LNO: Log Normal distribution for fitting in GAMLSS
In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape
LNO R Documentation
Log Normal distribution for fitting in GAMLSS
Description
The functions LOGNO and LOGNO2 define a gamlss.family distribution to fits the log-Normal distribution.
The difference between them is that while LOGNO retains the original parametrization for mu, (identical to the normal distribution NO) and therefore \mu=(-\infty,+\infty), the function LOGNO2 use mu as the median, so \mu=(0,+\infty).
The function LNO is more general and can fit a Box-Cox transformation
to data using the gamlss() function.
In the LOGNO (and LOGNO2) there are two parameters involved mu sigma, while in the
LNO there are three parameters mu sigma,
and the transformation parameter nu.
The transformation parameter nu in LNO is a 'fixed' parameter (not estimated) and it has its default value equal to
zero allowing the fitting of the log-normal distribution as in LOGNO.
See the example below on how to fix nu to be a particular value.
In order to estimate (or model) the parameter nu, use the gamlss.family
BCCG distribution which uses a reparameterized version of the the Box-Cox transformation.
The functions dLOGNO, pLOGNO, qLOGNO and rLOGNO define the density, distribution function, quantile function and random
generation for the specific parameterization of the log-normal distribution.
The functions dLOGNO2, pLOGNO2, qLOGNO2 and rLOGNO2 define the density, distribution function, quantile function and random
generation when mu is the median of the log-normal distribution.
The functions dLNO, pLNO, qLNO and rLNO define the density, distribution function, quantile function and random
generation for the specific parameterization of the log-normal distribution and more generally a Box-Cox transformation.
Usage
LNO(mu.link = "identity", sigma.link = "log")
LOGNO(mu.link = "identity", sigma.link = "log")
LOGNO2(mu.link = "log", sigma.link = "log")
dLNO(x, mu = 1, sigma = 0.1, nu = 0, log = FALSE)
dLOGNO(x, mu = 0, sigma = 1, log = FALSE)
dLOGNO2(x, mu = 1, sigma = 1, log = FALSE)
pLNO(q, mu = 1, sigma = 0.1, nu = 0, lower.tail = TRUE, log.p = FALSE)
pLOGNO(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
pLOGNO2(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qLNO(p, mu = 1, sigma = 0.1, nu = 0, lower.tail = TRUE, log.p = FALSE)
qLOGNO(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qLOGNO2(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rLNO(n, mu = 1, sigma = 0.1, nu = 0)
rLOGNO(n, mu = 0, sigma = 1)
rLOGNO2(n, mu = 1, sigma = 1)
Arguments
mu.link
Defines the mu.link, with "identity" or "log" link depending on te parametrization
sigma.link
Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse", "identity" ans "own"
x,q
vector of quantiles
mu
vector of location parameter values
sigma
vector of scale parameter values
nu
vector of shape parameter values
log, log.p
logical; if TRUE, probabilities p are given as log(p).
lower.tail
logical; if TRUE (default), probabilities are P[X <= x],
otherwise, P[X > x]
p
vector of probabilities.
n
number of observations. If length(n) > 1, the length is
taken to be the number required
Details
The probability density function in LOGNO is defined as
f(y|\mu,\sigma)=\frac{1}{y \sqrt{2\pi}\sigma} \exp [-\frac{1}{2 \sigma^2}(\log y-\mu)^2 ]
for y>0, -\infty<\mu<\infty and \sigma>0 see pp. 428-429 of Rigby et al. (2019).
The probability density function in LOGNO2 is defined as
f(y|\mu,\sigma)=\frac{1}{y \sqrt{2\pi}\sigma} \exp [-\frac{1}{2 \sigma^2}(\log y-\log\mu)^2 ]
for y>0, -\infty<\mu<\infty and \sigma>0 see pp. 429-430 of Rigby et al. (2019).
The probability density function in LNO is defined as
f(y|\mu,\sigma,\nu)=\frac{y^{\nu-1}}{\sqrt{2\pi}\sigma} \exp [-\frac{1}{2 \sigma^2}(z-\mu)^2 ]
where if \nu \neq 0 z =(y^{\nu}-1)/\nu else z=\log(y) and z \sim N(0,\sigma^2),
for y>0, \mu>0, \sigma>0 and \nu=(-\infty,+\infty). This is not a proper distribution see for example p. 447 of Rigby et al. (2019).
