-1=e^ipi

I've tried to sharpen / remove the exponential decay of the Volcanic Aerosol data to get a better approximation of Volcanic CO2 emissions. Using data from 1876 to 2009 I estimate that the 95% confidence interval for the decay time for Volcanic Aerosols is (2.04 +/- 0.42) years. Of course I used a 6 month lag so this might be a slight underestimate. It agrees with the literature, which suggests that the decay time is 2-3 years for volcanic aerosols. Removing the exponential lag allows you to see volcanic eruptions such as Mount Tarawera (1886) that were hidden in other eruptions (in this case Krakatoa (1883)).

See post #14. It gives a link. That value is ~1.15 C. You are definitely on the right path. With water vapor it comes to around 1.74 C (see link in post 14). Definately. Obviously it gets more complicated with lapse rate feedbacks, polar amplification effects, etc. The link in post #14 is obviously a simplification.

That's strange. I was under the impression that such a conclusion would be impossible regardless of the results of the regression because the regression being referred to (that you are quoting) is about explaining changes in atmospheric CO2 over time, not changes in temperature due to increased atmospheric CO2. That's sort of like doing a study on the effects of eating ice cream and then concluding that alligators cause cancer.

Okay, derived everything and implemented the code into matlab. I also figured out a way to 'sharpen' the volcanic aerosol data to remove the time delay. I would like to correct a mistake from earlier. Even after I calculate everything, it won't be possible to get the characteristic temperature of the Earth's heat sink because the factor C cannot be evaluated. However, I can still get C*(T-Ts), which means I can get the temperature difference between the heat sink temperature and the surface temperature up to a scalar constant. It is still useful. For example, if this value has decreased rapidly in the past 15 years compared to the 90's, then that would suggest that more heat has 'gone into the oceans' which would help explain the slowdown in global warming. It would also be interesting to see how this parameter compares to length of day oscillations, the El Nino Southern Oscillation, the Pacific Decadal Oscillation and the Atlantic Multidecadal oscillation. Of course I can use the computed C*(T-Ts) in a linear regression to explain temperature increase (since a transfer of heat from the surface to the heat sink corresponds with a reduction in surface temperature). Using that coefficient, I could compute the characteristic heat sink temperature provided I make an assumption about the heat capacities of the surface vs the heat sink (say by comparing the quantity of air in the troposphere with the amount of water on Earth). But what is ultimately relevant is the rate of heat transfer. The factor D, which is the characteristic decay rate of the heat sink to the surface temperature gets computed from the regression though, which is very useful.

@ eyeball - This isn't the appropriate thread to discuss my financial situation.

So I guess questions like 'what is climate sensitivity?', 'what is the polar amplification factor?', 'how will jetstreams be affected by increasing atmospheric CO2?', 'How much ocean acidification will there be?', 'how will the CO2 fertilization effect influence things?', 'what is the net-benefit of mitigation policies vs alternatives such as adaptation or geoengineering?', etc. do not matter. In that case I guess we should defund all climate science since the only relevant question is whether or not GW exists. *sarcasm* I can't afford a car nor do I drive.

Okay, after reading up on non-linear regressions I think I got it now. I'll spare you guys the calculations but basically, Perform OLS regression without the 3 restrictions. Modify OLS coefficients to satisfy the 3 restrictions such that the increase in the sum of squares of error is minimized. Apply Guass-Newton method for estimating non-linear regression.

Tried to solve for the estimates that minimize the variance from the first order conditions. After a while, I realized that to get the estimates, I would need to solve polynomials of order 5 or higher. Then I remembered that by Abel's impossibility theorem, this isn't possible for arbitrary coefficients. Heck even trying to get estimates for A and B from a simple regression such as Y = AX + BY + ABZ + error is not possible due to Abel's impossibility theorem. http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem I guess I need to take a numerical approach. http://en.wikipedia.org/wiki/Non-linear_least_squares

Nevermind. I can't do a simple linear regression because the restrictions are non-linear. So I would have to solve A, B, D, E, and F that minimize the variance directly from the 5 first order equations. Maybe I'll have time next week to do this.

