Package 'growthmodels'

growthmodels: Nonlinear Growth Models

Description

A compilation of nonlinear growth models.

Details

Package:	growthmodels
Version:	1.2.0
License:	GPL-3

Author(s)

Daniel Rodriguez [email protected]

References

D. Fekedulegn, M. Mac Siurtain, and J. Colbert, "Parameter estimation of nonlinear growth models in forestry," Silva Fennica, vol. 33, no. 4, pp. 327-336, 1999.

M. M. Kaps, W. O. W. Herring, and W. R. W. Lamberson, "Genetic and environmental parameters for traits derived from the Brody growth curve and their relationships with weaning weight in Angus cattle.," Journal of Animal Science, vol. 78, no. 6, pp. 1436-1442, May 2000.

A. Tsoularis and J. Wallace, "Analysis of logistic growth models.," Math Biosci, vol. 179, no. 1, pp. 21-55, Jul. 2002.

A. Khamiz, Z. Ismail, and A. T. Muhammad, "Nonlinear growth models for modeling oil palm yield growth," Journal of Mathematics and Statistics, vol. 1, no. 3, p. 225, 2005.

Michael J. Panik, "Growth Curve Modeling: Theory and Applications", John Wiley & Sons, December 2013.

http://en.wikipedia.org/wiki/Generalised_logistic_function

---

Blumberg growth model

Description

Computes the Blumberg growth model and its inverse

$y(t) = \frac{\alpha * (t + t_0)^m}{w_0 + (t + t_0)^m}$

Usage

blumberg(t, alpha, w0, m, t0 = 0)

blumberg.inverse(x, alpha, w0, m, t0 = 0)

Arguments

t	time
alpha	upper asymptote
w0	a reference value at t = t0
m	slope of growth
t0	time shift (default 0)
x	size

Author(s)

Daniel Rodriguez

References

A. Tsoularis and J. Wallace, "Analysis of logistic growth models.," Math Biosci, vol. 179, no. 1, pp. 21-55, Jul. 2002.

Examples

growth <- blumberg(0:10, 10, 2, 0.5)

# Calculate inverse function
time <- blumberg.inverse(growth, 12, 2, 0.5)

---

Brody growth model

Description

Computes the Brody growth model and its inverse

$y(t) = \alpha - (\alpha - w_0) exp(- k t)$

Usage

brody(t, alpha, w0, k)

brody.inverse(x, alpha, w0, k)

Arguments

t	time
alpha	upper asymptote
w0	the value at t = 0
k	growth rate
x	size

Author(s)

Daniel Rodriguez

References

M. M. Kaps, W. O. W. Herring, and W. R. W. Lamberson, "Genetic and environmental parameters for traits derived from the Brody growth curve and their relationships with weaning weight in Angus cattle.," Journal of Animal Science, vol. 78, no. 6, pp. 1436-1442, May 2000.

Examples

growth <- brody(0:10, 10, 5, 0.3)

# Calculate inverse function
time <- brody.inverse(growth, 10, 5, 0.3)

---

Chapman-Richards growth model

Description

Computes the Chapman-Richards growth model and its inverse

$y(t) = \alpha (1 - \beta exp(-k t)^{1/(1-m)})$

Usage

chapmanRichards(t, alpha, beta, k, m)

chapmanRichards.inverse(x, alpha, beta, k, m)

Arguments

t	time
alpha	upper asymptote
beta	growth range
k	growth rate
m	slope of growth
x	size

Author(s)

Daniel Rodriguez

References

D. Fekedulegn, M. Mac Siurtain, and J. Colbert, "Parameter estimation of nonlinear growth models in forestry," Silva Fennica, vol. 33, no. 4, pp. 327-336, 1999.

