# Justification for a Logistic Function — Experimental Data Analysis

As mentioned previously, we desire to investigate the form of a relationship for diffuse fraction as predicted by clearness index for several reasons. We propose a better one-predictor variable (kt) model for Australian conditions because the models developed elsewhere have not proven adequate for Australian conditions (Spencer, 1982). This then leads to a proposal of another one-predictor functional form for site-specific models. Also, we propose a parsimonious diffuse model, with fixed coefficients, for general application, which may be used to estimate diffuse solar radiation in the absence of measured data. Such a model may well be able to be parameterised to suit diverse climates.

To understand the relationship better, we construct a moving average of the dif­fuse fraction, as a function of the clearness index, rather than the standard version, a function of time. Specifically,

1 n

dave = N S di(kt) (8.2)

The functionality with respect to clearness index comes as a result of ordering the diffuse fraction for increasing clearness index. Figure 8.3 displays the moving average for a moving window of length N = 101 for the Adelaide data. It seems that a logistic type function will suit the modelling of the relationship as it can be spatially flexible for modelling purposes and requires a minimum number of parameters. Polynomials of orders 3 and 4 were checked for suitability, but gave a weaker fit than the logistic function.

A logistic growth model, growth being the most common phenomenon mod­elled in this fashion, resembles exponential growth in the early stages but in the later stages there is a reversal of concavity to reflect either a limited amount of re­sources or a maximum life span or size for example. However, logistic decay is also a possibility in the biological realm. For instance, in predator-prey models logistic decay can prevail in modelling predator numbers when multiple sources of food are present.

The concept of what drives logistic growth or decay was examined by Nash (1975). He describes a system where individuals occupy one of two states, one with growth level 0 and one with growth level h. The transition from level h to level 0 is not possible and the transition probability for 0 ^ h is proportional to

 Fig. 8.3 Moving average of diffuse fraction as a function of clearness index

the percentage of individuals in level h. We tend to get a so-called “band wagon” effect where the probability increases as the number of individuals increases, but of course, there is not an infinity of individuals available, so the transition probabil­ity must begin at some stage to decrease. It is essentially a non-stationary Markov process.

Perhaps we can envisage the dependence of diffuse fraction on clearness index in a similar “band wagon” manner, but with decay. As the atmosphere becomes clearer there is an increasing tendency to clear but of course there is a saturation effect here as well – the atmosphere can only tend towards perfectly clear. However, one would expect that this changing probability would progress in a smooth fashion, rather than in a piecewise linear version, as taken by the earliest diffuse fraction models. The subsequent section will describe the theoretical justification.