This question came to me as I was watching two of those drinking bird toys bobbing their heads in the same glass of water. I was wondering if it was possible to predict when they would both have their beaks in the glass at the same time.
Suppose that one bird bobs with a given period, like 1 bob every 10 minutes. The second bird bobs with a different period, like 1 bob every 13.77 minutes. If I know the last time that the each bird dipped, I can show that over a long enough time span in the future (long enough for both birds to dip at least once) there will be a closest convergence of the two birds dipping - that there will be a shortest interval between their dips. The interval may not be very short, and if the time span is long enough it may repeat itself, but there will be a shortest interval, and I can identify with certainty the time point(s) on which that interval(s) is (are) centered. Those time points are predictable with certainty if the periods are constant.
Now suppose that the periods for each bird, rather than being constant, vary slightly and randomly from dip to dip. The variability follows a normal distribution with a mean of zero and some standard deviation that is different for each bird. It is probable that the periods will hover around some mean over long time spans, but they could potentially drift far from their starting values. The variability is small compared to the starting period of each bird, though.
With variable periods like these, can I still predict what the shortest interval between dips will be over a particular time span? Can I still predict when that shortest interval will occur, perhaps with statistical methods? Does the fixed prediction that I get with constant periods become a distribution with a mean and standard deviation, or does the behavior become impossible to predict? Does it matter how far in the future I’m looking?
I’m hoping that even if nobody has an answer someone will be able to say ‘oh yeah that’s a type of xxyyzz problem’ and I’ll be able to read up on xxyyzz problems. As it stands, I’m not sure if this is a statistics problem, or a chaotic problem, or what.
Yes you can certainly do so with a couple of provisos.
The periods clearly can’t have a normal distribution since the normal has unbounded support so there would be positive probability that the period was negative, which is impossible. So you’re going to have to change that assumption or settle for an approximate answer. By this I mean even the distribution of the closest bobs will be approximate.
It’s not clear which of three interpretations you mean. One is that there is a period, say one minute, and that each bob is centered on the one minute marks starting from time zero + or - a mean zero error. Another is that each bob follows the latter by one minute = or - an error. A third is that the first bob comes at one minute to begin with, then the next is one minute + or - an error (say + 3 seconds) after that. then the next is 63 seconds later + or minus an error.
In the second case the inter-bob times are independent and identically distributed. In the third case the inter-bob times are following a random walk. In the second case you’re looking for the smallest of iid variables and you want to find out about order statistics. In the third case you want to find out about the minimum of a random walk. I don’t think the first case is what you have in mind, but what you want there isn’t obvious to me.
ETA: Actually let me think that through more. I forgot we’re dealing with two birds. I I assume you want those errors to be independent.
Oh no – – do NOT assume Poisson distribution, which can be applied to random uncorrelated events. OP has periodic events, but with a little noise in the periods. In fact, since OP specifies that period’s deviation is tiny with respect to mean, a Gaussian could be quite appropriate despite the flaw OldGuy correctly identifies. (For many purposes you needn’t even code it as a truncated Gaussian, though such discussion would be a digression here.)
I’m not sure exactly what OP’s question is. For noise-free periods, apparent behavior will depend on how close the ratio of periods is to a simple fraction. For the noisy period case, behavior will resemble statistically the case where ratio of periods is not a fraction of smallish of terms (though the latter is deterministic, the former not).
OldGuy - your third interpretation is the one that I was thinking of. The inter-bob times follow a random walk. I hadn’t thought of it in those terms, but that best fits my mental model of the situation. It’s the changes from one period to the next that follow a normal distribution, not the period itself. And yes, for the two birds the errors are not only independent, but the distributions of errors for each bird have different standard deviations. Let’s be arbitrary and say that the standard deviation for the errors for the first bird is about 1% of its initial period and for the second bird it’s 2% of the second bird’s initial period. The mean of the errors would be zero for both birds.
So, given that the two birds have this random-walk behavior in their dips, how can the *combination *of their behavior be described?
septimus- I’m afraid I don’t understand what “ratio of periods is not a fraction of smallish of terms” means. If you’re saying that the behavior is not deterministic, then I think I agree.
