- Thread starter
- #41
You didn't prove anything.
You're making a bogus claim.
One that you can disprove for yourself, if only you'd be a careful observer.
There is no error.
Go look at the Lorenz attractor CAREFULLY and tell me what you see
consciousness precedes real time
- Thread starter scruffy
- Start date
You didn't prove anything.
You're making a bogus claim.
One that you can disprove for yourself, if only you'd be a careful observer.
There is no error.
Go look at the Lorenz attractor CAREFULLY and tell me what you see
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I am not making the claim, mathematicians are, should I reject them and listen to you instead?
Don't you see how idiotic you look saying I made a bogus claim? I said chaos is deterministic and anyone who understand this knows that.
You are either out of your depth or very bad at explaining what it is you believe.
I see that it's deterministic:
So that clears it up I think, did you say you actually get paid for doing what you do?
Tell me how can deterministic equations generate random results? perhaps you have a bug in your modelling code, ever heard of GIGO?
Perhaps you're getting confused between chaos and pseudo-randomness, who knows, you're all over the place.
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- Thread starter
- #43
Do what I tell you.
Don't believe CNN, that's just stupid.
Go look at it with YOUR OWN EYES and tell me what you see.
Don't be a fallacious authoritarian.
Go look at the Lorenz attractor, then get back to me.
- Thread starter
- #44
Here, I will give you a quick course on chaos and randomness
First of all, let's start here. Here is the Lorenz attractor
mathematica.stackexchange.com
Now, look at the pretty picture, and tell me what you see with your own eyes.
You see the attractor changing shape, amirite?
To understand what this means, we will have to take a little trip.
Let us first understand that the Lorenz system is nonlinear snd the equations are static. Nevertheless it is a fractal (fractional) attractor with a Hausdorff dimension of just over 2.
It turns out we can REPLACE this system (map it) with stochastic differential equations with the same dimensionality. Why do this? Because it gives us MEASURES.
What this means in practice is we are replacing the random motion of particles entering the vortex, with random motions of the vortex itself.
Start here, to understand why we want to do this:
en.wikipedia.org
Scroll down to where it says Motivation 2, and the look just below where it says "Example". Read carefully.
We are substituting deterministic transition matrices with Markov matrices, and we can do this because there is a topological mapping from one to the other. Now keep reading where it says "formal definition". Note the verbiage about MEASURE PRESERVING TRANSFORMATIONS. That's why we can do this, because we're preserving the measure between the stochastic metric and the deterministic nonlinear metric.
From a topological standpoint, the two are one and the same. The formal treatment is given by the subsets of omega-limit sets in the phase space. The Lorenz attractor, it turns out, is not one trajectory, instead it is an infinite complex of surfaces with zero volume and infinite surface area.
You will notice in the simulation, that several times the attractor does a "reset", it's shape becomes discontinuous over a brief interval of time. This is exactly the same thing that happens in the financial markets, where stochastic differential equations are also used to map nonlinear systems. This:
looks exactly like the motion of a particle in a Lorenz vortex it is stochastic and also nonlinear, some would say it is random "and" chaotic but the truth is the two can not be divorced.
The key issue is SCALE. As you know from studying fractals, measured lengths change with the yardstick, and therefore areas and volumes change and so do dimensions. It is perfectly accurate to say a weather system is random, at the level of particles that enter the vortex - even though the global dynamic retains its shape - just as it is accurate to say the roll of dice is random, even though if you're being a stickler you could argue that it's a deterministic system that merely magnifies the initial conditions. The OUTCOME is random, just as the position of a particle in a Lorenz vortex is random. It can NOT be predicted in advance (I defy you to try lol).
It is most USEFUL to treat this as a stochastic system, this way you can write differential equations that will take you to the shape a lot faster than plotting points.
