nakagami {VGAM} | R Documentation |
Estimation of the two parameters of the Nakagami distribution by maximum likelihood estimation.
nakagami(lshape = "loge", lscale = "loge", eshape=list(), escale=list(), ishape = NULL, iscale = 1)
lshape, lscale |
Parameter link function applied to the
shape and scale parameters.
Log links ensure they are positive.
See Links for more choices.
|
eshape, escale |
List. Extra argument for each of the links.
See earg in Links for general information.
|
ishape, iscale |
Optional initial values for the shape and scale parameters.
For ishape , a NULL value means it is obtained in the
initialize slot based on the value of iscale .
For iscale , assigning a NULL means a value is obtained in the
initialize slot, however, setting another numerical
value is recommended if convergence fails or is too slow.
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The Nakagami distribution, which is useful for modelling wireless systems such as radio links, can be written
2 * (shape/scale)^shape * y^(2*shape-1) * exp(-shape*y^2/scale) / gamma(shape)
for y > 0, shape > 0, scale > 0. The mean of Y is sqrt(scale/shape) * gamma(shape+0.5) / gamma(shape) and these are returned as the fitted values. By default, the linear/additive predictors are eta1=log(shape) and eta2=log(scale). Fisher scoring is implemented.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
and vgam
.
The Nakagami distribution is also known as the Nakagami-m distribution, where m=shape here. Special cases: m=0.5 is a one-sided Gaussian distribution and m=1 is a Rayleigh distribution. The second moment is E(Y^2)=m.
If Y has a Nakagami distribution with parameters shape and scale then Y^2 has a gamma distribution with shape parameter shape and scale parameter scale/shape.
T. W. Yee
Nakagami, M. (1960) The m-distribution: a general formula of intensity distribution of rapid fading, pp.3–36 in: Statistical Methods in Radio Wave Propagation. W. C. Hoffman, Ed., New York: Pergamon.
n = 1000; shape = exp(0); Scale = exp(1) y = sqrt(rgamma(n, shape=shape, scale=Scale/shape)) fit = vglm(y ~ 1, nakagami, trace=TRUE, crit="c") y = rnaka(n, shape=shape, scale=Scale) fit = vglm(y ~ 1, nakagami(iscale=3), trace=TRUE) fitted(fit)[1:5] mean(y) coef(fit, matrix=TRUE) (Cfit = Coef(fit)) ## Not run: hist(sy <- sort(y), prob=TRUE, main="", xlab="y", ylim=c(0,0.6)) lines(sy, dnaka(sy, shape=Cfit[1], scale=Cfit[2]), col="red") ## End(Not run)