A graphical derivation and statistical evaluation of simplified polynomials to determine vapour pressure deficit for use in ultra-low power microcontroller applications

Abstract

The aim of this work has been to derive and statistically evaluate the accuracy of second-order and third-order polynomials to determine vapour pressure deficit (VPD). These polynomials take air temperature and relative humidity measurements to determine VPD without the use of an exponential function, as proposed by F W Murray in 1967. Replacing the exponential function with a 2nd or 3rd order polynomial may be beneficial in ultra-low power microcontroller-based measurement applications where; code size, memory usage and power requirements are critical design drivers. However, oversimplification may impact precision. This work presents alternative 2nd order and 3rd order equations that have been derived from a Murray equation dataset where VPD isothermal datasets were plotted against relative humidity. These linear relationships allow y = mx + c analysis where, (i) ‘c’ can be set to zero with a offset in the relative humidity data, and, (ii) ‘m’ can be derived from a 2nd of 3rd order polynomial where ‘m’ = f(T) and is derived using Excel-based fitting of the gradients from the isothermal datasets. The resulting ‘m’ = f(T) 2nd and 3rd order polynomials presented R 2 values of 0.998 04 and 0.999 98 respectively. A Bland-Altman statistical assessment was performed, where the Murray equation (reference) dataset is plotted against the difference between the reference and polynomial datasets using the same air temperature and relative humidity inputs. The difference datasets presented 2-sigma (95% confidence interval) variances for the 2nd and 3rd order polynomials as <±0.1 kPa and <±0.01 kPa respectively. The 2nd and 3rd order polynomials also resulted in a bias values of <0.0037 kPa and <0.0013 kPa respectively. These results suggest that a 3rd order polynomial equation could be used to determine VPD in ultra-low-power microcontroller measurement applications, with minimal impact on VPD measurement precision.