Guest Essay by Alan Welch – facilitated by Kip Hansen –14 May 2022
Nerem et al Paper, 2018, 4 Years on
by Dr Alan Welch FBIS FRAS, Ledbury, UK — May 2022
(Note: One new image has been added at the end of the essay.)
Abstract Having analysed the NASA Sea Level readings over the last 4 years it has been concluded that the accelerations derived by Nerem et al. are a consequence of the methodology used and are not inherent in the data. The analyses further predict that the perceived accelerations will drop to near zero levels in 10 to 20 years.
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It is now 4 years since the paper by Nerem et al. (2018) 1 was released. It spawned many disaster pictures, such as the Statue of Liberty with the sea lapping around her waist, and a proliferation in the use of Climate Crisis or Climate Catastrophe in place of Climate Change by the likes of the BBC and the Guardian.
It also kick-started my interest in Climate Change, not by what it presented, but by the unacceptable methodology used in determining an “acceleration”. I have inserted acceleration in quotes as care must be used in interpreting the physical meaning behind the coefficients derived in fitting a quadratic equation. In the paper by Nerem et al. there were 3 stages.
Mathematical – Coefficients are calculated for a quadratic equation that fits the data set.
Physical – Attaching a label – ”acceleration” – to 2 times the quadratic coefficient.
Unbelievable – Extrapolating to the year 2100.
The first is straightforward and acceptable. The second is very dependent on the quality of the data and the length of period involved. The third is fraught with danger as the quadratic term dominates the process when used outside the range of data. The last point is illustrated in Figure 1. This appeared in https://edition.cnn.com/2018/02/12/world/sea-level-rise-accelerating/index.html with the caption “Nerem provided this chart showing sea level projections to 2100 using the newly calculated acceleration rate”.
Figure 1
As a retired Civil Engineer with 40 years’ experience in engineering analysis I appreciate that curve fitting can, with care, be used to help understand a set of data. But to extrapolate 25 years’ worth of data for more than 80 years into the future is in my mind totally unacceptable. But it is this “acceleration” that generated the alarmist press following publication.
I will now discuss several aspects concerning the sea level data, including what could have been done differently in 2018, what the current situation is and what can be learnt from studying the last 10 years’ worth of data. Prior to 2012 the data, and any related analysis, were more erratic, but with time a steadier picture is emerging.
The Feb 2018 data were the first data analysed. The data used were derived from the NASA site https://climate.nasa.gov/vital-signs/sea-level/. These data do not include any of the adjustments introduced by Nerem et al. but calculated values for slope and “acceleration” are not too dissimilar. In discussing “acceleration” the process can be made simpler by taking the values of the straight-line fit, i.e., the slope, away from the actual readings and working with what are called the “residuals”. Using residuals or the full data results in the same “acceleration” but it is easier to see the trends using the residuals.
Figure 2 below shows quadratic and sinusoidal fits starting in Jan 1993 up to the Feb 2018 using the latest values of sea levels. (See Note 1 below)
Figure 2
Situation in 2021.
A later set of results refer the period from Jan 1993 up to Aug 2021. The above graph has been updated in Figure 3 to show that the x2 coefficient is now 0.0441.
Figure 3
This update shows that the sinusoidal form curve is still a reasonable alternative to the quadratic curve although the period could be extended to 24 or 25 years and the amplitude increased slightly. The 22-year period and the amplitude have been retained for the sake of continuity although the quadratic curve is reassessed at each update, which has the effect of slightly modifying the slope and residuals.
Study of the last 10 years set of data.
The NASA data were analysed over the last 10 years on a quarterly basis using the full Aug 2021 set of data. The “acceleration” was calculated for each time step using the data from 1993 up to each date. In tandem with this a second set of “accelerations” were derived by assuming the data followed the pure sinusoidal curve listed on the figures above. In the long-term these “accelerations” will approach near zero but when the wavelength and period being analysed are similar unrepresentative “accelerations” will be derived. Note 2 gives more detail for the sinusoidal curve to explain the process and results and illustrate the curve fitting process.
The results of these 2 sets of analyses are plotted in Figure 4 as “accelerations” against date NASA data set was released and analysed. For example, the two “accelerations” for 2018 would be those derived using a quadratic fit for both the NASA data and the pure sinusoidal curves over the period Jan 1993 to Jan 2018 respectively. The graph on the left shows the “acceleration” for the NASA data and the sinusoidal curve. Their shapes are very similar but offset by about 3 years. Shifting the sinusoidal curve over 3 years shows how closely the 2 curves follow each other. This close fit is of interest. The NASA “acceleration” peaked in about Jan 2020 and is reducing from then onwards dropping by about 8% over 2 years. Working backwards from the peak the “accelerations” keep reducing until at about Oct 2012 they were negative, i.e., deacceleration. The close fit with the shifted sinusoidal curve may be coincidental but there seems to be a clear message there, that is the high “acceleration” quoted by Nerem et al. is more an outcome of the method used and not inherent in the basic data.
Figure 4
The next few years will be telling as to whether the sinusoidal approach is more representative to the actual behaviour and if the NASA data continues to produce a reducing “acceleration”. If the actual “acceleration” curve follows the trend of the sinusoidal curve the perceived “acceleration” will have halved in about 6 years and reached near zero values in about 15 years.
1. Nerem, R. S., Beckley, B. D., Fasullo, J. T., Hamlington, B. D., Masters, D., & Mitchum, G. T. (2018). Climate-change-driven accelerated sea-level rise detected in the altimeter era. (full text .pdf) Proceedings of the National Academy of Sciences of the United States of America, 115(9). First published February 12, 2018
Note 1. The NASA data changes from month to month. Usually this is confined to the last month or two of data due to the method used in smoothing the readings. In July 2020 there was a major change to all the data by up to 2.5 mm which had little effect on the slope, but the “acceleration” was reduced by about 0.005 mm/yr2. I have been unable to ascertain the reason behind these adjustments, but they have little effect on the overall findings.
Note 2. The Sinusoidal Curve shown in figure 5 will be analysed.
Figure 5
The “accelerations” derived from analysing this sinusoidal curve over a range of periods from 2.5 years to 70 years are shown in figure 6.
Figure 6
The next 5 figures illustrate the curve fitting process for various time periods.
Figure 7 uses a short 5-year period and the fitted quadratic curve is very close to the actual sinusoidal curve and in this instant gives an “acceleration” of -0.2716 mm/yr2 very close to the curve’s maximum acceleration of 0.285 mm/yr2 obtained by differentiating the equation twice.
Figure 7
Figure 8 uses a 15-year period and the “acceleration” drops to -0.0566 mm/yr2.
Figure 8
Figure 9 is very close to the period used by Nerem et al. in that it uses 25 years. The resulting “acceleration” is 0.0904 mm/yr2 similar to that paraded by Nerem.
Figure 9
Figure 10 covers 35 years and results in a rapid reduction in “acceleration” to 0.0118 mm/yr2. (typo corrected with thanks to Steve Case)
Figure 10
Finally extending the period to 65 years, which is nearly 3 periods of 22 years, results in a near zero “acceleration” as shown in Figure 11.
Figure 11
The following image has been added at the request of Dr. Welch (15 May 2022, 9:00 am EST):
Fig 2 from Nerem 2022 with prediction by Welch
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About Dr. Alan Welch:
Dr. Welch received a B.Sc.(Hons 2A) Civil Eng. From the University of Birmingham and his PhD from the University of Southampton. He is a Chartered Civil Engineer (U.K.), a member of the Institution of Civil Engineers (U.K.) (retired), a fellow of the British Interplanetary Society, and a fellow of the Royal Astronomical Society.
Currently retired, he has over thirty years’ professional experience across many fields of engineering analysis. Complete CV here
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Comment from Kip Hansen:
Dr. Welch has been working on this analysis for years and has put his findings together at my suggestion as an essay here. The above is the result of many edited versions and is offered here as an alternative hypothesis to Nerem (2018) ( .pdf )and Nerem (2022). In a practical sense, Nerem (2022) did not change anything substantial from the 2018 paper discussed by Welch
On a personal note: This is not my hypothesis. I do not support curve fitting in general and an alternate curve fitting would not be my approach to sea level rise. I stand by my most recent opinions expressed in “Sea Level: Rise and Fall – Slowing Down to Speed Up”. Overall, my views have been more than adequately aired in my previous essay on sea levels here at WUWT.
I do feel that Dr. Welch’s analysis deserves to be seen and discussed.
Dr. Welch lives in the U.K. and his responses to comments on this essay will be occurring on British Summer Time : UTC +1.
Praise for his work in comments should be generous and criticism gentle.
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Well, may I suggest we continue to accumulate, and not “correct” the data for 1000 years then make a decision. Curve fitting pover such a small period, even less than one cycle, is bad practice
Robert ==> Well, at least another century or so….
So when exactly did the sea level stop rising in a linear fashion (the standard dogma around here is that that is a natural consequence of the end of the little ice age) and start varying sinusoidly as suggested by Dr. Welch? If you take his curve fitting and extrapolate back a hundred years or more you will see just how wrong it is.