FWIW, I think that I have a formula that describes the distributions of future dips as a function of the original period § and the standard deviation of the variability (s). Counting from a dip at t=0, the nth dip follows a distribution with a mean of np and standard deviation of s[sqrt([(n)(n+1)(2n+1)]/6)]. So the farther out you go, the broader the distributions become until eventually the predictions for one dip largely overlap those for the next dip. So I can describe the behavior of each bird individually but I don’t know how to combine the two.
Even without the noise, this system doesn’t behave like you think. If the ratio between the two periods is irrational (and it almost certainly will be), then there will be no “shortest interval between dips”. Every so often, there will be a very short interval, but no matter how short it is, if you wait long enough, you’ll eventually get a shorter one.
My intent was very similar to Chronos’. For the real-world ratio to truly be rational is essentially impossible, as Chronos’ implies, but I wrote “apparent behavior will depend on how close the ratio of periods is to a simple fraction,” because if the ratio is, say 0.50000001 it will take a long time for the bird-to-bird bobbing to differ from its starting condition.
I should have been clearer that I am searching over a finite time span - like ‘over the next two weeks starting now.’ Over any finite time span that’s long enough such that both birds dip at least once, there will be a shortest interval over that time span. If the search space was broadened, there may well be shorter intervals elsewhere. If the periods are fixed and the search space is very large, the shortest interval may show up repeatedly.
For Chronos and septimus - I don’t see how the ratio of periods can be anything but rational. Each individual period is a terminating number of seconds (my stopwatch doesn’t measure below Planck time units) and is nonzero, so the ratio of the two of them must be rational, yes?
Consider the following awesome visual aids:
Fixed periods Given fixed periods, and a finite search space (the length of the card) there is a unique shortest interval between birds A and B.
Variable periods When variability is introduced into the periods, they go from deterministic points to distributions that get broader the farther one looks into the future (the fixed points of the most recent dips are off to the left of the origin). So where is the shortest interval over this finite search range? Can such a thing even be calculated? Or simulated?
Really? Why would that be? That would require a huge coincidence to be true.
And nobody’s stopwatch goes down to Planck times, but there’s no real reason to believe that all time intervals are multiples of the Planck time, either. It’s conceivable that that’s maybe the way the world works, but even if so, a greatest common divisor of the intervals being of the order of the Planck time is, practically speaking, equivalent to the ratio being irrational.
I consulted the platinum rod I keep in the basement that has a mark on it that divides it into two lengths, the ratio of which encodes all truth and falsehood in all possible multiverses, and it said you were right. There will always be a shorter interval somewhere down the line. Sadly, I dropped the rod as I was pulling it from the reader and it broke into five pieces. I find the pieces hard to describe - I got woozy just looking at them - but fortunately I was able to reassemble them into two copies of the original. Would you like one?
I don’t think that invalidates the claim that there will be a shortest interval over a finite search period, though.
May I have one? That would be one of the coolest things on my trophy wall! Right up there with a blivet and a klein bottle!
A friend of mine is a fierce “constructive” mathematician. He insists, for instance, that there is an end to the useful series of approximations to pi. Once you’ve gone beyond the number of decimal places that could possibly refer to any measurement that could be taken in the known cosmos…further decimal places are meaningless.
This isn’t the most foolish interpretation ever. It has a crusty kind of pragmatic validity. 10^900 isn’t really a “number” because there are no real things in the cosmos that can be “counted” to that high a cardinality.
It’d be a little like dividing the United States into further refined Zip Codes: instead of Zip+4, Zip+900. You’d be addressing areas smaller than individual grains of sand on the beach.
The problem with strict constructivism, though, is that you can’t really draw the line between “possible” and “impossible” numbers. And any time you try, you’re likely to find that you’ve drawn it too low.
For instance, there are no real things in the cosmos that can be counted up to 10^900, but there are computer files that are more than 3000 bits, and a lot of practical things that are done with computer files are done by treating them as really big numbers.
– do NOT assume Poisson distribution, which can be applied to random uncorrelated events. OP has periodic events, but with a little noise in the periods. In fact, since OP specifies that period’s deviation is tiny with respect to mean, a Gaussian could be quite appropriate despite the flaw OldGuy correctly identifies. (For many purposes you needn’t even code it as a truncated Gaussian, though such discussion would be a digression here.)