What is required, therefore, is a simple 
-related weight that
has something like the characteristics (a)-(d) listed in the previous
section. Such a function can be constructed as follows. The function
increases from zero to an
asymptotic maximum in approximately the right fashion. The constant
determines the rate at which the increase drops off: with a large
constant, the function is approximately linear for small 
,
whereas with a small constant, the effect of increasing 
rapidly diminishes.
This function has an asymptotic maximum of one, so it needs to be
multiplied by an appropriate weight similar to equation 7.
Since we cannot estimate 7 directly, the obvious simple
alternative is the ordinary Robertson/Sparck Jones weight, equation
2, based on presence/absence of the term. Using the usual
estimate of 2, namely 
(equation 3), we obtain the
following weighting function:
where 
is an unknown constant.
The model tells us nothing about what kind of value to expect for
. Our approach has been to try out various values of 
(values around 1--2 seem to be about right for the TREC data---see the
results section 7 below).
However, in the longer term we hope to use regression methods to
determine the constant. It is not, unfortunately, in a form directly
susceptible to the methods of Fuhr or Cooper, but we hope to develop
suitable methods.
The shape of formula 8 differs from that of formula 5 in one important respect: 8 is convex towards the upper left, whereas 5 can under some circumstances (that is, with some combinations of parameters) be S-shaped, increasing slowly at first, then more rapidly, then slowly again. Averaging over a number of terms with different values of the parameters is likely to reduce any such effect; however, it may be useful to try a function with this characteristic. One such, a simple combination of 8 with a logistic function, is as follows:
where c>1 is another unknown constant. This function has not been tried in the present experiments.