IW
Alan is not suggesting that evidence exists for a natural or anthro sinusoidal variation, so much as showing that two different choices for curve fitting leads to a didactic conclusion. Geoff S
Geoff,
using the same analysis I could fit the data with any curve whose Taylor series expansion had a positive coefficient for the x^2 term. Which doesn’t mean anything except that everything look parabolic locally.
Furthermore Dr. Welch’s analysis also seems flawed since he uses one equation for his fit to the data and then a second different one to fit the acceleration. When he says that the sine wave is shifted by 3 years when discussing Fig. 4 he is saying that he needs two curves to fit the data, one for the values themselves and then a completely different one for the acceleration. If a sinusoid fit was valid then the same sinusoid would fit both the data and the acceleration derived from it.
The 3 year shift comes about due to the fact I stuck to the 22 year cycle first derived in 2018. Later I said the curve could be improved to a slightly longer period but I didn’t want to keep tuning all the while. It was the close fit of the shapes that is of interest. In Nerem 2022 paper he plots a similar graph (fig 2) but stops a year earlier saying it has leveled off.
I have added my sinusoidal based accelerations to this figure and sent a copy to Kip. I am unable to submit figures so I’ll contact Kip to see if he can upload it in a comment.
sorry for late reply due to time differences.
I am not implying SLR is rising sinusoidal but the readings may have a sinusoidal variation due to the method of measurement. I have already commented that the less than 100% coverage may have a bearing due to possible decadal ocean osculations. We are only talking of a 3.5mm amplitude but show that applying the methodology of Nerem et al leads to similar exaggerated “accelerations”. I have then predicted the trend (again dangerous) for the next few years – so watch this space.
Thank you Kip and Alan for perseverance with this analysis.
And Kip, for continuing to request proper error analysis and reporting that is a fault with many climate papers.
Which raises the point that the measurement accuracy of satellite distance measurements seems a larger error than that quoted in conclusions from papers. Reliance seems to be placed on old saws like central limit and laws of large numbers. Has this apparent discrepancy been resolved? Or do we have two scientific communities who use and accept different ways to calculate accumulated error? Both cannot be right.
Also, there needs to be more clarity in explanations why discrete tide gauge data shows no acceleration while satellite data are sometimes claimed to. Explanations for fundamental discrepancies are better than ignoring them. Geoff S
Geoff ==> There is simply no explanation as to why the oft-reported Sea Level Rise acceleration is not showing up at tide gauges around the world.
The most common response of the Sea Level Rise groups is to fudge to “global” tide gauge record to line up with the start of the satellite record and then say the whole thing is accelerating — despite no individual tide gauges showing any acceleration at all.
Kip, when the latest mean sea level [Feb 2022] is 99mm LOWER than the first MSL [May 1914] at probably the best Pacific Ocean tide gauge there is not only no acceleration, there is possibly no SLR [as Morner always said].
This is supported by the increase in Pacific atoll areas, too.
http://www.bom.gov.au/ntc/IDO70000/IDO70000_60370_SLD.shtml
Geoff may I throw in my pennies worth. The difference between Satellite and tidal gauges is they measure different things as their coverage is different and they are affected by decadal ocean oscillations differently. There may be a low background (0.01 mm/yr2) acceleration in all readings but this could easily be due to long term (millennium) variations that may exist in sea levels and temperatures.
See my post above LINK about the 0.01mm/yr² acceleration.
Climbing beans? Glad to know some people have a real life with real goals. I’ve tried Kentucky pole beans, and the local deer population just north of Milwaukee, WI loves them.
Steve & Alan ==>. We grow bush beans…… my wife picks them as I do not bend so easily.
Kip, you say tide gauge acceleration is not showing up any tide gauges at all, in particular since the start of the satellite record 30 yrs ago (if I have understood you correctly).
Looking at Sea Level Info (sidebar of this website) there are some tide gauges with records longer than 100 years. Of those practically all show acceleration in the last 30 years (except Scandanavia which is affected by Post Glacial Rebound) .
For example: Fremantle long term 1.75 mm/yr, 1992-2022 4.85mm/yr; Fort Denison 0.78, 3.34; The Battery 2.89,4.19; Honolulu 1.55,2.58; Trieste 1.32,3.3; Newlyn (UK) 1.91.4.00; Brest 1.03, 3.02; Key West 2.53,4.77; San Diego 2.21,3.3.
there are some tide gauges with records longer than 100 years. Of those practically all show acceleration in the last 30 years
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That spaghetti chart is a few years old now, but an update would probably show the same thing. Namely the rate of sea level rise over three decades or so tends to undulate porpoise or whatever you would like to name the 30 some odd years woggle.
Your post illustrates the point that extrapolating a mere 30 years or so of sea level rise out to 2100 is likely to produce large errors.
Moby ==> I fear, you are misunderstanding the forest and looking at this years growth on individual tree branches.
The Battery in NY continues it’s century long rise at about 3 mm per year. This includes the vertical land movement (downward) of about 1.3 – 1.5 mm/year. Like ALL tide gauge records, it has a lot of yer to year, decade to decade variation. Variation is no acceleration.
Moby ==> If you were to add the proper uncertainty range to the Battery graph, it would be between +/- 1 to 2 cm wide and swallows most of the variation.
Nerem makes the same mistake. Looks for acceleration in the annual/decadal variation.
The same is true of all the guages you mention.
Check back in 30-50 years and you’ll see.
What do you term the “proper uncertainty range”?
Phil ==> Tide Gauges have an original measurement error (uncertainty range) of +/- 2 centimeters — centimeters, not millimeters. Their 6 minute averages carry the same uncertainty range according to NOAA specs. And this refers to the most modern acoustic and laser tide gauges. Earlier tide gauges had much larger uncertainty ranges.
It is impossible to get rid of the uncertainty by averaging the time series….
See my essay: https://wattsupwiththat.com/2021/03/07/mind-over-math-throwing-out-the-numbers/
That’s the resolution of the instrument, they’re measuring a quantity with a daily fluctuation of about 2m multiple measurements certainly can yield a mean with better uncertainty than that.
Phil ==> The uncertainty of measurement is the uncertainty ….. it is pretense to claim a higher degree of accuracy or precision through “averaging” measurements taken at different times of a changing object (property).
Original measurement error can not be eliminated.
You’re calculating a mean value and the uncertainty of the mean does depend on the number of points:
“The average value becomes more and more precise as the number of measurements 𝑁 increases. Although the uncertainty of any single measurement is always ∆𝑥, the uncertainty in the mean ∆𝒙avg
becomes smaller (by a factor of 1/√𝑁) as more measurements are made.”
https://www.physics.upenn.edu/sites/default/files/Managing%20Errors%20and%20Uncertainty.pdf
I suggest you read it.
I think *YOU* are the only one that needs to read your link.
kip: “measurements taken at different times of a changing object (property).”
Uncertainty is a measure of accuracy, not precision. More measurements only defines the precision with which you can calculate the mean, it does not help with accuracy.
If you have *any* systematic error your precisely calculated mean from many measurements will still be inaccurate.
Look at “C” in your link. High precision, low accuracy. That’s what you get from measurements of different things by taking lots of measurements.
Look at “D”. High precision and high accuracy. You simply cannot get this from measuring different things.
Kip is correct here. Consider two piles of boards with an infinite number of boards in each pile. One pile has boards of length 4′ and the other 8′. You can take an infinite number of measurements, average them, and get a very precise mean, 6′.
But that mean will be very inaccurate because it won’t describe *any* of the boards in the distribution. Situation “C” – high precision but low accuracy.
(ps: this doesn’t even consider the uncertainty of the length of each individual board which will also affect the accuracy of that very precise mean you calculate)
That’s not what is being done here, you have a quantity which is varying continuously over time which is being measured at regular intervals (6 mins) and the average is calculated monthly. During that period the measurements are varying by about 2m to a resolution of 2cm.
How is that average calculated? (max-min)/2? You realize that is *not* an average of a sinusoid, right? Nor is the uncertainty of a set of measured values, each with an individual uncertainty, diminished by taking an average.
An average requires performing a sum. The uncertainty of a sum is additive, either directly or by root-sum-square. The uncertainty goes UP, not down. When you divide that sum by a constant the constant has no uncertainty so the uncertainty of the average remains the uncertainty of the sum.
The *only* way you can reduce the uncertainty is to ensure that *all* uncertainty is random and has a Gaussian distribution, i.e.. a +error for every -error. Then you can assume a complete cancellation of uncertainty. That is *very* difficult to justify for field measurements where the station can suffer from all kinds of problems such as hysteresis (i.e. the device reads differently going up than when coming down), non-linearity between max measurements vs min measurements, drift in calibration, etc.
When you are talking about trying to find an acceleration of mm/time^2 where the measurements have an uncertainty in cm, your signal gets lost in the uncertainty!
Phil ==> +/- 2 cm is the specification for the accuracy of each individual measurement by a modern tide gauge. Nothing to do with the variation of the tides. That accuracy defines the uncertainty range of the individual measurements, the 6 minute averages, he daily averages, the monthly averages, the annual averages…..
