I do not dispute a single point in the parent post, but I see that we differ on an assumption:
When you sneer at normal intelligent wordplay as ignorant, you're betraying your own ignorance.
From this I infer that you took the original poster's use of "gifted" as being an example of "normal intelligent wordplay"--whereas I took it as sophomoric mimicry of a joke from a television show, and reacted to it accordingly. As for the question of whether or not my reaction is typical or not, may I suggest that the whole reason "gifted" was used in the show in the first place was that (in a true example of intelligent word play) the writers of the show realized that it would sound stupid to most people. The very thing that makes it useful as a joke makes it risky to use when you aren't trying to sound silly.
You've never heard the word "regifted"? Shame on you, non-Seinfeld watcher!
To put a finer point on it, non-television watcher. But, if I may hazzard a guess, I would say that the word "regifted" was used on the show you mention specifically because the writers knew it would sound moronic to most of their veiwers.
So first my point was invalid because it (according to you) came from a textbook, and when I point out that it came from the heart it's "a personal prejudice" (and thus inconsequential). I'm afraid to ask where you'd like it to come from...
But to stick to the point, this isn't just my predjudice; 95% of the people asked by the Harper's Dictionary of Contemporary Usage (to pick one source sitting on my desk) about the use of "gifted" in this sense found it unacceptable. Their comments were frequently hash--"vulgar" "anti-charisma" "loathsome" and "highly objectionable" etc.--and so, should you choose to use it, you ought to expect that the majority of your readers will not be favourably impressed.
You can of course disregard all of our "prejudice" and (like Humpty Dumpty) decide that words mean whatever you want them to, and the rest of the world be damned. But you ought to at leat realize in doing so that most of the people who read what you have written will (perhaps unconciously) think less of you for it.
You keep saying you find "gifted" unclear, but you never justify the statement
Look, you can't keep saying that "language comes from the people, not from textbooks " and simultaniously demanding that I justify my views by reference to some external, objective source. They are my views for the simple reason that I hold them. Nothing more is needed.
I find "gifted" unclear. Whenever I encounter it, my first reaction is to reread the sentence, my second is to assume that it is a typo, and then I finally, reluctantly come to the conclusion that the person who used it isn't particularly well educated.
Why? Who knows. If you're willing to foot the bill I might consider letting you get a cat scan of my brain, but I doubt it would help much. While you're at it you might want to look into why I have the same reaction to "bling! bling!." "youse," and the use of numerals instead of letters.
A more constructive path, given that there are many people who feel roughly as I do, would be to simply try your best to write in standard English if you expect people to take you seriously.
Or go around saying you've been gifted with bling! bling! 'cause youse is ubber l33t, for all I care.
Where is it written that gift is a noun, and only a noun? And how is it ever unclear to use a noun as a verb?
Language comes from people, not textbooks.
Let me get this straight--you are telling me (a person, not a book) that I am wrong to object to a word that I find unclear (and, in fact, ugly) and demanding that I justify my objection by citing a textbook ("where is it written") and in the very same post you are telling me that "Language comes from people, not textbooks."
Well, I'm a native English speaker and I'm telling you flat out that "gifted" sounds moronic to my ears. So which is it? Do you respect the opinion of the people, or do you want to defend the useage on the basis of textbooks?
Conversely the fact that something absurd and contrary to the will of the people, deleterious to the common wellfare, and arguably unconstitutional may have been accepted in oral arguments by a court that explicitly denied having decided the issue doesn't seem to me to hold enough water to float a complete restucturing of our society--which is exactly what is being done on the "priciple" of corporate personhood, albeit with glacial slowness.
Communication, preferrably with grace, economy, and clarity, is a much higher priority
My point exactly. Trying to construct the past tense of a noun (gift, after all, is a noun) is neither graceful nor economical, and it is far from clear.
