#### Abstract

A high ambient temperature poses a serious threat to cattle. Above a certain threshold, an animal’s body temperature (Tb) appears to be driven by the hot cyclic air temperature (Ta) and hysteresis occurs. Elliptical hysteresis describes the output of a process in response to a simple harmonic input, and the trajectory forms a closed loop. The hysteresis loop shows a rotated elliptical pattern which depends on the lag between Tb and Ta. The objectives of this study are 1) to characterize hysteresis using bootstrapped ellipse specific nonlinear least squares 2) to reformulate models using the Bayesian method, and 3) to assess the contribution of the Bayesian approach by comparing the risks using two metrics: algebraic and geometric. Comparisons and illustrations are made using simulations over three levels of signal strength. For each method; bootstrap and Bayes, both algebraic and geometric distances are compared based on the root mean square distance (RMSE) from fitting the hysteresis loop. Data from a heat stressed steer in a field experiment was analyzed to illustrate and compare the results from each method.

#### Creative Commons License

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

#### Recommended Citation

Yang, F.; Parkhurst, A. M.; Zhang, S.; Lee, C. N.; Brown-Brandl, T. M.; Gebremedhin, K. G.; and Hillman, P. E.
(2012).
"A COMPARISON OF ANALYTIC AND BAYESIAN APPROACHES FOR CHARACTERIZING THERMAL HYSTERESIS IN CATTLE USING ALGEBRAIC AND GEOMETRIC DISTANCES,"
*Annual Conference on Applied Statistics in Agriculture*.
http://newprairiepress.org/agstatconference/2012/proceedings/6

A COMPARISON OF ANALYTIC AND BAYESIAN APPROACHES FOR CHARACTERIZING THERMAL HYSTERESIS IN CATTLE USING ALGEBRAIC AND GEOMETRIC DISTANCES

A high ambient temperature poses a serious threat to cattle. Above a certain threshold, an animal’s body temperature (Tb) appears to be driven by the hot cyclic air temperature (Ta) and hysteresis occurs. Elliptical hysteresis describes the output of a process in response to a simple harmonic input, and the trajectory forms a closed loop. The hysteresis loop shows a rotated elliptical pattern which depends on the lag between Tb and Ta. The objectives of this study are 1) to characterize hysteresis using bootstrapped ellipse specific nonlinear least squares 2) to reformulate models using the Bayesian method, and 3) to assess the contribution of the Bayesian approach by comparing the risks using two metrics: algebraic and geometric. Comparisons and illustrations are made using simulations over three levels of signal strength. For each method; bootstrap and Bayes, both algebraic and geometric distances are compared based on the root mean square distance (RMSE) from fitting the hysteresis loop. Data from a heat stressed steer in a field experiment was analyzed to illustrate and compare the results from each method.