None of the tide gauge measurement are more certain than “X” +/- 2 cm.
The rule that more measurements can provide more accuracy/precision applies ONLY when you are making multiple measurements of the exact same object that is not changing.
Multiple measurement of a changing object have and retain the uncertainty of the individual measurements.
“+/- 2 cm is the specification for the accuracy of each individual measurement by a modern tide gauge. Nothing to do with the variation of the tides. That accuracy defines the uncertainty range of the individual measurements, the 6 minute averages, he daily averages, the monthly averages, the annual averages…..”
Not true, a measurement is made every 6 mins, the daily average is the average of 240 such measurements, the standard error of that mean is the standard deviation of those measurements divided by √240, (~1/15), the monthly value would be divided by √7200 (~1/80). The accuracy of the mean is not limited to the resolution of the instrument. By the way I understand that the accuracy specified by GLOSS for modern tide gauges has been changed to +/-1cm.
Phil ==> I originally specified an uncertainty range of +/- 1 to 2 cm.
The desired accuracy range and the actual accuracy range are not the same. You must read the technical specs of the tide gauge itself.
You are still stuck on the “accuracy of the mean” which is a simplistic mathematical idea and not a real world physical property. The dumbest modern computer can turn any series of numbers into an average (mean) with near infinite “accuracy”.
It is impossible to turn “many inaccurate measurements of different conditions at different times” into an real-world accurate mean. The mean retains the uncertainty of the original measurement error.
You confuse statistical ideas with the real world — and make the same mistake that many others make. No matter how many decimal points you feel are justified on the end of your calculated mean, you still must tack on the rel world uncertainty from the original measurements.
For tide gauge data, this is +/- 2 cm from the technical pecs of the actual measuring instrument.
“You are still stuck on the “accuracy of the mean” which is a simplistic mathematical idea and not a real world physical property.”
It certainly is a ‘real world physical property’, it tells you how close you are to the real mean of the population you are measuring.
“The dumbest modern computer can turn any series of numbers into an average (mean) with near infinite “accuracy””.
Certainly can not without additional data.
“It is impossible to turn “many inaccurate measurements of different conditions at different times” into an real-world accurate mean. The mean retains the uncertainty of the original measurement error. ”
It does not.
“You confuse statistical ideas with the real world — and make the same mistake that many others make. No matter how many decimal points you feel are justified on the end of your calculated mean, you still must tack on the rel world uncertainty from the original measurements ”
No you do not, you quote it to the first significant figure of the standard error of the mean.
“For tide gauge data, this is +/- 2 cm from the technical pecs of the actual measuring instrument.”
No!
Phil ==> It seems we will not agree — Tim Gorman (and those actually trained in statistics) is quite right in his comment :
https://wattsupwiththat.com/2022/05/14/sea-level-rise-acceleration-an-alternative-hypothesis/#comment-3517810
Many measurements of the same object at the same time under the same conditions can be used to improve measurement — reduce uncertainty.
Many measurements of different object (surface level of the sea at Point A) at a different time under different conditions can not be used to reduce the uncertainty range.
For tide gauges being used for their original purposes, this point is moot — as +/- 2 cm is sufficient accuracy and precision….which only need be in feet or meters.
I have had a lot of practice arguing this point over the last decade. Yours is a very very common misunderstanding and I’m sure that you understand it the way it was taught — which just happens to be incorrect. The details of the rule of many measurements are almost always overlooked. It must be multiple measurements of the same thing — the errors in measurement must be know to be entirely random and fall in a normal distribution. None of this applies to tide gauge data which never measures the same thing (the property ‘the surface height of the sea at Point A’) twice, the surface height of the sea at Point A is constantly changing, the forces causing the differences are random only in a chaotic sense not known to be unbiased in a systematic way. There is no known “right” answer, there is no way to know if the mean is approaching reality, so all measurement of sea surface height at Point A is ALWAYS best represented as the range of uncertainty surrounding the mathematical mean.
As one who is well trained in statistics I can say that:
Many measurements of different object (surface level of the sea at Point A) at a different time under different conditions can be used to reduce the uncertainty range of the mean value.
“I have had a lot of practice arguing this point over the last decade.”
I’m sure you have, doesn’t make you right.
There can also be issues with the measurement such as bias, drift etc. but those have to be dealt separately.
E.G.
full-jtech-d-18-0235.1-f4.jpg
https://journals.ametsoc.org/view/journals/atot/36/10/jtech-d-18-0235.1.xml
You are describing what statisticians call standard error of the mean. That is the basic point of misunderstanding as to what it is.
It should be named “standard deviation of the sample means”.
The other misunderstanding that is common among statisticians is that the standard deviation of the sample means can substitute for the final uncertainty of the mean caused by uncertainty in the individual elements making up the data set.
I’ve purchased quite a number of statistics books and perused even more. They *all* ignore the uncertainty of the individual elements when calculating the standard deviation of the sample means – EVERY SINGLE ONE! There are two possible explanations – 1. They assume that all error contributed by the individual elements of the data set are only random and perfectly fit a Gaussian distribution so that all of the uncertainty in the individual elements cancel, or 2. they just ignore the uncertainty in the individual elements out of ignorance or because they just don’t care.
If you have a million data elements and each one is 50 +5 (not +/-, just +). So each measurement can be anywhere from 50 to 55. Now you select 100 samples of 100 elements each and calculate the mean of the stated values in each sample – and you get a mean of 50 for every sample. So what is the standard deviation of the sample means? ZERO! A seemingly accurate mean calculated from the means of the individual samples IF you use the standard deviation of the sample means as your measure of uncertainty.
Yet you know that can’t be right! All of the actual measurements can’t be 50 since they have identical uncertainty of +5 – a systematic error perhaps induced by a faulty calibration process.
So what you get from all the samples is a very PRECISE number for the mean but that mean calculated from just the stated value while ignoring the uncertainty associated with each stated value is *not* accurate.
At a minimum your mean will have an uncertainty of +5. But since uncertainties that are not totally random and Gaussian add, either directly or by root-sum-square, your total uncertainty will be far higher. Uncertainties that are not purely random and Gaussian stack. The more uncertain measurements you take the higher the uncertainty gets.
This should be obvious to a statistician but somehow it never is. If you think about adding two random variables what happens to the variance? The total variance goes UP. How is it calculated?
V_t = V_1 + V_2
or as standard deviations
(σ_t)^2 = (σ_1)^2 + (σ_2)^2
So the combined σ = sqrt[ (σ_1)^2 + (σ_2)^2 ]
Combining uncertainties of measurements of different things is done exactly the same way – root-sum-square. Root-sum-square assumes that *some* of the uncertainties will cancel but not all.
Think about building a beam using multiple 2″x4″ boards to span a foundation. Each board will have an uncertainty. That uncertainty may not be the same if you have different carpenters using different rulers. Let’s say you use three boards end-to-end to build the beam. What is the total uncertainty of the length of the beam?
You simply can’t assume that all the uncertainty is totally random and Gaussian. Your beam may come up short. If it’s too long you can always cut it but that’s not very efficient, is it? And it still doesn’t cancel the uncertainty you started with!
In this case assume all the uncertainties are equal and are +/- 1inch. In this case they uncertainties directly add and your beam will be X +/- 3in where X is the 3 times the stated value of the boards.
Again, this should all be easily understood by a statistician. But it just seems to elude each and every one! It can only be that they are trained to ignore uncertainty by the textbooks. And that *does* seem to be the case based on the statistics textbooks I’ve read. The only ones that seem to learn this are physical scientists and engineers who live and die by proper evaluation of uncertainty. It’s especially true for engineers where ignoring uncertainty has personal liability (i.e. money!) consequences!
Tim and Jim ==> Thanks for the support — it bambozzles me that this point has to be made over and over and over and fought so hard by those who should know better.
Well as long as you keep getting it wrong you will need to be rebutted.
We’re talking about making measurements to estimate the mean of a quantity and we get a rambling example of using three boards to build a beam! To make it worse the example uses a weird constraint on the uncertainty so that the beam can only be X+3, +1, -1 or -3, how is that relevant?
The other example: “If you have a million data elements and each one is 50 +5 (not +/-, just +). So each measurement can be anywhere from 50 to 55.”
So that would be properly described as 52.5±2.5.
I set that up on Excel as a random number between 50 and 55 and ran some simulations of 100 samples
mean: 52.347 52.435 52.554 52.551 52.688
sd: 1.38 1.47 1.35 1.40 1.52
sem: 0.138 0.147 0.135 0.140 0.152
All sample means comfortably within the range of ±2sem as expected.
Out of curiosity I repeated it for samples of 400 and got the significantly narrower range of values
mean: 52.50 52.56 52.56 52.48 52.52
sd: 1.48 1.49 1.46 1.50 1.46
sem: 0.074 0.074 0.073 0.075 0.073
again comfortably within the range of ±2sem
“Well as long as you keep getting it wrong you will need to be rebutted.”
He’s not getting it wrong.