--MarkusQ
P.S. For another example of the same principle, I think what you intended by the phrase "the guy means obvious..." was "the guy obviously means..."; note that while I am able to infer your meaning it would have been clearer and more graceful to user the more standard construction. I will conceed that it would be two letters longer and thus arguably less economical.
When defining a term, resorting to the idealized form is a reasonable thing to do. No actual physical "circle" has the properties we ascribe to mathematical circles, but that is a failing of the instances, not of the definition.
Likewise, just because people call themselves conservatives (or musicians, or gods, or anything else for that matter) I am not obligated to adjust my understanding of the term to include them. Instead I say (for example) that G. W. Bush is no conservative, and he can make all the claims to the contrary he wishes; I will not change my definition to suit his political goals, and I will judge him by his actions, dispite his rhetoric or that of his followers.
I can only answer for myself, but I hope my answer will serve nonetheless.
The core of true conservatism is to prevent any force from running amok; thus the power of government should be restricted by comparison to that of the individual, because there is more danger of governments abusing power (they have more, they live longer, etc.) This shows up in the chant of needing smaller government.
But the principle neither begins or ends there. Traditionally (AFAIK) private concentration of wealth went hand in hand with "conservatism" if for no other reason than the individuals who held it had no interest in seeing the boat rocked by others. This works so long as the wealth was acquired in a "conservative" fashon. Where it fails (e.g. slavery) the conservatives generally favour "rolling back" the disruptive concentration of wealth/power (remember, Lincoln was a Republican, and for the next hundred years the south was heavily Democratic). To be conservative is not mearly to support the status quo, but rather to support the mantanence of what is right and proper (to quote a famous fictional conservative, "but madest of all is to see life as it is, and not as it should be") and not support mere change-for-the-sake-of-change.
Gaming the system is a problem that cuts across the liberal/conservative spectrum and (like relative poverty) I fear it will always be with us.
Physicists are people too.
They (I'd be tempted to say "we" but I changed majors mid-stream) don't get as hung up on vi vs. emacs or BSD vs. Linux vs. Gnu/Linux, or open vs. closed vs. I'll-let-you-peek-but-then-I'll-have-to-kill-you source operating systems. And (the ones I know at least) have never watched (let alone voted for) any one on any American-Idle type television program.
But the fundemental personality trait is still there. It comes out in things like "is space quantized at the planck scale" or "are strings real/fundemental or is string theory an artifact of something deeper". Add beer and watch the fur fly.
Neocons are children who don't want their toys taken away, and won't clean their room because it isn't fun.
Thank you for recognizing that the distinction between the present domanant "neo-conservative" group think party in the US and true conservatives. True conservatives wouldn't have bought the toys in the first place (we're compulsive savers) and wouldn't have let the room get messy for fear of unspecified adverse consequences. True conservatives (I know, I am one) are more likely to avoid doing things because they are fun (on the principle that the more atractive the lure, the more likely it is to be bait).
Actually, corporations are legal persons and as such have some of the same independent rights that people ('natural persons') have.
No, in fact they are not. They have been treated as "persons" for many years based on a mistaken reading of a 19th century court rulling that did not in fact decide the issue.
Of course, they're not going to tell you that, are they?
I have nothing at this point (I'm working outside the US at this moment, I don't even have my notes) but it shouldn't be to hard to dig up again. Pre-google (Alta-vista + brick & mortar library) search took about a day.
Here's what I recall off the top of my head:
Pretty much every projection to/from a sphere is known art to cartographers, and has been for many decades. Look in a few old cartography books.
Likewise, the math behind them (called projective geometry) is old hat. We found a projective geometry book from 1900 or so that spelled out the transform
Artists in the 1800s or so used to do paintings (called anamorphoses, IIRC) that not only used the same tricks but for exactly the same purpose. It may go back much further, but (again, IIRC) the really compelling photos we found were from work in the mid 1800s.
Anyone who wants to is welcome to run with this, expand on it, and pass it on to anyone that it might help.
