Modelo de crecimiento
Jorge Mendez Gonzalez,UAAAN
2020-05-17
Crecimiento e incremento
El termino crecimiento tiene una variedad de acepciones en biología. En lo relativo a mediciones forestales, se le utiliza para señalar el aumento de tamaño ya sea a nivel de árbol individual o a nivel de rodal.
El crecimiento se expresa en un cambio en el tamaño de los individuos con el transcurso del tiempo. Normalmente, crecimiento o incremento se utilizan indistintamente para indicar un aumento de tamaño ocurrido en alguna variable durante un determinado lapso de tiempo, en tanto que la dimensión alcanzada hasta una edad especifica se refiere como crecimiento acumulado.
El crecimiento se evalúa dentro de un periodo de tiempo determinado (día, mes, año, u otro periodo más largo). La unidad de tiempo habitual para el estudio del crecimiento de un árbol o de un rodal es el año. De aquí surge la definicion del incremento corriente anual (ICA), que corresponde a la diferencia entre el tamaño al inicio y al final de un año de crecimiento. Si se considera un lapso de tiempo mayor a un año se habla de un periodo de crecimiento, y al incremento obtenido en ese lapso se le denomina incremento periódico (IP), esto es la diferencia entre el tamaño al final del periodo y el tamaño al inicio del periodo; ese tamaño, dividido por el número de años en el periodo, genera el incremento periódico anual (IPA). Al dividir el tamaño total acumulado entre el número de años (edad) se obtiene el incremento medio anual (IMA).
Curvas de Crecimiento. Cuando el tamaño de un organismo o una dimensión especifica de él (volumen, peso, altura, diámetro, área basal, etc.) es graficado con respecto a su edad, la curva aparecida se llama curva de crecimiento, aunque en muchas ocasiones la curva lo que muestre sea incrementos acumulativos a una edad determinada. Las curvas que resultan de todas estas variables, son sorprendentemente similares, presentando todas una forma sigmoidal (“s”).
Turno. Al período de tiempo que transcurre desde la formación de un bosque o rodal por siembra, plantación o cualquier otro medio, hasta su aprovechamiento final se le conoce como turno. Desde el punto de vista económico, el turno es un periodo de inversión, por eso deben considerarse los aspectos de rentabilidad y para poder decir cuál es un turno adecuado, se toman en cuenta factores de carácter biológico, estacional y socioeconómico que afectan la longitud del mismo.
Turno absoluto. Se define como la edad a la que se obtiene la máxima producción leñosa sin importar la naturaleza o calidad de los productos y se determina en base a la culminación del ICA en volumen.
Determinación del turno. Por medio del cálculo del ICA e IMA y la edad se obtiene un turno absoluto que se localiza en la intercepción de las líneas de ajuste de las curvas de ICA e IMA, también partiendo de este indicador es posible establecer un diámetro deseable de extracción para la industria forestal al que se le llama turno técnico, interceptando el diámetro con la línea de ajuste de la edad. Este cálculo ayuda a determinar una edad aproximada de corte y un ciclo de corta estándar que ayuda en la planeación de las actividades. Aunque también corresponde al valor máximo de la curva de IMA en altura, o sobre la curva guía en el punto en que una recta que parte del eje coordenado es tangente a la curva guía (Zepeda y Rivero, 1984).
Modelos de crecimiento
Existe una amplia gama de modelos matemáticos que pueden ser utilizados para ajustar a datos de crecimiento (e.g. Pienaar y Turnbull 1973, Nokoe 1978, Shifley y Brand 1984, Zeide 1993). En su mayoría son modelos no lineales en los parámetros y deben ajustarse mediante métodos numéricos. En el ajuste se utilizan métodos iterativos, los que requieren como punto de inicio un valor aproximado de los parámetros a estimar. De todos los parámetros de los modelos, el más fácilmente comprensible es la asíntota, esto es el valor al cual debiera tender la variable dependiente cuando la edad del individuo tiende a infinito. Ese valor puede aproximarse con facilidad al graficar la variable de interés en función de la edad.
Las librerias
Para ajustar este modelo, así como para hacer algunas predicciones de la variable dependiente, dar énfasis a las figuras, entre otras cosas, es necesario del uso de algunas librerías de r., se las enlisto a continuación.
library(pacman)
p_load(dplyr, tidyr, reshape, tidyverse, stats, graphics, propagate,
minpack.lm, lubridate, RColorBrewer, DT, datasets)Los datos y el modelo
En este ejemplo ajustaremos el modelo de crecimiento de Schumacher, el cual viene dado por la siguiente ecuación: \(y=exp(\beta _{0}+\beta _{1}/x)\) calcularemos es ICA, e IMA para determinar el turno absoluto. Los datos provienen de análisis trocales de diez arboles de la especie P. halepensis. la variable “y” es Biomasa (kg) y la variable “x” es edad (años).
rm(list=ls(all=TRUE))
setwd("G:/UAAAN/MATERIAS/2020/EPIDOMETRIA/Modelos de crecimiento/")
datos=read.csv("G:/UAAAN/MATERIAS/2020/EPIDOMETRIA/Modelos de crecimiento/Biomasa.csv") # Cargar los datos
attach(datos)
head(datos)## Edad Biomasa
## 1 1 0.0001731810
## 2 1 0.0001202640
## 3 1 0.0000769692
## 4 1 0.0004810580
## 5 1 0.0018472610
## 6 1 0.0002357180n=length(Edad) # Obtener el valor de nLo primero que haremos es una figura para ver cómo está la distribución de los datos de Biomasa y Edad, o ver si no hay datos atípicos, o datos erróneos etc
colores_0 <- factor(datos$Edad) #Haciendo una rampa de colores de edad
plot(Edad, Biomasa, xlab = "Edad (años)", ylab = "Biomasa (m)", col = colores_0, cex=2,
pch = 20) 
Vemos que todo está bien, no se ven datos erróneos o atípicos. Procedamos al siguiente paso, que es ajustar el modelo. Recodando que los modelos de crecimiento son no lineales. Muchos de estos modelos pueden linealizarse y ajustarse mediante regresión lineal, pero otros no.
Ajuste del modelo
Aquí, usaremos a0, y a1, para denotar los coeficientes del modelo de Schumacher. Como se indicó anteriormente, en los modelos no lineales, es necesario asignar valores iniciales a los coeficientes de regresión, aquí les daremos el valor de -0.4 y 4 respectivamente. Si el modelo no encuentra los valores de los estimadores, es necesario segur probando con otros valores iniciales.
# EL MODELO NO LINEAL ---------------------------------------------------------------
#$\mathrm{e}^{\frac{b}{x}+a}$ # El modelo
#$-\dfrac{b\mathrm{e}^{\frac{b}{x}+a}}{x^2}$ # La derivada de dy/dx
Fit1<- nlsLM(Biomasa ~ exp(a0+(a1/Edad)), # Ajustar el modelo
data = datos, # Nombre de los datos
start = list(a0 = -0.4, a1 = 4), # Valores iniciales
algorithm = "port", # Algoritmo
control = list(maxiter = 500))
summary(Fit1) # Resumen del modelo##
## Formula: Biomasa ~ exp(a0 + (a1/Edad))
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## a0 4.9473 0.1504 32.89 <2e-16 ***
## a1 -44.9721 2.6058 -17.26 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.388 on 245 degrees of freedom
##
## Algorithm "port", convergence message: Relative error in the sum of squares is at most `ftol'.Como se puede ver, con estos valores iniciales, se encontró la convergencia, es decir, la iteración matemática hasta encontrar los valores de los estimadores finalizo correctamente, y de alguna forma, minimizar la suma de cuadrados del error. Cuando no se encuentra la convergencia, es posible que encuentre el siguiente mensaje: Error in numericDeriv(form[[3L]], names (ind), env) : Missing value or an infinity produced when…
Tambien pueden ver en los resultados, el script no arroja los estadísticos de ajuste (Error, R2 o coeficiente de variación etc.), como en los modelos lineales. Es necesario calcularlos (ya lo saben todos en las lecciones anteriores). Para eso necesitaremos antes de todo: a) extraer los grados de libertad de los residuales, b) calcular el CME y c) calcular el CMT. Ya con esto podremos obtener el Error, R2 o coeficiente de variación. Primero voy a hacer un dataframe para extraer los valores de A0 y A1 del resumen, y ya después hacemos los cálculos.
df<-as.data.frame(coef(summary(Fit1))) # Hacer dataframe
df[1, 1] # Extraer valor de a0## [1] 4.947289df[2, 1] # Extraer valor de a1## [1] -44.97214df.residual(Fit1) # Extraer los GL de los residuales## [1] 245CMErr_Fit1<-sum((Biomasa-fitted(Fit1))^2)/(df.residual(Fit1)); CMErr_Fit1## [1] 5.703881CMTot_Fit1<-(sum((Biomasa-mean(Biomasa))^2))/((length(Biomasa)-1)); CMTot_Fit1## [1] 26.64969R2_Fit1<-1-(CMErr_Fit1)/(CMTot_Fit1); R2_Fit1 # La R2 ajustada## [1] 0.7859682Sxy_Fit1<-sqrt(CMErr_Fit1);Sxy_Fit1 # El Error error del modelo## [1] 2.38828CV_Fit1<-(Sxy_Fit1)/(mean(Biomasa))*100; CV_Fit1 # El Coeficiente de variacion## [1] 59.79015Según los cálculos, el ajuste es bueno, pues además de que los coeficientes de regresión son estadísticamente significativos (95 %), la R2 es de 0.7859682, es decir; la edad explica 78.59 % de la Biomasa. El error de estimación es de 2.38828 kg de biomasa y el coeficiente de variación es de 59.79015 %. Este último valor si es alto, pero es un estadístico eficiente para comparar el ajuste entre varios modelos, no así el error.
Observados y estimados
Una vez que ya tenemos los valores de los estimadores (a0 y a1) verifiquemos si el modelo está prediciendo correctamente la variable dependiente (Biomasa). Esto es muy sencillo: 1) hagamos la figura de Biomasa (eje y) vs Edad (eje x), y 2) Sobreponer en esa misma figura los estimados por el modelo.
plot(Edad, Biomasa, xlab = "Edad (años)", ylab = "Biomasa (m)", cex=1) # Hacer la figura
lines(Edad, fitted.values(Fit1), lwd=3, col="red") # Agregar los estimados a la plot
La figura superior demuestra que los valores de los estimadores (a0 y a1) son correctos. El modelo de Schumacher estima adecuadamente la Biomasa de P. halepensis en función de la Edad. Esto se constata pues los estimados pasan por el centro de los valores observados de Biomasa.