I figured out a way to estimate a characteristic heat sink temperature of the Earth. If we think about changes in CO2 concentrations over time, there are 4 main reasons why this can occur. Humans, Volcanoes, changes in the temperature of Oceans + Permafrost regions (which I will refer to as the characteristic heat sink of Earth), and other natural factors. Human Emissions, which are available: http://cdiac.ornl.gov/trends/emis/tre_glob.html Volcanic Emissions, which should be roughly proportional to volcanic aerosol emissions. Aerosol data is available here: http://data.giss.nasa.gov/modelforce/strataer/tau.line_2012.12.txt Other natural factors, which we can approximate with a constant term and a linear term proportional to CO2 concentrations. I.e. (dCO2/dt)_natural = A + B*CO2, where A and B are constants. A linear function (or a polynomial of higher order) is needed because we know that the Earth was in rough equilibrium at the end of the little ice age. So if there are no human emissions, no changes in Solar activity, no changes in the temperature of the Earth and its oceans then we get: 0 = dCO2/dt = (dCO2/dt)_natural + (dCO2/dt)_volcanic + (dCO2/dt)_human + (dCO2/dt)_heat_sink = A + B*CO2 + (dCO2/dt)_volcanic + 0 + 0 => A = -B*270 - mean((dCO2/dt)_volcanic) since the Earth had ~270 ppm CO2 in preindustrial times. mean((dCO2/dt)_volcanic) would be the mean CO2 emissions from volcanoes. The tricky part is the CO2 emissions due to the characteristic heat sink of the Earth (oceans + permafrost). Since CO2 solubility in water decreases with temperature, and the the change in global mean temperature since preindustrial times is relatively small (0.8 C), the CO2 emissions due to the characteristic heat sink should be roughly proportional to the change in temperature. So one gets (dCO2/dt)_heat_sink = C*dTs/dt, where C is a constant, and dTs/dt is the change in the characteristic heat sink temperature with respect to time. Furthermore, since the change in temperature of the characteristic heat sink of the earth is primarily due to heat exchange with the surface of the Earth, and it is known that for conduction and convection that the rate of heat transfer is roughly proportional to the temperature difference, we get: (dCO2/dt)_heat_sink = C*D*(T - Ts), where D is a constant, T is the surface temperature, and Ts is the characteristic heat sink temperature. Unfortunately, the characteristic heat sink temperature is not known. Using the fact that total change in CO2 is the sum of the effects of humans, volcanoes, changes in the heat sink and other natural factors, one gets: dCO2/dt - (dCO2/dt)_human - (dCO2/dt)_volcano - A - B*CO2 = C*D*(T - Ts). (1) Alternatively, one can write this as: dCO2/dt - (dCO2/dt)_human - (dCO2/dt)_volcano - A - B*CO2 = C*dTs/dt. (2) Integrating the above equation from time = 0 to time = t gives: CO2(t) - CO2(0) - Integral(0 to t; (dCO2/dt)_human*dt) - Integral(0 to t;(dCO2/dt)_volcano*dt) - At - B*Integral(0 to t;CO2(t)*dt) = C*(Ts(t) - Ts(0)). (3) Edit: I make a very critical mistake here, which I don't realize until post #149. So some of the conclusions I get until post #149 are nonsense. Substituting (3) into (1) gives: dCO2/dt - (dCO2/dt)_human - (dCO2/dt)_volcano - A - B*CO2(t) = D*(CO2(t) - CO2(0) - Integral(0 to t; (dCO2/dt)_human*dt) - Integral(0 to t;(dCO2/dt)_volcano*dt) - At - B*Integral(0 to t;CO2(t)*dt)) Rearranging this gives: dCO2/dt = (A - D*CO2(0)) + (B + D)*CO2(t) + (dCO2/dt)_human + (dCO2/dt)_volcano + D*Integral(0 to t; (dCO2/dt)_human*dt) + D*Integral(0 to t; (dCO2/dt)_volcano*dt) - (AD)*t - (BD)*Integral(0 to t;CO2(t)*dt)) Now the change in CO2 ppm due to Humans is going to be proportional to human emissions. So (dCO2/dt)_human = E*CO2_emissions, where E is a constant. And the change in CO2 ppm due to volcanoes is going to be roughly proportional to volcanic aerosol emissions. So (dCO2/dt)_volcano = F*Volcanic_Aerosols, where F is a constant. So one gets dCO2/dt = (A - D*CO2(0)) + (B + D)*CO2(t) + E*CO2_emissions + F*Volcanic_Aerosols + E*D*Integral(0 to t; CO2_emissions*dt) + F*D*Integral(0 to t; Volcanic_Aerosols*dt) - (AD)*t - (BD)*Integral(0 to t;CO2(t)*dt)) So that means that one can estimate these unknown parameters using a simple linear regression. Note that there are 8 dependent variables but the coefficients are all combinations of A, B, D, E and F. This means that one should do a restricted regression with 3 restrictions.