--MarkusQ
P.S. One further thing I recall, the laywer asked them something like "could you please specify what your patent covers--it obviously can't be the mathematics, and it can't be the technique san math, so...?"
A few years back I researched this very point for a client (I'm a programmer/math guy, not a laywer). We came up with prior art dating back to the 1800s (the very same technique was used in painting for "perspective lanterns" or some such). We turned over what we'd come up with to the lawyers, they wrote a letter, and we never heard from them again.
Of course I believe in observational science. What I don't believe in is magical algorithms that can "objectively" choose from an infinite set of possible functions the single correct interpretation of a finite set of noisy data.
I would have no problem with their results if they had said something like "assuming that there is an exponential increase in the Earth's temperature, here are the coefficients that best fit the attached data." But that's not the claim. The claim is that without any investigator bias they "objectively found" a "hocky stick" in the data--an exponential trend that can be used to extrapolate into the future. And that's just flat out not possible to do, with any algorithm.
I'm a firm believer in things like noticing that when hydogen and oxygen react they do so in a constant ratio, and using that to support the atomic theory of matter. Ditto using the luminocity of variable stars to cross-calibrate other ways of measuring distance, or double blind studies to test drugs, or pretty much any other instance of observational science you care to name.
But so far as I can see, this isn't observational science, dispite the way it's dressed up. It's voodoo. I distinguish them on this simple basis: are the researchers claiming to have extracted something from the data that could in principle be there, or are they claiming to have extracted information that in fact can not even in principle be extracted from the data.
To turn the point around, do you believe that the US stock markets are at record high valuations? How do you distinguish this "hockey stick" from the ones found by so many bull-analysts in the late 1990's, on the basis of which they "proved" that the Dow would be going up forever?
I just dispute its validity on principle. Why? Because it claims (or rather, its proponents claim) that the functions are chosen "objectively, by the data"
It's not a claim, it's a fact. The orthogonal functions are generated automatically by the algorithm.
Note that (according to the link you provided):
What is automatically produced is not arbitrary orthogonal functions but rather liniar combinations of the provided variables to which functions selected by some other mechanism could be applied; this seems quite reasonable, but it isn't what you (or other proponents claim).
Even so, it is a preselection (it presupposes a form for the data, e.g., something that can be well represented by combining arbitrary functions of individual orthogonal liniar combinations of the input variables). It would not work, say, to model data with cusps or in which something like hysteresis was significant.
It could not be the whole story for the original climate data analysis, since by itself it could not produce a "hocky stick" function. That had to come from somewhere, but it wasn't from PCA/PFA.
If, as you claim, I don't know what you're talking about, I would suggest that you don't either, since the link you provided does not support your position (that the hocky-stick function was objectively detected by a meaninful, unbiased automatic process). Nor does it undermine mine (that such a process is in priciple impossible).
with the implication that they are therefore "correct"
And that's a straw man argument.
If you are willing to posit that the results are incorrect I think we can come to an agrement rather quickly.
I know what it is, I just dispute its validity on principle. Why? Because it claims (or rather, its proponents claim) that the functions are chosen "objectively, by the data" (with the implication that they are therefore "correct").
This is clearly impossible on several levels; for one thing, even sticking to common everyday functions I can easily produce an f(t) and a g(t) such that:
Inside the interval t_min..t_max they are indistinguishable up to a noise term, and
for t > t_max they rapidly diverge or exhibit any sort of differnece you care to name--one may, for example, fluctuate between two modest limits while the other grows without bound, or converges to a constant, etc.
If I generate two identical (finite & noisy) samples from these functions and give you one of them, chosen at random, how in the heck can you tell me with a streight face that any algorithm can "objectively" look at the data and tell me which one (f or g) generated the sample?
For another thing, the whole notion of "objectively" choosing the functions is meaningless; if you are only choosing from among some limited subsets of all possible functions, the objectivity goes out the window by the very fact that you have pre-determined which functions are candidates and which are not. Conversely, if you truly make an objective choice between all possible functions, the clear winner will be some function like:
f(t) = sample(t) for all t in sample,
f(t) = 0 for all t not in sample
I think we are not understanding each other. At the very least, I am not understanding you.
