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DR. GLEN PETERS: “THE CARBON BUDGET CONCEPT IS SEDUCTIVELY SIMPLE BUT HAS LARGE UNCERTAINTIES AND MYSTERIOUS COMPLEXITIES”.

HERE WE SHOW THAT THESE UNCERTAINTIES AND COMPLEXITIES ARE THE PRODUCTS OF A FATAL STATISTICAL FLAW IN CARBON BUDGET MATHEMATICS. 

PART 1: HOW THE TCRE CARBON BUDGET IS USED IN CLIMATE SCIENCE

Carbon budget accounting is based on the TCRE  (Transient Climate Response to Cumulative Emissions). It is derived from the observed correlation between temperature and cumulative emissions. A comprehensive explanation of an application of this relationship in climate science is found in the  IPCC SR 15 2018. This IPCC description is quoted below in paragraphs #1 to #7 where the IPCC describes how climate science uses the TCRE for climate action mitigation of AGW in terms of the so called the carbon budget. Also included are some of difficult issues in carbon budget accounting and the methods used in their resolution.

1. Mitigation requirements can be quantified using the carbon budget approach that relates cumulative CO₂ emissions to global mean temperature in terms of the TCRE. Robust physical understanding underpins this relationship.
2. But uncertainties become increasingly relevant as a specific temperature limit is approached. These uncertainties relate to the transient climate response to cumulative carbon emissions (TCRE), non-CO₂ emissions, radiative forcing and response, potential additional Earth system feedbacks (such as permafrost thawing), and historical emissions and temperature. 
3. Cumulative CO₂ emissions are kept within a budget by reducing global annual CO₂ emissions to net zero. This assessment suggests a remaining budget of about 420 GtCO₂ for a two-thirds chance of limiting warming to 1.5°C, and of about 580 GtCO₂ for an even chance (medium confidence). 
4. The remaining carbon budget is defined here as cumulative CO₂ emissions from the start of 2018 until the time of net zero global emissions for global warming defined as a change in global near-surface air temperatures. Remaining budgets applicable to 2100 would be approximately 100 GtCO₂ lower than this to account for permafrost thawing and potential methane release from wetlands in the future, and more thereafter. These estimates come with an additional geophysical uncertainty of at least ±400 GtCO₂, related to non-CO₂ response and TCRE distribution. Uncertainties in the level of historic warming contribute ±250 GtCO₂. In addition, these estimates can vary by ±250 GtCO₂ depending on non-CO₂ mitigation strategies as found in available pathways. {2.2.2, 2.6.1}
5. Staying within a remaining carbon budget of 580 GtCO₂ implies that CO₂ emissions reach carbon neutrality in about 30 years, reduced to 20 years for a 420 GtCO₂ remaining carbon budget. The ±400 GtCO₂ geophysical uncertainty range surrounding a carbon budget translates into a variation of this timing of carbon neutrality of roughly ±15–20 years. If emissions do not start declining in the next decade, the point of carbon neutrality would need to be reached at least two decades earlier to remain within the same carbon budget. {2.2.2, 2.3.5} 
6. Non-CO₂ emissions contribute to peak warming and thus affect the remaining carbon budget. The evolution of methane and sulfur dioxide emissions strongly influences the chances of limiting warming to 1.5°C. In the near-term, a weakening of aerosol cooling would add to future warming, but can be tempered by reductions in methane emissions (high confidence). Uncertainty in radiative forcing estimates (particularly aerosol) affects carbon budgets and the certainty of pathway categorizations. Some non-CO₂ forcers are emitted alongside CO₂, particularly in the energy and transport sectors, and can be largely addressed through CO₂ mitigation. Others require specific measures, for example, to target agricultural nitrous oxide (N₂O) and methane (CH₄), some sources of black carbon, or hydrofluorocarbons. In many cases, non-CO₂ emissions reductions are similar in 2°C pathways, indicating reductions near their assumed maximum potential by integrated assessment models. Emissions of N₂O and NH₃ increase in some pathways with strongly increased bioenergy demand. {2.2.2, 2.3.1, 2.4.2, 2.5.3} 

