growthmodels: Nonlinear Growth Models

Description

A compilation of nonlinear growth models.

Details

Author(s)

Daniel Rodriguez daniel.rodriguez.perez@gmail.com

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)