# Hurst Persistence in UAH Temperature?

Posted September 27, 2018

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FIGURE 1: COMPARISON OF OLS TREND WITH MEAN 15YR TREND

FIGURE 2: DATA AND GAUSSIAN SIMULATION COMPARED

FIGURE 3: HURST EXPONENTS OF DATA AND GAUSSIAN SIMULATION COMPARED

**IS THERE HURST PERSISTENCE IN UAH ZONAL MEAN TEMPERATURE ANOMALIES?**

REFERENCE PAPER FULL TEXT DOWNLOAD: **[SSRN.COM]** **[ACADEMIA.EDU]**

- It was shown in a prior work (
**see links above**) that when the UAH lower troposphere temperatures Dec 1978 to Dec 2017 are studied on a monthly time scale (as a sequence of 481 months), evidence of Hurst persistence is found in the data. Here we present a further investigation into these data using an annual time scale and studying each calendar month separately for the seven calendar months January to July in study period 1979 to 2018. As of this writing data for the full time span are available for only these seven months. - Almost universally, the study of temperature trends are presented in terms of OLS (ordinary least squares) linear regression. The procedure contains some unforgiving assumptions that may not apply in time series field data such as the temperature data in climate studies but the use of OLS has gained such widespread acceptance that the limitations of the procedure imposed by assumptions are generally overlooked. The important assumption relevant here is the so called “iid” constraint. The procedure assumes that all occurrences of the time series are taken from an identical Gaussian distribution that differ only in magnitude and that each occurrence is independent of prior occurrences. Violations of these assumptions in time series of field data are common.
- A serious violation is that of persistence first discovered in the Nile River flow data by Edwin Hurst who was designing the Aswan Dam in Egypt (pictured above). He found that changes in the flow rate were not random but contained a persistence so that an increase was more likely to be followed by an increase than a decrease and a decrease was more likely to be followed by a decrease than an increase. This kind of behavior violates “iid” Gaussian randomness and invalidates OLS regression because OLS assumes Gaussian randomness. In 1950, Hurst published his first and only journal publication [Hurst, E. (1951). Long-term storage capacity of reservoirs. Trans. Amer. Soc. Civil Eng. , 116 (1951): 770-808]. In it he detailed a procedure by which such persistence in time series can be detected by tracking how distant (the range) the cumulative deviations from the mean can be measured as number of standard deviations. He proposed the ratio H = ln(range)/ln(sample size) as a measure of persistence where H = Hurst exponent. In a pure Gaussian iid series H=0.5 and in the case of persistence, H>0.5. As a practical matter, because the empirical setup can also affect the value of H (Granero, S. (2008). Some comments on Hurst exponent and the long memory processes on capital markets. Physica A: Statistical Mechanics and its applications , 387.22 (2008): 5543-5551), the best way to test for persistence is to compare the H-value of the test series with that of its Gaussian twin. If the test series H-value is statistically significantly greater than the H-value of its Gaussian twin, there is evidence of persistence, and otherwise not.
- This comparison is made in
**Figure 3**for all ten unique zonal regions for which satellite temperature anomaly data are published by the UAH. For each zonal region, we compute the value of H in the data and again in a Gaussian simulation of the data. The Gaussian simulation retains the standard deviation and OLS trend value and generates the actual values in a Monte Carlo simulation (labeled as SIMUL). Figure 3 shows that no significant difference is found between data and the iid Gaussian simulation. We conclude from this comparison that the Gaussian iid assumption is not violated in the UAH temperature data at an annual time scale. - This conclusion finds further support in
**Figure 1**where the OLS trend (the yellow line) is compared with the average of trends computed in a moving 15-year window that moves one year at a time from an end-year of 1993 to an end-year of 2018. If OLS assumptions are violated we should find significant differences between these two measures of overall trend. But no difference is evident in the graphic display. We therefore find no evidence of persistence or that the UAH data violate OLS assumptions. - Additional support for this conclusion is found in
**Figure 2**. Here the data (blue dots) and the corresponding Gaussian simulation (red dots) are compared directly in search for a visual incongruence that might identify non-Gaussian behavior. No such incongruence is found. - We conclude that the evidence of Hurst persistence at a monthly time scale reported in previous works (
**[SSRN.COM]****[ACADEMIA.EDU])**does not apply to the annual time scale when the calendar months are studied in isolation, one at a time.

1956: Hurst, Harold Edwin. “Methods of using long-term storage in reservoirs.” *Proceedings of the Institution of Civil Engineers*5.5 (1956): 519-543.

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