I’ve been looking at the Nikolov and Zeller paper again. Among other things, they claim to be able to calculate the surface temperature Ts of eight different planets and moons from knowing nothing more than the solar irradiation So and the surface pressure Ps for each heavenly body. Dr. Zeller refers to this as their MIRACLE equation. He says:
Why aren’t you all trying to disprove our MIRACLE equation rather than banging your heads against walls trying to prove or disprove who knows what and exclaiming you have problems with this or that? The question is how can we possibly have done it – there is no question that our equations work – if you haven’t verified that it works, why haven’t you? […] Why aren’t you thinking: “hmmmm, N&Z have given us an equation that lo-and-behold when we plug in the measured pressures and calculate Tgb as they suggest, gives us a calculated Ts that also matches measured values! You can’t disprove the equation? So maybe we are cooking the data books somehow, but how?
This is supposed to be evidence that their theory is correct, and people keep telling me ‘but they’ve got real evidence, they can make predictions of planetary temperatures, check it out”. Plus it’s hard to ignore an invitation like Dr. Zellers, so I checked it out.
Figure 1. These are not the equations you are looking for.
They first postulate something called the “Near-surface Atmospheric Thermal Enhancement” or “ATE” effect that makes the earth warmer than it would be without an atmosphere.
The “ATE effect” is measured by something called Nte(Ps), which is defined and estimated in their paper as follows.
where Nte(Ps) is a measure of the “Near-surface Atmospheric Thermal Enhancement” effect.
Nte(Ps) is defined as the actual average surface air temperature of the planet Ts divided by the theoretical “graybody” temperature of the planet Tgb calculated from the total solar insolation So of the planet. Nte(Ps) is estimated using a fitted function of the surface pressure of the planet Ps.
Let me simplify things a bit. Symbolically, the right part of equation (7) can be written as
Nte(Ps) = e^(t1 * Ps ^ t2 + t3 * Ps ^ t4) (7Sym)
where “e” is the base of natural logs and Ps is the surface pressure on the planet or moon. There are four tunable parameters (t1 through t4) that are “fitted” or tuned to the data. In other words, those values are repeatedly adjusted and tuned until the desired fit is obtained. This fitting can be easily done in Excel using the “Solve…” menu item. As you’d expect with four parameters and only eight datapoints, the fit is quite good, and their estimate is quite close to the actual value of Nte(Ps).
Amusingly, the result of equation (7) is then used in another fitted (tuned) equation, number (8). This is:
where So is total solar irradiation.
This is their piece de resistance, their MIRACLE equation, wherein they are saying the surface temperature of eight different planets and moons can be calculated from just two variables— Pr, the surface pressure, and So, the total Solar irradiation. This is what amazes the folks in the crowd so much that they write and tell me there is “evidence” that N&Z are right.
Obviously, there is another tuned parameter in equation (8), so we can rewrite this one symbolically as:
Ts = t5 * (Solar + adjustment ) ^ 1/4 * Nte(Ps). (8Sym)
Let me pause a minute and point something out about equation (8). The total solar irradiation Solar ranges from over 9,000 W/m2 for Mercury down to 1.51 W/m2 for Triton. Look at equation 8. How will adding the adjustment = 0.0001325 to any of those values before taking the fourth root make the slightest bit of difference in the result? That’s just bizarre, that is. They say they put it in so that the formula will be accurate when there is no solar, so it will give the background radiation of 3 Kelvins. Who cares? Truly, it changes Ts by a maximum of a thousandth of a degree for Triton. So for the moment let me remove it, as it makes no practical difference and it’s just confusing things.
Back to the tale. Removing the adjustment and substituting equation 7 into equation 8 we get:
Ts = t5 * Solar^0.25 * e^(t1 * Ps ^ t2 + t3 * Ps ^ t4) (eqn 9)
This is amazing. These guys are seriously claiming that with only eight datapoints and no less than five tunable parameters , they can calculate the surface temperature of the eight planets knowing only their surface pressure and solar irradiation. And with that many knobs to turn, I am sure they can do that. I did it on my own spreadsheet using their figures. I get about the same values for t1 through t5. But that proves nothing at all.
I mean … I can only stand in awe at the sheer effrontery of that claim. They are using only eight datapoints and five tunable parameters with a specially-designed ad-hoc equation with no physical basis. And they don’t think that’s odd in the slightest.
