the chaotic mathematical models unsettling when translated to the real world. Don’t you? The problem with accepting randomness as an inherent property instead of considering it a lack of complete knowledge is is conceding the game.
Just imagine: grown scientists sitting around and seriously suggesting that the ultimate quest of science is futile and their entire careers have been a waste of time! On top of that, the conjecture that events are random with NO underlying cause is an unprovable (therefore unscientific) concept. One might as well contend they are caused by the unpredictable hand of God.
—
“electrons and other fundamental particles have little tiny mechanisms … but I’m not holding my breath while I await their discovery.”
Yes, it’s so much more satisfying to believe the situation is the only thing you can think of. Life must go on.
Briggs,
“I don’t think we are all disagreeing here. It’s just important to be careful with definitions”
I agree but considering that the thrust of you post was “random just means unknown cause or insufficient information to calculate/predict to any precision” and “chaotic means insufficient information to calculate/predict to any precision,” the distinctions being made between random and chaotic are hardly important ones. When it comes to things like weather, the cause of the chaos is also unknown which makes the distinction between random weather systems and chaotic systems even less important.
Maybe I read too much into what you’ve said.
Briggs,
“chaotic systems (like the logistic function) but which might not naively appear so. ”
Really? I must be naive because many things follow the logistic function. It’s an expression of limitation. One can only squeeze so hard or run so fast or jump so high regardless of effort. Drag and available power puts limiting speeds on vehicles, etc. I would think that given enough information and time, one could indeed asymptotically determine the parameters of a logistic to any precision. But then, maybe we’re talking about different things. I’m thinking of sigmoid functions shaped like tanh. (I consider those having only positive results just as similar).
DAV
““perfectly deterministic but at the same time intractable in practice †==> unpredictable ==> random. One sense of ‘non-deterministic’ is ‘unpredictable’. One sense of ‘intractable’ is ‘too hard to calculate’. Defining ‘chaotic’ as ‘deterministic but intractable (therefore ‘unpredictable’)’ to distinguish it from ‘random’ (therefore ‘unpredictable’) is a lesson in tautology. ”
Sorry but this obviously translates ignorance of non linear dynamics which is the basis of chaotic systems .
This was not at all the meaning of my post but I admit that the word “intractable” was poorly chosen .
Then perhaps it will help for your understanding of temporal chaos if I start with the basics .
.
1)
The trajectory of a dynamical system in the phase space of dimension N is given by the solution of N ordinary first order differential equations with N time dependent variables . Everybody agrees that any such system system is trivially deterministic .
2)
These solutions as long as they are bounded may exhibit 3 different behaviours .
First is that the trajectory converges to an invariant point of the phase space and we have a classical equilibrium system .
Second is that the trajectory converges to a limit cycle and we have a steady state classical periodic movement . A more complex subcase are quasi periodic systems (N body problem) .
Third is that the trajectory stays within a subset of the phase space but is neither periodical nor quasi periodical . There is no constraint on the topology of this subset whose dimension may be fractal (think Mandelbrot set) .
3)
The chaotic systems belong to the third category . Note that they are perfectly deterministic because they verify 1) . Now if we observe the time evolution of a small volume in the phase space (think initial conditions) , this volume will be deformed during the time . This deformation is accomplished by a combination of stretching and folding . The latter is highly non linear and the cause of spreading of the trajectories in the invariant subset called attractor .
It is worth a special mention that this invariant set (attractor) is not necessarily stable because it depends on the coefficients of the ODE defined in 1) . Even an infinitesimally small variation of one of these coefficients may lead to the destruction of the attractor and apparition of another one which may even be non chaotic (f.ex a limit cycle) .
4)
The rate of divergence of trajectories (or deformation of volumes in the phase space what is the same thing) is given by the Eigenvalues of the Jacobian of the ODE system defined in 1) .
If at least 1 of these Eigenvalues is positive , the rate of divergence will increase exponentially with time .
And THIS is the rigorous definition of a chaotic system not some meaningless “euphemismes for random” . It has also nothing of a “tautology” because the sign of the Eigenvalue of a Jacobian is a highly non trivial property .
5)
There is a proven theorem that trajectories in the phase space NEVER cross . That’s why the question “What is the “probability” that the trajectory of a given system passes through a given point ?” doesn’t make any sense . Indeed there is exactly only ONE trajectory that will pass through this point . If the system happens to start on this particular trajectory then the “probability” that it will pass through this point is EXACTLY 1 . If it starts on any other of the infinities of trajectories regardless how “near” it starts to the special one then this “probability” is EXACTLY 0 . This is what William is saying when he points out that the system being deterministic , if we know where it starts , we know exactly where it will pass . This is also a proof that the system is not random and defines no probabilities . Chaotic systems have nothing to do with random systems .
6)
There are many systems that are deterministic (as defined in 1) and have at least 1 Eigenvalue of the Jacobian positive . These are the real chaotic systems we are talking about here . The unpredictability follows exactly from that property which leads to the exponential rate of divergence of trajectories . Now one must of course not naively assimilate this fundamental unpredictability with the notion that “nothing can be said about the behaviour of chaotic systems” . That’s why there is the chaos theory which has a lot to say about such systems .
