Just thought I would point out this page at Acaso Analytics, where Billy Boyle used my previous post on how to use a simple S-shaped curve to model the introduction of a new product on a market, and created a very cool interactive dashboard which illustrates how the curve looks like, and what happens to it when the parameters change. I am a big fan of quantitative models, and enjoyed his other posts as well, which are an eclectic collection of "illustrated" famous quantitative models. Nothing tells the story behind a mathematical model better than a good chart, or, better, an interactive one!

One of my clients recently asked me to modify an Excel model, so that the adoption of products entering the market would follow a S-curve. After some digging and googling, I came across this excellent post by Juan C. Mendez, where he proposes a clean and very practical way to use the logistic function, and calibrate it through 3 input parameters: the peak value, and the time at which the curve reaches 10% and 90% of its peak value.

The beauty of his approach is that his function is compact so it can be typed in easily in a worksheet cell, and the input very understandable. However, I found it a bit restrictive: transforming it for values other than 10% and 90% requires some recalibration, and more importantly, it cannot accomodate values that are not "symmetrical" around 50%.

So I set to work through a generalized solution to the following problem: find a S-Curve that fits any arbitrary value, rather than just 10% and 90%.

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