Since I don't have a math font I can't show you the topology. But you can find it if you look around a bit, and I can verbalize it. Attractors are the omega limits of nonempty inward sets Z in X such that dF^n(X,Z) < epsilon for every n. The "d" is why the differential form is useful, it stands for distance (not derivative), which is the MEASURE we started with. With stochastic differential equations you have a complete set of measures within the given dimensionality, limited only by the Hausdorff properties (like, the area of a line is zero).
Measure theory SOUNDS simple but it's not.
en.wikipedia.org
We call the outcome of a roll of dice "random" because that's how we perceive it, and that's how we measure it. Whether it is or is not truly random is completely beside the point. The only USEFUL way to treat it is as a random variable.
First of all, let's start here. Here is the Lorenz attractor
Animating the Lorenz Equations
I am trying to use the Animate command to vary a parameter of the Lorenz Equations in 3-D phase space and I'm not having much luck. The equations are: $\begin{align*} \dot{x} &= \sigma(y-x)...
mathematica.stackexchange.com
Now, look at the pretty picture, and tell me what you see with your own eyes.
You see the attractor changing shape, amirite?
To understand what this means, we will have to take a little trip.
Let us first understand that the Lorenz system is nonlinear snd the equations are static. Nevertheless it is a fractal (fractional) attractor with a Hausdorff dimension of just over 2.
It turns out we can REPLACE this system (map it) with stochastic differential equations with the same dimensionality. Why do this? Because it gives us MEASURES.
What this means in practice is we are replacing the random motion of particles entering the vortex, with random motions of the vortex itself.
Start here, to understand why we want to do this:
Random dynamical system - Wikipedia
Scroll down to where it says Motivation 2, and the look just below where it says "Example". Read carefully.
We are substituting deterministic transition matrices with Markov matrices, and we can do this because there is a topological mapping from one to the other. Now keep reading where it says "formal definition". Note the verbiage about MEASURE PRESERVING TRANSFORMATIONS. That's why we can do this, because we're preserving the measure between the stochastic metric and the deterministic nonlinear metric.
From a topological standpoint, the two are one and the same. The formal treatment is given by the subsets of omega-limit sets in the phase space. The Lorenz attractor, it turns out, is not one trajectory, instead it is an infinite complex of surfaces with zero volume and infinite surface area.
You will notice in the simulation, that several times the attractor does a "reset", it's shape becomes discontinuous over a brief interval of time. This is exactly the same thing that happens in the financial markets, where stochastic differential equations are also used to map nonlinear systems. This:
looks exactly like the motion of a particle in a Lorenz vortex it is stochastic and also nonlinear, some would say it is random "and" chaotic but the truth is the two can not be divorced.
The key issue is SCALE. As you know from studying fractals, measured lengths change with the yardstick, and therefore areas and volumes change and so do dimensions. It is perfectly accurate to say a weather system is random, at the level of particles that enter the vortex - even though the global dynamic retains its shape - just as it is accurate to say the roll of dice is random, even though if you're being a stickler you could argue that it's a deterministic system that merely magnifies the initial conditions. The OUTCOME is random, just as the position of a particle in a Lorenz vortex is random. It can NOT be predicted in advance (I defy you to try lol).
It is most USEFUL to treat this as a stochastic system, this way you can write differential equations that will take you to the shape a lot faster than plotting points.
Since I don't have a math font I can't show you the topology. But you can find it if you look around a bit, and I can verbalize it. Attractors are the omega limits of nonempty inward sets Z in X such that dF^n(X,Z) < epsilon for every n. The "d" is why the differential form is useful, it stands for distance (not derivative), which is the MEASURE we started with. With stochastic differential equations you have a complete set of measures within the given dimensionality, limited only by the Hausdorff properties (like, the area of a line is zero).
Measure theory SOUNDS simple but it's not.
Hausdorff measure - Wikipedia
We call the outcome of a roll of dice "random" because that's how we perceive it, and that's how we measure it. Whether it is or is not truly random is completely beside the point. The only USEFUL way to treat it is as a random variable.
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No.
Depends what they are saying.