“The other example: “If you have a million data elements and each one is 50 +5 (not +/-, just +). So each measurement can be anywhere from 50 to 55.”
So that would be properly described as 52.5±2.5.”
Nope, you just changed the stated value, i.e. the value you read off the tape measure from 50 to 52.5. Did you change tape measures?
“I set that up on Excel as a random number”
In other words you are confirming the consequent. A logical fallacy. You set your experiment up to prove what you wanted to prove,
You changed the example to work out exactly how you needed to in order to assume random, Gaussian error. Ransom errors provided by Excel are assured to be Gaussian!
The point of the uncertainty being only positive was to highlight the fact that there would be no cancellation of random error. You changed it so you could assume random, Gaussian error without having to justify the assumption.
When you are measuring different things at different times you simply cannot just assume a random, Gaussian error distribution. That is what statisticians do but it is wrong! It’s why statistic textbooks never show any uncertainty values for any data elements in any distribution set – only stated values!
(p.s. if you think you can’t have only positive or only negative error then try thinking about using a gauge block marked 10mm while actually being 9mm (or 11mm) because of an error in manufacturing. Nothing random about that error, nothing Gaussian about it)
“The other example: “If you have a million data elements and each one is 50 +5 (not +/-, just +). So each measurement can be anywhere from 50 to 55.”
So that would be properly described as 52.5±2.5.”
Nope, you just changed the stated value, i.e. the value you read off the tape measure from 50 to 52.5. Did you change tape measures?
“I set that up on Excel as a random number”
In other words you are confirming the consequent. A logical fallacy. You set your experiment up to prove what you wanted to prove, ”
No I set it up exactly as described, you said “each measurement can be anywhere from 50 to 55”, so I generated a series of numbers that met that description.
“You changed the example to work out exactly how you needed to in order to assume random, Gaussian error. Ransom errors provided by Excel are assured to be Gaussian!”
As said several times I did not use a Gaussian error, I used the function RANDBETWEEN which generates a series of numbers between the max and min values. The ones I used approximated a uniform distribution, e.g. one series was as follows:
50-51.25 25 values
51.25-52.5 24 values
52.5-53.75 27 values
53.75-55 24 values
“The point of the uncertainty being only positive was to highlight the fact that there would be no cancellation of random error. You changed it so you could assume random, Gaussian error without having to justify the assumption.”
No, as shown above I set it up exactly as described by you, you asserted that the mean would be 50, as I showed that it would converge on 52.5. I can only assume that your description of the example wasn’t what you intended.
“When you are measuring different things at different times you simply cannot just assume a random, Gaussian error distribution.”
Which I did not do, as pointed out multiple times
“(p.s. if you think you can’t have only positive or only negative error then try thinking about using a gauge block marked 10mm while actually being 9mm (or 11mm) because of an error in manufacturing. Nothing random about that error, nothing Gaussian about it)”
No and it’s what a real scientist/engineer would eliminate by calibration.
I read through the study you linked. The authors appear to have mixed “accuracy” and “uncertainty” all together. Most of what they did was to detect inaccuracies as compared to to a reference or bias due to time.
Uncertainty appears in each and every measurement, even those made with reference devices. It is partly due to the resolution whereby there is no way to know what the value beyond the resolution actually is. It can also be due to systematic things that reduce the ability to match conditions when the measuring device was calibrated along with any drift.
Jim ==> Yes, exactly. With tide gauges, even the latest and most sophisticated, the resolution of measurement is in a range of 20 millimeters. 95% sure the actual sea surface in within that 20 mm somewhere, 1 out of 20 chance of being outside that range.
I think the problem that you don’t perceive is that the “population” you are sampling is not a fixed population. It is a composite of many additive/subtractive astronomical sinusoids with periods of up to at least 19 years, (including 209 centuries for the solar perigee) and some random weather-related parameters. Depending on the interval of time over which you take several samples, one will get very different numbers that lead to a large variance when you calculate the standard deviation. The mean will drift (actually oscillate) depending on when the samples were taken.
I perceive that and understand it very well, I first had to deal with the misleading comments here made with reference to a ‘fixed’ population. So the idea that the mean does not get closer to the true mean as the number of samples is increased has been successfully rebutted, hopefully we won’t hear that nonsense again.
Regarding the sinusoidal variation of tides, the major period is twice a day, that is sampled every six minutes so 240 times a day and the average is usually reported monthly (so 7200 measurements).
“ So the idea that the mean does not get closer to the true mean as the number of samples is increased has been successfully rebutted, hopefully we won’t hear that nonsense again.”
It hasn’t been rebutted. You keep assuming all error cancels (random and Gaussian) when that just isn’t true in the real world.
You set your examples up so the error cancels. You failed to address where the error does *NOT* cancel. So you didn’t actually rebut anything.
You even admitted that systematic (i.e. NON RANDOM) error exists in the measurements but apparently you think that will cancel as well! Sorry, but it won’t!
You didn’t even rebut the fact that uncertainty in the measurements which form a sine wave winds up with the uncertainty showing up in the average! You even assume the average of a sine wave is zero – i.e. it doesn’t oscillate around a set value as opposed to zero. What does zero mean in sea level?
“You are still stuck on the “accuracy of the mean” which is a simplistic mathematical idea and not a real world physical property.”
The mean is not a real world physical property. There is no instrument that will measure a mean. You have to calculate it.
The persistence of this dumb idea that more samples does not improve the accuracy of the mean is bizarre. Anyone actually trying to find something out, as in drug testing, say, pays a lot of money to get many measurements. They know what they are doing.
Say you have a coin, and want to test whether it is biased (and by what amount). One toss won’t tell you. You need many.
If you score 1 for heads, 0 for tails, what you do is take the mean of tosses. The mean of 2 or 3 won’t help much either. But if you take the mean of 100, the standard error if unbiased is 0.05. Then it is very likely that the mean will lie between 0.4 and 0.6. If it doesn’t, there is a good chance of bias. More tosses will make that more certain, and give you a better idea of how much.
This is all such elementary statistics.
All ==> And there lies the problem — statisticians see the world as a coin toss. And coin tosses are always heads or tails — no in-betweenness allowed. Stats guys think this is the ultimate real world example.
Sailors know more about tides and their variation, and their importance than a million statisticians.
Engineers understand exactly what I am talking about and agree with me — because they deal in the real world.
A coin toss has two certain values, a die has 6 certain values, polls have certain answers. They have infinitely accurate and precise discrete values. Probablilities rule the day.
None of these are continuous physical phenomena that have uncertainties when they are measured. Measurements have a limit of their resolution. Nothing is exactly accurate and precise.
Mathematicians that have no engineering background or even mechanical expertise do not understand what this means. They would build a Formula 1 engine by assuming that each and every measurement was exact and ignore the tolerances involved. I’ll bet none of them understand why valve lapping is done in an engine.
Good point. I suspect you are correct. Not one understands valve lapping or what plastigauge is used for, all journal bearings are exact with no uncertainty.
“The persistence of this dumb idea that more samples does not improve the accuracy of the mean is bizarre. “
What is bizarre is that you and the rest of the statisticians on here think that all error is random and cancels!
You apparently can’t even conceive that someone might be doing measurements with a gauge block that is machined incorrectly. Every measurement made with it will have a fixed, systematic error that is either all positive or all negative – i.e. no cancellation possible.
It simply doesn’t matter how many measurements you take using that gauge block or how many samples you take from those measurements. The means you calculate *will* carry that systematic error with it. When you have multiple sample means that are all carrying that systematic error then the population mean you calculate using the sample means will also carry that systematic error. The mean you calculate will *NOT* be accurate. The more samples you take the more precisely you can calculate a mean but that is PRECISION of calculation, it is *NOT* accuracy of the mean! The standard deviation of the sample means only determines precision, not accuracy.
See the attached picture. The middle target shows high precision with low accuracy. *THAT* is what you get from systematic error. Systematic error doesn’t cancel. And *all* field measurements will have systematic error. And if you are measuring different things each time the error will most likely not be random either.
The problem with most statisticians is they have no skin in the game like engineers do. I assure you that if you have personal liability for the design of something that physically impacts either a client or the public you *will* learn quickly about uncertainty. If you don’t you will be in the poor house in a flash! And perhaps in jail for criminal negligence if someone dies.
“Say you have a coin, and want to test whether it is biased (and by what amount). One toss won’t tell you. You need many.”
This is probability, not uncertainty. You can’t even properly formulate an analogy.
““The persistence of this dumb idea that more samples does not improve the accuracy of the mean is bizarre. “
What is bizarre is that you and the rest of the statisticians on here think that all error is random and cancels!
You apparently can’t even conceive that someone might be doing measurements with a gauge block that is machined incorrectly. Every measurement made with it will have a fixed, systematic error that is either all positive or all negative – i.e. no cancellation possible.”
As I have pointed out multiple times that’s not the problem being discussed. Real scientists and engineers such as myself deal with that problem by calibration.
Calibration drifts over time. Does each measuring device have its own dedicated calibration lab that is used before each measurement?
No reply. I can only guess that you didn’t consider drift in the measuring device. And you call yourself an engineer?