The isomorphism I mentioned was from "there are an even/odd number of partitions not counting the trivial partition" to/from "there are an odd/even number of partitions counting the trivial partition. Since there is exactly one trivial partition (e.g., 1432 can be represented as the sum of one integer in exactly one way, namely, the integer 1432) including/excluding it just flips the parity. It's not in any way a deep or meaningful difference.
If you are representing numbers in a base X, and taking them modulo some prime P which is a factor of X, you only need to look at the last digit. In the case of base 16, P must be 2, whereas in base ten it can be 2 or 5. Saying that a decimal number "ends in 4 or 9" is the same thing as saying that it is congruent to -1 modulo 5.
The original poster asked if there was a similar pattern for hexidecimal numbers. Since 2 is the only prime factor of 16 (and because the discovery was that there was a pattern for the partitions of any number congruent to -1 modulo some prime) we can be sure that there is some pattern to the number of partitions of odd numbers.
No, they were doing principal components a.k.a. empirical orthogonal functions. In that method there is no assumed shape to the data. Look it up.
Given that there are infinitely many ways to decompose a finite sample into orthagonal functions (and in general they do not have the same extrapolation), simply picking one assumes a shape for the data.
It's a pretty interesting exercise to code this with synthetic data, actually.
And when you did this yourself, you found...what?
Of course as you say there's inevitably a nonzero projection of noisy data onto any arbitrary function. However, since this work passed peer review in Nature you might consider that nothing so stupid is likely to be happening here.
So, it's inevitable, but since the work passed peer review it didn't happen? I'm a big fan of peer review, but I have even more faith in logic.
The fact that sinusoids and exponential growth curves are both exponential functions on the complex plane may take you by surprise but I've been aware of it for decades.
I've known it for decades as well. I'm not sure why you think it took me by surprise, especially since (in your first reply) you didn't seem to know what I was refering to ('I am not sure what you mean by "imaginary periods"').
I fail to see the relevance of this fact even to what you seem to think is happening.
I'm sorry that you still don't see the relevance; I think I've already explained it rather clearly. Perhaps if you backed off from the ad hominem attitude it might be easier?
I was wrong, but not quite as wrong as you make it sound.
First off, ignoring the trivial partition just changes the parity, which would not affect any even/odd pattern beyond reverting the sense ("always even" <--> "always odd"). Using a trivial isomorphism is hardly a "tip-off that something is amiss".
Secondly, two is a prime and the proof in question AFAIK applies to all primes not just odd primes. The whole point of the discovery is that you can "reduce the bases that way" and you are guarenteed to "get something meaningful."
So there is some pattern to the partitions of odd vs. even numbers, modulo some base, I was just incorrect in guessing what it is.
You are correct, I incorrectly generalized the result. From the result, we know that there should be some pattern (two is a prime, and the proof is that such a pattern optains for all primes), but my off-the-cuff guess as to what it was is clearly wrong.
In math-speak, that would be "all x congruent to -1 modulo 5"; for hex the corresponding pattern of interest, (since 2 is prime) would be "ending in 1,3,5,7,9,B, or F", which in math-speak would be "all x congruent to -1 modulo 2"--in other words, what most of us would simply call "odd numbers."
If I'm thinking streight, the corresponding statement for numbers expressed in hex, octal, or binary would be "all odd numbers can be represented as the sum of smaller numbers an even number of ways."
Fourier uses sine/cosine, which are the complex duals of e^x (i.e. cosine(x) = e^(x*i), which is why e^(i*Pi) = -1). Conversely, cosine(-x*i) = e^x, which is what they were looking for (and found)--an exponential trend in the temperature data. This is what is being called "the hocky stick", presumably because there are more people who know what a hocky stick looks like than know what the graph of e^x looks like.