Update: I had some time so I got data sets on Temperature, CO2 concentrations, North Atlantic Multidecadal Oscillation, Pacific Decadal Oscillation, El Nino Southern Oscillation, Length of Day, Solar Irradiance and Stratospheric Aerosol Optical Thickness that all covered the years 1876-2014. I performed the regression and got an equilibrium climate sensitivity with a 95% confidence interval of (1.98 +/- 0.92) Celcius. I could probably do some minor tweaking (particularly get better solar irradiance data) to improve the estimate. The estimate of the characteristic decay time is way too low, so the assumption of constant decay might be biasing the climate sensitivity estimate downwards. I might have to start considering different decay times for different mechanisms; maybe I should look at the response functions of GCMs.

Depend what kind of supremacism and what the context is. I would argue that liberal democracies are superior to theocracies for example (which would make me a liberal-democratic supremacist).

No, I'm saying that one data set has a lower quality than another. Are you trying to suggest that you want statisticians to detect which data points are lying and which aren't or something? Yes, there are issues either way. But the inaccurate data is still better than no data. It depends what you are doing. If you are just trying to find an average or a percentage, then fine. But if you are trying to do some sort of regression that is trying to explain one factor using another and while controlling for additional factors, it can quickly become insufficient.

It doesn't mean that the 2006 data is good, it just indicates that the quality of the 2011 data is probably worse than the 2006 data.

It may be more accurate, but your sample size is smaller and it is less representative. If the non-response rate is correlated with a factor you are studying, then that can severely limit or skew results of a study. As for smaller sample, it is very simple. If you are trying to measure say the mean income of a population of people, then the standard deviation of your sample mean is roughly inversely proportional to the squareroot of the population. So larger sample -> more confident results.

I don't want to go into too much detail, but I'll give you a simple example. Let's say data on # of hours worked. There is variation and uncertainty on this data due to people lying. If you take this data and control for factors that can explain hours worked (say race, gender, location, education, profession, experience, etc.) and you find that the variance of the residual is much higher for the 2011 NHS than the 2006 Census, then is a strong indication that more people are lying or people are providing less accurate data.

Tim, I know researchers that deal a lot with income data from the census and have worked with it a bit myself. They know of these issues and there are ways of trying to account for them and to deal with them. They also acknowledge the limitations of the data. In any case, inaccurate data is far superior to no data, provided you understand the limitations of the data. It is better than no information.

Very dumb decision to abolish it. A few months ago I ended up having to use data from the 2006 census rather than the 2011 National Household Survey due to differences in data. Needless to say, time difference from 2006 to my other data made all my results were inconclusive (I performed many tests as to why things were inconclusive and it was due to having to use the 2006 census). Because it's no longer as representative of a sample.. Also, there are ways to test changes in the quality of data; you can look at changes in the variance of the responses. Oh yes. I know this so well when I was trying to collect various comparable data sets from both countries in a comparative study last semester. Literally takes 20x longer to find equivalent data on Canada, if it exists. Canada, why do you make data so hard to find?!

Really? So Han Supremacism and Sinocentrism do not exist? Perhaps you should ask historians in Korea, Vietnam or Tibet. As for 'muslim empires' it was more about muslim supremacism rather than racial supremacism. That is why non-muslims had to pay the jizya, could not build places of worship taller than nearby mosques, could only upgrade places of worship with permission of the Caliphate, muslim men could marry non-muslim women (and the children would become defined as muslim by the state) but non-muslim men could not marry muslim women, non-muslims could be slaves but muslims could not, and people were killed for converting away from islam but not killed for converting to islam. What does this even mean? Which group of people are you referring to as 'Arabs' and which 'African traits' do they have? Are you referring to Egyptians, who are both Arabs and Africans? No colonies? Ever heard of Korea, Manchuria, Vietnam, Tibet, Southern China (which wasn't originally han), Tibet, East Turkestan, Southern Mongolia, or Taiwan? Malthus' predictions have been falsified. Or maybe population increases were the result of increases in technology allowing more people to live in the same area and the desire for conquest was always there as it had been for thousands of years across many cultures. Yes. And now 'progressivism' is the dominant ideology that has completely infected the politics, education system, media and everything in Western Countries (with very few exceptions such as the US south). This statement goes completely against empirical evidence. Economic resources per capita was higher for Europeans at times of colonialism than times prior to that and also compared to other countries.

Also, I just realized that I made a mistake by saying that the assumption of constant decay towards equilibrium will result in an overestimation of the equilibrium climate sensitivity (ECS). It would result in an overestimation of earth system sensitivity (ESS) if I interpreted the equilibrium to mean the ESS rather than the ECS. Sorry, I must have confused ECS with ESS. In any case, there is no reason to expect that 2.95 C is an overestimate of ECS due to the assumption of constant decay towards equilibrium.