Predicciones para ciertos valores
Probemos el modelo, vamos a suponer que tenemos algunos datos de edad de esta especie, y que deseamos saber cuánta biomasa predice el modelo a esas edades. Como ven, ya no es necesario hace análisis troncales para conocer la Biomasa de cierto árbol, ya solo a partir de la edad, podremos saber cuánta biomasa almacena ese árbol.
min_x<-min(Edad) # Obtner el valor minim de los datos
max_x<-max(Edad) # Obtner el valor maximo de los datos
x0<-(min_x:max_x) # Generar un conjunto de valores desde el min al max
plot(Edad, Biomasa, xlab = "Edad (años)", ylab = "Biomasa (m)", cex=1)
#x0<-10:20 # Valores de Edad (de 10 a 20) para predecir Biomasa con el modelo
df[1, 1] # Extaer el valor de a0 del resumen## [1] 4.947289df[2, 1] # Extaer el valor de a1 del resumen## [1] -44.97214Prediccion_m<-exp(df[1, 1]+(df[2, 1]/x0)) # Substituir x0 en el modelo
Prediccion_m # Ver las predicciones## [1] 4.144085e-18 2.415486e-08 4.347068e-05 1.844134e-03 1.747230e-02
## [6] 7.823271e-02 2.282526e-01 5.095495e-01 9.515955e-01 1.568430e+00
## [11] 2.360593e+00 3.318825e+00 4.427760e+00 5.668891e+00 7.022691e+00
## [16] 8.469999e+00 9.992843e+00 1.157488e+01 1.320155e+01 1.486014e+01
## [21] 1.653962e+01 1.823059e+01points(x0, Prediccion_m, col="#CD1076", cex=1.8, pch=19) # Plotear las predicciones
Según los nuevos datos de Edad (de uno en uno hasta 22), el modelo predice bien. Por ejemplo, a una edad de 5 años, el modelo predice 0.0174723 (observación 5). A los 10, 15 y 20 años el modelo predice, 1.56843, 7.022691 y 14.86014 kg (ver valores del objeto Prediccion_m).
Calculo del Incremento Medio Anual (IMA)
Anteriormente se explicó que era el IMA. Una vez que hemos visto que el modelo predice bien, podemos calcular el IMA. Antes que nada, generemos una nueva base de datos de edad (de 1 hasta 50), este rango va a depender de la edad de los datos observados, pero la idea es ver el comportamiento del IMA más allá de la edad actual. Después, hay que aplicar el modelo, usando la nueva base de datos de edad, y finalmente, dividirlo entre la edad. ¡¡¡ es todo, ya obtuvo el IMA ¡¡¡¡¡¡¡
x= 1:50; x # Generar una nueva Edad (x) de 1 a 50## [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
## [26] 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50IMA<-exp(df[1, 1]+(df[2, 1]/x))/x; IMA # Calcular el IMA## [1] 4.144085e-18 1.207743e-08 1.449023e-05 4.610334e-04 3.494459e-03
## [6] 1.303879e-02 3.260752e-02 6.369369e-02 1.057328e-01 1.568430e-01
## [11] 2.145994e-01 2.765688e-01 3.405969e-01 4.049208e-01 4.681794e-01
## [16] 5.293749e-01 5.878143e-01 6.430487e-01 6.948185e-01 7.430068e-01
## [21] 7.876009e-01 8.286632e-01 8.663073e-01 9.006812e-01 9.319535e-01
## [26] 9.603042e-01 9.859175e-01 1.008977e+00 1.029661e+00 1.048143e+00
## [31] 1.064588e+00 1.079150e+00 1.091977e+00 1.103204e+00 1.112960e+00
## [36] 1.121362e+00 1.128521e+00 1.134538e+00 1.139507e+00 1.143514e+00
## [41] 1.146640e+00 1.148957e+00 1.150533e+00 1.151431e+00 1.151708e+00
## [46] 1.151416e+00 1.150604e+00 1.149318e+00 1.147597e+00 1.145479e+00plot(IMA, col="lightskyblue", pch=19, cex=1.5, xlab="Edad (años)", ylab="Incremento Medio Anual (kg)")
z<-round(IMA, 2)
text(IMA~x, labels=z, cex=0.7, font=1, pos=2)# Plotear IMA
arrows(45, 0, 45, 1.1, col = "#CD1076") # Colocar la flecha
Calculo del Incremento Corriente Anual (ICA)
Es turno de calcular el ICA. También es muy sencillo, pero para hacerlo aún más sencillo. Voy a extraer los valores de los estimadores, y les voy a llamar “a” y “b” respectivamente. Sabiendo que el modelo es \(y=exp(a+b/x)\), de aqui podemos obtener la derivada de dy/dx la cual queda de la siguientes manera: \(dy/dx=-\dfrac{b\mathrm{e}^{\frac{b}{x}+a}}{x^2}\). El ICA, simple y sencillamente es la derivada del modelo con respecto a “x” (dy/dx). Ve? Eso fue todo¡¡¡¡ así de simple.
#Creando objetos para simplificar la formula
a<-(df[1, 1]); a # Extaer del resumen a0 y llamarlo a## [1] 4.947289b<-(df[2, 1]); b # Extaer del resumen a1 y llamarlo b## [1] -44.97214ICA<--((b*(exp(((b/x)+a))))/(x^2)); ICA # Aplicar la derivada## [1] 1.863684e-16 2.715739e-07 2.172188e-04 5.183415e-03 3.143066e-02
## [6] 9.773035e-02 2.094900e-01 3.580552e-01 5.283368e-01 7.053566e-01
## [11] 8.773629e-01 1.036491e+00 1.178259e+00 1.300725e+00 1.403669e+00
## [16] 1.487945e+00 1.555016e+00 1.606626e+00 1.644604e+00 1.670730e+00
## [21] 1.686671e+00 1.693943e+00 1.693900e+00 1.687732e+00 1.676478e+00
## [26] 1.661036e+00 1.642179e+00 1.620566e+00 1.596761e+00 1.571242e+00
## [31] 1.544413e+00 1.516615e+00 1.488137e+00 1.459219e+00 1.430062e+00
## [36] 1.400835e+00 1.371676e+00 1.342700e+00 1.314002e+00 1.285657e+00
## [41] 1.257728e+00 1.230263e+00 1.203301e+00 1.176871e+00 1.150995e+00
## [46] 1.125688e+00 1.100961e+00 1.076818e+00 1.053263e+00 1.030293e+00plot(ICA, col="pink", pch=19, cex=1.5, xlab="Edad (años)", ylab="Incremento Corriente Anual (kg)")
z1<-round(ICA, 2)
text(ICA~x, labels=z1, cex=0.7, font=1, pos=2)# Plotear ICA
arrows(23, 0, 23, 1.5, col = "#CD1076") # Colocar la flecha
#Graficas de IMA e ICA
# COlores_1 <-factor(datos$Biomasa) # Hacer rampa de colores
# plot(x, IMA, col = COlores_1, cex=2, pch = 20, ylim=c(0, 2), xlab="", ylab="") # Graficar x (nueva edad) e IMA
# par (new = TRUE) # Sobreponer dos plots
# colores <- factor(datos$Edad) # Hacer rampa de colores
# plot(x, ICA, col = colores, cex=2, pch = 20, ylim=c(0, 2), xlab="Edad (años)", ylab="Incremento (kg/año)")El turno absoluto
Ahora vamos a colocar en una misma figura al ICA e IMA. Recordemos que el turno absoluto es donde se cruzan ambas. Esto es muy fácil, vea como… solo graficar ambos en una sola figura.
plot(x, ICA, type = "l", col="black", xlab="Edad (años)", ylab="ICA e IMA (kg)") # Graficas de ICA
lines(x, IMA, col="red") # Graficar x (nueva edad) e IMA
par (new = TRUE) # Sobreponer dos plots
lines(x, ICA, type = "l", col="black")
legend(0, 1.5, c("ICA", "IMA"), # Asignar leyendas a ICA e IMA
lty = rep(1, 2), bty = "n",
col = c(1, 2))
arrows(45, 0, 45, 1.2, col = "#CD1076")
Según los resultados, el máximo IMA se encuentra a le edad de 45 años, y tiene un valor de 1.152 kg. De la misma forma, el máximo ICA, se obtiene cuando el árbol llega a tener una edad de 23 años, registrando un valor de 1.694 kg. En este sentido, el Turno absoluto, es a la edad de 45 años.
Si desea ver los valores en una hoja de cálculo, corra las siguientes líneas. Aquí podrá verificar donde se encuentra el máximo IMA, máximo ICA.
x= 1:50 # Generar una nueva Edad (x) de 1 a 50
IMA<-exp(df[1, 1]+(df[2, 1]/x))/x # Calcular el IMA
ICA<--((b*(exp(((b/x)+a))))/(x^2)) # Aplicar la derivada
df_I <- data.frame(x, IMA, ICA) # Ponerlos en una tabla
df_II <-round(df_I,3) # Redondear a tres decimales
write.csv(df_II, file="Tabla de ICA e IMA.csv") # Exportar la tabla a la carpetaIntervalos de confianza (95 %) Simulacion Monte Carlo
Finalmente, es muy importante que además de predecir el valor puntual de la variable dependiente, también se hagan predicciones a cierto limites o intervalos de confianza. Esto lo hace muy sencillo el procedimiento en r. Esta misma técnica la puede aplicar más allá de los rangos de la variable dependiente, (no la usaremos aquí, pero puede ver mis rpubs, ahí encontrará la forma de hacerlos).