I will return to this question of the number of parameters in a bit, because even though it’s gobsmacking what they’ve done there, it’s not the best part of the story. Here’s the sting in the tale. We can also substitute equation (7) into equation (8) in a slightly different way, using the middle term in equation 7. This yields:
Ts = t5 * Solar^0.25 * Ts / Tgb (eqn 10)
This means that if we start out by knowing the surface temperature Ts on the right side of the equation, we can then calculate Ts on the left side … shocking, I know, who would have guessed. Let’s check the rest of the math in equation (10) to see why that works out.
Upon inspection it can be seen that the first part of the right side of equation (10),
t5 * Solar^0.25
is an alternate form of the familiar Stefan-Boltzmann equation relating temperature and radiation. The S-B equation can be written as
T = (Solar / c1) ^ 0.25.
where T is temperature and c1 is a constant equal to the S-B constant times the emissivity. We can rewrite this as
T = 1/(c1^0.25) * Solar^0.25
Setting another constant c2 equal to 1 / (c1^0.25) gives me the Stefan-Boltzmann equation as:
T = c2 * Solar^0.25
But this is exactly the form of the first part of the right side of equation 10. More to the point, it is an approximation of the graybody temperature of the planet Tgb.
We can check this by observing that if emissivity is .9 then constant c1 is 5.103E-8, and c2 is therefore about 66. However, that value will be reduced by the rotation of the planet. Per the N&Z formula in their latest post, that gives a value of about 27.
Their fitted value is 25, not far from the actual value. So curiously, what it turns out they’ve done is to estimate the Stefan-Boltzmann constant by a bizarre curve fitting method. And they did a decent job of that. Actually, pretty impressive considering the number of steps and parameters involved.
But since t5 * Solar^0.25 is an estimation of the graybody temperature of the planet Tgb, that means that Equation 10 reduces from
Ts = t5 * Solar^0.25 * Ts / Tgb (eqn 10)
to
Ts = Tgb * Ts / Tgb.
and finally to
Ts = Ts
TA-DA!
CONCLUSION
Let me recap the underlying effect of what they have done. They are looking at eight planets and moons.
1. They have used an equation
e^(t1 * Ps ^ t2 + t3 * Ps ^ t4)
with four free parameters to yield an estimate of Ts/Tgb based on surface pressure. As one would expect given the fact that there are half as many free parameters as there are data points, and that they are given free choice to pick any form for their equation without limit, this presents no problem at all, and can be done with virtually any dataset.
2. They have used an equation
t5 * Solar^0.25
with one free parameter in order to put together an estimate of Tgb based on total planetary insolation. Since Tgb does depend inter alia on planetary insolation, again this presents no problem.
3. They have multiplied the two estimates together. Since the result is an estimate of Tgb times an estimate of Ts/Tgb, of course this has the effect of cancelling out Tgb.
4. They note that what remains is Ts, and they declare a MIRACLE.
Look, guys … predicting Ts when you start out with Ts? Not all that hard, and with five free parameters and a choice of any equation no matter how non-physically based, that is no MIRACLE of any kind, just another case of rampant curve fitting …
Finally, there is a famous story in science about this kind of pseudo-scientific use of parameters and equations, told by Freeman Dyson:
We began by calculating meson–proton scattering, using a theory of the strong forces known as pseudoscalar meson theory. By the spring of 1953, after heroic efforts, we had plotted theoretical graphs of meson–proton scattering. We joyfully observed that our calculated numbers agreed pretty well with Fermi’s measured numbers. So I made an appointment to meet with Fermi and show him our results. Proudly, I rode the Greyhound bus from Ithaca to Chicago with a package of our theoretical graphs to show to Fermi.
When I arrived in Fermi’s office, I handed the graphs to Fermi, but he hardly glanced at them. He invited me to sit down, and asked me in a friendly way about the health of my wife and our newborn baby son, now fifty years old. Then he delivered his verdict in a quiet, even voice. “There are two ways of doing calculations in theoretical physics”, he said. “One way, and this is the way I prefer, is to have a clear physical picture of the process that you are calculating. The other way is to have a precise and self-consistent mathematical formalism. You have neither.
I was slightly stunned, but ventured to ask him why he did not consider the pseudoscalar meson theory to be a selfconsistent mathematical formalism. He replied, “Quantum electrodynamics is a good theory because the forces are weak, and when the formalism is ambiguous we have a clear physical picture to guide us. With the pseudoscalar meson theory there is no physical picture, and the forces are so strong that nothing converges. To reach your calculated results, you had to introduce arbitrary cut-off procedures that are not based either on solid physics or on solid mathematics.”