If you take a bunch of temperature data with enough resolution you will find all sorts of sine/cosine components--daily, yearly, 11-year (sun spots), etc. You can also look for exponential terms too (e.g., sine/cosine with an imaginary period--it's exactly the same thing) and if the data is at all noisy you will find them.
Slashdot Mirror
I do not dispute a single point in the parent post, but I see that we differ on an assumption: From this I infer that you took the original poster's use of "gifted" as being an example of "normal intelligent wordplay"--whereas I took it as sophomoric mimicry of a joke from a television show, and reacted to it accordingly. As for the question of whether or not my reaction is typical or not, may I suggest that the whole reason "gifted" was used in the show in the first place was that (in a true example of intelligent word play) the writers of the show realized that it would sound stupid to most people. The very thing that makes it useful as a joke makes it risky to use when you aren't trying to sound silly.
--MarkusQ
You've never heard the word "regifted"? Shame on you, non-Seinfeld watcher!
To put a finer point on it, non-television watcher. But, if I may hazzard a guess, I would say that the word "regifted" was used on the show you mention specifically because the writers knew it would sound moronic to most of their veiwers.
--MarkusQ
So first my point was invalid because it (according to you) came from a textbook, and when I point out that it came from the heart it's "a personal prejudice" (and thus inconsequential). I'm afraid to ask where you'd like it to come from...
But to stick to the point, this isn't just my predjudice; 95% of the people asked by the Harper's Dictionary of Contemporary Usage (to pick one source sitting on my desk) about the use of "gifted" in this sense found it unacceptable. Their comments were frequently hash--"vulgar" "anti-charisma" "loathsome" and "highly objectionable" etc.--and so, should you choose to use it, you ought to expect that the majority of your readers will not be favourably impressed.
You can of course disregard all of our "prejudice" and (like Humpty Dumpty) decide that words mean whatever you want them to, and the rest of the world be damned. But you ought to at leat realize in doing so that most of the people who read what you have written will (perhaps unconciously) think less of you for it.
--MarkusQ
Look, you can't keep saying that "language comes from the people, not from textbooks " and simultaniously demanding that I justify my views by reference to some external, objective source. They are my views for the simple reason that I hold them. Nothing more is needed.
I find "gifted" unclear. Whenever I encounter it, my first reaction is to reread the sentence, my second is to assume that it is a typo, and then I finally, reluctantly come to the conclusion that the person who used it isn't particularly well educated.
Why? Who knows. If you're willing to foot the bill I might consider letting you get a cat scan of my brain, but I doubt it would help much. While you're at it you might want to look into why I have the same reaction to "bling! bling!." "youse," and the use of numerals instead of letters.
A more constructive path, given that there are many people who feel roughly as I do, would be to simply try your best to write in standard English if you expect people to take you seriously.
Or go around saying you've been gifted with bling! bling! 'cause youse is ubber l33t, for all I care.
--MarkusQ
Let me get this straight--you are telling me (a person, not a book) that I am wrong to object to a word that I find unclear (and, in fact, ugly) and demanding that I justify my objection by citing a textbook ("where is it written") and in the very same post you are telling me that "Language comes from people, not textbooks."
Well, I'm a native English speaker and I'm telling you flat out that "gifted" sounds moronic to my ears. So which is it? Do you respect the opinion of the people, or do you want to defend the useage on the basis of textbooks?
--MarkusQ
Conversely the fact that something absurd and contrary to the will of the people, deleterious to the common wellfare, and arguably unconstitutional may have been accepted in oral arguments by a court that explicitly denied having decided the issue doesn't seem to me to hold enough water to float a complete restucturing of our society--which is exactly what is being done on the "priciple" of corporate personhood, albeit with glacial slowness.
--MarkusQ
Communication, preferrably with grace, economy, and clarity, is a much higher priority
My point exactly. Trying to construct the past tense of a noun (gift, after all, is a noun) is neither graceful nor economical, and it is far from clear.