Where?
Like a person who says "Do what I tell you"?
No.
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Right, that incorporates randomness, "noise" so it's not deterministic:Here, I will give you a quick course on chaos and randomness
First of all, let's start here. Here is the Lorenz attractor
Animating the Lorenz Equations
I am trying to use the Animate command to vary a parameter of the Lorenz Equations in 3-D phase space and I'm not having much luck. The equations are: $\begin{align*} \dot{x} &= \sigma(y-x)...
Now, look at the pretty picture, and tell me what you see with your own eyes.
You see the attractor changing shape, amirite?
To understand what this means, we will have to take a little trip.
Let us first understand that the Lorenz system is nonlinear snd the equations are static. Nevertheless it is a fractal (fractional) attractor with a Hausdorff dimension of just over 2.
It turns out we can REPLACE this system (map it) with stochastic differential equations with the same dimensionality. Why do this? Because it gives us MEASURES.
What this means in practice is we are replacing the random motion of particles entering the vortex, with random motions of the vortex itself.
Start here, to understand why we want to do this:
Random dynamical system - Wikipedia
Scroll down to where it says Motivation 2, and the look just below where it says "Example". Read carefully.
We are substituting deterministic transition matrices with Markov matrices, and we can do this because there is a topological mapping from one to the other. Now keep reading where it says "formal definition". Note the verbiage about MEASURE PRESERVING TRANSFORMATIONS. That's why we can do this, because we're preserving the measure between the stochastic metric and the deterministic nonlinear metric.
From a topological standpoint, the two are one and the same. The formal treatment is given by the subsets of omega-limit sets in the phase space. The Lorenz attractor, it turns out, is not one trajectory, instead it is an infinite complex of surfaces with zero volume and infinite surface area.
You will notice in the simulation, that several times the attractor does a "reset", it's shape becomes discontinuous over a brief interval of time. This is exactly the same thing that happens in the financial markets, where stochastic differential equations are also used to map nonlinear systems. This:
View attachment 1001143
looks exactly like the motion of a particle in a Lorenz vortex it is stochastic and also nonlinear, some would say it is random "and" chaotic but the truth is the two can not be divorced.
The key issue is SCALE. As you know from studying fractals, measured lengths change with the yardstick, and therefore areas and volumes change and so do dimensions. It is perfectly accurate to say a weather system is random, at the level of particles that enter the vortex - even though the global dynamic retains its shape - just as it is accurate to say the roll of dice is random, even though if you're being a stickler you could argue that it's a deterministic system that merely magnifies the initial conditions. The OUTCOME is random, just as the position of a particle in a Lorenz vortex is random. It can NOT be predicted in advance (I defy you to try lol).
It is most USEFUL to treat this as a stochastic system, this way you can write differential equations that will take you to the shape a lot faster than plotting points.
Since I don't have a math font I can't show you the topology. But you can find it if you look around a bit, and I can verbalize it. Attractors are the omega limits of nonempty inward sets Z in X such that dF^n(X,Z) < epsilon for every n. The "d" is why the differential form is useful, it stands for distance (not derivative), which is the MEASURE we started with. With stochastic differential equations you have a complete set of measures within the given dimensionality, limited only by the Hausdorff properties (like, the area of a line is zero).
Measure theory SOUNDS simple but it's not.
Hausdorff measure - Wikipedia
We call the outcome of a roll of dice "random" because that's how we perceive it, and that's how we measure it. Whether it is or is not truly random is completely beside the point. The only USEFUL way to treat it is as a random variable.
That's quite distinct from a chaotic system.
- Thread starter
- #47
Sigh.
Do you know what 1/f noise is?
It is created precisely by critical points in CHAOTIC systems.
You are woefully ignorant. You should go back to school, and learn about scale, and scope.
No, it isn't. It all depends on the scale. Very few systems are self similar in infinite dimensions.