CONC_Fit3 <- seq(n+-n, n+50, by = 1)
Pred_Fit3 <- predict(Fit1, newdata = data.frame(x = CONC_Fit3), nsim = 10000)
IC_95_Fit3 <- predictNLS(Fit1, newdata = data.frame(Edad = CONC_Fit3), nsim = 10000); ## predictNLS: Propagating predictor value #1...## predictNLS: Propagating predictor value #2...## predictNLS: Propagating predictor value #3...## predictNLS: Propagating predictor value #4...## predictNLS: Propagating predictor value #5...## predictNLS: Propagating predictor value #6...## predictNLS: Propagating predictor value #7...## predictNLS: Propagating predictor value #8...## predictNLS: Propagating predictor value #9...## predictNLS: Propagating predictor value #10...## predictNLS: Propagating predictor value #11...## predictNLS: Propagating predictor value #12...## predictNLS: Propagating predictor value #13...## predictNLS: Propagating predictor value #14...## predictNLS: Propagating predictor value #15...## predictNLS: Propagating predictor value #16...## predictNLS: Propagating predictor value #17...## predictNLS: Propagating predictor value #18...## predictNLS: Propagating predictor 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#240...## predictNLS: Propagating predictor value #241...## predictNLS: Propagating predictor value #242...## predictNLS: Propagating predictor value #243...## predictNLS: Propagating predictor value #244...## predictNLS: Propagating predictor value #245...## predictNLS: Propagating predictor value #246...## predictNLS: Propagating predictor value #247...## predictNLS: Propagating predictor value #248...## predictNLS: Propagating predictor value #249...## predictNLS: Propagating predictor value #250...## predictNLS: Propagating predictor value #251...## predictNLS: Propagating predictor value #252...## predictNLS: Propagating predictor value #253...## predictNLS: Propagating predictor value #254...## predictNLS: Propagating predictor value #255...## predictNLS: Propagating predictor value #256...## predictNLS: Propagating predictor value #257...## predictNLS: Propagating predictor value #258...## predictNLS: Propagating predictor value #259...## predictNLS: Propagating predictor value #260...## predictNLS: Propagating predictor value #261...## predictNLS: Propagating predictor value #262...## predictNLS: Propagating predictor value #263...## predictNLS: Propagating predictor value #264...## predictNLS: Propagating predictor value #265...## predictNLS: Propagating predictor value #266...## predictNLS: Propagating predictor value #267...## predictNLS: Propagating predictor value #268...## predictNLS: Propagating predictor value #269...## predictNLS: Propagating predictor value #270...## predictNLS: Propagating predictor value #271...## predictNLS: Propagating predictor value #272...## predictNLS: Propagating predictor value #273...## predictNLS: Propagating predictor value #274...## predictNLS: Propagating predictor value #275...## predictNLS: Propagating predictor value #276...## predictNLS: Propagating predictor value #277...## predictNLS: Propagating predictor value #278...## predictNLS: Propagating predictor value #279...## predictNLS: Propagating predictor value #280...## predictNLS: Propagating predictor value #281...## predictNLS: Propagating predictor value #282...## predictNLS: Propagating predictor value #283...## predictNLS: Propagating predictor value #284...## predictNLS: Propagating predictor value #285...## predictNLS: Propagating predictor value #286...## predictNLS: Propagating predictor value #287...## predictNLS: Propagating predictor value #288...## predictNLS: Propagating predictor value #289...## predictNLS: Propagating predictor value #290...## predictNLS: Propagating predictor value #291...## predictNLS: Propagating predictor value #292...## predictNLS: Propagating predictor value #293...## predictNLS: Propagating predictor value #294...## predictNLS: Propagating predictor value #295...## predictNLS: Propagating predictor value #296...## predictNLS: Propagating predictor value #297...## predictNLS: Propagating predictor value #298...#IC_95_Fit3$summary # Resumir las predicciones
H<-data.frame(IC_95_Fit3$summary) # Hacer un dataframe
df_III <-round(H,3); df_III # Redondear los valores a 3## Prop.Mean.1 Prop.Mean.2 Prop.sd.1 Prop.sd.2 Prop.2.5. Prop.97.5. Sim.Mean
## 1 0.000 0.000 NaN NaN NaN NaN Inf
## 2 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 3 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 4 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 5 0.002 0.002 0.001 0.001 0.000 0.004 0.002
## 6 0.017 0.019 0.007 0.007 0.005 0.032 0.019
## 7 0.078 0.081 0.022 0.023 0.036 0.127 0.081
## 8 0.228 0.234 0.051 0.052 0.131 0.337 0.234
## 9 0.510 0.518 0.091 0.092 0.336 0.699 0.520
## 10 0.952 0.961 0.136 0.137 0.691 1.232 0.963
## 11 1.568 1.579 0.180 0.181 1.222 1.935 1.579
## 12 2.361 2.371 0.217 0.218 1.942 2.799 2.370
## 13 3.319 3.328 0.243 0.243 2.848 3.807 3.329
## 14 4.428 4.435 0.256 0.256 3.931 4.940 4.437
## 15 5.669 5.675 0.257 0.257 5.168 6.181 5.673
## 16 7.023 7.027 0.250 0.250 6.534 7.520 7.026
## 17 8.470 8.474 0.245 0.245 7.991 8.957 8.472
## 18 9.993 9.996 0.255 0.255 9.494 10.498 9.994
## 19 11.575 11.579 0.293 0.293 11.002 12.155 11.581
## 20 13.202 13.207 0.362 0.362 12.493 13.920 13.204
## 21 14.860 14.867 0.459 0.459 13.964 15.771 14.857
## 22 16.540 16.550 0.577 0.577 15.414 17.686 16.548
## 23 18.231 18.244 0.711 0.711 16.844 19.645 18.245
## 24 19.925 19.944 0.858 0.859 18.252 21.635 19.949
## 25 21.616 21.640 1.016 1.016 19.638 23.642 21.634
## 26 23.299 23.329 1.182 1.182 21.000 25.658 23.343
## 27 24.968 25.005 1.354 1.355 22.335 27.674 24.989
## 28 26.620 26.664 1.532 1.533 23.643 29.684 26.676
## 29 28.251 28.303 1.714 1.716 24.924 31.683 28.310
## 30 29.860 29.921 1.900 1.902 26.175 33.666 29.917
## 31 31.444 31.514 2.087 2.090 27.398 35.630 31.511
## 32 33.002 33.081 2.277 2.279 28.591 37.570 33.057
## 33 34.533 34.621 2.467 2.470 29.755 39.486 34.552
## 34 36.035 36.133 2.658 2.661 30.891 41.375 36.153
## 35 37.509 37.617 2.849 2.853 31.998 43.236 37.681
## 36 38.954 39.072 3.039 3.044 33.077 45.067 39.095
## 37 40.369 40.498 3.229 3.234 34.129 46.868 40.525
## 38 41.755 41.895 3.417 3.423 35.153 48.637 41.866
## 39 43.112 43.263 3.604 3.611 36.151 50.375 43.252
## 40 44.441 44.602 3.790 3.797 37.124 52.081 44.615
## 41 45.741 45.913 3.974 3.981 38.071 53.755 45.921
## 42 47.012 47.196 4.156 4.164 38.994 55.398 47.159
## 43 48.256 48.451 4.336 4.345 39.893 57.009 48.401
## 44 49.473 49.679 4.514 4.523 40.769 58.589 49.717
## 45 50.663 50.880 4.690 4.700 41.623 60.137 50.794
## 46 51.827 52.055 4.863 4.874 42.455 61.655 51.986
## 47 52.965 53.204 5.034 5.046 43.266 63.143 53.259
## 48 54.078 54.329 5.203 5.215 44.057 64.601 54.336
## 49 55.167 55.429 5.369 5.382 44.828 66.029 55.491
## 50 56.232 56.504 5.533 5.547 45.579 67.430 56.493
## 51 57.274 57.557 5.695 5.709 46.313 68.801 57.472
## 52 58.293 58.587 5.854 5.868 47.028 70.146 58.640
## 53 59.290 59.595 6.010 6.026 47.726 71.463 59.610
## 54 60.265 60.581 6.164 6.180 48.407 72.754 60.469
## 55 61.220 61.546 6.316 6.333 49.072 74.019 61.615
## 56 62.154 62.490 6.465 6.483 49.721 75.259 62.548
## 57 63.068 63.415 6.612 6.630 50.355 76.474 63.350
## 58 63.963 64.320 6.757 6.776 50.974 77.665 64.338
## 59 64.839 65.206 6.899 6.918 51.579 78.833 65.297
## 60 65.697 66.074 7.039 7.059 52.170 79.978 66.035
## 61 66.537 66.924 7.176 7.197 52.747 81.100 66.785
## 62 67.359 67.756 7.312 7.333 53.312 82.200 67.726
## 63 68.165 68.572 7.445 7.467 53.864 83.280 68.625
## 64 68.954 69.371 7.576 7.599 54.403 84.338 69.335
## 65 69.728 70.154 7.705 7.728 54.931 85.376 70.066
## 66 70.486 70.921 7.832 7.856 55.447 86.394 70.888
## 67 71.228 71.673 7.956 7.981 55.953 87.393 71.659
## 68 71.957 72.410 8.079 8.104 56.447 88.373 72.299
## 69 72.670 73.133 8.200 8.226 56.931 89.335 73.063
## 70 73.370 73.842 8.318 8.345 57.405 90.279 73.924
## 71 74.057 74.537 8.435 8.462 57.869 91.205 74.533
## 72 74.730 75.219 