# Spurious Correlations in Climate Science

Posted May 27, 2018

on:FIGURE 1: CLIMATE SCIENTISTS

FIGURE 2: FEAR OF MELTING ICE AND SEA LEVEL RISE

FIGURE 3: SPURIOUS CORRELATIONS

**DETRENDED CORRELATION ANALYSIS OF TIME SERIES DATA**: Correlation between x and y in time series data derive from responsiveness of y to x at the time scale of interest and also from shared long term trends. These two effects can be separated by detrending both time series as explained by Alex Tolley in the video frame of**Figure 3**. When the trend effect is removed only the responsiveness of y to x remains. This is why detrended correlation is a better measure of responsiveness than source data correlation as explained very well by Alex Tolley in the video. The full video may be viewed on Youtube**[LINK]**. That spurious correlations can be found in time series data when detrended analysis is not used is demonstrated with examples at the Tyler Vigen Spurious Correlation website**[LINK]**. Spurious correlations are common in climate science where many critical relationships that support the fundamentals of anthropogenic global warming (AGW) are found to be based on spurious correlations.- EXAMPLE 1: For example, climate science assumes that
*changes in atmospheric CO2 concentration since pre-industrial times are due to fossil fuel emissions of the industrial economy*. This attribution is supported by a strong correlation between the rate of emissions and the rate of increase in atmospheric CO2 concentration in the time series of the source data. However, when the two time series are detrended, the correlation is not found. This result of detrended correlation analysis implies that the correlation seen in the source data derives from shared trends and not from responsiveness at an annual time scale. Details of this test are presented in a related post**[LINK]**. - EXAMPLE 2: A similar relationship is found in the ocean acidification hypothesis which claims that changes in the inorganic carbon concentration of oceans are driven by fossil fuel emissions. There, too the source data do show a strong correlation but that correlation vanishes when the two time series are detrended. As before, this pattern implies that the correlation in the source data derives from shared trends and not from responsiveness at an annual time scale.
**[LINK]**. - EXAMPLE 3: It is claimed that the observed rise in atmospheric methane concentration is due to human caused methane emissions in activities such as cattle ranching and dairy farming as well as rice cultivation and oil and gas production. Here too, a strong correlation is found in the time series of the source data but this correlation does not survive into the detrended series. This result implies that the correlation between human caused methane emissions and the rise in atmospheric methane derives from shared trends and not from responsiveness at an annual time scale. Such responsiveness is a necessary, though not sufficient, condition for causation. Details of this work may be found in a related post at this site
**[LINK]**. - EXAMPLE 4: A cornerstone of climate science is the effectiveness of proposed climate action in the form of reducing fossil fuel emissions. That the rate of warming can be attenuated by reducing fossil fuel emissions requires that the rate of warming must be responsive to the rate of emissions at the appropriate time scale for this causation to occur (thought to be a decade or perhaps longer (Ricke&Caldeira 2014). And in fact, we find a strong correlation between the rate of warming and the rate of emissions in the time series of the source data at five different time scales (10, 15, 20, 25, & 30 years). Both of these source time series show an upward trend such that the shared trend can create spurious correlations as in the Alex Tolley lecture. When the two time series are detrended, the correlation disappears. The absence of detrended correlation implies that the observed correlation was a faux relationship driven by shared trends and not by responsiveness at the time scales tested in the analysis as demonstrated in a related post
**[LINK]**. Thus no evidence is found in the data that reducing emissions will slow down the rate of warming. - EXAMPLE 5: It is also claimed in climate science that reducing emissions will slow down the rate of sea level rise. This relationship requires a responsiveness of the rate of sea level rise to the rate of emissions at the appropriate time scale for this causation. And in fact, we find a strong correlation between the rate of sea level rise and the rate of emissions in the time series of the source data at five different time scales ranging from 30 to 50 years. Both of these source time series show an upward trend such that the shared trend can create a faux correlation. When the two time series are detrended, the correlation disappears. The absence of detrended correlation implies that the observed correlation was a spurious relationship driven by shared trends and not by responsiveness at the time scales tested in the analysis. This work may be found in a related post
**[LINK]**. - EXAMPLE 6: Climate science supports the greenhouse gas heat trapping theory of atmospheric CO2 and the relevance of their climate models with a strong correlation between model projections of surface temperature and actual observations (see for example Santer 2019). However, this correlation is also between two time series with rising trends. In a related post it is shown that there is indeed a strong correlation between the source data but this correlation is not found in the detrended series
**[LINK]** - EXAMPLE 7: Arctic sea ice extent has played an important