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Chaotic is not the same as random, are you truly not able to bring yourself to admit this? Anyone here can check what I'm saying, you lose credibility every time you post about this now.
Here's a mathematician speaking all about it, she says at 0:46:
so everywhere I look I find this is the view, I can find nobody anywhere who says "Chaos is nothing to do with determinism" as you said.
Just click, I've even positioned the video at the point where she says this:
Once this has sunk in to your stubborn mind, click this next, where she explains the difference, have a pen and notepad handy:
She says "Chaos theory isn't random by the way, random and chaos are two very different things"
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- Thread starter
- #49
You're being an idiot.
Randomness is a DESCRIPTION, not a thing.
If you can't predict the outcome then a random variable is the best description.
Period.
We're done.
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How can making a true statement be the work of an idiot?
Whatever meaning you attribute to it, it's still true - chaotic is not the same as random.
You can interpret it that way if you want, but chaos is not randomness.
Wuwei
Gold Member
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The words random and chaotic are mostly used as adjectives involving a sequence of events. A random sequence means no prior events will influence or predict succeeding events. Chaos theory studies the underlying cause of seemingly random systems.
Once an underlying math is found, a system cannot be called random except in a colloquial sense. Functions with attractors can be accurately computed or predicted for a relatively small change from any chosen independent variables. So, by definition, a chaotic function cannot be called random.
- Thread starter
- #52
That'only in Markov land, with independent increments.
It doesn't cover the case of generators with memory (many exist, brains being examples).
No, for that you need to study the generators.
Chaos theory deals mainly with the attractors, which means outcomes.
False.
A stochastic generator has properties attached to it.
One of the important properties of "noise" is that it is self-similar at all scales. Many chaotic systems qualify with this description. That's one of the interesting properties of fractal attractors.
Sorry but the classical view is out of date. Suggest you study dimensionality, and measure theory. Hausdorff, Borel, and the modern topologists. There are mappings between noise and stochastic manifolds.
Consider a stochastic generator with non-zero Volterra kernels. Some of these will generate "noise", that qualifies in every physical sense.
Wuwei
Gold Member
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It seems that algorithmic generators with memory wouldn't produce a fully random sequence.
Why no? It seems you are generally agreeing, but are just being more specific.
I think we are talking about two different things. Yes, I have a classical view. And yes noise is chaotic in a colloquial sense, but true noise can't be algorithmically generated.
.
- Thread starter
- #54
What is "fully" random?
That's old vocabulary.
Random sequences have moments, means and variances and so on. They have distributions. There is all kinds of noise.
What is "fully" random?
Jeez - well yeah, specificity is important.
We were talking about pendulums.
You can use a nonlinear (chaotic) pendulum system to generate RANDOM numbers.
The "randomness" of those numbers will be BETTER THAN anything a computer can come up with.
Read NIST Special Publication 800-20.
Better than that.
"True" noise.... well, on that part we may agree. Maybe. It is difficult to impossible for a computer to generate "true" random numbers.
But to be specific, computer scientists use terms like "quasi-random". Meaning, "good enough" for most purposes
"Most purposes" is defined by SCALE. It is exceedingly rare to require randomness over 20 orders of magnitude. If you need that, you have to use a "true" random number generator, and there's only one place in the universe you can find that. (That I know of).
Most everything we deal with is quasi-random, which means it's only random within a window of scale. If you're doing computer generated fractals you're probably aware of this, I'm not telling you anything you don't already know.
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- Thread starter
- #55
I take it you're familiar with a Wiener process? Like, in Brownian motion?
The Wiener process is an abstraction, if you measure Brownian outcomes in real life you'll find they cease being entirely random, right around the size of water molecules. At that scale, the generator becomes more deterministic.
- Thread starter
- #56
Oh - the new NIST validation criteria are here:
csrc.nist.gov
You have to "test" any algorithm, which means measure. Yes?