Value
LNO() returns a gamlss.family object which can be used to fit a log-normal distribution in the gamlss() function.
dLNO() gives the density, pLNO() gives the distribution
function, qLNO() gives the quantile function, and rLNO()
generates random deviates.
Warning
This is a two parameter fit for \mu and \sigma while \nu is fixed.
If you wish to model \nu use the gamlss family BCCG.
Note
\mu is the mean of z (and also the median of y), the Box-Cox transformed variable and \sigma is the standard deviation of z
and approximate the coefficient of variation of y
Author(s)
Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou
References
Box, G. E. P. and Cox, D. R. (1964) An analysis of transformations (with discussion), J. R. Statist. Soc. B., 26, 211–252
Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion),
Appl. Statist., 54, part 3, pp 507-554.
Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019)
Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/9780429298547")}.An older version can be found in https://www.gamlss.com/.
Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R.
Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v023.i07")}.
Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017)
Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017)
Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}
(see also https://www.gamlss.com/).
See Also
gamlss.family, BCCG
Examples
LOGNO()# gives information about the default links for the log normal distribution
LOGNO2()
LNO()# gives information about the default links for the Box Cox distribution
# plotting the d, p, q, and r functions
op<-par(mfrow=c(2,2))
curve(dLOGNO(x, mu=0), 0, 10)
curve(pLOGNO(x, mu=0), 0, 10)
curve(qLOGNO(x, mu=0), 0, 1)
Y<- rLOGNO(200)
hist(Y)
par(op)
# plotting the d, p, q, and r functions
op<-par(mfrow=c(2,2))
curve(dLOGNO2(x, mu=1), 0, 10)
curve(pLOGNO2(x, mu=1), 0, 10)
curve(qLOGNO2(x, mu=1), 0, 1)
Y<- rLOGNO(200)
hist(Y)
par(op)
# library(gamlss)
# data(abdom)
# h1<-gamlss(y~cs(x), family=LOGNO, data=abdom)#fits the log-Normal distribution
# h2<-gamlss(y~cs(x), family=LNO, data=abdom) #should be identical to the one above
# to change to square root transformation, i.e. fix nu=0.5
# h3<-gamlss(y~cs(x), family=LNO, data=abdom, nu.fix=TRUE, nu.start=0.5)
gamlss.dist documentation built on Aug. 24, 2023, 1:06 a.m.
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| LNO | R Documentation |
Log Normal distribution for fitting in GAMLSS
Description
The functions LOGNO and LOGNO2 define a gamlss.family distribution to fits the log-Normal distribution.
The difference between them is that while LOGNO retains the original parametrization for mu, (identical to the normal distribution NO) and therefore \mu=(-\infty,+\infty), the function LOGNO2 use mu as the median, so \mu=(0,+\infty).
The function LNO is more general and can fit a Box-Cox transformation
to data using the gamlss() function.
In the LOGNO (and LOGNO2) there are two parameters involved mu sigma, while in the
LNO there are three parameters mu sigma,
and the transformation parameter nu.
The transformation parameter nu in LNO is a 'fixed' parameter (not estimated) and it has its default value equal to
zero allowing the fitting of the log-normal distribution as in LOGNO.
See the example below on how to fix nu to be a particular value.
In order to estimate (or model) the parameter nu, use the gamlss.family
BCCG distribution which uses a reparameterized version of the the Box-Cox transformation.
The functions dLOGNO, pLOGNO, qLOGNO and rLOGNO define the density, distribution function, quantile function and random
generation for the specific parameterization of the log-normal distribution.
The functions dLOGNO2, pLOGNO2, qLOGNO2 and rLOGNO2 define the density, distribution function, quantile function and random
generation when mu is the median of the log-normal distribution.
The functions dLNO, pLNO, qLNO and rLNO define the density, distribution function, quantile function and random
generation for the specific parameterization of the log-normal distribution and more generally a Box-Cox transformation.
