#### Abstract

A fractal dimension may be thought of as a measure of randomness. Fractal dimensions based on semivariograms have been used to determine degree of randomness in yearly crop yields. Through rescaled range analysis Hurst exponents also define fractal dimensions. This method of obtaining fractal dimensions gives more reasonable and sensitive measures than the semivariogram method. To address the inherent randomness due to yearly variations, global trends in yield must be removed before either method is applied. After detrending, a fractal dimension obtained from semivariogram is usually that of a random process. The Hurst method yields an exponent H, which results in a fractal dimension D = 2-H. The Hurst exponent H = 0.5 corresponds to a completely random system, H = 1 to a completely deterministic system. Natural processes such as river discharges, temperatures, precipitation, and tree rings have Hurst exponents about 0.72. The Hurst phenomenon is the occurrence of H greater than 0.5 corresponding to persistent Brownian motions, rather than equal to 0.5 corresponding to a random process. There is not much auto-correlation in the detrended crop yields, but the Hurst exponents from detrended yearly crop yields of Illinois soybean and wheat and of US soybean, wheat, and cotton are mainly between 0.5 and 1 suggesting long-term dependence similar to that of other natural processes. Illinois and US detrended yearly com yield have exponents less than 0.5, corresponding to anti-persistent Brownian motions. Com data from the Morrow Plots also have Hurst exponents less than 0.5 for two plots that were either over-fertilized or previously not treated, while an optimally treated (properly fertilized and previously manured) plot had an exponent greater than 0.5.

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This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

#### Recommended Citation

Aref, Susanne
(1998).
"HURST PHENOMENON AND FRACTAL DIMENSIONS IN LONG-TERM YIELD DATA,"
*Conference on Applied Statistics in Agriculture*.
http://newprairiepress.org/agstatconference/1998/proceedings/4

HURST PHENOMENON AND FRACTAL DIMENSIONS IN LONG-TERM YIELD DATA

A fractal dimension may be thought of as a measure of randomness. Fractal dimensions based on semivariograms have been used to determine degree of randomness in yearly crop yields. Through rescaled range analysis Hurst exponents also define fractal dimensions. This method of obtaining fractal dimensions gives more reasonable and sensitive measures than the semivariogram method. To address the inherent randomness due to yearly variations, global trends in yield must be removed before either method is applied. After detrending, a fractal dimension obtained from semivariogram is usually that of a random process. The Hurst method yields an exponent H, which results in a fractal dimension D = 2-H. The Hurst exponent H = 0.5 corresponds to a completely random system, H = 1 to a completely deterministic system. Natural processes such as river discharges, temperatures, precipitation, and tree rings have Hurst exponents about 0.72. The Hurst phenomenon is the occurrence of H greater than 0.5 corresponding to persistent Brownian motions, rather than equal to 0.5 corresponding to a random process. There is not much auto-correlation in the detrended crop yields, but the Hurst exponents from detrended yearly crop yields of Illinois soybean and wheat and of US soybean, wheat, and cotton are mainly between 0.5 and 1 suggesting long-term dependence similar to that of other natural processes. Illinois and US detrended yearly com yield have exponents less than 0.5, corresponding to anti-persistent Brownian motions. Com data from the Morrow Plots also have Hurst exponents less than 0.5 for two plots that were either over-fertilized or previously not treated, while an optimally treated (properly fertilized and previously manured) plot had an exponent greater than 0.5.