Cryptographic Algorithm Validation Program | CSRC
The NIST Cryptographic Algorithm Validation Program (CAVP) provides validation testing of Approved (i.e., FIPS-approved and NIST-recommended) cryptographic algorithms and their individual components. Cryptographic algorithm validation is a prerequisite...
csrc.nist.gov
You have to "test" any algorithm, which means measure. Yes?
- Thread starter
- #57
AFAIK, Kolmogorov is still the king of random numbers. His defining theorems include existence and continuity. USUALLY, when a forward or backward process breaks down it's for one of those reasons. The theorems relate to limits of point sets. At some lower bound (Heisenberg) you can no longer define a point, so it becomes useless to measure. At the other end there is the Law of Large Numbers, where everything becomes Gaussian. In between, is the useful range, where the math works and you can use a Feller process and the Levy criteria and all that.
- Thread starter
- #58
Here, I came across an interesting take -
"Truly random" is when no one relevant can predict it.
",Perfectly random" is when no one in any context can predict it.
Still want to know what "fully random" means.
As near as I can figure it you're alluding to a flat distribution.
A random variable is a MEASURE, not an outcome. To determine whether a generator is random we MEASURE outcomes.
Once we have measured, we can calculate an expectation. The expectations have relationships with the outcomes.
Those relationships may be highly nonlinear. One of the ways we can tell is if the evolution depends on the current distribution.
Classical generators have independent and stationary increments, neither of which are necessary for "true randomness". A nonlinear evolution can still be "truly random" in the sense that it is entirely unpredictable.
In fact there's a well known set of generators where the distribution itself changes by random FUNCTIONS (which change the shape of the entire distribution all at once). The outcomes of such processes are definitely truly random.
"Truly random" is when no one relevant can predict it.
",Perfectly random" is when no one in any context can predict it.
Still want to know what "fully random" means.
As near as I can figure it you're alluding to a flat distribution.
A random variable is a MEASURE, not an outcome. To determine whether a generator is random we MEASURE outcomes.
Once we have measured, we can calculate an expectation. The expectations have relationships with the outcomes.
Those relationships may be highly nonlinear. One of the ways we can tell is if the evolution depends on the current distribution.
Classical generators have independent and stationary increments, neither of which are necessary for "true randomness". A nonlinear evolution can still be "truly random" in the sense that it is entirely unpredictable.
In fact there's a well known set of generators where the distribution itself changes by random FUNCTIONS (which change the shape of the entire distribution all at once). The outcomes of such processes are definitely truly random.
- Thread starter
- #59
Not many people realize, that Brownian motion isn't caused by collisions.
It's caused by "transimpact", which approximately means an orbital jumping ship.
Nevertheless the molecules still have to be "close enough" for this to happen.
Close "enough" isn't well defined, it's kind of random. Statistically though, there's a big peak around atomic/molecular size.
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Here, I came across an interesting take -
"Truly random" is when no one relevant can predict it.
",Perfectly random" is when no one in any context can predict it.
Still want to know what "fully random" means.
As near as I can figure it you're alluding to a flat distribution.
A random variable is a MEASURE, not an outcome. To determine whether a generator is random we MEASURE outcomes.
Once we have measured, we can calculate an expectation. The expectations have relationships with the outcomes.
Those relationships may be highly nonlinear. One of the ways we can tell is if the evolution depends on the current distribution.
Classical generators have independent and stationary increments, neither of which are necessary for "true randomness". A nonlinear evolution can still be "truly random" in the sense that it is entirely unpredictable.
In fact there's a well known set of generators where the distribution itself changes by random FUNCTIONS (which change the shape of the entire distribution all at once). The outcomes of such processes are definitely truly random.
Wuwei already explained "random" to you, it is when a preceding event has no influence over a subsequent event. As he also explained, chaotic systems are often predictable to an extent, in the short term, that is a preceding event does influence a subsequent event and therefore such a system is not to be described as random. I'm paraphrasing him here but I think that's the gist of what he explained to you.
If you dispute this then say why and cite meaningful sources.
