#### Abstract

Thermal hysteresis in cattle becomes visible when the phase diagram of body temperature (Tb) vs ambient temperature (Ta) exhibits a loop. The hysteresis loop shows a rotated elliptical pattern which depends on the lag between Tb and Ta. The area of the loop can be used to quantify the amount of heat stress during thermal challenge. Three methods to estimate the area and lag of the elliptical hysteresis loop are: linear least squares method, ellipse-specific nonlinear least squares method, and Lapshin’s analytical method. Linear least squares method uses residual least squares to estimate the coefficients of the ellipse for which the sum of the squares of the distances to the observations is minimal. The estimated coefficients can be used to calculate both the rotated angle and area of the ellipse. The ellipse-specific method is based on quadratic constrained least mean squares fitting to simultaneously determine the best elliptical fit for a set of scattered data. It provides estimates of the rotated angle and semi-major and semi-minor axes to calculate the area of the ellipse. Lapshin’s analytical method is a two-stage procedure that fits a sinusoidal function to the input and then the output. It provides parameters in addition to lag and area which further characterize the hysteresis loop. The area and lag along with their standard errors are compared for the three methods using the delta method and bootstrapping. The delta method is used to calculate the standard errors of the derived parameter estimates and bootstrapping is used to assess the appropriateness of the delta method.

#### Keywords

Delay-Relay Model, Ta-Tb phase diagram, Thermo-regulatory response, Eigenvalue-eigenvector, Heat load, Tb-Ta correlation, Energy dissipation, Farm animals

#### Creative Commons License

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

#### Recommended Citation

Yang, F. and Parkhurst, A. M.
(2011).
"ESTIMATING AREA AND LAG ASSOCIATED WITH THERMAL HYSTERESIS IN CATTLE,"
*Annual Conference on Applied Statistics in Agriculture*.
http://newprairiepress.org/agstatconference/2011/proceedings/14

ESTIMATING AREA AND LAG ASSOCIATED WITH THERMAL HYSTERESIS IN CATTLE

Thermal hysteresis in cattle becomes visible when the phase diagram of body temperature (Tb) vs ambient temperature (Ta) exhibits a loop. The hysteresis loop shows a rotated elliptical pattern which depends on the lag between Tb and Ta. The area of the loop can be used to quantify the amount of heat stress during thermal challenge. Three methods to estimate the area and lag of the elliptical hysteresis loop are: linear least squares method, ellipse-specific nonlinear least squares method, and Lapshin’s analytical method. Linear least squares method uses residual least squares to estimate the coefficients of the ellipse for which the sum of the squares of the distances to the observations is minimal. The estimated coefficients can be used to calculate both the rotated angle and area of the ellipse. The ellipse-specific method is based on quadratic constrained least mean squares fitting to simultaneously determine the best elliptical fit for a set of scattered data. It provides estimates of the rotated angle and semi-major and semi-minor axes to calculate the area of the ellipse. Lapshin’s analytical method is a two-stage procedure that fits a sinusoidal function to the input and then the output. It provides parameters in addition to lag and area which further characterize the hysteresis loop. The area and lag along with their standard errors are compared for the three methods using the delta method and bootstrapping. The delta method is used to calculate the standard errors of the derived parameter estimates and bootstrapping is used to assess the appropriateness of the delta method.