Actually, there is a really simple way to combine the approach I did in the original post with the CSALT approach. If we start with the basic assumption that temperature change is proportional to the difference between the equilibrium temperature and the current temperature: dT/dt = k(Te(t) - T(t)), where k is some constant, t is time, Te is the equilibrium temperature, and T is the current temperature. And then we suppose that the equilibrium temperature depends linearly on the logarithm of CO2, volcanic aerosols, solar irradiance, the Atlantic MultiDecadal Oscillation, and the Pacific Decadal Oscillation (so we complicate my model in the original post by adding the other explanatory factors mentioned in CSALT) then we get: Te(t) = A + B*ln(CO2(t)) + C*Irradiance(t) + D*Aerosols(t) + E*AMO(t) + F*PDO(t), where CO2 is the atmospheric CO2, Irradiance is the solar irradiance, Aerosols is the volcanic aerosols, AMO is the Atlantic MultiDecadal Oscillation, PDO is the Pacific Decadal Oscillation, and A, B, C, D, E, and F are all constants. So then you get dT/dt = G + H*ln(CO2(t)) + I*Irradiance(t) + J*Aerosols(t) + K*AMO(t) + L*PDO(t) - kT(t) , where G, H, I, J, K and L are constants. This is basically the CSALT model derived from the Gibbs energy formulation but with temperature added as an explanatory variable. So all you would need to do is perform a simple linear regression to estimate this model. And you can directly get the equilibrium climate sensitivity from this; it is H/k*ln(2). Not to mention the error of this estimate is really easy to obtain. So you don't even need fancy general circulation models that run on super computers, try to directly evaluate the effects of various feedback mechanisms (such as clouds or the lapse rate, which can be difficult), or do crazy differential calculus that I was doing earlier in order to obtain a good estimate of climate sensitivity. Just perform a simple linear regression then divide two numbers and multiply by ln(2). It's so simple! In addition, you could probably use this model to make predictions about the future given an emission scenario. For the natural fluctuations in solar irradiance, the Pacific Decadal Oscillation, or the Atlantic Multidecadal Oscillation, you could just perform fourier analysis on each of these, take note of the peak frequencies, fit sinusoidal waves of these peak frequencies to the natural fluctuations to use as an estimate of future natural fluctuations, and then put that in your model to get future global temperature.

As for the CSALT model, I think you miss its relevance here. It shows that the majority of the variability in the climate that isn't explained by the Loehle paper or by my corrections can be explained by things like volcanic activity, variations in solar irradiance, North Atlantic Multidecadal Oscillation and the Pacific Decadal Oscillation. Now arguably, Loehle does take into account most of the two ocean oscillations using his fitting technique for 20-year and 60-year sinusoidal waves; but as Cawley pointed out, the 20-year and 60-year periods may not be accurate and as the CSALT model shows, it is probably far better to input the empirical data directly for these oscillations. I do have a few issues with the CSALT model though. The first is that the choice of lags is somewhat arbitrary and does not have much physical justification. The second is that I think that the temperature of the Earth should go into the regression model since the closer you get to the equilibrium temperature, the lower the magnitude of temperature change. The third is that putting in both the length of day variation and the Southern Oscillation Index are very strongly correlated; so I'm a bit skeptical of putting them both in the regression model.

Interestingly, 'curve-fitting exercise' is exactly what Cawley et al. said about the Loehle paper. http://www.sciencedirect.com/science/article/pii/S0304380014004876 But why do you consider it a lower bound? Just because it is lower than what other papers say? It is hard to refute the data, and if the methodology were good and relatively simple then it would make sense to reject the other papers (which have a lot more assumptions that may bias their estimates upwards or underestimate the error of these estimates) and accept the Loehle paper. Also, even if you did consider Loehle a lower estimate, that doesn't suggest you should treat 1.99 C as the lower bound. Rather you should do that for the lower end of the 95% confidence interval (which would be 1.51 C assuming that Cawley is right about Loehle understating the error by a factor of 2). I think it's better to directly look at the science and the assumptions made. Loehle being a skeptic doesn't mean he is biased or wrong. Rather, the simplifying assumptions make the paper biased and inaccurate as I showed.

Well more accurately, I would say that the consensus estimates are validated. The uncertainty should be large enough such that any difference between an equilibrium climate sensitivity of ~3C and the corrections to Loehle's estimate is statistically insignificant. Of course, I would have to use solar irradiance data to make the correction to Loehle to see if this is the case.

@ TimG - If you liked the Loehle approach/model then you might like this CSALT model since it also tries to approach the question of climate sensitivity via time series data, but the analysis is better and it uses more empirical data. http://contextearth.com/2013/10/26/csalt-model/ Edit: It also gets an equilibrium climate sensitivity of approximately 3 C.