In desperation I asked Fermi whether he was not impressed by the agreement between our calculated numbers and his measured numbers. He replied, “How many arbitrary parameters did you use for your calculations?”
I thought for a moment about our cut-off procedures and said, “Four.”
He said, “I remember my friend Johnny von Neumann used to say, with four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” With that, the conversation was over. I thanked Fermi for his time and trouble, and sadly took the next bus back to Ithaca to tell the bad news to the students.
The Nikolov and Zeller equation contains five parameters and only eight data points. I rest my case that it is not a MIRACLE that they can make the elephant wiggle his trunk, but an expected and trivial result of their faulty procedures.
My regards to everyone,
w.
PS—There is, of course, a technical term for what they have done, as there are no new mistakes under the sun. It is called “overfitting”. As Wikipedia says, “Overfitting generally occurs when a model is excessively complex, such as having too many parameters relative to the number of observations.” Five parameters is far, far too many relative to eight observations, that is a guaranteed overfit.
PPS—One problem with N&Z’s MIRACLE equation is that they have not statistically tested it in any way.
One way to see if their fit is even remotely valid is to leave out some of the datapoints and fit it again. Of course with only eight datapoints to start with, this is problematic … but in any case if the fitted parameters come out radically different when you do that, this casts a lot of doubt on your fit. I encourage N&Z to do this and report back on their results. I’d do it, but they don’t believe me, so what’s the point?
Aother way to check their fit is to divide the dataset in half, do the fit on one half, and then check the results on the other half. This is because fitted equations like they are using are known to perform very poorly “out of sample”, that is to say on data not used to fit the parameters. Given only eight data points and four parameters for equation 7, of course this is again problematic, since if you divide the set in half you end up with as many parameters as data points … you’d think that might be a clue that the procedure is sketchy but what do I know, I was born yesterday. In any case I encourage N&Z to perform that test as well. My results from that test say that their fit is meaningless, but perhaps their test results will be different.
[UPDATE] One of the commenters below said:
Willis – go ahead – fit an elephant. Please!
Seriously N&Z are only demonstrating in algebra what has been observed in experiments, that heating a gas in a sealed container increases both pressure and temperature.
OK, here’s my shot at emulating the surface temperature using nothing but the data in the N&Z chart of planetary body properties:
Figure 1. Willis’s emulation of the surface temperature of the planetary bodies.
My equation contains one more variable and two less parameters than the N&Z equation. Remember their equation was:
Ts = 25.3966 * Solar^0.25 * e^(0.233001 * Pressure ^ 0.0651203 + 0.0015393 * Pressure ^ 0.385232)
My equation, on the other hand, is:
Ts = 0.8 * Tgb + 6.9 * Density + 0.2 * Gravity)
Note that I am absolutely not making any claim that temperature is determined by density and gravity. I am merely showing that fitting a few points with a few variables and a few parameters is not all that difficult. It also shows that one can get the answer without using surface pressure at all. Finally, it shows that neither my emulation nor N&Z’s emulation of the planetary temperatures are worth a bucket of warm spit …
[UPDATE 2] I figured that since I was doing miracles with the N&Z miracle equation, I shouldn’t stop there. I should see if I could beat them at their own game, and make a simpler miracle. Once again, their equation:
Ts = 25.3966 * Solar^0.25 * e^(0.233001 * Pressure ^ 0.0651203 + 0.0015393 * Pressure ^ 0.385232)
My simplified version of their equation looks like this:
Ts = 25.394 * Solar^0.25 * e^(0.092 * Pressure ^ 0.17)
Curiously, my simplified version actually has a slightly lower RMS error than the N&Z version, so I did indeed beat them at their own game. My equation is not only simpler, it is more accurate. They’re free to use my simplified miracle equation, no royalties necessary. Here are the fits:
Figure 2. A simpler version of the N&Z equation 8
Again, I make no claim that this improves things. The mere fact that I can do it with two less tuned parameters (three instead of five) than N&Z used does not suddenly mean that it is not overfitted.
Both the simplified and the complex version of the N&Z equations are nothing but curve fitting. This is proven by the fact that we already have three simple and very different equations that hindcast the planetary temperatures. That’s the beauty of a fitted equation, if you are clever you can fit a lot using only a little … but THAT DOESN’T MEAN THAT PRESSURE DETERMINES TEMPERATURE.
