Monday, 24 November 2008

Technology's S-curve and parameter standard errors

Technology is often proposed to spread slowly at first, then become more widely accepted at a faster rate, then slow down in its spread as most people become familiar with it or its utility declines. Graphically, it follows an S-curve over time, like this:

The S-curve may be modelled by fitting a time dependent function like the logistic function:

Technology use = a0/(1+exp{-(a1+a2*time)})

where the as are constants to be estimated. They can be estimated by non-linear least squares, which minimises the sum of (Observed values - predicted values)^2. The usual procedure is to approximate the predicted values by their Taylor series linearisation, or its numerical approximation, so we have to minimise the sum of

(Observed Values - b0 + a0*f1(t) + a1*f2(t) + a2*f3(t))^2

where the fs are functions of t, time.

A complication arises in estimation of standard errors. Because the coefficients are functions of time, their cross-product matrix coefficients do not all converge at the usual least squares rate equal to the sample size, and so the estimates of least squares standard errors will be divergent.

Accurate convergent standard errors could be calculated by working out the order of magnitude of the sums of the fs, and premultiplying the cross-product matrix by a suitable rebasing matrix. The precise forms of the sums may not be neat, but the orders of magnitude should be accessible without too much difficulty.

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