What you are discussing as Standard Error is often taught incorrectly and used incorrectly. Read this site:
https://byjus.com/maths/standard-error/
Let me summarize where some of the misunderstandings originate.
1) You must declare whether your data is the entire population or if it consists of a sample. This is important.
2) If it is a sample, then the mean of those samples is a “sample mean”. This is an estimate of the population mean.
3) The standard deviation of the sample(s) IS THE STANDARD ERROR. More importantly you DO NOT subdivide this standard deviation of the sample(s) by a number called N in order to reduce it.
4) Another name for Standard Error is Standard Error of the sample Mean, i.e. SEM.
5) The SEM/SE provides an INTERVAL within which the the population may lay. The Sample Mean is only an estimate of the population mean, not the true mean.
6) Now here is the key mathematically.
SEM/SE = σ / √N where
SEM/SE –> standard deviation of the sample distribution
σ –> Standard Deviation of the population
N –> is the sample size
7) Now that we have said that we have a sample, we can calculate the population statistical parameters.
Population Mean = Sample mean
Population σ = SEM/SE * √N
Note: the Central Limit Theory says that with sufficient sample size and number of samples, you will get a sample means distribution that is normal regardless of the distribution of the population data.
8) “N” is never the number of data points when dealing with a sample unless you have only one sample. In that case to find the population standard deviation you would multiply by the square root of the number of data points in your sample, say √4000.
9) If you declare your data to be the entire population then there is no reason to perform sampling at all. IOW, you don’t need to have a sample size. You just calculate the population mean and standard deviation as usual. The number of data points only enter into the standard deviation.
Here is another site to read:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1255808/
Play with this site to find out what happens with sampling.
Sampling Distributions (onlinestatbook.com)
“What you are discussing as Standard Error is often taught incorrectly and used incorrectly.”
Maybe so, however I was using it correctly, I consistently referred to the standard error of the mean.
“Maybe so, however I was using it correctly, I consistently referred to the standard error of the mean.”
Except the standard error of the mean is not the uncertainty of the mean. And it is truly better described as the standard deviation of the sample means in order to emphasize that it is not the uncertainty of the sample means.
No it is best described as a measure of how closely the sample mean will be to the actual mean, see the calculated example above.
“No it is best described as a measure of how closely the sample mean will be to the actual mean, see the calculated example above.”
If you don’t include the uncertainty of the individual elements in your calculation of each sample mean then you have no idea of how accurate that sample mean actually is. You must carry that uncertainty into the calculation of the sample means.
If your data is “stated-value” +/- uncertainty then your sample mean should be of the same form – “stated-value” +/- uncertainty.
Not doing this means you have made an assumption that all error is random and Gaussian. It may be an unstated assumption but it is still an assumption. It means you assume that the mean you calculate from the sample means is 100% accurate.
When you are measuring different things you have no guarantee that all error is random and Gaussian so you can’t just assume that the calculated mean of a sample is 100% accurate. Therefore you *must* carry that uncertainty through all calculations using that uncertainty.
I can only point you back to my example of an infinite number of measurements, each which come back as 50 + 5. If you pull 100 samples each with 100 elements and calculate the means of each sample using only the stated value of 50, then the standard deviation of the sample means will be zero. But there is no way that mean can be accurate.
You keep conflating precision with accuracy. They are not the same.
I showed the result of the calculation above, it included the uncertainty and showed the expected reduction in the standard error of the mean. Also that didn’t use a Gaussian distribution, it was ~uniform.
“I showed the result of the calculation above, it included the uncertainty and showed the expected reduction in the standard error of the mean. Also that didn’t use a Gaussian distribution, it was ~uniform.”
You changed the error into a random, Gaussian distribution using Excel. In other words you set your experiment up to give the answer you wanted.
You set it up so the error cancelled and you didn’t have to propagate it into the calculation of the sample mean. You assumed each sample mean to be 100% accurate.
That simply doesn’t work in the real world, only in the statistician world.
“You changed the error into a random, Gaussian distribution using Excel. In other words you set your experiment up to give the answer you wanted.”
You’ve shown yourself to have comprehension difficulties so I’ll repeat again it was not a Gaussian distribution, just a random value between the stated uncertainty limits.
“You set it up so the error cancelled and you didn’t have to propagate it into the calculation of the sample mean. You assumed each sample mean to be 100% accurate.”
No I used the error in the calculation of the sample mean, no such assumption was made.
Kip stated that the measured value could be any value between 50 and 55 so I generated a random series of numbers between 50 and 55, exactly as he specified.
I then calculated the mean and standard deviation of that series of 100 numbers.
mean: 52.347 52.435 52.554 52.551 52.688
sd: 1.38 1.47 1.35 1.40 1.52
sem: 0.138 0.147 0.135 0.140 0.152
All sample means comfortably within the range of ±2sem as expected.
Out of curiosity I repeated it for samples of 400 numbers and got the significantly narrower range of values
mean: 52.50 52.56 52.56 52.48 52.52
sd: 1.48 1.49 1.46 1.50 1.46
sem: 0.074 0.074 0.073 0.075 0.073
“Kip stated that the measured value could be any value between 50 and 55″
Sorry Kip, it should have been ‘Tim’
Phil, Jim, Tim, Nick ==> This little discussion has gone far afield. The mathematics of finding a mean are not in questions.
The real issue is that when Tide Gauges take a measurement, the number recorded is a representation of a range, 4 cm wide (two cm higher than the number recorded and 2 cm lower then the number recorded. That is notated as X +/- 2 cm.
Thus, tide gauge records can be subjected to “find the mean” — but it has to be a mean of the ranges, not the mid-points. You cannot pretend the the original measurement was a discrete number — it was a range — indicating that the instantaneous sea level was within that 4 cm range somewhere, but only 95% of the time. The recorded digits are used to represent a 4 cm wide range in which the true level of the could probably be found — 5% of the time it is fully expected to be outside the range.
Every mean that is a “mean of ranges” is itself a range. There is no known scientific way to narrow the range — ever. To claim so is pretense.
“Phil, Jim, Tim, Nick ==> This little discussion has gone far afield. The mathematics of finding a mean are not in questions.”
Actually they have been and I’ve been trying to keep the discussion to that question and avoid issues of calibration and building beams.
“The real issue is that when Tide Gauges take a measurement, the number recorded is a representation of a range, 4 cm wide (two cm higher than the number recorded and 2 cm lower then the number recorded. That is notated as X +/- 2 cm.
Thus, tide gauge records can be subjected to “find the mean” — but it has to be a mean of the ranges, not the mid-points.”
Which is exactly what I did, and as shown the uncertainty of the mean decreases as the number of samples taken increases.
The resolution of an instrument says that any value in the range X+/- 2 will be recorded as X. As long as the range of variation in the quantity being measured is greater than that resolution then the mean can be determined to an accuracy which depends on the number of readings taken. In the case of the Battery tide gauge the tidal range is about 2m which is measured every 6 minutes.
Phil ==> You may have your “uncertainty of the mean” — but you can’t get rid of the fact that no matter how many decimals you add to the end of your now very “certain” mean, you still have a number that represents only the mid-point of a range +/- 2 cm — or 40 millimeters wide.
Why this point makes a difference is that claims are made about single digit millimeters of “rise” or “acceleration”, which for decades all have overlapping uncertainty ranges….meaning that they are, for scientific purposes, considered to be the same….
So if you measure a tide with range of 2m using a measure which is marked every 4cm and reported as the midpoint of each 4cm interval it would be impossible to determine the mean more accurately than ±2cm?
Phil ==> One last time — you can determine the mean any way you like — to any precision your computer will allow — but you then are left with a result to which the uncertainty must be added — +/- 2 cm. which will represent the best estimate, from measurements, of the actual mean level of the sea surface at that location over that period of time.
OK i’ll use a ruler to measure sin(x)+1 with that resolution, equivalent to a tide of 2m from 0 to 2.
1.00 1.05 1.10 1.16 1.21 1.26 1.31 1.36 ……..
which will be measured as:
1.00 1.04 1.08 1.16 1.20 1.24 1.32 1.36 ……
So when I measure sin(x)+1 from 0-2π in 120 intervals with that resolution I get a mean of 0.999 instead of 1.000.
“So when I measure sin(x)+1 from 0-2π in 120 intervals with that resolution I get a mean of 0.999 instead of 1.000.”
You didn’t include an uncertainty interval for each measurement nor did you properly propagate the uncertainty to the mean.
I did it exactly the way one would do it if using the pole method of tide measurement with 4cm intervals as I defined it above. Recovers the mean of the sine wave to within 1mm.
inline-jtech-d-18-0235.1-f1.jpg
“I did it exactly the way one would do it if using the pole method of tide measurement with 4cm intervals as I defined it above. Recovers the mean of the sine wave to within 1mm.”
So what? You didn’t properly state the measurements using a stated value +/- uncertainty nor did you properly propagate the uncertainty into the mean.
