The Gompertz curve or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). It is a sigmoid function which describes growth as being slowest at the start and end of a given time period. The right-side or future value asymptote of the function is approached much more gradually by the curve than the left-side or lower valued asymptote. This is in contrast to the simple logistic function in which both asymptotes are approached by the curve symmetrically. It is a special case of the generalised logistic function. The function was originally designed to describe human mortality, but since has been modified to be applied in biology, with regard to detailing populations.
History Benjamin Gompertz (1779–1865) was an actuary in London who was privately educated. He was elected a fellow of the Royal Society in 1819. The function was first presented in his June 16, 1825 paper at the bottom of page 518. The Gompertz function reduced a significant collection of data in life tables into a single function. It is based on the assumption that the mortality rate increases exponentially as a person ages. The resulting Gompertz function is for the number of individuals living at a given age as a function of age. Earlier work on the construction of functional models of mortality was done by the French mathematician Abraham de Moivre (1667–1754) in the 1750s. However, de Moivre assumed that the mortality rate was constant. An extension to Gompertz's work was proposed by the English actuary and mathematician William Matthew Makeham (1826–1891) in 1860, who added a constant background mortality rate to Gompertz's exponentially increasing one.
Formula
f ( t ) = a e − b e − c t {\displaystyle f(t)=a\mathrm {e} ^{-b\mathrm {e} ^{-ct}}}
where
a is an asymptote, since lim t → ∞ a e − b e − c t = a e 0 = a {\textstyle \lim _{t\to \infty }a\mathrm {e} ^{-b\mathrm {e} ^{-ct}}=a\mathrm {e} ^{0}=a}
b sets the displacement along the x-axis (translates the graph to the left or right). c sets the growth rate (y scaling) e is Euler's Number (e = 2.71828...)
Properties The curve f ( t ) = a e − b e − c t = a e − e − c t + ln b {\textstyle f(t)=a\mathrm {e} ^{-b\mathrm {e} ^{-ct}}=ae^{-e^{-ct+\ln b}}} has the same shape as f ( t ) = e − e − t {\displaystyle f(t)=e^{-e^{-t}}} after an affine transformation. The halfway point is found by solving f ( t ) = a / 2 {\textstyle f(t)=a/2} for t.
t h w p = ln ( b ) − ln ( ln ( 2 ) ) c {\displaystyle t_{hwp}={\frac {\ln(b)-\ln(\ln(2))}{c}}}
The point of maximum rate of increase ( 0.368 a {\textstyle 0.368a} ) is found by solving d 2 d t 2 f ( t ) = 0 {\textstyle {\frac {d^{2}}{dt^{2}}}f(t)=0} for t.
t m a x = ln ( b ) / c {\displaystyle t_{max}=\ln(b)/c}
The increase at t m a x {\textstyle t_{max}} is
max ( d f d t ) = a c e {\displaystyle \max \left({\frac {df}{dt}}\right)={\frac {ac}{e}}}
Derivation The function curve can be derived from a Gompertz–Makeham law of mortality, which states the rate of absolute mortality (decay) falls exponentially with current size. Mathematically,
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