The Gompertz growth curve fits a registration campaign the logistic cannot

Somebody fits an S-curve to a show's registration history, extrapolates it, and gets a number at doors that is 13 per cent below what the show actually does. They refit with better starting values, get the same answer, and conclude the campaign is unpredictable.

The campaign was fine. The curve was symmetric and the campaign was not, and no amount of parameter tuning fixes a shape assumption.

The Gompertz growth curve is the standard fix, and knowing exactly where it differs from the logistic is enough to decide whether it is the one you need.

The one number that separates the two families

Both curves have three constants, both start near zero and flatten toward an upper asymptote, and both have a single point of inflection where growth is fastest. They disagree about where that point sits.

The logistic puts it at exactly half the asymptote. The Gompertz puts it at the asymptote divided by e, which is 0.36788, or a shade under 37 per cent.

Winsor worked through the mathematics of this in the Proceedings of the National Academy of Sciences in 1932, in a paper on the Gompertz curve as a growth curve. He derives the inflection ordinate as the asymptote over e and gives the practical reading directly: the curve is a good choice for growth data whose inflection falls in the early part of the cycle, "when approximately 35 to 40 per cent of the total growth has been realized". He also sets the two families side by side in a table, where the symmetry row records the logistic as symmetrical about its inflection and the Gompertz as asymmetrical.

The function itself is older. Winsor traces it to Gompertz's 1825 paper in the Philosophical Transactions of the Royal Society on the law of human mortality, where the underlying idea is that a person loses equal proportions of their remaining resistance to death in equal intervals of time. Applied to a registration campaign the same idea reads as a growth rate that decays by a constant proportion each week, which is a plausible description of what an audience acquisition team's outbound effort actually does.

What does the asymmetry look like in daily arrivals?

The clearest way to see it is the arrival rate, which is the derivative of the cumulative curve.

For a logistic the arrival curve is symmetric about the peak. The number of registrations arriving 40 days before the peak day equals the number arriving 40 days after it, exactly, for every choice of parameters. That is a strong claim and it is testable without fitting anything.

For a Gompertz the arrival curve leans right. Take a curve with an asymptote of 11,000, a steepness of 35 days and its inflection 70 days before doors, on a campaign that opened 210 days out. The peak arrival rate is 116 registrations a day. Thirty-five days earlier, at 105 days out, the rate is 56 a day. Thirty-five days later, at 35 days out, it is 80 a day.

Those two flanking rates differ by 24 registrations a day, which is 42 per cent of the smaller one. A logistic requires them to be equal. That single comparison, run on a closed edition's daily arrivals, tells you which family you are in before you fit anything.

The cumulative counts on that same curve are 726 registrations at 105 days out, 4,047 at the peak, 7,614 at 35 days out and 9,608 at doors. The count at the peak is 4,047 against an asymptote of 11,000, which is 0.368 of it, exactly as Winsor's derivation says.

What does the wrong shape cost you?

Fit a logistic to that curve using only the data through 30 days before doors, then extrapolate to doors.

The best-fitting logistic comes out with an asymptote of 8,504 and a steepness of 16.4, and it forecasts 8,374 registrations at doors. The curve it was fitted to reaches 9,608. The error is 1,234 registrations, or 12.8 per cent low, and the residuals through the fitted region are small enough that nothing on the diagnostic plots looks wrong.

That is the failure mode worth internalising. A symmetric curve fitted to an asymmetric campaign compensates by shrinking its steepness parameter, which lets it track the observed portion closely and forces the asymptote down. The fit looks good and the extrapolation is badly biased, and the bias runs in the direction that makes you order too few badges.

Fit the same data with the right family and the asymptote comes back correct, because the shape being extrapolated is the shape the data came from. That is the entire argument for choosing the family deliberately.

The orientation nobody mentions

A registration campaign with a slow climb and a spike in the closing fortnight is a common pattern, and the standard Gompertz is the wrong curve for it.