“1.00 1.05 1.10 1.16 1.21 1.26 1.31 1.36 ……..
which will be measured as:
1.00 1.04 1.08 1.16 1.20 1.24 1.32 1.36 ……”
Where is your uncertainty?
Revised:
1.00 +/- 2cm, 1.05 +/- 2cm, 1.10 +/- 2cm, 1.16 +/- 2cm, 1.21 +/- 2cm, 1.26 +/- 2cm, 1.31 +/- cm, 1.36 +/- cm, …..
When you determine the mean the mean will come out as an example 1.18 +/- sqrt[ 8 * 2cm] = 1.18 +/- 4cm
Reading problems again?
The first row was the ideal sine series, no uncertainty:
1.00 1.05 1.10 1.16 1.21 1.26 1.31 1.36 ……..
The second row is the result that would be obtained if measured using a ruler with ±2cm resolution:
1.00 1.04 1.08 1.16 1.20 1.24 1.32 1.36 ……
The resulting mean of the measurements was 0.999 as opposed to the actual 1.000.
“Reading problems again?
The first row was the ideal sine series, no uncertainty:
1.00 1.05 1.10 1.16 1.21 1.26 1.31 1.36 ……..
The second row is the result that would be obtained if measured using a ruler with ±2cm resolution:
1.00 1.04 1.08 1.16 1.20 1.24 1.32 1.36 ……
The resulting mean of the measurements was 0.999 as opposed to the actual 1.000.”
You STILL didn’t do it right! Uncertainty doesn’t directly add or subtract from the stated value. The uncertainty is uncertainty and is handled separately from the stated value!
So if your ideal sine wave values were 1.00, 1.05, 1.10, 1.16, 1.21, 1.26, 1.31, and 1.36 then the *actual* measurements should be stated as I laid out – stated value +/- uncertainty!
1.00 +/- 2cm, 1.05 +/- 2cm, 1.10 +/- 2cm, 1.16 +/- 2cm, 1.21 +/- 2cm, 1.26 +/- 2cm, 1.31 +/- cm, 1.36 +/- cm, …..
And you get the result I gave: 1.18 +/- 4cm.
Remember, uncertainty is an INTERVAL. You don’t know the exact value within that uncertainty interval so you can’t add/subtract it from the stated value, it just remains an interval throughout.
You created a random, Gaussian error distribution and used that as the absolute error value when that is impossible in the real world where systematic error exists along side random error. That is why uncertainty is an interval and not a value.
“Which is exactly what I did, and as shown the uncertainty of the mean decreases as the number of samples taken increases.”
That is probably *NOT* what you did. If you used the RAND function you generated a Gaussian distribution of random error. In other words you set up a scenario in which all the error cancels! Exactly what doesn’t happen with a field measurement device that has both random and systematic error.
“The resolution of an instrument says that any value in the range X+/- 2 will be recorded as X.”
Resolution is *NOT* the same as uncertainty. The gauge can have a resolution of 0.1mm and still have an uncertainty of +/- 2cm. It’s stated value +/- uncertainty.
“As long as the range of variation in the quantity being measured is greater than that resolution then the mean can be determined to an accuracy which depends on the number of readings taken.”
You *still* have to follow the rules of significant digits. You are still confusing precision with accurace.
““Which is exactly what I did, and as shown the uncertainty of the mean decreases as the number of samples taken increases.”
That is probably *NOT* what you did. If you used the RAND function you generated a Gaussian distribution of random error.”
As stated multiple times that is exactly what did, I did not use the RAND function, learn to read!
“You’ve shown yourself to have comprehension difficulties so I’ll repeat again it was not a Gaussian distribution, just a random value between the stated uncertainty limits.”
What kind of distribution do you suppose Excel gives a set of random values?
“No I used the error in the calculation of the sample mean, no such assumption was made.”
You can’t do that! Uncertainty is an interval, not a value. How do you add uncertainty into a mean? You calculate the mean of the stated value and then propagate the uncertainty onto the mean. E.g. the mean winds up being X_sv +/- u_p where X_sv is the mean of the stated values and u_p is the propagated uncertainty!
“Kip stated that the measured value could be any value between 50 and 55 so I generated a random series of numbers between 50 and 55, exactly as he specified.”
Which tries to identify a specific value of error within the uncertainty interval. The problem is that you simply do not know the value of the error! That’s why uncertainty is given as an interval and not a value! When you generate a random variable within a boundary in Excel you *are* generating a Gaussian distribution which is guaranteed to cancel. If you think you didn’t create a normal distribution then tell us what Excel function you used because RAND gives a normal distribution.
““You’ve shown yourself to have comprehension difficulties so I’ll repeat again it was not a Gaussian distribution, just a random value between the stated uncertainty limits.”
What kind of distribution do you suppose Excel gives a set of random values?”
You really do have comprehension difficulties don’t you, the same one I’ve told I was using at least twice!
Randbetween, I even presented the distribution of a sample I used.
“When you generate a random variable within a boundary in Excel you *are* generating a Gaussian distribution which is guaranteed to cancel. If you think you didn’t create a normal distribution then tell us what Excel function you used because RAND gives a normal distribution”
I know I didn’t generate a Gaussian as stated above. I did say which Excel function I used and even quoted a sample, try reading.
https://wattsupwiththat.com/2022/05/14/sea-level-rise-acceleration-an-alternative-hypothesis/#comment-3519801
“I know I didn’t generate a Gaussian as stated above. I did say which Excel function I used and even quoted a sample, try reading.”
Actually I believe I misspoke earlier, both RAND and RANDBETWEEN appear to generate a uniform distribution, RAND between 0 and 1 and RANDBETWEEN between a bottom number and a top number and can generate negative numbers.
A RANDBETWEEN generated list, with a uniform distribution around zero, will do the exact same thing a Gaussian distributed list will do – the errors will cancel.
So, once again, you have proven that a symmetric set of error values will cancel.
And, once again, uncertainty is *NOT* a value, it is an interval. You *must* propagate uncertainty as uncertainty. You simply don’t know that all the errors will cancel, especially when you are measuring different things. It’s exactly like adding variances when combining two independent, random variables – the variances add.
Jim,
I think that an important point to be made is that you are basically talking about a constant, or some objects that have a nominal value. That is, 1″ ball bearings cluster around a fixed specification with a normal distribution.
However, variables such as temperatures or sea level surfaces change with time and a sub-sampled time-series will give a mean that varies depending on the time of sampling, and will change if the length of sampling time is changed. One can calculate a mean for a variable, but one must ask just what does the mean mean? Fundamentally, it becomes a smoothing operation if a moving window of time is used. One can’t improve precision if the target is moving.
Unless the time series encompasses the longest period affecting the tide (~20 years) there will be a drift in the calculated mean. That is, averaging over short periods may produce less accuracy than an instantaneous reading.
Does Nyquist ring a bell?
If a fixed object or parameter is measured many times with the same instrument, by the same observer, with all environmental variables held constant, then the precision can be improved by cancelling random measuring error.
However, if everything is in flux, the best you can do is an instantaneous measurement, which is limited by the resolution of the measuring instrument. You can average several measurements, but if the parameter is varying with time (as with tides) the mean will drift, not converge.
Try it and see, take a sinewave, period of 12 hrs, amplitude 2m, uncertainty ±2cm, sampling every 6 mins, you’ll find it’s fairly stable after 2 days.
It may be stable but it may not be accurate. If that +/- 2cm includes any systematic error, such as sensor drift over time, you may get a stable reading but it’s accuracy can’t be guaranteed.
If your curve is a true sine wave then each measurement will have a value of:
sin(x) ± u
Integrate that function to find the area under the curve from 0 to π/2
∫sin(x)dx ± ∫u dx
This gives:
-cos(x) ± ux evaluated from 0 to π/2
For the first term
-cos( π/2) = 0, the second term is just u(π/2)
-cos(0) = -1, ux = 0;
Second term subtracted from first term :
±u(π/2) – (-1) = 1 ± u(π/2)
Multiply by 4 to get the integral of the entire curve and you get
4 ± 2πu
divide by 2π to get the average and you wind up with
2/π ± u
You can’t just ignore the uncertainty in each measurement. It just keeps showing up, even in the average. In order to cancel error you would have to have several measurements at the same point in time and have the generated error values be random and Gaussian distributed. Since that doesn’t happen the error can’t be assumed to cancel.
“It may be stable but it may not be accurate. If that +/- 2cm includes any systematic error, such as sensor drift over time, you may get a stable reading but it’s accuracy can’t be guaranteed.”
We’re not discussing systematic errors, it has already been stated that error in calibration, drift and bias are different issues that have to be addressed in any measurement system. Stick to the subject, which is what is the error of the mean when measuring a quantity over time when there is an instrument uncertainty in the measurement.
“Multiply by 4 to get the integral of the entire curve and you get
4 ± 2πu”
Get the basic maths right, the integral of sin(x) from 0 to 2π is 0!
Not 4 times the integral of sin(x) from 0 to π/2.
If you did the maths correctly you get:
0+∑u/N where N is the number of measurements.