8.550 8.578 58.323 92.115 75.348
## 73 75.390 75.888 8.663 8.692 58.768 93.008 75.885
## 74 76.038 76.544 8.774 8.803 59.204 93.884 76.482
## 75 76.674 77.188 8.884 8.913 59.631 94.745 77.163
## 76 77.297 77.820 8.991 9.022 60.050 95.591 77.970
## 77 77.910 78.441 9.097 9.128 60.461 96.421 78.442
## 78 78.511 79.050 9.202 9.233 60.863 97.237 79.113
## 79 79.101 79.648 9.304 9.337 61.258 98.038 79.646
## 80 79.680 80.235 9.406 9.438 61.645 98.826 80.194
## 81 80.249 80.812 9.505 9.538 62.025 99.600 80.701
## 82 80.808 81.379 9.603 9.637 62.397 100.360 81.464
## 83 81.357 81.935 9.699 9.734 62.763 101.108 81.976
## 84 81.896 82.482 9.794 9.829 63.121 101.843 82.473
## 85 82.426 83.020 9.888 9.923 63.473 102.566 82.991
## 86 82.947 83.548 9.980 10.016 63.819 103.276 83.674
## 87 83.459 84.067 10.071 10.107 64.158 103.975 84.190
## 88 83.962 84.577 10.160 10.197 64.492 104.662 84.465
## 89 84.457 85.079 10.248 10.286 64.819 105.338 85.233
## 90 84.943 85.572 10.335 10.373 65.141 106.003 85.438
## 91 85.422 86.057 10.420 10.459 65.457 106.658 86.270
## 92 85.892 86.534 10.504 10.543 65.767 107.301 86.555
## 93 86.355 87.004 10.587 10.627 66.072 107.935 87.077
## 94 86.810 87.465 10.669 10.709 66.372 108.559 87.526
## 95 87.257 87.919 10.749 10.790 66.667 109.172 87.935
## 96 87.698 88.366 10.829 10.870 66.956 109.777 88.381
## 97 88.131 88.806 10.907 10.948 67.241 110.371 88.660
## 98 88.558 89.239 10.984 11.026 67.521 110.957 89.392
## 99 88.978 89.665 11.060 11.102 67.797 111.534 89.451
## 100 89.391 90.085 11.135 11.178 68.068 112.102 90.180
## 101 89.798 90.498 11.209 11.252 68.335 112.661 90.455
## 102 90.199 90.905 11.281 11.325 68.597 113.212 91.069
## 103 90.594 91.305 11.353 11.398 68.855 113.755 91.380
## 104 90.982 91.700 11.424 11.469 69.109 114.290 91.628
## 105 91.365 92.088 11.494 11.539 69.359 114.817 92.247
## 106 91.742 92.471 11.563 11.609 69.606 115.336 92.464
## 107 92.114 92.848 11.631 11.677 69.848 115.848 92.762
## 108 92.480 93.219 11.698 11.744 70.087 116.352 93.236
## 109 92.840 93.586 11.764 11.811 70.322 116.850 93.699
## 110 93.196 93.946 11.829 11.877 70.553 117.340 93.881
## 111 93.546 94.302 11.893 11.941 70.781 117.823 94.494
## 112 93.891 94.652 11.957 12.005 71.005 118.299 94.740
## 113 94.231 94.998 12.020 12.069 71.227 118.769 94.998
## 114 94.567 95.338 12.082 12.131 71.445 119.232 95.129
## 115 94.897 95.674 12.143 12.192 71.659 119.689 95.608
## 116 95.224 96.005 12.203 12.253 71.871 120.140 95.983
## 117 95.545 96.332 12.262 12.313 72.079 120.585 96.272
## 118 95.862 96.654 12.321 12.372 72.285 121.023 96.774
## 119 96.175 96.972 12.379 12.430 72.488 121.456 96.878
## 120 96.483 97.285 12.437 12.488 72.687 121.883 97.335
## 121 96.788 97.594 12.493 12.545 72.884 122.304 97.536
## 122 97.088 97.899 12.549 12.601 73.079 122.720 97.906
## 123 97.384 98.200 12.604 12.657 73.270 123.130 98.095
## 124 97.677 98.497 12.659 12.712 73.459 123.535 98.469
## 125 97.965 98.790 12.712 12.766 73.645 123.934 98.757
## 126 98.250 99.079 12.765 12.819 73.829 124.329 99.080
## 127 98.531 99.364 12.818 12.872 74.010 124.718 99.334
## 128 98.808 99.646 12.870 12.924 74.189 125.103 99.735
## 129 99.082 99.924 12.921 12.976 74.366 125.483 99.777
## 130 99.352 100.199 12.972 13.027 74.540 125.858 100.298
## 131 99.619 100.470 13.022 13.077 74.712 126.228 100.417
## 132 99.882 100.737 13.071 13.127 74.881 126.594 100.619
## 133 100.142 101.002 13.120 13.176 75.049 126.955 100.960
## 134 100.399 101.263 13.168 13.225 75.214 127.311 101.200
## 135 100.653 101.520 13.216 13.273 75.377 127.664 101.650
## 136 100.903 101.775 13.263 13.320 75.538 128.012 101.717
## 137 101.151 102.026 13.310 13.367 75.697 128.356 102.118
## 138 101.395 102.275 13.356 13.414 75.854 128.696 102.296
## 139 101.637 102.520 13.401 13.460 76.009 129.031 102.677
## 140 101.875 102.763 13.447 13.505 76.162 129.363 102.473
## 141 102.111 103.002 13.491 13.550 76.313 129.691 103.113
## 142 102.344 103.239 13.535 13.594 76.463 130.015 103.383
## 143 102.574 103.473 13.579 13.638 76.610 130.335 103.354
## 144 102.801 103.704 13.622 13.681 76.756 130.652 103.536
## 145 103.026 103.932 13.664 13.724 76.900 130.965 104.050
## 146 103.248 104.158 13.706 13.767 77.042 131.274 104.165
## 147 103.468 104.381 13.748 13.809 77.183 131.580 104.320
## 148 103.685 104.602 13.789 13.850 77.321 131.882 104.604
## 149 103.899 104.820 13.830 13.891 77.459 132.181 105.107
## 150 104.112 105.035 13.870 13.932 77.594 132.477 105.406
## 151 104.321 105.249 13.910 13.972 77.728 132.769 105.337
## 152 104.529 105.459 13.950 14.012 77.861 133.058 105.605
## 153 104.734 105.668 13.989 14.051 77.992 133.344 106.160
## 154 104.936 105.874 14.027 14.090 78.121 133.627 105.570
## 155 105.137 106.078 14.066 14.128 78.249 133.906 106.079
## 156 105.335 106.279 14.103 14.166 78.376 134.183 106.015
## 157 105.531 106.479 14.141 14.204 78.501 134.456 106.521
## 158 105.725 106.676 14.178 14.241 78.624 134.727 106.943
## 159 105.917 106.871 14.214 14.278 78.747 134.995 106.663
## 160 106.107 107.064 14.251 14.315 78.868 135.260 107.062
## 161 106.294 107.255 14.287 14.351 78.987 135.522 107.460
## 162 106.480 107.443 14.322 14.387 79.106 135.781 107.613
## 163 106.664 107.630 14.357 14.422 79.223 136.038 107.530
## 164 106.846 107.815 14.392 14.457 79.338 136.292 107.945
## 165 107.026 107.998 14.427 14.492 79.453 136.543 107.934
## 166 107.204 108.179 14.461 14.527 79.566 136.792 108.290
## 167 107.380 108.358 14.495 14.561 79.678 137.038 108.294
## 168 107.554 108.535 14.528 14.594 79.789 137.282 108.496
## 169 107.727 108.711 14.561 14.628 79.899 137.523 108.991
## 170 107.898 108.884 14.594 14.661 80.007 137.762 108.933
## 171 108.067 109.056 14.627 14.693 80.115 137.998 109.024
## 172 108.234 109.227 14.659 14.726 80.221 138.232 109.327
## 173 108.399 109.395 14.691 14.758 80.326 138.464 109.324
## 174 108.563 109.562 14.722 14.790 80.430 138.693 109.348
## 175 108.726 109.727 14.753 14.821 80.533 138.920 109.602
## 176 108.886 109.890 14.784 14.852 80.635 139.145 109.964
## 177 109.046 110.052 14.815 14.883 80.736 139.367 110.061
## 178 109.203 110.212 14.845 14.914 80.836 139.588 110.591
## 179 109.359 110.371 14.876 14.944 80.935 139.806 110.475
## 180 109.514 110.528 14.905 14.974 81.033 140.023 110.807
## 181 109.666 110.683 14.935 15.004 81.130 140.237 110.945
## 182 109.818 110.838 14.964 15.033 81.226 140.449 110.859
## 183 109.968 110.990 14.993 15.063 81.321 140.659 110.890
## 184 110.117 111.141 15.022 15.092 81.415 140.867 111.027
## 185 110.264 111.291 15.050 15.120 81.509 141.073 111.391
## 186 110.410 111.439 15.079 15.149 81.601 141.278 111.633
## 187 110.554 111.586 15.107 15.177 81.692 141.480 111.620
## 188 110.697 111.732 15.134 15.205 81.783 141.680 111.875
## 189 110.839 111.876 15.162 15.232 81.872 141.879 111.807
## 190 110.979 112.018 15.189 15.260 81.961 142.076 111.923
## 191 111.118 112.160 15.216 15.287 82.049 142.271 111.779
## 192 111.256 112.300 15.243 15.314 82.136 142.464 112.226
## 193 111.392 112.439 15.269 15.341 82.223 142.655 112.535
## 194 111.528 112.577 15.295 15.367 82.308 142.845 112.337
## 195 111.662 112.713 15.321 15.393 82.393 143.033 112.655
## 196 111.795 112.848 15.347 15.419 82.477 143.219 112.790
## 197 111.926 112.982 15.373 15.445 82.560 143.404 112.902
## 198 112.057 113.115 15.398 15.471 82.642 143.587 113.049
## 199 112.186 113.246 15.423 15.496 82.724 143.768 113.115
## 200 112.314 113.376 15.448 15.521 82.805 143.948 113.220
## 201 112.441 113.506 15.473 15.546 82.885 144.126 113.452
## 202 112.567 113.634 15.497 15.571 82.964 144.303 113.638
## 203 112.692 113.761 15.522 15.595 83.043 144.478 113.778
## 204 112.815 113.886 15.546 15.619 83.121 144.652 114.028
## 205 112.938 114.011 15.570 15.643 83.198 144.824 113.935
## 206 113.059 114.135 15.593 15.667 83.275 144.994 113.819
## 207 113.180 114.257 15.617 15.691 83.351 145.163 114.226
## 208 113.299 114.379 15.640 15.714 83.426 145.331 114.158
## 209 113.418 114.499 15.663 15.738 83.501 145.497 