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PART 2: THE TCRE PROPORTIONALITY IS A SPURIOUS STATISTIC AND THEREFORE THE CARBON BUDGET IS AN ILLUSORY TOOL OF CLIMATE ACTION

1. The computation of cumulative values requires the use of the same source data item in the computation of more than one cumulative value. This replication implies that there is an information overlap among the cumulative values. The information lost in this manner can be represented in terms of the multiplicity in the use of the time series data when constructing the series of cumulative values. The repeated use of the same source data for computing more than one cumulative value reduces the effective value of the sample size N. Here we present a procedure for estimating the average value of the multiplicity for any sample size and thereby to estimate the effective sample size EFFN in the cumulative series.
2. The last value in the source series is used only once but all of the other values are used more than once. Under these conditions the statistical significance of the correlation between two time series cannot be evaluated regardless of the magnitude of the correlation coefficient.
3. If the summation starts at K=2, the series of cumulative values of a time series X of length N is computed as Σ(X1 to X2), Σ(X1 to X3), Σ(X1 to X4), Σ(X1 to X5) … Σ(X1 to XN-3), Σ(X1 to XN-2), Σ(X1 to XN-1), Σ(X1 to XN).
4. In these N-K+1=N-1 cumulative values, X(N) is used once, X(N-1) is used twice, X(N-2) is used three times, X(N-3) is used four times, X(4) is used N-3 times, X(3) is used N-2 times, X(2) is used N-1 times , and X(1) is also used N-1 times.
5. In general, each of the first K data items will be used N-K+1 times. Thus, the sum of the multiples for the first K data items may be expressed as K*(N-K+1). The multiplicities of the remaining N-K data items form a sequence of integers from one to N-K and their sum is (N-K)*(N-K+1)/2. Therefore, the average multiplicity of the N data items in the computation of cumulative values may be expressed as:
6. AVERAGE-MULTIPLE = [(K*(N-K+1) + (N-K)*(N-K+1)/2]/N. Since multiplicity of use reduces the effective value of the sample size we can express the effective sample size as: EFFN = N/(AVERAGEMULTIPLE) = (N^2)/(K*(N-K+1) + (N-K)*(N-K+1)/2).
7. The usual procedure in the TCRE computation is to use K=2 in which case, EFFN = 1.988. This means that the effective  value of the sample size is less than two  EFFN<2, and that therefore the time series of cumulative values has no degrees of freedom computed as DF=EFFN-2. This is the essence of the statistics issue with the TCRE and the carbon budget procedure that uses the TCRE.
8. The anomalies and oddities with the carbon budget that climate science struggles to rationalize in terms of “radiative forcings of non-CO₂ emissions and Earth system feedbacks such as permafrost thawing are really the creation of a statistics error consisting of the failure to correct the sample size for multiplicity. This failure creates a faux statistical significance of the correlation coefficient and the TCRE regression coefficient that does not actually exist. 
9. To be able to determine the statistical significance of the correlation coefficient it is necessary that the degrees of freedom (DF) computed as EFFN -2 should be a positive integer. This condition is not possible for a sequence of cumulative values that begins with Σ(X1 to X2) where EFFN=1.988 and therefore DF=1.988-2≈0. Thus a time series of the cumulative values of another time series has no degrees of freedom. Also, since all time spans from unity to the full span are used in the computation, the time series of cumulative values has no time scale. The essential statistics issue in the TCRE and its carbon budget implications is that a time series of the cumulative values of another time series has neither time scale nor degrees of freedom.
10. EFFN can be increased to values higher than two only by beginning the cumulative series at a later point K>2 in the time series so that the first summation is Σ(X1 to XK) where K>2. In that case, the total multiplicity is reduced and this reduction increases the value of EFFN somewhat but the sample size is reduced by (K-2) and nothing is gained.
11. These relationships provide the theoretical underpinning for for the spuriousness of correlations between cumulative values. The results show that the time series consisting of the cumulative values of another time series contains neither degrees of freedom not time scale. Therefore both the TCRE correlation and carbon budgets derived from that correlation are spurious and illusory. Numerical results derived from such analysis have no interpretation in terms of the phenomena of nature being studied.

