My introduction to Chaos theory started with Michael Crichton’s book “Jurassic Park” where he explained the Butterfly effect and how Non linear systems are hard to model. Understand that the book was fiction but it inspired me to read up a bit on Chaos theory.
Is the assertion “Long term weather cannot be predicted” proven mathematically or is it just taken as an empirical truth ? Proven mathematically to me and would mean something like “it’s proven that there are infinite prime numbers”.
Also : does Chaos theory apply to Autonomous systems and/or AI ? Are there mathematical limits on Autonokous systems or AIs ?
I’ve never heard anyone suggest that it can be mathematically proven that “long-term weather cannot be predicted”. What can be proven mathematically is that in chaotic systems, extremely small changes to inputs can have major effects on outputs. Meaning that in practice there is currently no prospect of us knowing all inputs to the extent necessary to definitively predict outputs, for long-term weather.
Not sure what you mean by that. Model predictive controls are fairly common systems in Chemical Plants and we have been doing them for a few decades now.
Isn’t “long term weather” synonymous with “climate”? We can and have certainly predicted that based on changing conditions on our planet. Obviously, the farther down the road that you predict, the more chance for error because circumstances affecting weather or climate have a greater chance to change over an extended period of time.
I took some physics classes during my undergraduate years and one such function was Dirac-delta function. IIRC it was a function such that it had a very narrow width and extremely long sides such that its area was one.
Effectively a small number on the x-axis and and extremely large number on the y-axis, and that was used to describe the mathematics of some systems.
So if Dirac delta function can solve some problems with small x/ values resulting in large y- values, why can’t there be math in the future that can deal with Chaotic systems ?
I maybe understanding this totall wrong and thank you for your patience
Grab a tree leaf and drop it from a height of 3 meters inside an environmentally-controlled lab. Where will it land?
Before the experiment, feel free to use any kind of analysis you want. Perform a laser scan or X-ray of the leaf to measure its shape. Perform material analysis on the leaf to understand its material properties. Feel free to install dozens of sensors to measure air speed, air temperature, etc. in the vicinity of the experiment. And then use the fastest supercomputer to predict where the leaf will land.
The leaf will probably land a couple feet from where it is predicted to land, if you’re lucky.
I think Climate is 30 year average. Long term weather is like 15 days in advance. But you are generally correct
Agree. I am asking if the uncertainty can be quantified and proven to be extremely large over some time. You know like if you look at professional publications - there are error bars (uncertainty bars) (3 standard deviations) on measurements and predictions.
A prediction or measurement is useless if the uncertainty is comparable to the value of the measurement.
“Chaos theory tells us that long-term weather prediction will never be possible” is a back-of-the-envelope version of Ed Lorenz’s work. Unpacking that a little bit from memory, Lorenz was working with a very simple model of weather with just a tiny handful of variables (maybe just three); I think he was loosely planning on gradually adding more complexity but was first looking to see if he could generate formulas that would predict the model’s behavior at any given time x.
He wanted to re-run the model from a certain elapsed-time, so he looked up the values for his three variables from the computer screen and input them as starting points and hit the “Go” button. But from that starting point the model diverged from what it had run from that point previously.
After a bit of investigation he found out that the computer display had only shown the values of his parameters to three decimal points’ worth of precision, but the computer’s own internal registers were actually storing them to six decimal point’s precision, and that tiny difference was enough to generate a completely different outcome. So that’s the “sensitive dependence on initial conditions” thingie right there.
The logical leap he made was that if in even this simple model a measuring difference that small meant the difference between an accurate and a worthless prediction, we’ve got a problem on our hands.
To use a non-Lorenz example, computers actually work with binary, but display the decimal equivalents to us. There’s often a minor discrepancy between a value displayed in decimal and a value stored in binary, or vice versa, and if dependence on exact initial conditions is sufficiently sensitive, that could be enough difference to get entirely different results.
Now let’s throw in Heisenberg. To get the starting values in the first place, we have to measure the current weather (apparently to a ridiculously sensitive degree of precision). But if it’s that sensitive, the act of measuring it could make it vary.
ETA: Please note that this was about measuring and predicting the equivalent of “on April 3 2023 the temp will be 61.5°, with a light fog in the Terrence Park suburb of Cincinnati turning to rain at 10:43 AM”. Not “we think the average weather in 2030 will be 1°F higher on average than it was in 1965”, the latter of which is aggregate averaged stuff and not precision forecasting.