114.805
## 210 113.535 114.619 15.686 15.761 83.575 145.662 114.296
## 211 113.651 114.737 15.709 15.784 83.648 145.826 114.859
## 212 113.767 114.854 15.731 15.806 83.721 145.988 115.115
## 213 113.881 114.971 15.754 15.829 83.793 146.149 115.233
## 214 113.995 115.086 15.776 15.851 83.865 146.308 115.229
## 215 114.107 115.201 15.798 15.873 83.935 146.466 115.227
## 216 114.219 115.314 15.820 15.895 84.006 146.623 115.313
## 217 114.330 115.427 15.841 15.917 84.075 146.779 115.727
## 218 114.439 115.539 15.863 15.939 84.144 146.933 115.738
## 219 114.548 115.649 15.884 15.960 84.213 147.086 115.478
## 220 114.656 115.759 15.905 15.981 84.281 147.238 115.903
## 221 114.763 115.868 15.926 16.003 84.348 147.388 115.824
## 222 114.869 115.976 15.947 16.023 84.415 147.538 115.760
## 223 114.975 116.083 15.967 16.044 84.481 147.686 116.532
## 224 115.079 116.190 15.988 16.065 84.547 147.833 116.420
## 225 115.183 116.295 16.008 16.085 84.612 147.978 116.291
## 226 115.286 116.400 16.028 16.106 84.677 148.123 116.323
## 227 115.388 116.504 16.048 16.126 84.741 148.267 116.553
## 228 115.489 116.607 16.068 16.146 84.804 148.409 116.682
## 229 115.589 116.709 16.088 16.166 84.867 148.550 116.536
## 230 115.689 116.810 16.107 16.185 84.930 148.690 116.586
## 231 115.788 116.911 16.127 16.205 84.992 148.829 116.746
## 232 115.886 117.011 16.146 16.224 85.054 148.967 116.847
## 233 115.983 117.110 16.165 16.244 85.115 149.104 117.104
## 234 116.080 117.208 16.184 16.263 85.175 149.240 117.093
## 235 116.175 117.305 16.203 16.282 85.235 149.375 117.503
## 236 116.270 117.402 16.222 16.300 85.295 149.509 117.211
## 237 116.365 117.498 16.240 16.319 85.354 149.642 117.556
## 238 116.458 117.593 16.259 16.338 85.413 149.773 117.549
## 239 116.551 117.688 16.277 16.356 85.471 149.904 117.616
## 240 116.643 117.782 16.295 16.374 85.529 150.034 117.979
## 241 116.735 117.875 16.313 16.392 85.587 150.163 117.740
## 242 116.826 117.967 16.331 16.410 85.644 150.291 118.005
## 243 116.916 118.059 16.349 16.428 85.700 150.418 118.204
## 244 117.005 118.150 16.366 16.446 85.756 150.544 118.364
## 245 117.094 118.240 16.384 16.464 85.812 150.669 118.413
## 246 117.182 118.330 16.401 16.481 85.867 150.793 118.448
## 247 117.270 118.419 16.418 16.499 85.922 150.916 118.473
## 248 117.356 118.507 16.435 16.516 85.976 151.038 118.488
## 249 117.443 118.595 16.452 16.533 86.030 151.160 118.763
## 250 117.528 118.682 16.469 16.550 86.084 151.280 118.728
## 251 117.613 118.769 16.486 16.567 86.137 151.400 118.800
## 252 117.697 118.854 16.503 16.584 86.190 151.519 118.837
## 253 117.781 118.940 16.519 16.600 86.242 151.637 119.114
## 254 117.864 119.024 16.536 16.617 86.294 151.754 118.950
## 255 117.947 119.108 16.552 16.633 86.346 151.870 118.983
## 256 118.029 119.192 16.568 16.649 86.397 151.986 119.054
## 257 118.110 119.274 16.584 16.666 86.448 152.100 118.941
## 258 118.191 119.357 16.600 16.682 86.499 152.214 119.292
## 259 118.271 119.438 16.616 16.698 86.549 152.327 119.657
## 260 118.351 119.519 16.632 16.713 86.599 152.440 119.588
## 261 118.430 119.600 16.647 16.729 86.648 152.551 119.692
## 262 118.508 119.680 16.663 16.745 86.697 152.662 119.926
## 263 118.586 119.759 16.678 16.760 86.746 152.772 119.947
## 264 118.664 119.838 16.693 16.776 86.795 152.881 120.137
## 265 118.740 119.916 16.709 16.791 86.843 152.989 119.871
## 266 118.817 119.994 16.724 16.806 86.890 153.097 120.029
## 267 118.893 120.071 16.739 16.821 86.938 153.204 119.945
## 268 118.968 120.148 16.754 16.836 86.985 153.310 120.112
## 269 119.043 120.224 16.768 16.851 87.032 153.416 120.483
## 270 119.117 120.299 16.783 16.866 87.078 153.521 120.518
## 271 119.191 120.374 16.798 16.881 87.124 153.625 120.736
## 272 119.264 120.449 16.812 16.896 87.170 153.728 120.654
## 273 119.337 120.523 16.827 16.910 87.216 153.831 120.423
## 274 119.409 120.597 16.841 16.924 87.261 153.933 120.281
## 275 119.481 120.670 16.855 16.939 87.306 154.034 120.741
## 276 119.552 120.742 16.869 16.953 87.350 154.135 120.910
## 277 119.623 120.815 16.883 16.967 87.394 154.235 120.849
## 278 119.694 120.886 16.897 16.981 87.438 154.334 121.017
## 279 119.763 120.957 16.911 16.995 87.482 154.433 120.696
## 280 119.833 121.028 16.925 17.009 87.525 154.531 120.955
## 281 119.902 121.098 16.939 17.023 87.569 154.628 121.176
## 282 119.970 121.168 16.952 17.037 87.611 154.725 121.559
## 283 120.039 121.238 16.966 17.050 87.654 154.821 121.060
## 284 120.106 121.306 16.979 17.064 87.696 154.917 121.431
## 285 120.173 121.375 16.992 17.077 87.738 155.012 121.117
## 286 120.240 121.443 17.006 17.091 87.780 155.106 121.554
## 287 120.307 121.510 17.019 17.104 87.821 155.200 121.867
## 288 120.373 121.578 17.032 17.117 87.862 155.293 121.251
## 289 120.438 121.644 17.045 17.130 87.903 155.385 121.673
## 290 120.503 121.710 17.058 17.143 87.944 155.477 122.066
## 291 120.568 121.776 17.071 17.156 87.984 155.569 121.787
## 292 120.632 121.842 17.084 17.169 88.024 155.659 121.809
## 293 120.696 121.907 17.096 17.182 88.064 155.750 121.766
## 294 120.759 121.971 17.109 17.194 88.104 155.839 122.227
## 295 120.823 122.036 17.121 17.207 88.143 155.928 122.036
## 296 120.885 122.099 17.134 17.220 88.182 156.017 122.104
## 297 120.947 122.163 17.146 17.232 88.221 156.105 122.275
## 298 121.009 122.226 17.158 17.245 88.259 156.192 122.652
## Sim.sd Sim.Median Sim.MAD Sim.2.5. Sim.97.5.
## 1 NaN 0.000 0.000 0.000 Inf
## 2 0.000 0.000 0.000 0.000 0.000
## 3 0.000 0.000 0.000 0.000 0.000
## 4 0.000 0.000 0.000 0.000 0.000
## 5 0.001 0.002 0.001 0.001 0.005
## 6 0.007 0.017 0.006 0.008 0.036
## 7 0.024 0.078 0.023 0.044 0.138
## 8 0.053 0.229 0.050 0.146 0.354
## 9 0.093 0.513 0.090 0.359 0.724
## 10 0.140 0.953 0.138 0.716 1.263
## 11 0.185 1.571 0.184 1.243 1.967
## 12 0.220 2.360 0.219 1.966 2.829
## 13 0.244 3.321 0.246 2.875 3.823
## 14 0.257 4.429 0.255 3.955 4.952
## 15 0.259 5.667 0.260 5.182 6.199
## 16 0.249 7.025 0.247 6.548 7.513
## 17 0.245 8.469 0.239 7.993 8.960
## 18 0.255 9.992 0.257 9.501 10.494
## 19 0.293 11.577 0.298 11.018 12.155
## 20 0.360 13.204 0.355 12.502 13.927
## 21 0.462 14.854 0.458 13.969 15.767
## 22 0.583 16.543 0.573 15.433 17.724
## 23 0.715 18.233 0.716 16.859 19.662
## 24 0.865 19.930 0.852 18.321 21.709
## 25 1.026 21.601 1.022 19.669 23.721
## 26 1.202 23.291 1.187 21.119 25.809
## 27 1.372 24.941 1.366 22.464 27.820
## 28 1.519 26.630 1.545 23.852 29.778
## 29 1.727 28.254 1.732 25.056 31.874
## 30 1.902 29.869 1.865 26.326 33.782
## 31 2.081 31.468 2.099 27.640 35.704
## 32 2.318 32.927 2.299 28.873 37.828
## 33 2.474 34.477 2.455 29.936 39.640
## 34 2.652 36.047 2.648 31.205 41.606
## 35 2.893 37.588 2.869 32.335 43.685
## 36 3.092 38.951 3.098 33.393 45.546
## 37 3.250 40.419 3.213 34.485 47.253
## 38 3.410 41.740 3.373 35.571 49.042
## 39 3.626 43.114 3.606 36.537 50.772
## 40 3.847 44.463 3.811 37.506 52.671
## 41 3.959 45.771 3.905 38.592 54.161
## 42 4.215 46.982 4.202 39.394 55.796
## 43 4.338 48.214 4.341 40.502 57.461
## 44 4.539 49.559 4.510 41.388 59.150
## 45 4.738 50.586 4.700 42.074 60.654
## 46 4.895 51.761 4.900 43.059 62.128
## 47 5.048 53.025 4.989 44.081 63.789
## 48 5.317 54.114 5.245 44.547 65.587
## 49 5.476 55.224 5.459 45.457 66.918
## 50 5.595 56.212 5.516 46.209 68.307
## 51 5.784 57.133 5.631 46.983 69.683
## 52 5.983 58.321 5.845 47.866 71.179
## 53 6.081 59.294 6.066 48.626 72.249
## 54 6.178 60.220 6.152 49.228 73.177
## 55 6.431 61.303 6.315 49.797 75.240
## 56 6.540 62.184 6.451 50.617 76.284
## 57 6.767 62.993 6.758 51.275 77.676
## 58 6.874 63.988 6.774 52.001 78.718
## 59 6.917 64.971 6.779 52.676 79.824
## 60 7.153 65.647 7.058 53.170 81.214
## 61 7.257 66.336 7.171 53.938 82.340
## 62 7.315 67.323 7.161 54.416 83.327
## 63 7.526 68.165 7.393 55.016 84.870
## 64 7.653 68.858 7.534 55.667 85.691
## 65 7.799 69.684 7.686 56.007 86.395
## 66 7.919 70.511 7.808 56.443 87.661
## 67 8.086 71.280 7.763 56.948 89.059
## 68 8.166 71.938 8.194 57.583 89.319
## 69 8.299 72.560 8.168 