For example, I can do the same thing without using pressure at all, but using density instead. Here’s that equation:
Ts = 25.491 * Solar^0.25 * e^(0.603 * Density ^ 0.201)
And here’s the results:
Figure 3. An emulation of the planetary temperatures, using density instead of pressure.
Does this now mean that the planetary temperature is really controlled by density? Of course not, this whole thing is an exercise in curve fitting.
w.
@DeWitt Payne
> Here’s the only one I could find quickly.
Got it. Thanks.
John Day says:
January 31, 2012 at 1:34 pm
@Phil.
> For the moon you’d expect more like 100K (they plot it from
> NASA in their response), that’s the result of not including surface heat capacity.
So, I missed the previous message traffic on this. are you saying that N&Z have never responded at all to this issue?
Not as far as I’m aware, Anthony even followed up on it, they have responded on here since but not addressed heat capacity.
Seems rather crucial (now that I understand where you’re coming from, wrt heat capacity. Sorry for the confusion). I guess a faster rotating airless planet would have an even warmer backside.
I would expect that too for the same heat capacity and heat transfer coefficient.
@me
>… are you saying that N&Z have never responded at all to this issue?
I think I found it in their first response (below). Looking at the sentence I bolded, I can see why you object. Actually, they do seem to address heat capacity, in the sense that temperature distribution would become more uniform. So, yes, the average temp should rise, given the same insolation. I don’t see how spinning faster would change the effective power input over time.
So they have never responded to this objection? Hmm.
😐
@N&Z
> would only facilitate a more efficient spatial distribution of the absorbed solar energy
Those words above occur just before the bolded section. This seems to be their explanation for why the average temp would not rise, i.e. due to “more efficient spatial distribution”.
What do they mean by that?
@me
> What do they mean by that?
(Talking to myself, sorry). Maybe they mean higher power outflux according to S-B because of the fourth power of temperature creating some kind of exponential gain?
John Day says:
Personally, I am less concerned about what is a realistic value for T_sb than I am that they are making people believe that there is 133 K that needs to be explained when in fact we already understand the explanation of 100 K of it: I.e., that there can be a variety of different temperature distributions with different average temperatures that all satisfy radiative balance.
If they want their T_sb to represent a planet with such a slow rotation rate and small heat capacity that their approximation for the temperature distribution is valid, then fine. But, don’t pretend it is any big mystery why real planets have a higher average temperature than this (because they have a more uniform temperature distribution)!
No…What they are saying regarding the more efficient spatial distribution is simply that they agree that such a planet would not have as non-uniform a distribution as they imagined with their approximation that the instantaneous local temperature is just determined by radiative balance with the instantaneous local insolation. However, they still think that the average temperature would be the same. They explain clearly why they think this:
And, this statement makes it clear that they are clueless as to how to apply conservation of energy. They think that a planet receiving gobs of energy from the sun and emitting gobs of energy back out into space can be treated as if it were isolated. Hence, they think that the constraint on the planet is that its thermal energy can’t change.
In fact, the planet is nowhere close to being an isolated system and the constraint on it is that it must emit back out into space the same amount as it absorbs from the sun.
This is a huge and very fundamental error that they make!
N+Z as quoted above are incorrect
” However, we will point out that increasing the mean equilibrium temperature of a physical body always requires a net input of extra energy. ”
Increasing the mean amount of radiated energy always requires a net input of extra energy, but increasing the mean temperature does not, because the T^4 dependence. This is easily shown by example. Consider a planet which for some reason had one hemisphere at 200K, and the other at 400K. The warmer hemisphere is emitting 16 times as much energy. Now even out the temperatures such that the hemispheres are equal. At what temperature is the same amount of energy emitted? Well it is not 300K – it is in fact 341K. Any evening out of the temperature distribution results in a higher mean temperature, provided the input energy remains constant.
jimmi_the_dalek says:
January 31, 2012 at 3:42 pm
N+Z as quoted above are incorrect
” However, we will point out that increasing the mean equilibrium temperature of a physical body always requires a net input of extra energy. ”
Increasing the mean amount of radiated energy always requires a net input of extra energy, but increasing the mean temperature does not, because the T^4 dependence. This is easily shown by example. Consider a planet which for some reason had one hemisphere at 200K, and the other at 400K. The warmer hemisphere is emitting 16 times as much energy. Now even out the temperatures such that the hemispheres are equal. At what temperature is the same amount of energy emitted? Well it is not 300K – it is in fact 341K. Any evening out of the temperature distribution results in a higher mean temperature, provided the input energy remains constant.