Since the uncertainty is both positive and negative then they tend to cancel out resulting in a value lower than the instrument uncertainty. If random and a symmetrical distribution (either Gaussian or uniform) then for sufficiently large N the error reduces to zero.
When discussing uncertainty how do you separate random error from systematic error? If you don’t know how to separate the two then you *are* discussing systematic error.
“Stick to the subject, which is what is the error of the mean when measuring a quantity over time when there is an instrument uncertainty in the measurement.”
The standard deviation of the sample means (what you call the error of the mean) is meaningless if you don’t know the uncertainty of the means you use to calculate the standard deviation of the sample means.
Not 4 times the integral of sin(x) from 0 to π/2.”
And just how does make the integral of uncertainty (u) equal to zero? “u” is positive through both the negative and positive part of the cycle. Or it is negative throughout the entire sine wave.
Even at the zero point you will still have 0 +/- u.
BTW, the average of a sine wave during the positive part of the cycle *is* .637 * Peak_+ value. During the negative part of the cycle it is -.637 * Peak_-. What is the average rise and fall of the sea level? Is it zero? If it is zero then how do you know what the acceleration in the rise might be?
This is a common mistake that statisticians make.
If each measured value is x +/-ẟx and y = (∑x)/N
then the uncertainty of the average is ∑ẟu + ẟN. Since the uncertainty ẟN = 0, the uncertainty of the mean is just ∑ẟu.
“Since the uncertainty is both positive and negative then they tend to cancel out resulting in a value lower than the instrument uncertainty.”
Again, uncertainty only cancels *IF* they are random and Gaussian. You’ve already admitted that the errors include systematic error such as drift, etc. Therefore the measurement errors cannot cancel. The error at t-100 won’t have the same error as t+100 because of drift, hysteresis, etc. The error distribution won’t be Gaussian, it will be skewed.
The ∑ẟu might be done using root-sum-square if you think there will be some cancellation of error but it can’t be zero. The error will still grow with each measurement. Primarily because you are measuring different things with each measurement. Errors only cancel when you are measuring the same thing with the same thing multiple times. And even then only if there is no systematic error.
If you are measuring different things, especially over time, error can only be Gaussian by coincidence.
Nor is uncertainty considered to be uniform. That’s another mistake statisticians make. In an uncertainty interval there is ONE AND ONLY ONE value that can be the true value. That value has a probability of 1. All the other values can’t be the true value because you can’t have more than one “true value”. Thus all the other values have a probability of zero. The issue is that you don’t know which value has the probability of 1. Thus the uncertainty interval.
If you are going to truly claim that sea level measurements only have random and Gaussian error then you probably are going to have to justify it somehow and retract your agreement that the measurement devices do have systematic error.
When discussing uncertainty how do you separate random error from systematic error? If you don’t know how to separate the two then you *are* discussing systematic error.”
It’s called calibration, I’ve already referenced a paper evaluating that for sea level measuring systems.
Kip’s false assertion was that the uncertainty of the mean evaluated by making multiple measurements of a quantity could not be reduced below the instrumental uncertainty of the instrument. That’s what’s being discussed so stop trying to change the subject.
This is a common mistake that statisticians make.”
Rubbish! ∑u is the sum of all the measurement errors and is divided by N to find the average.
“And just how does make the integral of uncertainty (u) equal to zero? “u” is positive through both the negative and positive part of the cycle. Or it is negative throughout the entire sine wave.”
The measurement was stated to have an uncertainty between -u and +u so for each measurement it can be either positive or negative. Also we’re talking about the mean of a sinewave not the mean of ∣sin(x)∣ which you were doing in your faulty maths.
“ Errors only cancel when you are measuring the same thing with the same thing multiple times.”
That’s exactly what we are doing, measuring the sea level with the same instrument which has the same range of uncertainty. Unless all the errors are positive or negative they must cancel to a certain extent. The more measurements one makes the closer to zero it gets.
“It’s called calibration, I’ve already referenced a paper evaluating that for sea level measuring systems.”
Calibration of field equipment is fine but the field equipment *never* stays in calibration. Even in the Argo floats, with calibrated sensors, the uncertainty of the float itself is +/- 0.6C because of device variations, drift, etc. You can’t get away from aging.
“Kip’s false assertion was that the uncertainty of the mean evaluated by making multiple measurements of a quantity could not be reduced below the instrumental uncertainty of the instrument. That’s what’s being discussed so stop trying to change the subject.”
Kip is correct. You simply can’t obtain infinite resolution in any physical measurement device. The numbers past the resolution of the instrument are forever unknowable. That resolution ability remains the *minimum* uncertainty. You simply cannot calculate resolution finer than that. It is a violation of the use of significant figures. No amount of averaging can help.
You are still stuck in statistician world where a repeating decimal has infinite resolution!
“The measurement was stated to have an uncertainty between -u and +u so for each measurement it can be either positive or negative.”
You *forced* that to be the case by restating the example I provided. There is simply no guarantee in the real world that error is random and Gaussian. You must *prove* that is the case in order to assume cancellation. You failed to even speak to the use of a gauge block that is machined incorrectly – how does that error become + and – error?
“Also we’re talking about the mean of a sinewave not the mean of ∣sin(x)∣ which you were doing in your faulty maths.”
And *YOUR* faulty math assumes the sine wave oscillates around zero. With a systematic bias the sine wave does *NOT* oscillate around zero.
“That’s exactly what we are doing, measuring the sea level with the same instrument which has the same range of uncertainty.”
Just like temperature, sea level changes over time. Thus you are *NOT* measuring the same thing each time. It is *exactly* like collecting a pile of boards randomly picked up out of the ditch or the land fill. You wind up measuring different things with each measurement. There is absolutely no guarantee that the average of the length of those boards will even physically exist. The average of those different things will give you *NO* expectation of what the length of the next randomly collected board will be. That is exactly like adding two random variables together – the variance of the combined data set increases and thus the standard deviation does as well, just like you do with uncertainty.
σ_total = sqrt[ (σ_1)^2 + (σ_2)^2 ]
Why would you think something different will happen with uncertainty associated with different things like sea level or temperature?
“Unless all the errors are positive or negative they must cancel to a certain extent. The more measurements one makes the closer to zero it gets.” (bolding mine, tpg)
“Certain extent” is *NOT* complete. If you don’t get complete cancellation then you simply cannot approach a true value using more measurements. The more measurements you have the more the uncertainty grows. If your error is +/- 0.5 and you cancel all but +/- .1 of it then the total uncertainty will be +/- 1 for ten measurements and +/- 10 for 100 measurements.
“Calibration of field equipment is fine but the field equipment *never* stays in calibration.”
That’s why you have maintenance and recalibration.
“You simply can’t obtain infinite resolution in any physical measurement device. The numbers past the resolution of the instrument are forever unknowable. That resolution ability remains the *minimum* uncertainty.”
But you can determine the ‘mean’ of a sample of readings of that device to better uncertainty than the device resolution. Which appears to be the point you fail to understand and bring in unrelated issues such as drift and bias.
““The measurement was stated to have an uncertainty between -u and +u so for each measurement it can be either positive or negative.”
You *forced* that to be the case by restating the example I provided.”
No you said: “If you have a million data elements and each one is 50 +5 (not +/-, just +). So each measurement can be anywhere from 50 to 55.”, which is exactly what I calculated.
“You failed to even speak to the use of a gauge block that is machined incorrectly – how does that error become + and – error?”
I certainly did address it I pointed out that such an operator error would be eliminated by calibration, and it’s irrelevant to the issue under discussion.
““Also we’re talking about the mean of a sinewave not the mean of ∣sin(x)∣ which you were doing in your faulty maths.”
And *YOUR* faulty math assumes the sine wave oscillates around zero. With a systematic bias the sine wave does *NOT* oscillate around zero.”
You stated that the integral of sin(x) from 0 to π/2 was 1 (which is correct) but then made an elementary error that the integral from 0 to 2π was four times that i.e. 4 which is nonsense, it is 0 as I stated. The bias/uncertainty term was a separate term in your equation the sine wave certainly does oscillate around zero.
“Just like temperature, sea level changes over time. Thus you are *NOT* measuring the same thing each time. It is *exactly* like collecting a pile of boards randomly picked up out of the ditch or the land fill. You wind up measuring different things with each measurement. There is absolutely no guarantee that the average of the length of those boards will even physically exist.”
Which is a terribly wrong analogy, measuring a continuously varying quantity in a time series is nothing like sampling from a collection of unrelated items. In the continuously varying quantity the mean certainly does exist.
“Certain extent” is *NOT* complete. If you don’t get complete cancellation then you simply cannot approach a true value using more measurements. The more measurements you have the more the uncertainty grows. If your error is +/- 0.5 and you cancel all but +/- .1 of it then the total uncertainty will be +/- 1 for ten measurements and +/- 10 for 100 measurements.”
But the error of the ‘mean’ which is what we are discussing will tend towards +/-0.1 (e.g. 10/100) which is less than the originally quoted uncertainty of +/- 0.5, so you appear to have proved my point.