Ergodic theory which is fundamental to the understanding of the dynamics can’t be developped in the limited space of a blog post but it continues on this basis too .
In any case even if you don’t like it , you will have to live with the fact that chaotic systems are not random .
Vonk,
You seem to be talking about mathematical concepts and not physical phenomena. Just because there are chaotic mathematical entities, it doesn’t follow that they have any connection to the real world. I think you would be hard put to demonstrate any physical phenomenon is chaotic in the mathematical sense (vs. ‘random’) especially when presented with only a time plot (like we get with weather). If I gave you two plots and told you one was generated by an unspecified non-mathematically chaotic process and the other by a mathematical chaotic process then I think you would not be able to tell me which is which.
Anticipating agreement, this is why I object to anyone insisting a physical system like weather is chaotic vs. ‘random’ and then proceed to explain its properties because it IS one. It is also why I asked “when did weather became deterministic”. I maintain that is really substituting one word for another, which is an uninformative distinction and, at best, indicates a mindset similar to those who insisted that people are ‘challenged’ vs. ‘disabled’.
So N-S is supposed to explain weather as a mathematical model and all that’s missing are those pesky parameters? Well until someone stumbles across them and can demonstrably explain weather and climate, I maintain that is an unproven conjecture. I suspect that may take a while if only because I also suspect that weather is a bit more complex than the dynamics of air. But that’s just my ignorance showing.
Sorry, but I don’t get it.
There IS randomness and there IS chaos. They are being treated here as mutually exclusive. Folks have opined that weather isn’t random, it’s chaotic… it’s just that we don’t know the starting states. That’s true, but that’s not the only factor. The reality is that we don’t know all the things that can affect weather and we don’t model them well. Heck, there’s uncertainty about the effect of clouds! I’ve not seen models of weather that actually include everything that could affect weather. For example, anyone know of a GCM that includes possible meteor strikes? We know meteors exist. We know they have struck the earth in the past. We believe that meteors have impacted weather and climate in the past. Yet there is no mention of the possibility of meteor strikes in the models. We don’t know when there’s going to be a meteor strike… it’s effectively random.
Same thing hods in basketball and coin flipping…
“This statement is confused. Each time a “properly weighted coin†is tossed something makes it fall heads or tails, some physical cause. “Randomness†does not make the coin choose a side. Spin and momentum cause it to land on one side or the other. There is nothing “random†in a coin toss: there is only physics. If you knew the amount of force propelling the coin upwards, and the amount of spin imparted, you can predict with certainty the outcome of the flip. (Persi Diaconis and Ed Jaynes—both non-traditional statisticians—have written multiple papers on this subject.)”
Yes, there is plenty of randomness in a REAL coin toss (not the artificial model one uses to think about the steps of a coin toss). It’s true that physics explains the rate of spin, the fall, etc… But, it doesn’t matter because physics can’t be used to predict what was going on with the individual (e.g., what was he thinking about at the moment of the flip or the free throw, did he or she happen to have a heart attack at that moment, did a storm cause a power outage that affected the individuals concentration, etc.). Randomness occurs. One can explain the physical processes involved in the random action after the fact, but one cannot reliably predict what will happen or when it will happen.
Bruce
Actually, BDAABAT, everything you say is false. Individual coin tosses can be predicted perfectly if the initial conditions are known. Further, I can toss a quarter (only a quarter, so far) and make it show any side I like. Why? Because I have learned, as anybody can with hours of practice, to manipulate the initial conditions.
Weather/climate might be, probably is, chaotic. There is no proof of that, of course. There is proof that some of the dynamic equations we used to model weather and climate display chaos. This allows us to infer that the real weather and climate are likely chaotic.
Keep in mind that “random” means unknown. What might be random to you might not be random to me if I have different information that you. Simple as that.
Briggs: one word… poppycock!
The reason? You can’t control ALL of the conditions. If you can, then it’s a model of reality and not reality. It’s not a question of “knowing” vs. not knowing.
Implicit in your statement is that the outcome of the coin flip is determined SOLELY by skill. That’s a pretty silly argument. Think about it.
What if, as you are in the process of flipping the coin, you have a heart attack? Would you really be able to predict the outcome of that flip under those conditions??? What if a large chunk of metal that broke off of an airplane 35,000 feet up and suddenly crashed through your apartment striking your hand and the exact moment you were flipping? What if an earthquake happened to start at the moment of your flip???
The “what ifs” aren’t part of your initial conditions… they aren’t part of your model and cannot be accounted for. Yet, they can and do happen in real life. Randomness happens! Which is one reason why highly paid professionals can’t always sink seemingly easy free throws in big games.
Bruce
Strong word, Bruce old son. Sure you want to stick with it?
What you have stumbled upon, and succeeded in redefining, is the idea of contingency (I’ll let you look it up). Roughly, it means that if an event depends on the universe being in a certain way, than that event (possibly conditional on additional evidence) has a probability between 0 and 1, but never strictly 0 or 1.
Chaos is a mathematical property, which contains infinities of examples where we know (we certainty; it’s logical sense) everything.
I can’t have a heart attack. I had that organ removed back in the 70s. Some say this is why I am unable to appreciate the Beatles.