Examples

growth <- chapmanRichards(0:10, 10, 0.5, 0.3, 0.5)

# Calculate inverse function
time <- chapmanRichards.inverse(growth, 10, 0.5, 0.3, 0.5)

---

Generalised Logistic growth model

Description

Computes the Generalised Logistic growth model

$y(t) = A + \frac{U - A}{1 + \beta exp(-k (t- t_0))}$

Usage

generalisedLogistic(t, A, U, k, beta, t0)

generalisedLogistic.inverse(x, A, U, k, beta, t0 = 0)

Arguments

t	time
A	the lower asymptote
U	the upper asymptote
k	growth range
beta	growth range
t0	time shift (default 0)
x	size

Author(s)

Daniel Rodriguez

References

http://en.wikipedia.org/wiki/Generalised_logistic_function

Examples

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)

---

Generalised Richard growth model

Description

Computes the Generalised Richard growth model and its inverse

$y(t) = A + \frac{U - A}{(1 + \beta exp(-k (t - t_0)))^{(1/m)} }$

Usage

generalisedRichard(t, A, U, k, m, beta, t0)

generalisedRichard.inverse(x, A, U, k, m, beta, t0 = 0)

Arguments

t	time
A	the lower asymptote
U	the upper asymptote
k	growth range
m	slope of growth
beta	growth range
t0	time shift (default 0)
x	size

Author(s)

Daniel Rodriguez

References

http://en.wikipedia.org/wiki/Generalised_logistic_function

Examples

growth <- generalisedRichard(0:10, 5, 10, 0.3, 0.5, 1, 3)

time <- generalisedRichard.inverse(growth, 5, 10, 0.3, 0.5, 1, 3)

---

Gompertz growth model

Description

Computes the Gompertz growth model and its inverse

$y(t) = \alpha exp(-\beta exp(-k^t))$

Usage

gompertz(t, alpha, beta, k)

gompertz.inverse(x, alpha, beta, k)

Arguments

t	time
alpha	upper asymptote
beta	growth displacement
k	growth rate
x	size

Author(s)

Daniel Rodriguez

References

D. Fekedulegn, M. Mac Siurtain, and J. Colbert, "Parameter estimation of nonlinear growth models in forestry," Silva Fennica, vol. 33, no. 4, pp. 327-336, 1999.

Examples

growth <- gompertz(0:10, 10, 0.5, 0.3)

# Calculate inverse function
time <- gompertz.inverse(growth, 10, 0.5, 0.3)

---

Janoschek growth model

Description

Computes the Janoschek growth model and its inverse

$y(t) = \alpha *(\alpha - \beta) \exp(-b * t^c))$

Usage

janoschek(t, alpha, beta, b, c)

janoschek.inverse(x, alpha, beta, b, c)

Arguments

t	time
alpha	upper asymptote
beta	lower asymptote
b	growth parameter
c	shape parameter
x	size

Author(s)

Daniel Rodriguez

References

Michael J. Panik, "Growth Curve Modeling: Theory and Applications", John Wiley & Sons, December 2013.

Examples

growth <- janoschek(0:10, 10, 2, 0.5, 2)

# Calculate inverse function
time <- janoschek.inverse(growth, 12, 2, 0.5, 2)

---

Logistic growth model

Description

Computes the Logistic growth model

$y(t) = \frac{\alpha}{1 + \beta exp(-k t)}$

Usage

logistic(t, alpha, beta, k)

logistic.inverse(x, alpha, beta, k)

Arguments

t	time
alpha	upper asymptote
beta	growth range
k	growth rate
x	size

Author(s)

Daniel Rodriguez

References

D. Fekedulegn, M. Mac Siurtain, and J. Colbert, "Parameter estimation of nonlinear growth models in forestry," Silva Fennica, vol. 33, no. 4, pp. 327-336, 1999.

Examples

growth <- logistic(0:10, 10, 0.5, 0.3)

# Calculate inverse function
time <- logistic.inverse(growth, 10, 0.5, 0.3)

---

Log-logistic growth model

Description

Computes the Log-logistic growth model

$y(t) = \frac{\alpha}{1 + \beta exp(-k log(t)}$

Usage

loglogistic(t, alpha, beta, k)

loglogistic.inverse(x, alpha, beta, k)

Arguments

t	time
alpha	upper asymptote
beta	growth range
k	growth rate
x	size

Author(s)

Daniel Rodriguez

References

A. Khamiz, Z. Ismail, and A. T. Muhammad, "Nonlinear growth models for modeling oil palm yield growth," Journal of Mathematics and Statistics, vol. 1, no. 3, p. 225, 2005.

