Fitting a booking curve to five editions with a three parameter logistic

Somebody in commercial asks for the whole shape of the campaign. They can already see the registration count sitting at 3,140 with a hundred days to go. What they want is the rest of it: the week the curve steepens, and the count the early bird deadline will be working against.

Fitting a booking curve answers both from one object, and it answers the number at doors as a by-product. The catch that stops most teams is a belief that a curve needs more history than the show has. Five editions feels like too little to fit anything. It is plenty, once you notice which parameters are actually being estimated from which data.

The three parameters and what each one does

Write the cumulative registration count as a function of days before doors. Call that variable t, counting down, so t of 180 is six months out and t of 0 is opening day.

The three parameter logistic has the form: count at t equals A divided by one plus e raised to the power of t minus m, all over s.

A is the asymptote, the total the curve flattens toward. The midpoint m is the days-to-open value at which the count reaches exactly half of A, and it is also, for this curve, the day on which registrations arrive fastest. The steepness s controls how compressed the transition is, and it has a direct reading: the count moves from about 12 per cent of A to about 88 per cent of A across four units of s.

Pearl and Reed set this out in the Proceedings of the National Academy of Sciences in 1920, fitting it to United States census counts from 1790 onwards and arguing for exactly three constants. Their curve has the properties that matter here: it starts near zero, it flattens toward an upper limit, and its point of inflection sits at half the asymptote. Their fitted inflection came out at 98,637,000 people against an asymptote of 197,274,000, which is half of it to the person, because the logistic is symmetric by construction.

Solving for the asymptote from a partial curve

The move that makes five editions enough is splitting the parameters by where they come from.

Fit the full three parameter curve to each closed edition, where you have the whole cumulative series and the final total. Take the average of the five midpoints and the average of the five steepnesses. Those two describe your show's campaign shape, and shape is the thing that repeats. Then hold them fixed and let the current edition supply only A.

Suppose the five closed editions give an average midpoint of 74 days out and an average steepness of 26. Today is 100 days out and the file holds 3,140 registrations.

Substitute. The exponent is 100 minus 74 over 26, which is exactly 1. So the fraction of A reached today is 1 over 1 plus e, which is 1 over 3.7183, or 0.26894.

Divide. A equals 3,140 over 0.26894, which is 11,675.

That is the asymptote and it is not the forecast. Registration closes on opening day, and at t of 0 the curve reads A over one plus e to the minus 74 over 26, which is 11,675 over 1.0581, or 11,035. The difference of 640 registrations is the tail the curve would have collected if registration had stayed open. Reporting A as the show open forecast overstates by that amount, and it is a mistake I have seen survive several rounds of review because both numbers look plausible.

Why does one edition give you 180 observations?

Because the observation is the curve.

The instinct with five editions is to count five data points, which is right if the thing you are modelling is the final total. It is wrong here. A show with a 180 day registration window produces 180 daily cumulative counts per edition, so five closed editions produce 900 points, and the three parameters of a logistic are estimated from those 900 rather than from the five finals.

The framing comes from functional data analysis, where each curve is treated as a single observation of a function and the analysis happens on curves. Ramsay and Silverman's Functional Data Analysis, in its second edition from Springer in 2005, is the standard treatment. What matters operationally is the accounting: shape parameters are cheap because within-edition data is abundant, and level parameters are expensive because you get one final total a year.

The 900 points are not 900 independent observations, and pretending otherwise will make every standard error you compute far too small. Two adjacent days inside one campaign are nearly the same measurement. That is a real limit on inference and it changes nothing about identification. The shape is determined. The confidence you may claim about it is much lower than the point count suggests, and choosing a smoothing penalty by holding out whole editions rather than individual days is O6's answer to the same problem.

How wrong can the asymptote be?

Wrong enough to matter, and the sensitivity runs through the midpoint.

Redo the calculation with a midpoint of 70 days instead of 74, which is well inside the spread you would expect across five editions. The exponent becomes 100 minus 70 over 26, which is 1.1538. Raise e to that and you get 3.1702, so the fraction of A reached today is 1 over 4.1702, or 0.23979. Dividing 3,140 by that gives 13,095.

A four day change in an averaged parameter moved the asymptote from 11,675 to 13,095, which is 12 per cent, or 1,420 registrations. Nothing about the current edition's data changed.

Two things follow. Report the range implied by the five individual midpoints rather than by their mean, because that range is the honest width. And prefer to fit later in the campaign, since the same four day error at 60 days out moves the answer far less: the curve is steeper there, so the count in hand pins A down harder.

The general shape of this is that a curve fitted early is mostly extrapolation dressed as estimation. At 150 days out you are observing the flat foot of the curve, which is consistent with a very wide range of asymptotes, and the model will still return a single confident number.

The ceiling that Pearl and Reed published

Their paper is worth keeping in mind for a reason beyond the algebra.

Having fitted their logistic to more than a century of census counts, they read off the upper asymptote and put the maximum population of continental United States at 197,274,000, roughly twice the population at the time. They were careful about it and said plainly that an empirical curve does not demonstrate an underlying law. The United States Census Bureau counted 203,211,926 residents in 1970, fifty years later, and the count has gone up every decade since.

The lesson transfers directly. An asymptote is a property of a functional form, and the functional form is a hypothesis about the future, so a fitted asymptote is a forecast even when it is presented as a parameter. When somebody asks what the ceiling on your show's audience is and you answer with A, you are making the same move Pearl and Reed made and inheriting the same exposure. The defensible version of that answer names the assumption: the ceiling implied by this curve, fitted to these five editions, on the assumption that the campaign keeps its current shape.

Where this stops

The logistic buys you a lot for three parameters and it hands you one specific assumption you cannot switch off.

It is symmetric about its midpoint. Half the registrations arrive before m and half after, and the daily arrival rate on the tenth day before m equals the rate on the tenth day after. Registration campaigns often break that badly, with a slow build and a spike in the final fortnight, and no choice of A, m or s will produce an asymmetric curve. The fix is a different functional form with the inflection somewhere other than the halfway point, and the Gompertz alternative is O8's subject.

The second limit is that a least squares fit to a cumulative series can dip. Cumulative registrations never fall, so a fitted curve that goes backwards anywhere is wrong by construction, and imposing that constraint is worth doing explicitly, which is O5's territory.

The third is the one that bites in practice. A parametric curve fitted to five editions assumes those five editions came from the same process. One venue move, one date change or one edition that opened registration eleven weeks late, and you are averaging midpoints that are not measuring the same thing. Check that first. A show whose five midpoints are 71, 73, 74, 75 and 77 has a shape. A show whose five midpoints are 52, 74, 79, 68 and 97 has a history of disruptions, and the average of those is a number with no referent.

Pull the daily cumulative registration series for one closed edition, plot it against days to open, and mark the day on which the count crossed half its final total and the day on which daily arrivals peaked. If those two days are within a week of each other, a logistic is the right family and the rest is fitting. If they are a month apart, stop and read O8 before you fit anything, and keep the wider set of forecasting methods in view while you do.