58.262 90.760
## 70 8.333 73.367 8.300 58.900 91.659
## 71 8.598 74.137 8.355 59.123 92.923
## 72 8.740 74.936 8.563 59.820 94.276
## 73 8.786 75.377 8.654 59.938 94.567
## 74 8.753 76.068 8.507 60.674 94.924
## 75 9.082 76.617 8.941 60.932 96.196
## 76 9.135 77.460 9.114 61.431 97.411
## 77 9.193 78.010 8.904 61.891 98.190
## 78 9.409 78.654 9.294 62.158 99.280
## 79 9.454 79.147 9.330 62.635 99.872
## 80 9.526 79.590 9.539 63.379 100.148
## 81 9.594 80.102 9.579 63.667 100.760
## 82 9.773 80.863 9.618 64.231 102.384
## 83 9.835 81.467 9.755 64.368 102.407
## 84 9.897 81.961 9.841 64.367 103.304
## 85 10.120 82.276 9.829 65.183 104.758
## 86 10.155 83.030 9.890 65.353 105.198
## 87 10.147 83.548 9.975 66.117 105.952
## 88 10.328 83.907 10.246 66.022 106.469
## 89 10.519 84.578 10.321 66.711 107.607
## 90 10.419 84.945 10.337 66.561 107.479
## 91 10.479 85.672 10.288 67.416 108.417
## 92 10.576 85.805 10.406 67.696 109.298
## 93 10.847 86.518 10.867 67.455 110.189
## 94 10.898 86.883 10.887 68.379 110.985
## 95 10.688 87.251 10.445 68.960 110.484
## 96 10.946 87.686 10.562 69.008 112.129
## 97 11.083 87.959 10.887 68.881 112.160
## 98 11.248 88.776 11.157 69.373 113.237
## 99 11.303 88.683 11.258 69.268 113.251
## 100 11.535 89.341 11.392 69.926 115.176
## 101 11.352 89.829 11.049 70.060 114.559
## 102 11.365 90.449 10.897 70.811 114.788
## 103 11.569 90.664 11.309 70.996 115.860
## 104 11.553 91.055 11.512 70.833 115.791
## 105 11.606 91.669 11.534 71.519 117.257
## 106 11.801 91.725 11.698 71.643 117.771
## 107 11.857 91.897 11.805 72.016 117.705
## 108 11.864 92.498 11.653 71.906 118.440
## 109 11.929 92.849 11.868 72.497 118.915
## 110 12.059 93.049 11.816 72.340 120.119
## 111 12.009 93.931 11.770 73.394 120.277
## 112 12.117 93.891 12.003 73.388 120.343
## 113 12.150 94.190 12.081 73.616 120.892
## 114 12.239 94.416 12.093 73.627 121.233
## 115 12.486 94.807 12.177 73.589 122.631
## 116 12.350 95.149 12.115 73.908 122.176
## 117 12.235 95.428 12.052 74.661 122.248
## 118 12.530 95.842 12.167 74.527 123.721
## 119 12.551 96.091 12.454 74.680 123.483
## 120 12.700 96.476 12.476 75.130 124.443
## 121 12.759 96.671 12.699 75.105 124.711
## 122 12.587 97.124 12.475 75.064 124.226
## 123 12.770 97.366 12.416 75.528 125.798
## 124 12.850 97.705 12.558 75.367 126.102
## 125 12.864 98.096 12.710 75.716 125.832
## 126 13.028 98.100 12.534 75.610 127.513
## 127 13.068 98.402 12.684 76.298 127.788
## 128 13.106 98.891 13.089 76.394 127.664
## 129 13.050 98.912 12.791 76.717 127.655
## 130 13.268 99.466 13.051 76.763 128.477
## 131 13.181 99.623 12.917 77.067 128.552
## 132 13.370 99.741 12.959 76.719 128.944
## 133 13.464 100.088 13.082 77.066 129.996
## 134 13.216 100.294 13.198 78.079 129.287
## 135 13.440 100.722 13.090 78.004 130.169
## 136 13.530 100.973 13.405 77.652 130.844
## 137 13.577 101.202 13.356 78.287 131.080
## 138 13.538 101.510 13.362 77.957 130.813
## 139 13.614 101.757 13.188 78.659 131.821
## 140 13.560 101.589 13.496 78.433 130.968
## 141 13.493 102.295 13.226 78.793 131.924
## 142 13.792 102.441 13.348 78.650 133.052
## 143 13.737 102.552 13.500 78.887 132.819
## 144 13.852 102.547 13.688 79.056 133.310
## 145 14.147 103.013 13.738 79.141 134.424
## 146 14.186 103.100 13.998 79.396 135.076
## 147 13.972 103.340 13.785 79.651 134.714
## 148 13.947 103.644 13.626 80.253 134.710
## 149 14.147 104.198 13.848 79.726 135.254
## 150 14.185 104.646 13.966 80.377 136.286
## 151 14.269 104.157 13.745 80.165 136.819
## 152 14.287 104.609 13.907 80.243 135.902
## 153 14.417 105.182 14.035 80.759 137.220
## 154 13.986 104.681 13.759 80.684 135.609
## 155 14.292 105.110 13.801 80.566 136.281
## 156 14.381 105.045 14.336 80.906 136.741
## 157 14.370 105.597 13.948 81.458 137.689
## 158 14.559 105.866 14.381 81.641 138.350
## 159 14.407 105.822 14.205 81.088 137.757
## 160 14.674 105.900 14.242 81.405 138.670
## 161 14.432 106.483 14.270 81.901 137.733
## 162 14.680 106.627 14.294 81.907 139.443
## 163 14.540 106.470 14.331 81.649 138.762
## 164 14.718 107.153 14.435 81.804 139.358
## 165 14.782 106.930 14.346 81.477 139.188
## 166 14.918 107.268 14.387 81.757 140.668
## 167 14.656 107.291 14.550 82.513 139.497
## 168 14.675 107.457 14.513 82.920 140.707
## 169 14.835 108.113 14.443 82.905 141.245
## 170 14.785 108.123 14.550 82.478 140.430
## 171 14.851 107.945 14.610 82.683 141.058
## 172 14.902 107.983 14.753 83.645 141.774
## 173 15.121 108.413 14.762 82.623 141.809
## 174 14.896 108.273 14.372 83.329 141.563
## 175 14.961 108.721 14.839 83.208 141.457
## 176 15.105 109.055 14.726 83.002 142.008
## 177 15.200 109.187 14.750 82.890 142.473
## 178 15.231 109.442 14.965 83.805 143.674
## 179 15.077 109.363 14.780 84.017 142.670
## 180 15.282 109.756 15.131 84.505 143.572
## 181 15.040 109.819 15.006 84.791 142.229
## 182 15.177 109.607 14.854 83.903 144.064
## 183 15.107 109.924 14.966 84.640 143.097
## 184 15.350 109.886 14.998 83.881 143.813
## 185 15.507 110.590 15.003 83.381 144.581
## 186 15.178 110.490 14.864 84.811 144.277
## 187 15.246 110.704 15.036 84.546 143.985
## 188 15.466 110.908 14.987 84.495 144.857
## 189 15.175 110.849 14.901 85.319 144.618
## 190 15.536 110.909 15.576 84.423 145.064
## 191 15.320 110.659 15.217 84.548 144.974
## 192 15.514 111.185 15.162 84.560 146.014
## 193 15.455 111.410 15.342 85.366 144.897
## 194 15.529 111.381 15.136 84.451 145.590
## 195 15.553 111.499 15.172 85.401 146.259
## 196 15.640 111.621 15.247 85.627 146.539
## 197 15.521 111.833 15.528 85.249 145.657
## 198 15.389 111.925 15.100 86.083 146.640
## 199 16.001 112.083 15.753 85.460 148.133
## 200 15.453 112.175 15.110 85.807 146.591
## 201 15.652 112.549 15.603 85.754 147.089
## 202 15.862 112.496 15.579 85.418 148.025
## 203 15.835 112.513 15.405 86.056 147.720
## 204 15.932 112.901 15.643 86.341 148.590
## 205 15.836 112.719 15.620 86.626 148.050
## 206 15.635 112.566 15.278 86.229 147.470
## 207 15.953 113.067 15.662 86.425 148.252
## 208 15.772 113.292 15.705 86.009 147.516
## 209 16.089 113.745 15.727 86.454 149.480
## 210 16.057 113.288 15.642 86.213 148.689
## 211 16.191 113.611 15.699 86.653 149.028
## 212 16.121 113.852 15.625 86.991 150.048
## 213 16.142 114.152 15.710 87.145 149.682
## 214 16.020 114.099 15.780 86.921 149.740
## 215 16.257 113.910 15.818 86.939 149.989
## 216 16.183 114.198 15.929 87.146 150.288
## 217 16.070 114.685 15.536 87.255 150.065
## 218 16.353 114.612 16.133 87.039 150.799
## 219 16.098 114.424 16.025 87.034 149.610
## 220 16.564 114.575 16.319 86.786 152.482
## 221 16.234 114.691 15.873 87.177 150.624
## 222 16.231 114.561 15.934 86.968 150.386
## 223 16.640 115.233 16.787 87.819 152.654
## 224 16.241 115.339 16.311 87.721 151.115
## 225 16.380 115.151 16.020 87.699 151.772
## 226 16.210 115.186 15.779 87.336 151.170
## 227 16.140 115.329 15.595 87.790 151.529
## 228 16.366 115.701 16.050 87.606 152.002
## 229 16.259 115.418 15.765 87.534 151.890
## 230 16.456 115.547 16.177 87.603 152.480
## 231 16.352 115.518 15.930 87.989 151.524
## 232 16.593 115.709 16.285 88.026 152.696
## 233 16.148 116.182 15.970 88.238 151.052
## 234 16.509 115.895 16.363 88.051 152.369
## 235 16.452 116.407 16.108 88.214 152.693
## 236 16.318 115.825 15.904 88.549 152.298
## 237 16.672 116.232 16.193 87.827 153.269
## 238 16.373 116.647 16.385 88.369 152.606
## 239 16.443 116.644 16.387 88.940 153.265
## 240 16.496 116.855 16.087 89.196 153.773
## 241 16.787 116.413 16.294 88.690 154.227
## 242 16.816 116.670 16.517 88.643 153.793
## 243 16.770 116.890 16.216 88.679 154.471
## 244 16.557 117.164 16.232 89.048 154.025
## 245 16.638 117.149 16.309 88.848 154.146
## 246 16.469 117.259 16.194 89.908 153.704
## 247 16.622 117.329 16.161 89.000 154.496
## 248 16.721 117.234 16.197 89.568 155.423
## 249 16.804 117.619 16.667 89.035 154.988
## 250 16.935 117.612 16.654 88.958 155.564
## 251 16.630 117.660 16.329 89.513 154.411
## 252 16.966 117.871 16.749 89.009 155.437
## 253 16.942 117.903 16.356 89.311 156.101
## 254 16.774 117.984 16.600 