Property of logistic data exposed with Gompertz model and resistance to noise in actual data

Abstract

A property of data on the exact solution of the logistic curve model was exposed by behavior of upper limits estimated with an unsuitable model (the Gompertz curve model). The property is that the upper limits estimated with the Gompertz curve model for logistic data change with the downward convex for the data as the data size increases, i.e., time elapses. This contributes to identifying a suitable model because the behavior is independent of noise included in actual data. A suitable model is indispensable for correct forecasts. The property was proved and the proof was accomplished thanks to difference equations that have exact solutions. Moreover, we analyzed how resistant the downward convex and monotonic decrease of the estimated upper limit were to noise included in actual data sets.

Appendix: Derivation of Eq. (77)

We show derivation of Eq. (77) [23]. From the property of simple linear regression, the following equation

$$\begin{aligned} -B_n V(x_n) = Cov(x_n,y_n) \end{aligned}$$

(90)

holds because $-B_n$ is the slope of the regression equation (Eq. (30)). Since function $\log (\alpha +\beta \exp x)$ is strictly monotonically decreasing as x increases,

$$\begin{aligned} Cov(x_n,y_n)<0. \end{aligned}$$

(91)

We obtain

$$\begin{aligned} 0 < B_n, \end{aligned}$$

(92)

because of Eqs. (35), (90), and (91). We obtain

$$\begin{aligned} -(n+1)B_{n+1} V(x_{n+1}) = (n+1)Cov(x_{n+1},y_{n+1}) \end{aligned}$$

(93)

from Eq. (90). The left-hand side of Eq. (93) is rewritten as

$$\begin{aligned}&-(n+1)B_{n+1}V(x_{n+1})\nonumber \\&\quad =-B_{n+1}\sum _{i=1}^{n+1}(x_i-\bar{x}_{n+1})^2\nonumber \\&\quad =-B_{n+1}\left( \sum _{i=1}^{n}(x_i-\bar{x}_n+\bar{x}_n-\bar{x}_{n+1})^2+(x_{n+1}-\bar{x}_{n+1})^2\right) \nonumber \\&\quad =-B_{n+1}\left( \sum _{i=1}^{n}(x_i-\bar{x}_{n})^2+n(\bar{x}_{n+1}-\bar{x}_{n})^2+(x_{n+1}-\bar{x}_{n+1})^2\right) \nonumber \\&\quad =-B_{n+1}\left( \sum _{i=1}^{n}(x_i-\bar{x}_{n})^2+n(\bar{x}_{n+1}-\bar{x}_{n})^2+n^2(\bar{x}_{n+1}-\bar{x}_{n})^2\right) \nonumber \\&\quad =-B_{n+1}\left( \sum _{i=1}^{n}(x_i-\bar{x}_{n})^2+n(n+1)(\bar{x}_{n+1}-\bar{x}_{n})^2\right) \nonumber \\&\quad =-nB_{n+1}\left\{ V(x_n)+(n+1)(\bar{x}_{n+1}-\bar{x}_{n})^2\right\} , \end{aligned}$$

(94)

where we use Eq. (55). The right-hand side of Eq. (93) is rewritten as

$$\begin{aligned}&(n+1)Cov(x_{n+1},y_{n+1})\nonumber \\&\quad =\sum _{i=1}^{n}(x_i-\bar{x}_n+\bar{x}_n-\bar{x}_{n+1})(y_i-\bar{y}_n+\bar{y}_n-\bar{y}_{n+1})\nonumber \\&\qquad +(x_{n+1}-\bar{x}_{n+1})(y_{n+1}-\bar{y}_{n+1})\nonumber \\&\quad =\sum _{i=1}^{n}(x_i-\bar{x}_{n})(y_i-\bar{y}_{n})+n(\bar{x}_{n+1}-\bar{x}_{n})(\bar{y}_{n+1}-\bar{y}_n)\nonumber \\&\qquad +(x_{n+1}-\bar{x}_{n+1})(y_{n+1}-\bar{y}_{n+1})\nonumber \\&\quad =\sum _{i=1}^{n}(x_i-\bar{x}_{n})(y_i-\bar{y}_{n})+n(\bar{x}_{n+1}-\bar{x}_{n})(\bar{y}_{n+1}-\bar{y}_n)\nonumber \\&\qquad +n^2(\bar{x}_{n+1}-\bar{x}_{n})(\bar{y}_{n+1}-\bar{y}_n)\nonumber \\&\quad =-B_n \sum _{i=1}^{n}(x_i-\bar{x}_n)^2+n(n+1)(\bar{x}_{n+1}-\bar{x}_{n})(\bar{y}_{n+1}-\bar{y}_n)\nonumber \\&\quad =-n\left( B_n V(x_{n})-(n+1)(\bar{x}_{n+1}-\bar{x}_{n})(\bar{y}_{n+1}-\bar{y}_n)\right) , \end{aligned}$$

(95)

where we also use Eqs. (55) and (56) in Eq. (95). Thus, we obtain

$$\begin{aligned} B_{n+1}\left( V(x_n)+(n+1)(\bar{x}_{n+1}-\bar{x}_{n})^2\right) =B_n V(x_n)-(n+1)(\bar{x}_{n+1}-\bar{x}_{n})(\bar{y}_{n+1}-\bar{y}_n)\nonumber \\ \end{aligned}$$

(96)

from Eqs. (94) and (95). Eq. (96) is rewritten as

$$\begin{aligned}&(B_{n+1}-B_n)\left( V(x_n)+(n+1)(\bar{x}_{n+1}-\bar{x}_{n})^2\right) \nonumber \\&\quad =-(n+1)\left\{ B_n(\bar{x}_{n+1}-\bar{x}_{n})^2+(\bar{x}_{n+1}-\bar{x}_n)(\bar{y}_{n+1}-\bar{y}_n)\right\} \nonumber \\&\quad =-(n+1)(\bar{x}_{n+1}-\bar{x}_n)\left\{ B_n(\bar{x}_{n+1}-\bar{x}_n)+\bar{y}_{n+1}-\bar{y}_n\right\} \nonumber \\&\quad =-(\bar{x}_{n+1}-\bar{x}_n)\left\{ B_n(x_{n+1}-\bar{x}_n)+y_{n+1}-\bar{y}_n\right\} \nonumber \\&\quad =(\bar{x}_{n+1}-\bar{x}_n)\left\{ A_n-B_nx_{n+1}-y_{n+1}\right\} , \end{aligned}$$

(97)

where we use

$$\begin{aligned} \bar{y}_n=A_n-B_n\bar{x}_n, \end{aligned}$$

(98)

and Eqs. (55) and (56).