DEMONSTRATION OF THE ANOMALOUS NATURE OF THE TCRE

1. An example TCRE computation appears below in Figure 1. The data are annual emissions and mean annual HadCRUT4 temperature reconstructions 1851 to 2015. In the first chart, a value of TCRE=0.0027C/gigaton of emissions is indicated in the source data at an annual time scale but no correlation is found to support that relationship.
2. The relationship between cumulative values is shown in the next chart of Figure 1 where TCRE=0.0024C/gigaton of emissions is indicated and supported by a near perfect correlation of ρ=0.9017. The absence of correlation in the source data appears to have been overcome by the use of cumulative values.
3. However, the tabulation under the charts shows that the apparent gain in statistical power is illusory. The effective sample size of a time series of the cumulative values of another time series is EFFN=1.988 yielding negative degrees of freedom of DF=1.988-2 = -0.012. Therefore no conclusion can be made from the spurious and illusory correlation between cumulative values because the time series of cumulative values has neither time scale not degrees of freedom.  It is noted that temperature is cumulative annual warming.
4. Figure 2 shows that a the effective sample size can be increased to EFFN=2.05, thereby yielding a positive degrees of freedom if the summation starts at n=30 but the t-statistic is too low to reject the null hypothesis that the hypothesized correlation does not exist.
5. Figure 3  is a split half test of the correlation and TCRE of cumulative values in Figure 1. In the split half test, the first half, the second half. and the mid-half are compared with the full span in terms of correlation and the value of the TCRE coefficient. The comparison shows that there is no agreement between the four TCRE and correlation values for full span, first half, last half, and mid-half. The TCRE is highest for the mid-half (TCRE=0.0035) and lowest for the first half (TCRE=0.0018). The other two values lie somewhere in the middle with full span (TCRE=0.0024) somewhat higher than the second half (TCRE=0.0020). Very little correlation is found in the first half (ρ=0.1784) but the other three spans show statistically significant correlations of ρ=0.9017 for the full span, ρ=0.8860 for the second half and a lower but statistically significant  value of ρ=0.6526 for the mid-half.
6. These differences in the value of the TCRE coefficient at different locations along the time series of cumulative values provide further evidence of its spuriousness and also offers a simple and more realistic explanation for the so called Remaining Carbon Budget (RCB) issue in climate science. The RCB issue is that the carbon budget accounting over the period of the budget is not linear so that for example, if the budget for the next 20 years is 200 gigatons of carbon dioxide, and after 10 years 80 gigatons of cumulative emissions have been produced, the RCB can’t be assumed to be 200-80=120 gigatons because a fresh computation of a carbon budget for the next 10 years typically yields an entirely different non-proportional figure.

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8. The correlation and the TCRE value between cumulative warming and cumulative emissions derives from the fact that during a time of long term warming trend, the annual warming values are mostly positive and further that if the rate of warming is rising over the long term the second half of the carbon budget period will contain a greater fraction of positive annual warming values. These values are driven by the fraction of the annual warming values that are positive. In a random series this fraction cannot be assumed to be the same in the two halves of the time series. This is the mundane statistical explanation of the remaining carbon budget issue.
9. In climate science, the Remaining Carbon Budget  issue is interpreted and resolved in esoteric ways and in terms of climate science variables of convenience. These include non CO2 forcings and the so called “earth system factors” such as release of permafrost methane. The analysis presented here suggests a more banal explanation that goes to the heart of the spuriousness of the TCRE correlation. In terms of the statistics presented here, the RCB anomaly is understood simply in terms of the spuriousness of the TCRE coefficient itself and therefore of its non-proportionality across time time span increments.
10. A summary of the correlations observed in the data is shown in Figure 3A. The data do not show a correlation at an annual time scale. The somewhat higher correlation seen at a decadal time scale is found to be spurious and driven by shared trends and not at responsiveness at a decadal time scale. The apparent correlation does not survive into the detrended series. The strong correlation between cumulative values does survive into the detrended series but, as shown above, correlations between cumulative values have no interpretation because of the absence of degrees of freedom and time scale in the data.