--MarkusQ
P.S. For another example of the same principle, I think what you intended by the phrase "the guy means obvious..." was "the guy obviously means..."; note that while I am able to infer your meaning it would have been clearer and more graceful to user the more standard construction. I will conceed that it would be two letters longer and thus arguably less economical.
I was recently gifted with $100
I'd suggest a dictionary. You could learn lots of really useful words ("given" comes to mind...).
--MarkusQ
When defining a term, resorting to the idealized form is a reasonable thing to do. No actual physical "circle" has the properties we ascribe to mathematical circles, but that is a failing of the instances, not of the definition.
Likewise, just because people call themselves conservatives (or musicians, or gods, or anything else for that matter) I am not obligated to adjust my understanding of the term to include them. Instead I say (for example) that G. W. Bush is no conservative, and he can make all the claims to the contrary he wishes; I will not change my definition to suit his political goals, and I will judge him by his actions, dispite his rhetoric or that of his followers.
--MarkusQ
I can only answer for myself, but I hope my answer will serve nonetheless.
The core of true conservatism is to prevent any force from running amok; thus the power of government should be restricted by comparison to that of the individual, because there is more danger of governments abusing power (they have more, they live longer, etc.) This shows up in the chant of needing smaller government.
But the principle neither begins or ends there. Traditionally (AFAIK) private concentration of wealth went hand in hand with "conservatism" if for no other reason than the individuals who held it had no interest in seeing the boat rocked by others. This works so long as the wealth was acquired in a "conservative" fashon. Where it fails (e.g. slavery) the conservatives generally favour "rolling back" the disruptive concentration of wealth/power (remember, Lincoln was a Republican, and for the next hundred years the south was heavily Democratic). To be conservative is not mearly to support the status quo, but rather to support the mantanence of what is right and proper (to quote a famous fictional conservative, "but madest of all is to see life as it is, and not as it should be") and not support mere change-for-the-sake-of-change.
Gaming the system is a problem that cuts across the liberal/conservative spectrum and (like relative poverty) I fear it will always be with us.
--MarkusQ
Physicists are people too. They (I'd be tempted to say "we" but I changed majors mid-stream) don't get as hung up on vi vs. emacs or BSD vs. Linux vs. Gnu/Linux, or open vs. closed vs. I'll-let-you-peek-but-then-I'll-have-to-kill-you source operating systems. And (the ones I know at least) have never watched (let alone voted for) any one on any American-Idle type television program.
But the fundemental personality trait is still there. It comes out in things like "is space quantized at the planck scale" or "are strings real/fundemental or is string theory an artifact of something deeper". Add beer and watch the fur fly.
--MarkusQ
Neocons are children who don't want their toys taken away, and won't clean their room because it isn't fun.
Thank you for recognizing that the distinction between the present domanant "neo-conservative" group think party in the US and true conservatives. True conservatives wouldn't have bought the toys in the first place (we're compulsive savers) and wouldn't have let the room get messy for fear of unspecified adverse consequences. True conservatives (I know, I am one) are more likely to avoid doing things because they are fun (on the principle that the more atractive the lure, the more likely it is to be bait).
--MarkusQ
No, in fact they are not. They have been treated as "persons" for many years based on a mistaken reading of a 19th century court rulling that did not in fact decide the issue.
Of course, they're not going to tell you that, are they?
--MarkusQ
Great post! I just used my last real mod point earlier today, but I'll give you +2 insightful on the MQR standard.
--MarkusQ
I have nothing at this point (I'm working outside the US at this moment, I don't even have my notes) but it shouldn't be to hard to dig up again. Pre-google (Alta-vista + brick & mortar library) search took about a day.
Here's what I recall off the top of my head:
- Pretty much every projection to/from a sphere is known art to cartographers, and has been for many decades. Look in a few old cartography books.