I just ran a simulation of a series of measurements of a mean value of 10 +0.5/-0.4, after 10 measurements the mean was 9.95, after 30 10.03, 60 10.05 with the standard deviation stabilized at 0.06. So the asymmetric error cancellation does lead to a small bias in the mean but less than the uncertainty of the instrument.
“That’s why you have maintenance and recalibration.”
And just how often is the measuring device taken out of service and sent to a certified calibration lab for recalibration?
What happens in between recalibrations? Increased systematic uncertainty?
“But you can determine the ‘mean’ of a sample of readings of that device to better uncertainty than the device resolution. Which appears to be the point you fail to understand and bring in unrelated issues such as drift and bias.”
No, you can’t, not if you use significant figure rules. The average can only be stated to the resolution of the values used to determine the average. Anything else is assuming precision you can’t justify!
“So each measurement can be anywhere from 50 to 55.”, which is exactly what I calculated.”
No, you changed it to 52.5 +/- 2 in order to get a Gaussian distribution. Did you forget what you posted or just try to slip it by?
“I certainly did address it I pointed out that such an operator error would be eliminated by calibration, and it’s irrelevant to the issue under discussion.”
What happens to anything the gauge is used for before the next calibration? It is totally relevant and you are just using the fallacy Argument by Dismissal to avoid having to address it.
“4 which is nonsense, it is 0 as I stated”
Like I said, if you have systematic error the sine wave will not oscillate around zero. You keep making assumptions like a statistician and not a physical scientist or engineer working in the real world!
“Which is a terribly wrong analogy, measuring a continuously varying quantity in a time series is nothing like sampling from a collection of unrelated items. In the continuously varying quantity the mean certainly does exist.”
Another use of the fallacy of Argument by Dismissal. Since each measurement is independent you have the same situation – a collection of unrelated terms – which can have varying systematic errors based on measuring device parameters like hysteresis.
I’ve seen barges on the Mississippi make waves that lap at the shore for literally minutes after it passes. I’m sure the same thing happens to coastal sea level measuring devices. That alone will cause deviations from the sine wave in your measurements. So will storm fronts with winds that cause choppy water. The exact same things that keep temperatures from exactly matching the sine wave the sun follows in its path on the earth!
“And just how often is the measuring device taken out of service and sent to a certified calibration lab for recalibration?
What happens in between recalibrations? Increased systematic uncertainty?”
That would depend on the design of the equipment and it’s location, but it has nothing to do with the question of the propagation of errors when determining the mean.
““So each measurement can be anywhere from 50 to 55.”, which is exactly what I calculated.”
No, you changed it to 52.5 +/- 2 in order to get a Gaussian distribution. Did you forget what you posted or just try to slip it by?”
No it was a comment that that is how a scientist would describe it.
As I stated before I set it up exactly as described, you said “each measurement can be anywhere from 50 to 55”, so I generated series of numbers that met that description.
I used the function RANDBETWEEN which generates a series of numbers between the max and min values. The ones I used approximated a uniform distribution, e.g. one series was as follows:
50-51.25 25 values
51.25-52.5 24 values
52.5-53.75 27 values
53.75-55 24 values
Clearly not Gaussian!
“Like I said, if you have systematic error the sine wave will not oscillate around zero. You keep making assumptions like a statistician and not a physical scientist or engineer working in the real world!”
But no Physical scientist or engineer would multiply the integral of sin(x) from 0-π/2 by 4 to get the integral from 0-2π, they wouldn’t be that stupid. That’s what you did to get the value of 4, nothing to do with systematic error, just operator error!
““Which is a terribly wrong analogy, measuring a continuously varying quantity in a time series is nothing like sampling from a collection of unrelated items. In the continuously varying quantity the mean certainly does exist.”
Another use of the fallacy of Argument by Dismissal. Since each measurement is independent you have the same situation – a collection of unrelated terms”
They are a series off measurements made of a continuously varying quantity, so they are related.
“I’ve seen barges on the Mississippi make waves that lap at the shore for literally minutes after it passes. I’m sure the same thing happens to coastal sea level measuring devices. That alone will cause deviations from the sine wave in your measurements.”
Which design of the apparatus and siting of it would be done to minimize such events, of course such a wave would introduce an oscillating error. The data is presented as a monthly average of readings taken every 6 minutes. We’re discussing the effect of the resolution of the measuring instrument on the error of that mean, your introducing irrelevant issues shows the weakness of your case.
“But the error of the ‘mean’ which is what we are discussing will tend towards +/-0.1 (e.g. 10/100) which is less than the originally quoted uncertainty of +/- 0.5, so you appear to have proved my point.”
The issue is that the final error *will* grow if you don’t have complete cancellation. By the time you have five measurements you will be back to the original +/- 0.5! Add in more measurements and you mean becomes more and more uncertain! And how do you separate out random and systematic error represented by the uncertainty interval? Don’t use the copout of recalibration because you won’t know what the systematic error was during the interval between calibrations. That is why you use root-sum-square to add the uncertainties instead of direct addition – the assumption that *some* cancellation will occur.
“The issue is that the final error *will* grow if you don’t have complete cancellation. By the time you have five measurements you will be back to the original +/- 0.5! Add in more measurements and you mean becomes more and more uncertain!”
Nonsense, we’re calculating the ‘mean’, so we divide by N so the error of the ‘mean’ will be +/- 0.1.
“If you are going to truly claim that sea level measurements only have random and Gaussian error then you probably are going to have to justify it somehow and retract your agreement that the measurement devices do have systematic error.”
I made no such claim. Also I stated that systematic errors in a measuring device if they exist should be removed by calibration.
There is no requirement that the measurement error have a Gaussian distribution, they just need to be symmetrical.
“Also I stated that systematic errors in a measuring device if they exist should be removed by calibration.”
And I’ll repeat: Field equipment never remains in calibration. That’s why periodic re-calibration is required for anything associated with field measurements. Even lab equipment in a controlled environment ages and loses calibration – the re-calibration interval may be longer but re-calibration is still necessary.
“There is no requirement that the measurement error have a Gaussian distribution, they just need to be symmetrical.”
What distributions have symmetrical error where the mean and the median are equal?
“What distributions have symmetrical error where the mean and the median are equal?”
Uniform, symmetrical triangular, Bates and of course Gaussian are some which are used which come to mind
And are you claiming that your all measurements fit into Gaussian, uniform, and triangular distributions?
No, I was just answering your question. In my experience the Gaussian is the most common, sometimes uniform.
Kip,
I’ll speculate that there are possibly some changes in the satellite orbit and/or aging of the electronics is causing a drift in calibration — not unheard of with electronic systems.
In fitting a polynomial to data, the order of the polynomial is chosen by the person doing the fitting. A quadratic is a second order polynomial which means that it has only 1 bend and that both its ends tend off to infinity. Also, its second derivative (acceleration) is constant.
This might work for interpolation, but extrapolation 80 years into the future is laughable. If nothing else, the point of Nerem et al is that SLR acceleration is new, because of global warming. But quadratics have a constant second derivative (acceleration doesn’t change). A quadratic might fit a limited section of the curve, but YOU CANNOT USE IT TO EXTRAPOLATE.
Thanks Roger. In full agreement with you.
When Nerem 2022 came out you may have heard Kip and myself laughing what ever side of the Pond you are on. At least in this, as I said previously, his period of extrapolation is reducing – 80 years in 2018, 30 years in 2022. Soon, hopefully, he may be down to zero extrapolation and be able to concentrate on the real Physics.
If nothing else, the point of Nerem et al is that SLR acceleration is new,
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BINGO! Climate science and the reports in the media imply that every bump and wiggle in the data for polar bears, tornados, hurricanes, droughts, floods, forest fires, ice caps, hardiness zones, prostitution etc. are new and caused by climate change i.e., carbon dioxide in the air which they call carbon.
for those who are wondering
y = x^2
first derivative = 2x
second derivative = 2 ; i.e. a constant
Yep – and there, in secondary school maths, endeth the validity of Nerem et al.
Kip Hansen has corrected my Typo when I was referring to Fig 10.
He has also added a figure after fig 11 based on Fig 2 in Nerem’s 2022 paper (Extrapolating Empirical Models of Satellite-Observed Global Mean Sea Level to Estimate Future Sea Level Change) where he shows the historic values of “acceleration” based on quadratic curve fitting from 1993 up to year ends. I have added the Red dots which are the identical process applied to a good fitting sinusoidal curve. This set of dots is NOT a sinusoidal curve but a portion of that shown in figure 6 of the above essay. Prior to 2012 the data were influenced by possible El Nino and La Nina influences such that “accelerations” would be unrepresentative.
Nerem stops at the start of 2021 (I think his axis refers to start of years as the “accelerations” peaked at start of 2020) whereas analyzing up to start 2022 shows another few % reduction. He does refer to the curve leveling off.
The red dots drop by about 8% over period start 2020 to start 2022 as did the analysis of the NASA data and could drop another 12% or more over the next 2 years. This new figure can be compared with the RHS of Figure 4 in the essay.
Prediction is a dangerous process but I’m too long in the tooth to worry about that – hope to see if I’m right in 2024!!