Indeed that’s the whole point about Holder’s inequality, it is the mean T^4 which will be constant regardless of the distribution of T. The max mean T corresponds to a uniform distribution and for a radiatively heat planet (from a single source) the N&Z gives the minimum mean T (also the most unrealistic unlike the uniform case which is not a bad assumption for the earth).
I’m studying the paper related to this (Arthur Smith 2008), where Smith is explaining the non-rotating planet bit, in section IIIA. He assumes that, for his particular example, emissivity is 1, but albedo is independent, if I’m reading the words correctly.
[Possibly dumb question] Isn’t emissivity tied to albedo through Kirchoff’s Law of Thermal Radiation, which asserts that emissivity must be equal absorptivity?
In any case, it’s strange that N&Z rejected Smith’s paper here as ‘beyond the scope’ of their piece. Hard to imagine any rebuttal so complicated or subtle that it couldn’t be summarized (or at least catalogued by variety) in a sentence or two. Does look like a bluff.
😐
> Isn’t emissivity tied to albedo
At a given frequency. LW emmisivity is indep of SW albedo. Sfc LW emissivity is usually ~1. SW albedo varies with sfc type (and wavelength too, if you want detail).
@John Day: Just to expand on what William said…
First of all “LW” stands for “long wave” (meaning wavelengths associated with typical terrestrial radiation, i.e., from a body having a temperature of around 300 K) and “SW” stands for “short wave” (meaning wavelengths associated with typical solar radiation, i.e., from a body having a temperature of around 6000 K).
The fact that absorptivity and emissivity depend on wavelength is what allows for the possibility of a greenhouse effect, because it allows for the fact that our atmosphere is more transparent to shortwave radiation than it is to longwave radiation.
@Stoat
> albedo varies with sfc type (and wavelength too.
That’s immaterial here (pun intended). These coefficients relate incoming and outgoing energy with respect to the solar constant S, which Smith clearly defines as the “integral over all wavelengths and propagation directions”. So, frequencies have been marginalized here.
I’m thinking it might time-related, in today-out tomorrow, so related to potential energy. But over long enough time window, Kirchofff must hold, right?
Or perhaps Smith is being a little loose and general here. But what I’ve read so far is drum-tight, seems to be a well-written monograph.
Joel, Phil. or DeWitt, can you help me out here?
😐
Solar power is negligible in the LW, compared to the SW. He is probably using one albedo (=1-emissivity) across all the SW, and another for the LW (=infra red, =terrestrial thermal radiation). It is what everyone else does, at that level of simplification.
@Stoat
> Solar power is negligible in the LW, compared to the SW.
True statement, but immaterial in the Smith paper because the frequency variable has been marginalized. Only ‘x’ (surface location) and ‘t’ (time) appear as parameters in his equations 1 through (12), after which we encounter this statement (pg 3) that I was pondering (and probably misinterpreting):
“Emissivity is assumed to be 1 everywhere; like S and a are uniform”
Doesn’t e=1 imply a perfect black body, hence a=0?
Just asking.
😐
It would probably be best to post a link to, or full ref to, the Smith paper you’re trying to understand.
Oops, make that “likewise S and a are uniform”
> …best to post a link …
Smith, A. 2008. Proof of the atmospheric greenhouse effect. Atmos. Oceanic Phys. arXiv:0802.4324v1 [physics.ao-ph]
http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.4324v1.pdf
It’s the “(Smith, 2008)” paper that N&Z referenced in their Part I reply, in the part that I quoted above.
Looks like Smith wrote this in 2008 as a rebuttal to yet another theory similar to N&Z.
Thanks. Smith says “Emissivity is assumed to be 1 everywhere; likewise S and a are uniform”.
e=1 means the LW (= infra-red = terrestrial heat radiation) emissivity is assumed to be 1. As I said, that is what people usually assume. So the LW albedo of the sfc is zero, if you like to think of it that way.
“Define the albedo a of the planet as the fraction of incoming irradiance that is reflected”
So “the albedo” means the SW (=visible) albedo. “incoming irradiance” means SW, because (as I said) so little of it overlaps with the LW that it can be neglected.
Note that Smith is ignoring variations of the albedo with frequency; he is effectively assuming that all the incoming radiation is at a single freq (or equivalently, that the albedo is constant across the SW band), which is an acceptable simplification for the level he is looking at. Don’t confuse that with his integration across space.
