Wednesday, May 13, 2009

Wednesday Math, Vol. 71: The Logistic Curve

Two types of growth patterns that show up in natural and man-made settings are the linear model and the exponential model. If we use the linear model, it draws a line. For example, If your cable bill is $70 a month, it will cost 12*($70) = $840 a year. The increase of the total from one month to the next is found by addition. The exponential model instead acts like multiplying by a constant. If you invest $100 and get 5% interest on your investment in a year, you will have $105 at the end of that year. If instead you had invested $1,000, you would have $1,050 at the end. The larger the amount you start with, the larger the increase, unlike the constant size of the increase with the linear model.

While both of these models would tend towards infinity as time passes, the exponential model gets bigger faster. As I stated a few weeks ago, nature abhors the infinite. In the real world, a system showing exponential growth will eventually have to cool off. The idea is that a natural system will have some kind of limit to the size to which it can grow, and eventually instead of growing more and more as time goes forward, the growth rate cools off and the size of the thing being measured levels off, unable to get larger than the natural limiting value, which may be difficult to measure precisely. For example, the spread of a disease or a rumor cannot go beyond 100% of the population, so the curve seen here might model that growth rate. In reality, the rumor or disease may only reach a percentage of the population less than 100%, and the top of the curve doesn't reach 1.

Regardless of the highest value attained, this growth pattern is called the logistic curve. All curves with a shape like an S are also called sigmoid curves, but I like logistic better. I can't think of two syllables put together in English that have a creepier sound than "sigmoid". This is completely unrelated to the meaning of the word, it's just a disturbing sound.

There are different kinds of logistic curves. The one above is defined by an equation found in most books, but the great French mathematician Cauchy came up with an equation that created a more radical logistic curve, though the picture I found of it on the internet gives the credit to Edwin Mansfield. In the first picture, if we drew the tangent line to the curve near the middle, we would get a line at an angle of 45 degrees, which is to say a growth rate of 1. The tangent at the middle of this logistic curve would be a vertical line, which is to say the growth rate for one instant is technically infinite.

If you see a prospectus for an investment that boasts the growth rate for the most recent year is more than the growth rate for the previous year, and even more still that the growth rate for the year prior to that, be very wary. All such investments must eventually level off, and it must be getting closer to the time when the growth rate will start to get smaller, and eventually the growth will be effectively zero.

Before it flattens out, and it always flattens out, logistic curve growth is the cause of much irrational exuberance in the markets.

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