GENLOGIS                package:nsRFA                R Documentation

_T_h_r_e_e _p_a_r_a_m_e_t_e_r _g_e_n_e_r_a_l_i_z_e_d _l_o_g_i_s_t_i_c _d_i_s_t_r_i_b_u_t_i_o_n _a_n_d _L-_m_o_m_e_n_t_s

_D_e_s_c_r_i_p_t_i_o_n:

     'GENLOGIS' provides the link between L-moments of a sample and the
     three parameter generalized logistic distribution.

_U_s_a_g_e:

     f.genlogis (x, xi, alfa, k)
     F.genlogis (x, xi, alfa, k)
     invF.genlogis (F, xi, alfa, k)
     Lmom.genlogis (xi, alfa, k)
     par.genlogis (lambda1, lambda2, tau3)
     rand.genlogis (numerosita, xi, alfa, k)

_A_r_g_u_m_e_n_t_s:

       x: vector of quantiles

      xi: vector of genlogis location parameters

    alfa: vector of genlogis scale parameters

       k: vector of genlogis shape parameters

       F: vector of probabilities

 lambda1: vector of sample means

 lambda2: vector of L-variances

    tau3: vector of L-CA (or L-skewness)

numerosita: numeric value indicating the length of the vector to be
          generated

_D_e_t_a_i_l_s:

     See <URL: http://en.wikipedia.org/wiki/Logistic_distribution> for
     an introduction to the Logistic Distribution.

     *Definition*

     Parameters (3): xi (location), alpha (scale), k (shape).

     Range of x: -infty < x <= xi + alpha / k if k>0; -infty < x <
     infty if k=0; xi + alpha / k <= x < infty if k<0.

     Probability density function:

          f(x) = frac{alpha^{-1} e^{-(1-k)y}}{(1+e^{-y})^2}

     where y = -k^{-1}log{1 - k(x - xi)/alpha} if k ne 0, y =
     (x-xi)/alpha if k=0.

     Cumulative distribution function:

                         F(x) = 1/(1+e^{-y})


     Quantile function: x(F) = xi + alpha[1-{(1-F)/F}^k]/k if k ne 0,
     x(F) = xi - alpha log{(1-F)/F} if k=0.

     k=0 is the logistic distribution.

     *L-moments*

     L-moments are defined for -1<k<1.


             lambda_1 = xi + alpha[1/k - pi / sin (k pi)]


                  lambda_2 = alpha k pi / sin (k pi)


                              tau_3 = -k


                         tau_4 = (1+5 k^2)/6


     *Parameters*

     k=-tau_3, alpha = frac{lambda_2 sin (k pi)}{k pi},  xi = lambda_1
     - alpha (frac{1}{k} - frac{pi}{sin (k pi)}).

     'Lmom.genlogis' and 'par.genlogis' accept input as vectors of
     equal length. In 'f.genlogis', 'F.genlogis', 'invF.genlogis' and
     'rand.genlogis' parameters ('xi', 'alfa', 'k') must be atomic.

_V_a_l_u_e:

     'f.genlogis' gives the density f, 'F.genlogis' gives the
     distribution function F, 'invF.genlogis' gives the quantile
     function x, 'Lmom.genlogis' gives the L-moments (lambda_1,
     lambda_2, tau_3, tau_4), 'par.genlogis' gives the parameters
     ('xi', 'alfa', 'k'), and 'rand.genlogis' generates random
     deviates.

_N_o_t_e:

     For information on the package and the Author, and for all the
     references, see 'nsRFA'.

_S_e_e _A_l_s_o:

     'rnorm', 'runif', 'EXP', 'GENPAR', 'GEV', 'GUMBEL', 'KAPPA',
     'LOGNORM', 'P3'; 'DISTPLOTS', 'GOFmontecarlo', 'Lmoments'.

_E_x_a_m_p_l_e_s:

     data(hydroSIMN)
     annualflows
     summary(annualflows)
     x <- annualflows["dato"][,]
     fac <- factor(annualflows["cod"][,])
     split(x,fac)

     camp <- split(x,fac)$"45"
     ll <- Lmoments(camp)
     parameters <- par.genlogis(ll[1],ll[2],ll[4])
     f.genlogis(1800,parameters$xi,parameters$alfa,parameters$k)
     F.genlogis(1800,parameters$xi,parameters$alfa,parameters$k)
     invF.genlogis(0.7697433,parameters$xi,parameters$alfa,parameters$k)
     Lmom.genlogis(parameters$xi,parameters$alfa,parameters$k)
     rand.genlogis(100,parameters$xi,parameters$alfa,parameters$k)

     Rll <- regionalLmoments(x,fac); Rll
     parameters <- par.genlogis(Rll[1],Rll[2],Rll[4])
     Lmom.genlogis(parameters$xi,parameters$alfa,parameters$k)