89.012 154.547
## 255 16.778 117.943 16.421 89.080 154.761
## 256 16.714 117.926 16.382 89.006 154.958
## 257 16.778 117.866 16.621 89.569 155.948
## 258 16.832 118.409 16.416 89.611 155.718
## 259 17.007 118.741 16.727 89.392 156.084
## 260 16.817 118.445 16.268 89.814 155.034
## 261 17.030 118.467 16.575 89.849 156.710
## 262 17.102 118.724 16.560 90.017 156.918
## 263 17.202 118.699 16.726 89.869 157.132
## 264 17.254 118.809 16.939 89.866 157.637
## 265 16.915 118.676 16.554 90.006 156.099
## 266 17.112 118.671 16.645 90.687 157.479
## 267 16.898 119.065 16.388 90.104 155.916
## 268 17.107 118.931 16.577 89.967 156.861
## 269 17.058 119.242 16.301 90.430 157.744
## 270 17.243 119.316 17.036 90.377 156.941
## 271 17.067 119.295 16.487 91.006 158.835
## 272 17.308 119.388 16.964 89.972 157.779
## 273 17.022 119.260 16.880 90.623 157.404
## 274 16.876 119.200 16.466 90.393 156.861
## 275 17.174 119.404 17.128 90.539 156.988
## 276 17.299 119.836 16.842 90.317 158.281
## 277 17.018 119.792 16.760 90.940 157.153
## 278 17.220 119.909 17.117 90.487 157.662
## 279 17.212 119.465 16.936 90.741 157.301
## 280 17.239 119.800 16.786 90.151 157.870
## 281 17.311 119.795 17.073 90.423 158.576
## 282 17.303 120.514 16.906 91.236 158.929
## 283 17.387 119.699 16.812 90.242 158.149
## 284 17.252 120.346 16.831 91.507 158.576
## 285 17.234 120.042 17.209 90.647 158.291
## 286 17.447 120.265 17.204 91.234 158.506
## 287 17.295 120.736 17.060 91.374 158.917
## 288 17.436 120.293 17.074 90.750 158.822
## 289 17.310 120.594 16.994 91.206 158.479
## 290 17.309 120.736 16.802 91.631 159.676
## 291 17.397 120.542 17.332 91.364 159.261
## 292 17.586 120.518 17.156 90.803 160.168
## 293 17.495 120.525 17.327 90.837 158.679
## 294 17.366 120.921 17.143 92.020 159.358
## 295 17.710 120.683 17.485 91.361 160.792
## 296 17.545 121.064 17.585 91.425 159.862
## 297 17.332 121.077 17.052 91.523 159.594
## 298 17.660 121.382 17.381 91.387 160.919write.csv(df_III, file="Simulacion_ IC.csv") # Exportar los resultados
z<-1:298 # Generar z de 1 hasta "n" predichos
c<-(df_III[, 1]) # Extraer columna 1 de la simulacion (Predichos)
d<-(df_III[, 5]) # Extraer el IC 2.5 (Columna 5 de la simulacion)
e<-(df_III[, 6]) # Extraer el IC 97.5 (Columna 6 de la simulacion)
mx<- max(Edad) # Obtener el valor maximo de Edad
my<- max(Biomasa) # Obtener el valor maximo de Biomasa
#lines(z, c, lwd=3, col="red") # Agregar los estimados a la plot
plot(z,c, xlim=c(0, mx), ylim=c(0, my), pch = 20, cex=1.5,
xlab = "Edad (años)", ylab = "Biomasa (m)") # Limitar ejes "x" y "y" a valores max observados
par (new = TRUE)
#plot(z,d, xlim=c(0, 20), ylim=c(0, 15))
lines(z, d, lwd=3, col="red", lty = "dotted") # Agregar los IC (inferior, 2.5) simulados a la plot
par (new = TRUE)
#plot(z,e, xlim=c(0, 20), ylim=c(0, 15))
lines(z, e, lwd=3, col="red", lty = "dotted") # Agregar los IC (superior, 97.5) simulados a la plot
points(Edad, Biomasa) # Agregar los datos observados a la plot
En la figura anterior, puede ver que ahora además de la predicción puntual (círculos negros) también se presentan los límites de predicción. En este sentido, r genera siete columnas, 1) Prop.Mean.1; 2) Prop.Mean.2; 3) Prop.sd.1; 4) Prop.sd.2; 5) Prop.2.5.; 6) Prop.97.5; y 7) Sim.Mean. Las cuales su nombre mismo indica su significado. Anteriormente, se había indicado que la predicción del modelo a los 20 años era de 14.86014 kg. Como puede ver, en la observación 1, no se predice. El valor de la edad de 20 años lo encuentra en la línea 21, cuya predicción puntual es 14.860 (columna Prop.Mean.1), en este mismo sentido, su intervalo de predicción va desde 13.964 hasta 15.771 kg.
Varios estadísiticos
coef(Fit1) # Coeficientes## a0 a1
## 4.947289 -44.972139#confint(Fit1) # Intervalos de Confianza de los Coeficientes
deviance(Fit1) # La devianza## [1] 1397.451df.residual(Fit1) # GL de los residuales## [1] 245fitted(Fit1) # Estimados## [1] 4.144085e-18 4.144085e-18 4.144085e-18 4.144085e-18 4.144085e-18
## [6] 4.144085e-18 4.144085e-18 4.144085e-18 4.144085e-18 4.144085e-18
## [11] 4.144085e-18 4.144085e-18 4.144085e-18 2.415486e-08 2.415486e-08
## [16] 2.415486e-08 2.415486e-08 2.415486e-08 2.415486e-08 2.415486e-08
## [21] 2.415486e-08 2.415486e-08 2.415486e-08 2.415486e-08 2.415486e-08
## [26] 2.415486e-08 4.347068e-05 4.347068e-05 4.347068e-05 4.347068e-05
## [31] 4.347068e-05 4.347068e-05 4.347068e-05 4.347068e-05 4.347068e-05
## [36] 4.347068e-05 4.347068e-05 4.347068e-05 4.347068e-05 1.844134e-03
## [41] 1.844134e-03 1.844134e-03 1.844134e-03 1.844134e-03 1.844134e-03
## [46] 1.844134e-03 1.844134e-03 1.844134e-03 1.844134e-03 1.844134e-03
## [51] 1.844134e-03 1.844134e-03 1.747230e-02 1.747230e-02 1.747230e-02
## [56] 1.747230e-02 1.747230e-02 1.747230e-02 1.747230e-02 1.747230e-02
## [61] 1.747230e-02 1.747230e-02 1.747230e-02 1.747230e-02 1.747230e-02
## [66] 7.823271e-02 7.823271e-02 7.823271e-02 7.823271e-02 7.823271e-02
## [71] 7.823271e-02 7.823271e-02 7.823271e-02 7.823271e-02 7.823271e-02
## [76] 7.823271e-02 7.823271e-02 7.823271e-02 2.282526e-01 2.282526e-01
## [81] 2.282526e-01 2.282526e-01 2.282526e-01 2.282526e-01 2.282526e-01
## [86] 2.282526e-01 2.282526e-01 2.282526e-01 2.282526e-01 2.282526e-01
## [91] 2.282526e-01 5.095495e-01 5.095495e-01 5.095495e-01 5.095495e-01
## [96] 5.095495e-01 5.095495e-01 5.095495e-01 5.095495e-01 5.095495e-01
## [101] 5.095495e-01 5.095495e-01 5.095495e-01 5.095495e-01 9.515955e-01
## [106] 9.515955e-01 9.515955e-01 9.515955e-01 9.515955e-01 9.515955e-01
## [111] 9.515955e-01 9.515955e-01 9.515955e-01 9.515955e-01 9.515955e-01
## [116] 9.515955e-01 9.515955e-01 1.568430e+00 1.568430e+00 1.568430e+00
## [121] 1.568430e+00 1.568430e+00 1.568430e+00 1.568430e+00 1.568430e+00
## [126] 1.568430e+00 1.568430e+00 1.568430e+00 1.568430e+00 1.568430e+00
## [131] 2.360593e+00 2.360593e+00 2.360593e+00 2.360593e+00 2.360593e+00
## [136] 2.360593e+00 2.360593e+00 2.360593e+00 2.360593e+00 2.360593e+00
## [141] 2.360593e+00 2.360593e+00 2.360593e+00 3.318825e+00 3.318825e+00
## [146] 3.318825e+00 3.318825e+00 3.318825e+00 3.318825e+00 3.318825e+00
## [151] 3.318825e+00 3.318825e+00 3.318825e+00 3.318825e+00 3.318825e+00
## [156] 3.318825e+00 4.427760e+00 4.427760e+00 4.427760e+00 4.427760e+00
## [161] 4.427760e+00 4.427760e+00 4.427760e+00 4.427760e+00 4.427760e+00
## [166] 4.427760e+00 4.427760e+00 4.427760e+00 4.427760e+00 5.668891e+00
## [171] 5.668891e+00 5.668891e+00 5.668891e+00 5.668891e+00 5.668891e+00
## [176] 5.668891e+00 5.668891e+00 5.668891e+00 5.668891e+00 5.668891e+00
## [181] 5.668891e+00 5.668891e+00 7.022691e+00 7.022691e+00 7.022691e+00
## [186] 7.022691e+00 7.022691e+00 7.022691e+00 7.022691e+00 7.022691e+00
## [191] 7.022691e+00 7.022691e+00 7.022691e+00 7.022691e+00 7.022691e+00
## [196] 8.469999e+00 8.469999e+00 8.469999e+00 8.469999e+00 8.469999e+00
## [201] 8.469999e+00 8.469999e+00 8.469999e+00 8.469999e+00 8.469999e+00
## [206] 8.469999e+00 8.469999e+00 8.469999e+00 9.992843e+00 9.992843e+00
## [211] 9.992843e+00 9.992843e+00 9.992843e+00 9.992843e+00 9.992843e+00
## [216] 9.992843e+00 9.992843e+00 9.992843e+00 1.157488e+01 1.157488e+01
## [221] 1.157488e+01 1.157488e+01 1.157488e+01 1.157488e+01 1.157488e+01
## [226] 1.157488e+01 1.157488e+01 1.157488e+01 1.320155e+01 1.320155e+01
## [231] 1.320155e+01 1.320155e+01 1.320155e+01 1.320155e+01 1.320155e+01
## [236] 1.486014e+01 1.486014e+01 1.486014e+01 1.486014e+01 1.486014e+01
## [241] 1.486014e+01 1.486014e+01 1.653962e+01 1.653962e+01 1.653962e+01
## [246] 1.823059e+01 1.823059e+01
## attr(,"label")
## [1] "Fitted values"formula(Fit1) # El modelo ## Biomasa ~ exp(a0 + (a1/Edad))logLik(Fit1) # ## 'log Lik.' -564.5054 (df=3)print(Fit1) # El modelo y coeficientes## Nonlinear regression model
## model: Biomasa ~ exp(a0 + (a1/Edad))
## data: datos
## a0 a1
## 4.947 -44.972
## residual sum-of-squares: 1397
##
## Algorithm "port", convergence message: Relative error in the sum of squares is at most `ftol'.profile(Fit1) # Datos del modelo## $a0
