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Modelo de Crecimiento Logístico Generalizado

La función o curva logística generalizada (generalizada) , también conocida como curva de Richards , desarrollada originalmente para el modelado del crecimiento, es una extensión de las funciones logísticas o sigmoideas , lo que permite curvas en forma de S más flexibles: ### \[y(t) = \frac{A +(U - A)}{1+\beta exp(-k(t-t_0))}\] t tiempo x tamaño A La asintota inferior U La asintota superior ß Rango de crecimiento tø Tiempo inicial ( Predeterminado Ø)

Ajuste del modelo no lineal

Donde:

t tiempo

A La asintota inferior

U La asintota superior

ß Rango de crecimiento = 1

tø Tiempo inicial ( Predeterminado Ø) = Ø

\[y(t) = A+ \frac{ (U - A)}{1+ 1exp(-t)}\]

Derivadas parciales

Con respecto a A

\[ \frac{\delta y}{\delta A} = 1 - \frac{ 1}{1+ 1exp(-t)} \]

respecto a U

\[ \frac{\delta y}{\delta U} = \frac{ 1}{1+ 1exp(-t)} \]

set.seed(1972)
t=rep(c(0,7,14,21,28,35,42,49,56,63,70,77,84),each=4);t
##  [1]  0  0  0  0  7  7  7  7 14 14 14 14 21 21 21 21 28 28 28 28 35 35 35 35 42
## [26] 42 42 42 49 49 49 49 56 56 56 56 63 63 63 63 70 70 70 70 77 77 77 77 84 84
## [51] 84 84
mediast=tapply(as.numeric(t), as.factor(t), mean);mediast
##  0  7 14 21 28 35 42 49 56 63 70 77 84 
##  0  7 14 21 28 35 42 49 56 63 70 77 84
A<-((runif(4,5,20))*t/(runif(4,1,6)+t))+rnorm(52,0.2,0.6);length(A)
## [1] 52
mediasW=tapply(A,as.factor(t),mean);mediasW
##         0         7        14        21        28        35        42        49 
## 0.2283996 5.5903261 6.3799559 7.0768154 7.2503005 7.2730990 7.1370564 7.5852693 
##        56        63        70        77        84 
## 7.3219973 7.2647365 7.4997753 7.0568071 7.2974621
U=((runif(4,6,27)*t)/(runif(4,1,5)+t^2)+rnorm(52,0.2,0.8))
plot(t,A,pch=19,cex=0.6,ylab="peso Seco (g)",xlab="tiempo (días)")
points(mediast,mediasW,col="darkblue",pch=19)
points(7,mediasW[2],col="darkgreen",cex=1.2,pch=19)
points(14,mediasW[2],col="darkgreen",cex=1.2,pch=19)
points(14,mediasW[3],col="darkgreen",cex=1.2,pch=19)
xx = c(7,14,14)
yy = c(mediasW[2],mediasW[2],mediasW[3])
polygon(xx,yy,col="lightblue")
text(12,(mediasW[2]+mediasW[3])/2, "A",cex=0.9)
text(3,mediasW[2], bquote(paste('(t'['2']*',','A'['2'],')')),cex=0.75)
text(19,mediasW[2], bquote(paste('(t'['3']*',','A'['2'],')')),cex=0.75)
text(10,mediasW[3], bquote(paste('(t'['3']*',','A'['3'],')')),cex=0.75)
lines(mediast,mediasW,lty=1,col="darkred",lwd=1);grid(10,10)

library(growthmodels)
growth <- generalisedLogistic(0:10, 5, 10, 0.3, 0.5, 3)
# Calculate inverse function
time <- generalisedLogistic.inverse(growth, 5, 10, 0.3, 0.5, 3)
plot(growth)

plot(time)

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