FIGURE 1: CORRELATIONS AND TCRE AT K=2

FIGURE 2: CORRELATIONS AND TCRE AT K=30

FIGURE 3: SPLIT HALF TEST OF THE TCRE

FIGURE 3A: CORRELATION TABLE

1. The anomalous behavior of correlation and the TCRE demonstrated with Figure 1, Figure 2, and Figure 3 above can be understood in terms of how the correlation and TCRE values are generated by the time series. The tabulations in Figure 1 and Figure 2 demonstrate that temperature is cumulative annual warming. It can be shown that the high correlation seen between cumulative emissions and cumulative warming is not an indication of correlation but of a fortuitous sign pattern of annual emissions and annual warming. The sign pattern is that annual emissions are always positive; and, at a time of an overall long term warming trend, annual warming is mostly positive. It is this sign pattern and not real responsiveness of warming to emissions that creates the faux correlation of the TCRE and the faux statistical significance of the TCRE itself. For this reason these values have no interpretation in terms of the phenomena of nature being studied in terms of a responsiveness of warming to emissions.
2. The role of the sign pattern in the TCRE correlation is demonstrated in Figure 4 and Figure 5. The left frame of these figures shows two time series of random numbers X and Y generated by a random number generator such that the X-values in both Figure 4 and Figure 5 are all positive (as in emissions). The Y-values in Figure 4 are completely random with no sign bias. However, the Y-values in Figure 5 are contain a small bias for positive values in the random number generator. The differences between Figure 4 and Figure 5 are therefore understood only in terms of this only difference between them.
3. In both Figure 4 and Figure 5, the left frame shows no sign of correlation between X and Y at an annual time scale verifying that these are random numbers independently generated. However, a significant difference is seen between Figure 4 and Figure 5 in their right frames where the cumulative values of the random numbers are tested graphically for correlation. Here we find a strong positive correlation in Figure 5, where a common positive sign pattern exists with X-values all positive and Y-values with a bias for positive values. No such evidence of a positive correlation exists in Figure 4 where though the X-values are all positive, no positive bias exists in Y-values. Therefore the difference in visual correlation between the two figures is understood on that basis.
4. This demonstration supports the statement above that the TCRE correlation is a creation of a sign pattern. The sign pattern is that emissions are always positive and annual warming is mostly positive during a time of overall warming.
5. that do not show any correlation at all. Their right frames show the relationship between the cumulative values of these completely uncorrelated random numbers.
6. The only difference between Figure 4 and Figure 5 is sign pattern. In Figure 4, the numbers are completely random with no bias for positive values but in Figure 5, a small bias of 5% for positive values is inserted into the random number generator.
7. The strong TCRE linear correlation between cumulative values is seen in Figure 5 but not in Figure 4.  This difference is thus interpreted in terms of the bias for positive values in Figure 5. We conclude from the demonstration in Figure 4 and Figure 5 that the TCRE correlation seen in emissions and warming data is a spurious and illusory creation of a sign pattern. The other, and perhaps more important interpretation of the strong visual correlations seen in Figure 5 is that, under the same conditions and using the same procedures that yield the strong proportionality and statistically significant TCRE value in climate data, the same strong proportionality and statistical significance is seen in random numbers.
8. The only information content of the TCRE is the sign pattern. If the the two time series have a common sign bias, either both positive or both negative, the correlation will be positive. If the the two time series have different sign biases, one positive and the other negative, the correlation will be negative. If the the two time series have no sign bias, no correlation will be found. Therefore, the only information content of the TCRE is the sign pattern and no rational interpretation of such a proportionality exists in terms of a causal relationship that can be used in the construction of carbon budgets. The TCRE carbon budgets of climate science is a a meaningless exercise with an illusory statistic.

FIGURE 4: RANDOM NUMBERS WITHOUT SIGN BIAS 

FIGURE 5: RANDOM NUMBERS WITH SIGN BIAS

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