Examples

growth <- loglogistic(0:10, 10, 0.5, 0.3)

# Calculate inverse function
time <- loglogistic.inverse(growth, 10, 0.5, 0.3)

---

Mitcherlich growth model

Description

Computes the Mitcherlich growth model

$y(t) = (\alpha - \beta k^t)$

Usage

mitcherlich(t, alpha, beta, k)

mitcherlich.inverse(x, alpha, beta, k)

Arguments

t	time
alpha	upper asymptote
beta	growth range
k	growth rate
x	size

Author(s)

Daniel Rodriguez

References

D. Fekedulegn, M. Mac Siurtain, and J. Colbert, "Parameter estimation of nonlinear growth models in forestry," Silva Fennica, vol. 33, no. 4, pp. 327-336, 1999.

Examples

growth <- mitcherlich(0:10, 10, 0.5, 0.3)

# Calculate inverse function
time <- mitcherlich.inverse(growth, 10, 0.5, 0.3)

---

Morgan-Mercer-Flodin growth model

Description

Computes the Morgan-Mercer-Flodin growth model

$y(t) = \frac{(w_0 \gamma + \alpha t^m)}{\gamma} +t^m$

Usage

mmf(t, alpha, w0, gamma, m)

mmf.inverse(x, alpha, w0, gamma, m)

Arguments

t	time
alpha	upper asymptote
w0	the value at t = 0
gamma	parameter that controls the point of inflection
m	growth rate
x	size

Author(s)

Daniel Rodriguez

References

A. Khamiz, Z. Ismail, and A. T. Muhammad, "Nonlinear growth models for modeling oil palm yield growth," Journal of Mathematics and Statistics, vol. 1, no. 3, p. 225, 2005.

Examples

growth <- mmf(0:10, 10, 0.5, 4, 1)

# Calculate inverse function
time <- mmf.inverse(growth, 10, 0.5, 4, 1)

---

Monomolecular growth model

Description

Computes the monomolecular growth model

$y(t) = \alpha ( 1 - \beta exp(-k t))$

Usage

monomolecular(t, alpha, beta, k)

monomolecular.inverse(x, alpha, beta, k)

Arguments

t	time
alpha	upper asymptote
beta	growth range
k	growth rate
x	size

Author(s)

Daniel Rodriguez

References

D. Fekedulegn, M. Mac Siurtain, and J. Colbert, "Parameter estimation of nonlinear growth models in forestry," Silva Fennica, vol. 33, no. 4, pp. 327-336, 1999.

Examples

growth <- monomolecular(0:10, 10, 0.5, 0.3)

# Calculate inverse function
time <- monomolecular.inverse(growth, 10, 0.5, 0.3)

---

Negative exponential growth model

Description

Computes the negative exponential growth model

$y(t) = \alpha ( 1 - exp(-k t))$

Usage

negativeExponential(t, alpha, k)

negativeExponential.inverse(x, alpha, k)

Arguments

t	time
alpha	upper asymptote
k	growth rate
x	size

Author(s)

Daniel Rodriguez

References

D. Fekedulegn, M. Mac Siurtain, and J. Colbert, "Parameter estimation of nonlinear growth models in forestry," Silva Fennica, vol. 33, no. 4, pp. 327-336, 1999.