The Gompertz rises fastest early and then has a long, slowly decaying tail. Its inflection at 37 per cent of the asymptote means the peak arrival day comes when just over a third of the eventual file is in. A show with a late spike has its peak arrival day when two thirds or more of the file is in, which is the opposite skew.

The fix is the reflection. Flip the Gompertz in time and its inflection moves to one minus one over e, which is 63.2 per cent of the asymptote, and the curve becomes a slow climb to a late peak followed by a short finish. It has the same three constants and the same amount of skewness, pointing the other way.

So the practical family is three fixed shapes: inflection at 37 per cent, at 50 per cent, or at 63 per cent of the asymptote. Measure the peak arrival day on your closed editions, work out what fraction of the eventual file was in hand on that day, and pick the shape closest to it. On the show above that fraction was 42 per cent of the final, and 37 per cent of the asymptote, and knowing which denominator you are using matters because registration closes before the asymptote is reached.

The denominator problem has a cheap workaround. The gap between the final count and the asymptote is the tail the campaign would have collected if registration had stayed open, and you can bound it from the arrival rate on the last day. If the show was taking 60 registrations a day when the doors opened and the curve had another month of decay in it, the asymptote is a few hundred above the final. If it was taking 400 a day at close, the asymptote is a long way above and the fraction-of-final figure will understate the fraction-of-asymptote badly. Print the closing-day arrival rate next to the diagnostic so nobody reads the fraction without it.

Testing the reflection costs one extra fit. Reverse the days-to-open axis, fit the same Gompertz form to the reversed series, and compare the residual sum of squares with the forward fit. Whichever direction wins is the orientation your campaign has, and the two fits take the same code with one sign changed.

Where the three-shape menu runs out

Winsor is blunt about the limitation and it is the honest thing to quote him on. He notes that the degree of skewness in the Gompertz is just as fixed as it is in the logistic, and that introducing a variable degree of skewness into a growth curve requires at least four constants. He also declines to claim any general superiority for the Gompertz over any other three-constant S-shaped curve.

That has a direct operational reading. If your show's peak arrival day sits at 48 per cent of the eventual file, the logistic is close enough. If it sits at 55 per cent, none of the three shapes is right and you have two choices: add a fourth parameter and pay for it out of a very short history, or stop using a parametric family and fit a penalised spline instead, which is O6's subject and buys flexibility without committing to a shape.

The four parameter route deserves a warning. Adding a shape parameter to a curve fitted across five editions means estimating it from five editions, and the shape parameter is the one most sensitive to a single unusual campaign. I would use it only where the peak arrival fraction has been stable across at least eight editions.

Where this stops

Choosing the right S-curve fixes one class of error and leaves several standing.

The curve says nothing about a discontinuity. Every price tier deadline puts a step in the arrival rate, and a step is not a feature any three constant smooth curve can represent. If your campaign has three tier boundaries, the arrival curve has three spikes on it and you are fitting a smooth shape to something with corners. The fitted asymptote absorbs the mismatch and the fit is worse near the deadlines, where forecast attention is usually highest.

It also assumes the shape parameters transfer between editions. Fitting the Gompertz to five closed editions and averaging the inflection positions makes sense only if those five campaigns were run the same way, and a change of agency, a new pre-registration flow or a launch three weeks later moves the inflection without moving anything else.

The comparison between families needs a protocol, and eyeballing the fit is the wrong one. Score both forms on editions the fit never saw, at the days-to-open point that matters to your operations calendar, and the way to organise that scoring is O25's subject. A single held-out edition is one observation and a family that wins on it may lose on the next.

Take one closed edition, compute the daily registration arrivals, smooth them lightly, and find the day the rate peaked. Then read off the cumulative count on that day and divide by the final total. If that fraction is near 0.4 the Gompertz is your family, if it is near 0.5 the symmetric logistic that O4 fits is fine, and if it is above 0.6 you want the reflection. It is one number per edition, and five of them settle an argument that otherwise runs every year alongside the rest of the forecasting methods work.