This continues the series of experiments to compare Euler’s method and an extrapolation method for calculating WBGT.
Previously, I had come up with a function that can be used to extrapolate a value for Tw, necessary for the calculation of WBGT. Currently we use Euler’s iterative method to do this, and this is a time-consuming process on large datasets. Initial tests to compare thse two methods, showed that the extrapolation method agreed closely with Eueler’s method most of the time, but in some cases it was significantly out. In this experiment we aim to observe what is happening in each iteration cycle of Euler’s method that causes it to deviate from the proposed function for the extrapolation method.
Extrapolation of values is a process whereby you can extend a series of known data points to predict new values. Continuous functions, where the equation of the curve is known are ideal for extrapolation e.g.:
All of the lines above fit to a nice polynomial function. Using these functions, we could determine any pair of (x, y) values along these graphs.
WBGT by Euler’s Method
The intermediate values of Tw were captured at each iteration of Euler’s method. A general sample is graphed below:
The line is painfully close to being linear, which would be the ideal scenario for extrapolating from! However, as you can see there is a slight decay on the curve, which gives rise to the deviation between the iterative approach and the extrapolation method.
Extrapolating WBGT is not entirely accurate. The next step is to find out how often the extrapolation method actually agrees / disagrees with Euler’s Method and try to determine exactly what inputs are “acceptable” vs. “unacceptable”. To test this, both methods were used to calculate WBGT over 6 years of climate data (2006, …, 2011). The time to calculate, percentage fits and maximum deviation were recorded for each year and are summarised below:
Time to calculate WBGT:
The first thing we can say is that calculating WBGT by extrapolation is much much faster, and consistently under 80 seconds. This is fantastic!
The extrapolation method agrees with Euler’s Method in the large majority of cases. However, it deviates significantly in a fair amount of the data (as little as 3%, but as much as 24 % of the data).
Observing the iterations of Euler’s Method (and the graph above), it seems that the extrapolation method does not fit well when there are a large number of iterations. Clearly, the amount of decay accumulates with increasing iterations. This occurs when there is a large difference in the initial values of Ta and Td (large difference being > 15 degrees Celsius).
- Can we factor in the decay?
- Can we ‘adjust’ the extrapolation by a factor proportional to Ta : Td?