# A Probability Distribution Function of the Transient Climate Response to Cumulative Emissions

Posted on: November 10, 2020

THIS POST IS A CRITICAL REVIEW OF Spafford&MacDougall 2020: A Probability Distribution of the Transient Climate Response to Cumulative Emissions.

Spafford, Lynsay (THE DOCTORAL STUDENT), and Andrew H. MacDougall (THE PROFESSOR). “Quantifying the probability distribution function of the transient climate response to cumulative CO2 emissions.” Environmental Research Letters 15.3 (2020): 034044. ABSTRACT: The Transient Climate Response to Cumulative CO2 Emissions (TCRE) is the proportionality between global temperature change and cumulative CO2 emissions. The TCRE implies a finite quantity of CO2 emissions, or carbon budget, consistent with a given temperature change limit. The uncertainty of the TCRE is often assumed be normally distributed, but this assumption has yet to be validated. We calculated the TCRE using a zero-dimensional ocean diffusive model and a Monte-Carlo error propagation (n = 10 000 000) randomly drawing from probability density functions of the climate feedback parameter, the land-borne fraction of carbon, radiative forcing from an e-fold increase in CO2 concentration, effective ocean diffusivity, and the ratio of sea to global surface temperature change. The calculated TCRE has a positively skewed distribution, ranging from 1.1 to 2.9 K EgC−1 (5%–95% confidence), with a mean and median value of 1.9 and 1.8 K EgC−1. The calculated distribution of the TCRE is well described by a log-normal distribution. The CO2-only carbon budget compatible with 2 °C warming is 1100 PgC, ranging from 700 to 1800 PgC (5%–95% confidence) estimated using a simplified model of ocean dynamics. Climate sensitivity is the most influential Earth System parameter on the TCRE, followed by the land-borne fraction of carbon, radiative forcing from an e-fold increase in CO2, effective ocean diffusivity, and the ratio of sea to global surface temperature change. While the uncertainty of the TCRE is considerable, the use of a log-normal distribution may improve estimations of the TCRE and associated carbon budgets. LINK TO FULL TEXT PDF: https://iopscience.iop.org/article/10.1088/1748-9326/ab6d7b/pdf

PART-1: RELATED POSTS ON THE TCRE AND TCRE CARBON BUDGETS.

1. Illusory statistical power in time series analysis: LINK: https://tambonthongchai.com/2019/04/30/illusory-statistical-power-in-time-series-analysis/
2. Statistical issues in the TCRE: https://tambonthongchai.com/2018/05/06/tcre/

2. TCRU, a parody of the TCRE: https://tambonthongchai.com/2018/12/03/tcruparody/

3. A mathematical inconsistency: https://tambonthongchai.com/2020/08/26/a-mathematical-inconsistency/

4. The TCRE and its carbon budgets: https://tambonthongchai.com/2019/08/06/tcrebudget/

5. Carbon budget mysteries created by the TCRE: https://tambonthongchai.com/2020/10/13/carbon-budget-mystery/

6. Statistical flaws create climate science confusion: https://tambonthongchai.com/2020/04/09/climate-statistics/

7. Earth system models derived from the TCRE: https://tambonthongchai.com/2020/08/25/earth-system-models-and-carbon-budgets/

8. A SIGNIFICANT IMPLICATION OF THESE RELATIONSHIPS FOR CLIMATE SCIENCE IS THAT A TIME SERIES OF THE CUMULATIVE VALUES OF ANOTHER TIME SERIES, AS IN THE TCRE, HAS NEITHER DEGREES OF FREEDOM NOR TIME SCALE. THERFORE THE TCRE DOES NOT CONTAIN USEFUL INFORMATION.

PART-2: THE MULTIPLICITY ISSUE

1. In correlation and regression analysis of time series data each data value has a unique place and role in that time series sequence. This structure is dramatically altered when the same data item appears more than once in the series. For example, in the famous Kerry Emanuel paper on the PDI measure of the destructiveness of hurricanes he used a 30-year study period 1975 to 2004 but the rising trend that he was looking for did not have a statistically significant trend. So instead of annual mean PDI, he used a moving decadal window that moved through the time series one year at a time and computed a time series of decadal means. He lost the first 9 years of the time series but he still had a sample size of n=30-9=21 years and a smooth decadal mean PDI curve with very little variance and that gave him the the statistical significance he was looking for. Or so he thought.
2. The statistical power that had apparently been gained without the help of additional information, derives from multiplicity of use whereby the same data item is used more than once when the time series is pre-processed to create a time series of moving averages.
3. Preprocessing of time series data in this way serves a useful purpose in time series analysis but the further use of the preprocessed series for computing probability in hypothesis tests or for constructing confidence intervals reduces the degrees of freedom by virtue of multiplicity.
4. Multiplicity derives from repeated use of the same data item in the source data series for the computation of multiple items in the filtered series. This issue is described in terms of hurricane frequency data shown below. Here, the red line is a 5-year moving average of the source data in black.