Usage
LNO(mu.link = "identity", sigma.link = "log")
LOGNO(mu.link = "identity", sigma.link = "log")
LOGNO2(mu.link = "log", sigma.link = "log")
dLNO(x, mu = 1, sigma = 0.1, nu = 0, log = FALSE)
dLOGNO(x, mu = 0, sigma = 1, log = FALSE)
dLOGNO2(x, mu = 1, sigma = 1, log = FALSE)
pLNO(q, mu = 1, sigma = 0.1, nu = 0, lower.tail = TRUE, log.p = FALSE)
pLOGNO(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
pLOGNO2(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qLNO(p, mu = 1, sigma = 0.1, nu = 0, lower.tail = TRUE, log.p = FALSE)
qLOGNO(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qLOGNO2(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rLNO(n, mu = 1, sigma = 0.1, nu = 0)
rLOGNO(n, mu = 0, sigma = 1)
rLOGNO2(n, mu = 1, sigma = 1)
Arguments
mu.link |
Defines the |
sigma.link |
Defines the |
x,q |
vector of quantiles |
mu |
vector of location parameter values |
sigma |
vector of scale parameter values |
nu |
vector of shape parameter values |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x] |
p |
vector of probabilities. |
n |
number of observations. If |
Details
The probability density function in LOGNO is defined as
f(y|\mu,\sigma)=\frac{1}{y \sqrt{2\pi}\sigma} \exp [-\frac{1}{2 \sigma^2}(\log y-\mu)^2 ]
for y>0, -\infty<\mu<\infty and \sigma>0 see pp. 428-429 of Rigby et al. (2019).
The probability density function in LOGNO2 is defined as
f(y|\mu,\sigma)=\frac{1}{y \sqrt{2\pi}\sigma} \exp [-\frac{1}{2 \sigma^2}(\log y-\log\mu)^2 ]
for y>0, -\infty<\mu<\infty and \sigma>0 see pp. 429-430 of Rigby et al. (2019).
The probability density function in LNO is defined as
f(y|\mu,\sigma,\nu)=\frac{y^{\nu-1}}{\sqrt{2\pi}\sigma} \exp [-\frac{1}{2 \sigma^2}(z-\mu)^2 ]
where if \nu \neq 0 z =(y^{\nu}-1)/\nu else z=\log(y) and z \sim N(0,\sigma^2),
for y>0, \mu>0, \sigma>0 and \nu=(-\infty,+\infty). This is not a proper distribution see for example p. 447 of Rigby et al. (2019).
Value
LNO() returns a gamlss.family object which can be used to fit a log-normal distribution in the gamlss() function.
dLNO() gives the density, pLNO() gives the distribution
function, qLNO() gives the quantile function, and rLNO()
generates random deviates.
Warning
This is a two parameter fit for \mu and \sigma while \nu is fixed.
If you wish to model \nu use the gamlss family BCCG.
Note
\mu is the mean of z (and also the median of y), the Box-Cox transformed variable and \sigma is the standard deviation of z
and approximate the coefficient of variation of y
Author(s)
Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou
References
Box, G. E. P. and Cox, D. R. (1964) An analysis of transformations (with discussion), J. R. Statist. Soc. B., 26, 211–252
Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.
Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/9780429298547")}.An older version can be found in https://www.gamlss.com/.
Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v023.i07")}.
Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}
(see also https://www.gamlss.com/).
See Also
gamlss.family, BCCG
Examples
LOGNO()# gives information about the default links for the log normal distribution
LOGNO2()
LNO()# gives information about the default links for the Box Cox distribution
# plotting the d, p, q, and r functions
op<-par(mfrow=c(2,2))
curve(dLOGNO(x, mu=0), 0, 10)
curve(pLOGNO(x, mu=0), 0, 10)
curve(qLOGNO(x, mu=0), 0, 1)
Y<- rLOGNO(200)
hist(Y)
par(op)
# plotting the d, p, q, and r functions
op<-par(mfrow=c(2,2))
curve(dLOGNO2(x, mu=1), 0, 10)
curve(pLOGNO2(x, mu=1), 0, 10)
curve(qLOGNO2(x, mu=1), 0, 1)
Y<- rLOGNO(200)
hist(Y)
par(op)
# library(gamlss)
# data(abdom)
# h1<-gamlss(y~cs(x), family=LOGNO, data=abdom)#fits the log-Normal distribution
# h2<-gamlss(y~cs(x), family=LNO, data=abdom) #should be identical to the one above
# to change to square root transformation, i.e. fix nu=0.5
# h3<-gamlss(y~cs(x), family=LNO, data=abdom, nu.fix=TRUE, nu.start=0.5)
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