role in climate change fear based activism because of periods of diminishing summer minimum sea ice extent in September and the forecasts of “ice free Arctic” that these trends have engendered. The underlying fear of human caused climate change causing Arctic sea ice melt was thus created. The evidence for the causal connection for this causation is a correlation between the rate of warming and the rate of summer sea ice decline; but detrended correlation analysis shows that this correlation is spurious as no year to year responsiveness of September Arctic sea ice extent to the rate of warming is found in the detrended series
**[LINK]**. - EXAMPLE 8: With the assumption that the observed rise in atmospheric CO2 concentration is driven by fossil fuel emissions (discussed in example 1) the effect of higher atmospheric CO2 concentration on climate is then established in terms of climate sensitivity, that is the responsiveness of surface temperature to the logarithm of atmospheric CO2 concentration. The validity of the climate sensitivity function can be shown with strong and statistically significant correlations between the climate model temperature series and observations. However, as shown in a related post
**[LINK]**, this correlation does not survive into the detrended series and is therefore a spurious correlation, similar to the Tyler Vigen examples, that derives from shared trends and not from responsiveness at an annual or other fixed and finite time scale. **CORRELATION BETWEEN CUMULATIVE VALUES OF TIME SERIES DATA.**When moving averages or moving sums of a time series are used to construct a derived time series, care must be taken to correct for the effective sample size (EFFN) in hypothesis tests because*multiplicity*(the use of the same data point more than once) reduces the*effective sample size*. An extreme case of such multiplicity is the construction of a time series of the cumulative values of another time series. In these cases it can be shown that the effective sample size is always EFFN=2 so that the degrees of freedom in hypothesis tests is DF=0. This relationship is described in an online paper**[LINK]**with the relevant text reproduced in paragraph#8 below. It should also be noted that the time series of the cumulative values of another time series does not contain a time scale. Thus, without either time scale or degrees of freedom, it is not possible to test for the statistical significance of any statistic for a time series of the cumulative values of another time series. The spuriousness of such correlations is demonstrated with Monte Carlo simulation in paragraph#9 below.- EFFECTIVE SAMPLE SIZE OF THE CUMULATIVE VALUES OF A TIME SERIES. If the summation starts at K=2, series 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). In these N-K+1 cumulative values, XN is used once, XN-1 is used twice, XN-2 is used three times, XN-3 is used four times, X4 is used N-3 times, X3 is used N-2 times, X2 is used N-1 times , X1 is used N-1 times. 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. The average multiplicity of the N data items in the computation of cumulative values may be expressed as 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: EffectiveN = N/(AVERAGE-MULTIPLE) = N2/(K*(N-K+1) + (N-K)*(N-K+1)/2). To be able to determine the statistical significance of the correlation coefficient it is necessary that the degrees of freedom (DF) computed as effectiveN -2 should be a positive integer. This condition is not possible for a sequence of cumulative values that begins with Σ(X1 to X2). Effective-N 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 effectiveN somewhat but not enough to reach values much greater than two.
- MONTE CARLO SIMULATION OF SPURIOUS CORRELATION BETWEEN CUMULATIVE VALUES OF TIME SERIES DATA
- EXAMPLE 9: An example of the use of cumulative values in climate science is the so called TCRE or Transient Climate Response to Cumulative Emissions. It is the correlation between cumulative emissions and cumulative warming (note that temperature = cumulative warming). This relationship shows a nearly perfect proportionality that is thought to provide convincing evidence of a causal relationship between emissions and temperature and provides a convenient metric for the computation of the so called remaining “carbon budget”, that is the amount of additional emissions possible for a given constraint on the amount of warming. The spuriousness of the TCRE proportionality is described in a related post on this site
**[LINK]**and its spuriousness is further supported with a parody of the procedure that shows that UFO visitations are the real cause of global warming**[LINK]**. A related post shows that when a finite time scale is inserted into the TCRE, the correlation disappears**[LINK]**. - EXAMPLE 10: A paper by Peter Clark of Oregon State University extended the TCRE methodology to sea level rise to provide empirical evidence that fossil fuel emissions cause sea level rise and that climate action in the form of reducing fossil fuel emissions should moderate the rate of sea level rise. (Clark, Peter U., et al. “Sea-level commitment as a gauge for climate policy” Nature Climate Change 8.8 2018: 653). In a related post on this site it is shown that this correlation is spurious
**[LINK]**. In another, we show that when finite time scales are inserted so that both time scale and degrees of freedom are available for carrying out hypothesis tests, the correlation seen in the cumulative series is not found**[LINK]**.

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### 15 Responses to "Spurious Correlations in Climate Science"

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