Conceptually it’s like this, although the details of chaos math are not the same under the hood …
If you could put a small (say 1%) error bar on each of the one jillion parameters you’ve measured about the weather right now, the error bars on the resulting predicted values for the weather 1 minute from now will be bigger. Lather rinse repeat for a few hundred iterations to get a prediction for an hour from now and the accumulated ever-growing error bars overwhelm the plausible range of the output data. When the predicted temperature at your house is 25C plus/minus 30C, you’ve not gotten a useful answer.
Now in the real world you can’t measure 1 jillion parameters to high but knowable precision and exactly at the same moment. More like a few thousand or maybe a million. At somewhat differing times. With differing uncertainties and calibration errors. All that further muddies the predictive water, ensuring the error bar explosion happens even sooner.
It’s doubtful you could ever come up with a mathematical model that predicts unexpected atmospheric changes. As it is, long-range forecasts for 10 days and beyond are said to be accurate about 50% of the time, so flip a coin.
Math is useful in some applications, however.
Captain Queeg: “Ahh, but the strawberries! That’s - that’s where I had them. They laughed at me and made jokes, but I proved beyond the shadow of a doubt and with - geometric logic - that a duplicate key to the wardroom icebox DID exist!”
Your Y,X example is interesting. Realistically the X values would have to contain some level of random noise because that’s the way things are in the real world. Even in the precision of numbers. So the plot of Y in your example would not have been smooth. It would have been jagged as a result of the noise and how jagged depends on how much noise. A single small change of any X value would not have made a noticeable difference in the output because it would have appeared normal within the noise level. Just one of the squiggles.
The same is true of the consequences of removing an individual (butterfly) from an environment containing predators. Fun theater but no biggie.
But that’s almost tautological. What is unexpected was not predicted.
The point is, the reason that some changes are unexpected, is because the weather is a chaotic system, meaning that tiny differences in input variables (indeed, any difference) will ultimately lead to a different result set. You can only run the models so far before they start to diverge from reality.
With perfect input data you could hypothetically predict the weather arbitrarily far into the future, but it becomes a practical impossibility to even get a week into the future with high confidence, and a theoretical impossibility at some point beyond that (probably very far beyond that, but the point is, there are physical limits on what information we can measure).
“(Ernest) Zebrowski proposed that the answer (to why seemingly harmless atmospheric fluctuations lead to major hurricanes while other apparently identical ones do not) might lie in the science of ‘nonlinear dynamics’: chaos theory and the famous butterfly effect. He framed the question this way: 'Could a butterfly in a West African rain forest, by flitting to the left of a tree rather than to the right, possibly set into motion a chain of events that escalates into a hurricane striking coastal South Carolina a few weeks later?”
The Dirac delta function isn’t just very narrow, it’s defined as infinitely narrow and infinitely tall with an integral of one. It’s not really relevant to the challenge of chaotic systems. We could predict weather just fine with more detailed inputs, but the practical limits happen to be so that it will never be possible to give a reliable answer as to whether it will rain in a specific spot at a specific time in two months.
It looks to me like there are three things you would need in order to predict the weather arbitrarily far into the future.
One is “perfect input data”: knowledge of the values of all the relevant parameters to infinite precision.
The second is knowledge of all the relevant laws and formulas that govern the weather, so that you know what calculations to perform on that perfect input data.
And the third is the ability to perform those calculations in a reasonable amount of time. There are problems that can theoretically be solved given unlimited computing time, but for which there is no known algorithm that can solve them in any reasonable amount of time.
The problem is, this kind of phrasing leads to the popular misconception of chaos theory.
That that single butterfly is “special” in some sense, and literally caused the future event.
The actual meaning though, is that the difference of the butterfly going left or right is a difference in input data, and that difference will ultimately lead to a different result set. But you can say the same thing about any entity on earth – take entity X out of the picture, and ultimately you would have a different results set.
However, “ultimately” is likely to be a very, very long time for most entities. Not weeks. Though it’s hard for us to ever know.
It also depends on your definition of “predict the weather”. If you need precision like “how much snow will fall on Chicago”, then you get a couple of days at best. If you are looking at broader categories like “Will the spring be warmer than normal? Will the spring be wetter than normal?”, we’re already there. No, we can’t say that there will be rain in Chicago on May 23, but we can say “Spring in Chicago will be colder and wetter than normal.”
Even then, those long-range forecasts are worded more like, “Spring in Chicago is likely to be colder and wetter than normal.” And, often, the model is only showing that there’s a 20% or 40% chance that it’ll actually deviate from the normal.