- Likewise, the math behind them (called projective geometry) is old hat. We found a projective geometry book from 1900 or so that spelled out the transform
- Artists in the 1800s or so used to do paintings (called anamorphoses, IIRC) that not only used the same tricks but for exactly the same purpose. It may go back much further, but (again, IIRC) the really compelling photos we found were from work in the mid 1800s.
Anyone who wants to is welcome to run with this, expand on it, and pass it on to anyone that it might help.--MarkusQ
P.S. One further thing I recall, the laywer asked them something like "could you please specify what your patent covers--it obviously can't be the mathematics, and it can't be the technique san math, so...?"
A few years back I researched this very point for a client (I'm a programmer/math guy, not a laywer). We came up with prior art dating back to the 1800s (the very same technique was used in painting for "perspective lanterns" or some such). We turned over what we'd come up with to the lawyers, they wrote a letter, and we never heard from them again.
--MarkusQ
Of course I believe in observational science. What I don't believe in is magical algorithms that can "objectively" choose from an infinite set of possible functions the single correct interpretation of a finite set of noisy data.
I would have no problem with their results if they had said something like "assuming that there is an exponential increase in the Earth's temperature, here are the coefficients that best fit the attached data." But that's not the claim. The claim is that without any investigator bias they "objectively found" a "hocky stick" in the data--an exponential trend that can be used to extrapolate into the future. And that's just flat out not possible to do, with any algorithm.
I'm a firm believer in things like noticing that when hydogen and oxygen react they do so in a constant ratio, and using that to support the atomic theory of matter. Ditto using the luminocity of variable stars to cross-calibrate other ways of measuring distance, or double blind studies to test drugs, or pretty much any other instance of observational science you care to name.
But so far as I can see, this isn't observational science, dispite the way it's dressed up. It's voodoo. I distinguish them on this simple basis: are the researchers claiming to have extracted something from the data that could in principle be there, or are they claiming to have extracted information that in fact can not even in principle be extracted from the data.
To turn the point around, do you believe that the US stock markets are at record high valuations? How do you distinguish this "hockey stick" from the ones found by so many bull-analysts in the late 1990's, on the basis of which they "proved" that the Dow would be going up forever?
--MarkusQ
Note that (according to the link you provided):
- What is automatically produced is not arbitrary orthogonal functions but rather liniar combinations of the provided variables to which functions selected by some other mechanism could be applied; this seems quite reasonable, but it isn't what you (or other proponents claim).
- Even so, it is a preselection (it presupposes a form for the data, e.g., something that can be well represented by combining arbitrary functions of individual orthogonal liniar combinations of the input variables). It would not work, say, to model data with cusps or in which something like hysteresis was significant.
- It could not be the whole story for the original climate data analysis, since by itself it could not produce a "hocky stick" function. That had to come from somewhere, but it wasn't from PCA/PFA.
If, as you claim, I don't know what you're talking about, I would suggest that you don't either, since the link you provided does not support your position (that the hocky-stick function was objectively detected by a meaninful, unbiased automatic process). Nor does it undermine mine (that such a process is in priciple impossible). If you are willing to posit that the results are incorrect I think we can come to an agrement rather quickly.--MarkusQ
I know what it is, I just dispute its validity on principle. Why? Because it claims (or rather, its proponents claim) that the functions are chosen "objectively, by the data" (with the implication that they are therefore "correct").
This is clearly impossible on several levels; for one thing, even sticking to common everyday functions I can easily produce an f(t) and a g(t) such that:
If I generate two identical (finite & noisy) samples from these functions and give you one of them, chosen at random, how in the heck can you tell me with a streight face that any algorithm can "objectively" look at the data and tell me which one (f or g) generated the sample?
For another thing, the whole notion of "objectively" choosing the functions is meaningless; if you are only choosing from among some limited subsets of all possible functions, the objectivity goes out the window by the very fact that you have pre-determined which functions are candidates and which are not. Conversely, if you truly make an objective choice between all possible functions, the clear winner will be some function like:
which will tell you nothing.--MarkusQ
I think we are not understanding each other. At the very least, I am not understanding you.