## tau par.vals.a0 par.vals.a1
## 1 0 4.947289 -44.972139
##
## $a1
## tau par.vals.a0 par.vals.a1
## 1 0 4.947289 -44.972139
##
## attr(,"original.fit")
## Nonlinear regression model
## model: Biomasa ~ exp(a0 + (a1/Edad))
## data: datos
## a0 a1
## 4.947 -44.972
## residual sum-of-squares: 1397
##
## Algorithm "port", convergence message: Relative error in the sum of squares is at most `ftol'.
## attr(,"summary")
##
## Formula: Biomasa ~ exp(a0 + (a1/Edad))
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## a0 4.9473 0.1504 32.89 <2e-16 ***
## a1 -44.9721 2.6058 -17.26 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.388 on 245 degrees of freedom
##
## Algorithm "port", convergence message: Relative error in the sum of squares is at most `ftol'.
##
## attr(,"class")
## [1] "profile.nls" "profile"residuals(Fit1) # Residuales## [1] 1.731810e-04 1.202640e-04 7.696920e-05 4.810580e-04 1.847261e-03
## [6] 2.357180e-04 4.810580e-04 7.696920e-05 7.696920e-05 3.078770e-04
## [11] 3.078770e-04 1.731810e-04 7.696920e-05 6.926988e-04 6.926988e-04
## [16] 3.078528e-04 1.826092e-02 4.156313e-03 9.974968e-04 2.770867e-03
## [21] 6.926988e-04 4.637370e-03 1.558602e-03 9.428488e-04 4.810338e-04
## [26] 4.810338e-04 1.188036e-03 2.284847e-03 1.515155e-03 5.039060e-02
## [31] 7.345572e-03 2.284847e-03 7.653449e-03 3.307253e-02 4.267564e-02
## [36] 4.286047e-03 1.880759e-03 1.188036e-03 1.515155e-03 8.009621e-05
## [41] 1.427425e-01 4.078959e-02 1.321316e-01 4.486775e-02 4.390371e-03
## [46] 1.556733e-01 7.765062e-02 9.677265e-02 1.174774e-01 3.394582e-03
## [51] 2.846730e-02 4.048820e-03 -8.986443e-03 2.707004e-01 7.117457e-02
## [56] 2.105321e-01 1.405058e-01 8.884141e-02 3.132740e-01 1.564829e-01
## [61] 1.556555e-01 2.793209e-01 -5.445859e-03 5.309403e-02 -5.445859e-03
## [66] 1.596436e-02 3.281358e-01 1.309888e-01 4.192337e-01 2.900360e-01
## [71] 1.196383e-01 5.652731e-01 3.350726e-01 3.373048e-01 6.653812e-01
## [76] 1.432810e-01 6.574299e-02 1.163839e-01 3.731996e-02 3.500939e-01
## [81] 1.627292e-01 5.131243e-01 3.601721e-01 1.077420e-01 1.058149e+00
## [86] 3.717475e-01 5.385578e-01 1.176224e+00 1.184984e-01 2.428330e-02
## [91] 1.563817e-01 5.594132e-01 3.661267e-01 1.937445e-01 3.978765e-01
## [96] 4.938210e-01 2.527098e-02 1.983021e+00 3.558681e-01 7.363365e-01
## [101] 1.584402e+00 2.110022e-02 -1.959364e-02 1.813886e-01 9.014741e-01
## [106] 2.800079e-01 4.007726e-01 2.858047e-01 3.831274e-01 -1.582463e-01
## [111] 3.304772e+00 4.030877e-01 8.256955e-01 2.121123e+00 -4.157557e-03
## [116] -1.799324e-01 3.665021e-01 3.669886e-01 2.794248e-01 4.491250e-01
## [121] 9.959253e-04 3.852563e-02 -4.851848e-01 4.327170e+00 1.221791e-01
## [126] 7.903680e-01 2.594420e+00 3.640993e-01 -2.963698e-01 5.386547e-01
## [131] 3.315480e-01 4.198232e-01 5.446007e-02 -4.677760e-01 -3.838422e-01
## [136] -4.402667e-01 5.071985e+00 -1.747205e-01 5.048069e-01 2.907409e+00
## [141] 1.051910e+00 -1.008686e-01 1.219803e+00 -2.442308e-01 8.972465e-02
## [146] -3.507893e-01 -1.070213e+00 -9.713415e-01 -6.086855e-01 5.678990e+00
## [151] -6.139350e-01 -2.437498e-02 2.985163e+00 2.123619e+00 -2.824094e-01
## [156] 1.824334e+00 -7.774582e-01 -1.438298e-01 -8.620847e-01 -1.838331e+00
## [161] -1.621391e+00 -1.019588e+00 6.043520e+00 -1.294050e+00 -5.653444e-01
## [166] 2.916757e+00 3.179568e+00 -4.681082e-01 2.267339e+00 -1.470243e+00
## [171] -6.262342e-01 -1.423626e+00 -2.803274e+00 -2.467654e+00 -1.521208e+00
## [176] 6.226794e+00 -2.060170e+00 -1.315229e+00 2.630054e+00 3.109482e+00
## [181] -7.307921e-01 2.187676e+00 -1.873859e+00 -1.302682e+00 -1.793031e+00
## [186] -3.812551e+00 -3.463231e+00 -2.221881e+00 6.657934e+00 -2.784305e+00
## [191] -2.063869e+00 2.405675e+00 3.332494e+00 -1.125388e+00 2.700852e+00
## [196] -2.379629e+00 -2.148923e+00 -2.618560e+00 -4.853996e+00 -4.269641e+00
## [201] -3.041525e+00 6.617938e+00 -3.645147e+00 -2.439650e+00 2.628525e+00
## [206] 3.951424e+00 -2.319138e+00 2.926848e+00 -3.060646e+00 -2.285912e+00
## [211] -3.650343e+00 -3.788866e+00 6.466231e+00 -4.461098e+00 2.123961e+00
## [216] 4.327344e+00 -1.858280e+00 3.032150e+00 -3.956614e+00 -2.727555e+00
## [221] -4.656567e+00 -4.555056e+00 6.701494e+00 -5.156590e+00 1.589821e+00
## [226] 6.190356e+00 -2.766394e+00 3.585223e+00 -5.159454e+00 -3.430023e+00
## [231] -5.541267e+00 5.920696e-01 6.063020e+00 -3.782237e+00 3.484183e+00
## [236] -5.023038e+00 -3.312148e+00 -6.132771e+00 9.591330e-01 5.893509e+00
## [241] -4.353122e+00 3.099058e+00 6.720512e+00 -4.026207e+00 3.410601e+00
## [246] 9.953492e+00 4.494033e+00
## attr(,"label")
## [1] "Residuals"summary(Fit1) # Resumen del modelo##
## Formula: Biomasa ~ exp(a0 + (a1/Edad))
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## a0 4.9473 0.1504 32.89 <2e-16 ***
## a1 -44.9721 2.6058 -17.26 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.388 on 245 degrees of freedom
##
## Algorithm "port", convergence message: Relative error in the sum of squares is at most `ftol'.update(Fit1) # .....## Nonlinear regression model
## model: Biomasa ~ exp(a0 + (a1/Edad))
## data: datos
## a0 a1
## 4.947 -44.972
## residual sum-of-squares: 1397
##
## Algorithm "port", convergence message: Relative error in the sum of squares is at most `ftol'.vcov(Fit1) # Mariz de covarianza de los coeficienets## a0 a1
## a0 0.02263175 -0.3865491
## a1 -0.38654909 6.7903115weights(Fit1) # Ponderaciones## NULLEjercicios
Aqui, https://1drv.ms/u/s!Agbm4NfF2M7JxSmXJdVPKTcy1_1B?e=uLBj7Z tiene acceso a datos de analisis troncales de una especie del noreste de mexico. Use el script https://rpubs.com/jorge_mendez/601482 para obtener una muestra de 100 datos de forma aleatoria. Con ellos, ajuste los siguientes modelos de crecimiento, para obtener el turno absoluto de esa especie, en: Diametro, Area Basal y Volumen.
- Logistico \(y=\frac{k}{1+b\cdot exp^{-a\cdot x}}\)
- Gompertz \(y=k\cdot b^{-a\cdot c^{-x}}\)
- Weibull \(y=k(1-exp^{-a\cdot x^{b}})\)
- Weibull modificada \(y=1-exp^{-ax^{^{b}}}\)
- Chapman-Richard’s \(y=a[1-exp^{-bx}]^{c}\)
- Logistico \(y=\frac{a}{d+exp^{b-cx}}\)
- Gompertz \(y=aexp^{-exp^{b-cx}}\)
- Exponencial tipo II \(y=exp^{a-bx}\)
- Exponencial tpo III \(y=aexp^{\frac{b}{x}}\)
En esos modelos, “y”" es el tamaño acumulado alcanzado a la edad “x” (Diametro, Altura, Area Basal, Volumen, Biomasa etc); “x” es la edad; a, b, c, d, k, son los coeficientes de regresion.
Bibliografia
Pienaar, L.V. and K.J. Turnbull. 1973. The Chapman-Richards generalization of Von Bertalanffy´s growth model for basal area growth and yield in even-aged stands. For. Sci. 19(1):2-22.
Nokoe, S. 1978. Demonstrating the flexibility of the Gompertz function as a yield model using mature species data. Commonw. For. Rev. 51(1):35-42.
Shifley, S.R., and G.J. Brand. 1984. Chapman-Richards growth function constrained for maximum tree size. For. Sci. 30(4):1066-1070.
Zeide, B. 1993. Analysis of growth equations. For. Sci. 39(3):594-616.
Zepeda B., E. M. y D. P. Rivero B. 1984. Construcción de curvas anamórficas de índice de sitio: ejemplificación del método de la curva guía. Ciencia Forestal. 51(9): 1-38.