The visual indication in the chart above is that the filtered series indicated by the red line contains less uncertainty and more information than the source data indicated by the black line. The apparent reduction in uncertainty and the implied gain in information and statistical power is illusory because our information is unchanged. No new information was gathered in the moving average process. This illusion of increased statistical power was created by multiplicity in data usage.

When moving windows are used in this way, the first and last data points are used only once but the other data values in the middle are used more than once. Therefore, an adjustment of the effective sample size and degrees of freedom in the filtered time series is necessary to account for multiplicity.

A moving window of length λ advancing by an increment of one time unit through a time series of length N will generate a total of N-λ+1 windows. Since each window contains λ numbers, a total of λ(N-λ+1) numbers are used by the moving window. Yet, there are only N numbers in the time series. Therefore, the average multiplicity is M = (λ/N)(N-λ+1). Each number in the series is used M times on average. The effective value of N is then computed as ξ = N/M.

PART 3: AN EXTREME FORM OF MULTIPLICITY

A TIME SERIES OF THE CUMULATIVE VALUES OF ANOTHER TIME SERIES.

A special case of the multiplicity issue is the construction of a time series of the cumulative values of another time series. This issue has arisen in climate science by way of the TCRE parameter described as the Transient Climate Response to Cumulative Emissions proposed as empirical evidence of anthropogenic global warming theory that relates warming directly to fossil the fuel emissions of humans by way of a strong correlation between
cumulative fossil fuel emissions and surface temperature. The TCRE correlation is shown graphically below. Here we use Monte Carlo simulation to test the validity of the TCRE correlation between two time series that are the cumulative values of other time series data.

MONTE CARLO SIMULATION OF THE TCRE

Monte Carlo simulation is used to compare the correlation between normally distributed random numbers with that between their cumulative values under various conditions. The Microsoft Excel function RAND() generates uniformly distributed random numbers from zero to one. The Excel function NORMSINV(RAND()) serves to create normally distributed numbers with RAND() serving as the probability value. Monte Carlo simulation requires a large number of random numbers to be generated.
Typically 10,000 values are generated for each variable. Screenshots of the values generated and the Excel formulas used are shown below.

1. The Monte Carlo simulation begins with two random series taken from a normal distribution and placed in columns A and B. Since these values are random, they have a 50% chance of being negative and 50%
chance of being positive.
2. Columns C and D impose a fixed bias toward positive numbers. Four different bias levels are tested. They are: no bias, 2% bias, 4% bias, and 6% bias. For example, if the bias is 2%, there will be 2% more positive numbers in the series than there would have been with no bias. Columns C and D are intended to correspond with observed changes as C=Δx and
D=Δy and the values in columns E and F represent their cumulative values as E=x and F=y. An imposed bias ensures that changes are more likely to be positive than negative.
3. Cell G2 contains the correlation between columns C and D and cell H2 contains the correlation between columns E and F. A special condition arises if one of the variables is known to be always positive. This is the case for fossil fuel emissions which are never negative. This condition is imposed with the ABS() function. The corresponding screen shots for this model appear as Figure 4 and Figure 5. Figure 4 differs
from Figure 2 only in that column C contains only positive numbers. Figure 5 shows the use of the ABS() function to impose this constraint on the values in column C.
4. The correlations in columns G and H are computed for all 13088 numbers generated for the simulation. Since the RAND() function is at the root of the numbers, each recalculation of the sheet by Excel creates
a new set of numbers and a new pair of correlation values.
These recalculations are carried out fifteen times for each condition to be tested and the fifteen pairs of correlations are assumed to represent the
propensity of the condition being tested to create spurious correlations.
5. Since the data are random and not correlated in their construction, all correlations observed are spurious. It should be noted that the summary tables and charts presented below may not exactly correspond because there may have been a recalculation in the interval before their capture for display in this presentation.
6. BIAS FOR POSITIVE NUMBERS: The screenshots shown in Figures 2, 3, 4, and 5 are taken from the worksheet that uses a 2% bias toward positive values imposed by the RAND()>0.98 condition seen in Figures 3 and 5. This condition changes to RAND()>0.96 for the 4% bias case and to RAND()>0.94 for a 6% bias. The condition is absent in the no bias case. The bias for positive numbers is the key to the source of the TCRE correlation that has been interpreted as causation in climate science.
7. The results of the Monte Carlo simulations at the different bias levels are summarized in the charts and tabulations below.