- The isomorphism I mentioned was from "there are an even/odd number of partitions not counting the trivial partition" to/from "there are an odd/even number of partitions counting the trivial partition. Since there is exactly one trivial partition (e.g., 1432 can be represented as the sum of one integer in exactly one way, namely, the integer 1432) including/excluding it just flips the parity. It's not in any way a deep or meaningful difference.
- If you are representing numbers in a base X, and taking them modulo some prime P which is a factor of X, you only need to look at the last digit. In the case of base 16, P must be 2, whereas in base ten it can be 2 or 5. Saying that a decimal number "ends in 4 or 9" is the same thing as saying that it is congruent to -1 modulo 5.
The original poster asked if there was a similar pattern for hexidecimal numbers. Since 2 is the only prime factor of 16 (and because the discovery was that there was a pattern for the partitions of any number congruent to -1 modulo some prime) we can be sure that there is some pattern to the number of partitions of odd numbers.I will admit, I don't know what it is.
--MarkusQ
Given that there are infinitely many ways to decompose a finite sample into orthagonal functions (and in general they do not have the same extrapolation), simply picking one assumes a shape for the data.
And when you did this yourself, you found...what?
So, it's inevitable, but since the work passed peer review it didn't happen? I'm a big fan of peer review, but I have even more faith in logic.
I've known it for decades as well. I'm not sure why you think it took me by surprise, especially since (in your first reply) you didn't seem to know what I was refering to ('I am not sure what you mean by "imaginary periods"').
I'm sorry that you still don't see the relevance; I think I've already explained it rather clearly. Perhaps if you backed off from the ad hominem attitude it might be easier?
--MarkusQ
I was wrong, but not quite as wrong as you make it sound.
First off, ignoring the trivial partition just changes the parity, which would not affect any even/odd pattern beyond reverting the sense ("always even" <--> "always odd"). Using a trivial isomorphism is hardly a "tip-off that something is amiss".
Secondly, two is a prime and the proof in question AFAIK applies to all primes not just odd primes. The whole point of the discovery is that you can "reduce the bases that way" and you are guarenteed to "get something meaningful." So there is some pattern to the partitions of odd vs. even numbers, modulo some base, I was just incorrect in guessing what it is.
--MarkusQ
You are correct, I incorrectly generalized the result. From the result, we know that there should be some pattern (two is a prime, and the proof is that such a pattern optains for all primes), but my off-the-cuff guess as to what it was is clearly wrong.
--MarkusQ
In math-speak, that would be "all x congruent to -1 modulo 5"; for hex the corresponding pattern of interest, (since 2 is prime) would be "ending in 1,3,5,7,9,B, or F", which in math-speak would be "all x congruent to -1 modulo 2"--in other words, what most of us would simply call "odd numbers."
If I'm thinking streight, the corresponding statement for numbers expressed in hex, octal, or binary would be "all odd numbers can be represented as the sum of smaller numbers an even number of ways."
- 1 (0 ways)
- 3 = 1+1+1 or 1+2 (2 ways)
- 5 = 1+1+1+1+1, 1+1+1+2, 1+1+3, 1+2+2, 1+4, or 2+3 (6 ways)
- ...and so forth
--MarkusQFourier uses sine/cosine, which are the complex duals of e^x (i.e. cosine(x) = e^(x*i), which is why e^(i*Pi) = -1). Conversely, cosine(-x*i) = e^x, which is what they were looking for (and found)--an exponential trend in the temperature data. This is what is being called "the hocky stick", presumably because there are more people who know what a hocky stick looks like than know what the graph of e^x looks like.
If you take a bunch of temperature data with enough resolution you will find all sorts of sine/cosine components--daily, yearly, 11-year (sun spots), etc. You can also look for exponential terms too (e.g., sine/cosine with an imaginary period--it's exactly the same thing) and if the data is at all noisy you will find them.
--MarkusQ