Examples

growth <- negativeExponential(0:10, 1, 0.3)

# Calculate inverse function
time <- negativeExponential.inverse(growth, 10, 0.3)

---

Richard growth model

Description

Computes the Richard growth model and its inverse

$y(t) = \frac{\alpha}{(1 + \beta exp(-k t))^{(1/m)}}$

Usage

richard(t, alpha, beta, k, m)

richard.inverse(x, alpha, beta, k, m)

Arguments

t	time
alpha	upper asymptote
beta	growth range
k	growth rate
m	slope of growth
x	size

Author(s)

Daniel Rodriguez

References

D. Fekedulegn, M. Mac Siurtain, and J. Colbert, "Parameter estimation of nonlinear growth models in forestry," Silva Fennica, vol. 33, no. 4, pp. 327-336, 1999.

Examples

growth <- richard(0:10, 10, 0.5, 0.3, 0.5)

time <- richard.inverse(growth, 10, 0.5, 0.3, 0.5)

---

Schnute growth model

Description

Computes the Schnute growth model

$y(t) = \left[ r_0 + \beta exp(k t) \right]^m$

Usage

schnute(t, r0, beta, k, m)

schnute.inverse(x, r0, beta, k, m)

Arguments

t	time
r0	reference value
beta	growth displacement
k	growth rate
m	slope of growth
x	size

Author(s)

Daniel Rodriguez

References

A. Khamiz, Z. Ismail, and A. T. Muhammad, "Nonlinear growth models for modeling oil palm yield growth," Journal of Mathematics and Statistics, vol. 1, no. 3, p. 225, 2005.

Examples

growth <- schnute(0:10, 10, 5, .5, .5)

# Calculate inverse function
time <- schnute.inverse(growth, 10, 5, .5, .5)

---

Stannard growth model

Description

Computes the Stannard growth model

$y(t) = \alpha \left[ 1 + exp(-(\beta + k t)/m) \right]^{-m}$

Usage

stannard(t, alpha, beta, k, m)

stannard.inverse(x, alpha, beta, k, m)

Arguments

t	time
alpha	upper asymptote
beta	growth displacement
k	growth rate
m	slope of growth
x	size

Author(s)

Daniel Rodriguez

References

A. Khamiz, Z. Ismail, and A. T. Muhammad, "Nonlinear growth models for modeling oil palm yield growth," Journal of Mathematics and Statistics, vol. 1, no. 3, p. 225, 2005.

Examples

growth <- stannard(0:10, 1, .2, .1, .5)

# Calculate inverse function
time <- stannard.inverse(growth, 1, .2, .1, .5)

---

von Bertalanffy growth model

Description

Computes the von Bertalanffy growth model

$y(t) = (\alpha^(1-m) - \beta * exp(-k t))^(1/(1-m))$

Usage

vonBertalanffy(t, alpha, beta, k, m)

vonBertalanffy.inverse(x, alpha, beta, k, m)

Arguments

t	time
alpha	upper asymptote
beta	growth range
k	growth rate
m	slope of growth
x	size

Author(s)

Daniel Rodriguez

References

D. Fekedulegn, M. Mac Siurtain, and J. Colbert, "Parameter estimation of nonlinear growth models in forestry," Silva Fennica, vol. 33, no. 4, pp. 327-336, 1999.

Examples

growth <- vonBertalanffy(0:10, 10, 0.5, 0.3, 0.5)

# Calculate inverse function
time <- vonBertalanffy.inverse(growth, 10, 0.5, 0.3, 0.5)

---

Weibull growth model

Description

Computes the Weibull growth model

$y(t) = \alpha - \beta exp(-k * t^m)$

Usage

weibull(t, alpha, beta, k, m)

weibull.inverse(x, alpha, beta, k, m)

Arguments

t	time
alpha	upper asymptote
beta	growth range
k	growth rate
m	slope of growth
x	size

Author(s)

Daniel Rodriguez

References

D. Fekedulegn, M. Mac Siurtain, and J. Colbert, "Parameter estimation of nonlinear growth models in forestry," Silva Fennica, vol. 33, no. 4, pp. 327-336, 1999.

Examples

growth <- weibull(0:10, 10, 0.5, 0.3, 0.5)

# Calculate inverse function
time <- weibull.inverse(growth, 10, 0.5, 0.3, 0.5)

---