CASE#1: THE NO BIAS CASE

CASE#2: THE IMPACT OF BIAS FOR POSITIVE VALUES

FIFTEEN OBSERVED CORRELATION PAIRS FOR EACH OF FOUR BIAS CONDITIONS AS: 0%, 2%, 4%, and 6%

SUMMARY AND CONCLUSIONS

The Monte Carlo simulation results for the two cases are summarized in the chart above. The left panel shows the results for the unconstrained case where emissions are allowed to have negative values. In this case the bias toward positive values of 2%, 4%, and 6% are applied to both Δx and Δy.

The right panel shows the constrained case with emissions are restricted to positive values as is the case in the AGW data. In this constrained case, the bias toward positive values of 0.02, 0.04, and 0.06 are applied only to Δy. The correlation between cumulative values is shown in blue and their standard deviation in red.

In both the constrained and unconstrained cases, as the bias is increased the value of the correlation rises and that of its standard deviation falls. This relationship is the key to understanding the TCRE correlation. Both the probability and magnitude of a spurious correlation rises with bias. In the constrained case, where emissions are always positive and which mimics the TCRE condition, these changes progress more rapidly.

THE RESULTS IMPLY THAT IN THE CLIMATE SCIENCE CASE, WITH EMISSIONS ALWAYS POSITIVE, EVEN A 2% BIAS FOR ANNUAL WARMING CAN CREATE STRONG AND STATISTICALLY SIGNIFICANT CORRELATIONS BETWEEN CUMULATIVE ANNUAL WARMING AND CUMULATIVE ANNUAL EMISSIONS. A 4% BIAS CREATES ALMOST PERFECT TCRE CORRELATIONS. WE CONCLUDE THAT THE TCRE IS A SPURIOUS CORRELATION WITH NO CAUSATION INTERPRETATION. IT IS A CREATION OF THE SIGN PATTERNS IN THE DATA WHERE ANNUAL EMISSIONS ARE ALWAYS POSITIVE AND ANNUAL WARMING HAS A POSITIVE BIAS. THIS CORRELATION HAS NO CAUSATION IMPLICATION.

THE SPURIOUSNESS OF CORRELATIONS BETWEEN CUMULATIVE VALUES UNDER THESE SIGN CONVENTIONS IS DEMONSTRATED IN A RELATED POST WHERE WE SHOW THAT UFO SIGHTINGS WORK JUST AS WELL AS FOSSIL FUEL EMISSIONS SIMPLY BY VIRTUE OF BEING POSITIVE VALUES.

In another related post we show that the TCRE correlation is not found at decadal time scales. The spuriousness of the TCRE correlation is underscored in a related work where we find no TCRE correlation when a decadal time scale is used: LINK: https://tambonthongchai.com/2020/11/12/a-decadal-tcre/

THE EFFECTIVE SAMPLE SIZE ISSUE IN TIME SERIES ANALYSIS

Multiplicity also reduces the effective sample size of the time series. This issue is discussed in some detail in a related post: In the case of the the cumulative values of a time series as used in the TCRE, the effective sample size is N=1. This means that the time series of the cumulative values of another time series has neither time scale nor degrees of freedom. Therefore the time series of the cumulative values of another time series does not contain useful information. This issue is discussed in some detail in a related post on illusory statistical power in time series analysis.

EXCERPT FROM THE RELATED POST: All moving window processes in time series analysis involve repeated use of the same data value. If the same data value is used multiple times, it creates a false sense of information because this piece of data brings with it new information only in the first use. It is therefore proposed that the information content of a filtered series and therefore its degrees of freedom must be adjusted for multiplicity. A procedure is presented for estimating the average multiplicity in the use of the source data series in generating the filtered series. The average multiplicity is used to estimate an effective sample size and the effective degrees of freedom. Hypothesis tests must be checked to ensure that rejection of H0 survives when the degrees of freedom are adjusted for multiplicity.

THE MATHEMATICAL INCONSISTENCY ISSUE

in a related post: LINK: https://tambonthongchai.com/2020/08/26/a-mathematical-inconsistency/ we note that since the ECS is a logarithmic relationship between emissions and warming and the TCR is a linear relationship between emissions and warming, the two measures are inconsistent and incompatible. The use of both measures of warming to explain the same phenomenon of nature is not possible.