Actually, he says explicitly “S is an integral over all wavelengths”; because the fraction of LW in solar is negligible, that is effectively an integral across SW only.
It would be helpful if Smith explicitly stated that he is treating LW and SW as non-overlapping; this is such a common procedure that he probably just forgot. The situation is indeed as I said at 2012 at 11:13 am.
William M. Connolley says:
February 1, 2012 at 2:31 pm
Thanks. Smith says “Emissivity is assumed to be 1 everywhere; likewise S and a are uniform”.
e=1 means the LW (= infra-red = terrestrial heat radiation) emissivity is assumed to be 1. As I said, that is what people usually assume. So the LW albedo of the sfc is zero, if you like to think of it that way.
“Define the albedo a of the planet as the fraction of incoming irradiance that is reflected”
So “the albedo” means the SW (=visible) albedo. “incoming irradiance” means SW, because (as I said) so little of it overlaps with the LW that it can be neglected.
Note that Smith is ignoring variations of the albedo with frequency; he is effectively assuming that all the incoming radiation is at a single freq (or equivalently, that the albedo is constant across the SW band), which is an acceptable simplification for the level he is looking at. Don’t confuse that with his integration across space.
Actually, he says explicitly “S is an integral over all wavelengths”; because the fraction of LW in solar is negligible, that is effectively an integral across SW only.
It would be helpful if Smith explicitly stated that he is treating LW and SW as non-overlapping; this is such a common procedure that he probably just forgot. The situation is indeed as I said at 2012 at 11:13 am.
I posted similar to this referring to Bond albedo but it got lost, note that it’s the “fraction of incoming irradiance that is reflected”, so only the incoming wavelengths are considered. Even if 10 micron radiation and above constituted 0.1% of the incoming and had an emissivity of 0.98 its contribution to the albedo is negligible (0.001* 0.02).
Ok, I was overlooking that Kirchoff’s Law is frequency dependent. Makes sense now. Thanks.
I’m not trying to be a jerk about it, and yes I can — without hardly being a math expert — at least follow it written this way, but don’t you have a math program you can create images in where you can use multiple lines and more standard notation?
And thus the walkback starts — beginning with a stunning admission!
This isn’t very logical of you, Ira.
Even if Willis is totally right, he recounts a famous story where Freemon Dyson, of all people, proudly brought gibberish mathematical physics to the attention of Enrico Fermi. Yet we’ve still heard of Dyson and he did good work.
So your statement about what you will allow yourself to consider in the future is unsound.
Christoph Dollis says:
February 10, 2012 at 5:36 am
The difference is that when it was pointed out to him that he was just curve fitting, Dyson acknowledged his mistake and learned from it.
When it was pointed out to N&Z that they were just curve fitting, N&Z refused to either acknowledge their mistake or learn from it.
As a result, Ira’s statement is perfectly reasonable.
w.
Willis stated:
” When it was pointed out to N&Z that they were just curve fitting, N&Z refused to either acknowledge their mistake or learn from it. As a result, Ira’s statement is perfectly reasonable. w”
Of course, you never seem to consider that they weren’t CURVE FITTING. Did you actually ask them politiely whether they had tried other values for the parameters before settling on what they have?? They may have done the calculations and came up with a “miraculous” result. YOU did the curve fitting finding other equations that worked that is simply SOP.
Christoph Dollis says:
February 10, 2012 at 3:30 am
I haven’t a clue what non-standard notation you think I’m using … more detail in the post would be useful. As I have said many times, QUOTE MY WORDS if you disagree with them. Otherwise, as in this case, people may have no clue what you are on about.
w.
Christoph Dollis says:
February 10, 2012 at 4:57 am
So … what was the “stunning admission!”??
Was it where I admitted that four parameters is enough to fit an elephant? Was it where I admitted that I doubted that the fifth number was not a parameter?
Once again, your writing is totally opaque, Christoph.
w.
PS—”And thus the walkback starts” sounds like a line from some movie I haven’t watched, but I don’t see any sign of anyone walking back. I said above, and I still say, that Equation 8 is just a trivial exercise in curve fitting. I’m not walking back an inch from that true statement.
And N&Z continue to insist that fitting an elephant, with four (or five) parameters, only 8 points to fit, and free choice of equation, is a MIRACLE! They’re not walking back from their inanity.
If I understand your posts, they have one more rube totally fooled … if you run over to Tallbloke’s Talkshop, they’re still peddling their snake oil there, you can get in on the fun.