chiplot                 package:evd                 R Documentation

_D_e_p_e_n_d_e_n_c_e _M_e_a_s_u_r_e _P_l_o_t_s

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

     Plots of estimates of the dependence measures chi and chi-bar for
     bivariate data.

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

     chiplot(data, nq = 100, qlim = NULL, which = 1:2, conf = 0.95, lty = 1,
         cilty = 2, col = 1, cicol = 1, xlim = c(0,1), ylim1 = NULL,
         ylim2 = c(-1,1), main1 = "Chi Plot", main2 = "Chi Bar Plot", xlab =
         "Quantile", ylab1 = "Chi", ylab2 = "Chi Bar", ask = nb.fig <
         length(which) && dev.interactive(), ...)

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

    data: A matrix or data frame with two columns. Rows (observations)
          with missing values are stripped from the data before any
          computations are performed.

      nq: The number of quantiles at which the measures are evaluated.

    qlim: The limits of the quantiles at which the measures are
          evaluated (see *Details*).

   which: If only one plot is required, specify '1' for chi and '2' for
          chi-bar.

    conf: The confidence coefficient of the plotted confidence
          intervals.

lty, cilty: Line types for the estimates of the measures and for the
          confidence intervals respectively. Use zero to supress.

col, cicol: Colour types for the estimates of the measures and for the
          confidence intervals respectively.

xlim, xlab: Limits and labels for the x-axis; they apply to both plots.

   ylim1: Limits for the y-axis of the chi plot. If this is 'NULL' (the
          default) the upper limit is one, and the lower limit is the
          minimum of zero and the smallest plotted value.

   ylim2: Limits for the y-axis of the chi-bar plot.

main1, main2: The plot titles for the chi and chi-bar plots
          respectively.

ylab1, ylab2: The y-axis labels for the chi and chi-bar plots
          respectively.

     ask: Logical; if 'TRUE', the user is asked before each plot.

     ...: Other arguments to be passed to 'matplot'.

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

     These measures are explained in full detail in Coles, Heffernan
     and Tawn (1999). A brief treatment is also given in Section 8.4 of
     Coles(2001). A short summary is given as follows. We assume that
     the data are _iid_ random vectors with common bivariate
     distribution function G, and we define the random vector (X,Y) to
     be distributed according to G.

     The chi plot is a plot of q against empirical estimates of

        chi(q) = 2 - log(Pr(F_X(X) < q, F_Y(Y) < q)) / log(q)

     where F_X and F_Y are the marginal distribution functions, and
     where q is in the interval (0,1). The quantity chi(q) is bounded
     by

                2 - log(2u - 1)/log(u) <= chi(q) <= 1

     where the lower bound is interpreted as '-Inf' for q <= 1/2 and
     zero for q = 1. These bounds are reflected in the corresponding
     estimates.

     The chi bar plot is a plot of q against empirical estimates of

      chibar(q) = 2log(1-q)/log(Pr(F_X(X) > q, F_Y(Y) > q)) - 1

     where F_X and F_Y are the marginal distribution functions, and
     where q is in the interval (0,1). The quantity chibar(q) is
     bounded by -1 <= chibar(q) <= 1 and these bounds are reflected in
     the corresponding estimates.

     Note that the empirical estimators for chi(q) and chibar(q) are
     undefined near q=0 and q=1. By default the function takes the
     limits of q so that the plots depicts all values at which the
     estimators are defined. This can be overridden by the argument
     'qlim', which must represent a subset of the default values (and
     these can be determined using the component 'quantile' of the
     invisibly returned list; see *Value*).

     The confidence intervals within the plot assume that observations
     are independent, and that the marginal distributions are estimated
     exactly. The intervals are constructed using the delta method;
     this may lead to poor interval estimates near q=0 and q=1.

     The function chi(q) can be interpreted as a quantile dependent
     measure of dependence. In particular, the sign of chi(q)
     determines whether the variables are positively or negatively
     associated at quantile level q. By definition, variables are said
     to be asymptotically independent when chi(1) (defined in the
     limit) is zero. For independent variables, chi(q) = 0 for all q in
     (0,1). For perfectly dependent variables, chi(q) = 1 for all q in
     (0,1). For bivariate extreme value distributions, chi(q) =
     2(1-A(1/2)) for all q in (0,1), where A is the dependence
     function, as defined in 'abvpar'. If a bivariate threshold model
     is to be fitted (using 'fbvpot'), this plot can therefore act as a
     threshold identification plot, since e.g. the use of 95% marginal
     quantiles as threshold values implies that chi(q) should be
     approximately constant above q = 0.95.

     The function chibar(q) can again be interpreted as a quantile
     dependent measure of dependence; it is most useful within the
     class of asymptotically independent variables. For asymptotically
     dependent variables (i.e. those for which chi(1) < 1), we have
     chibar(1) = 1, where chibar(1) is again defined in the limit. For
     asymptotically independent variables, chibar(1) provides a measure
     that increases with dependence strength. For independent variables
     chibar(q) = 0 for all q in (0,1), and hence chibar(1) = 0.

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

     A list with components 'quantile', 'chi' (if '1' is in 'which')
     and 'chibar' (if '2' is in 'which') is invisibly returned. The
     components 'quantile' and 'chi' contain those objects that were
     passed to the formal arguments 'x' and 'y' of 'matplot' in order
     to create the chi plot. The components 'quantile' and 'chibar'
     contain those objects that were passed to the formal arguments 'x'
     and 'y' of 'matplot' in order to create the chi-bar plot.

_A_u_t_h_o_r(_s):

     Jan Heffernan and Alec Stephenson

_R_e_f_e_r_e_n_c_e_s:

     Coles, S. G., Heffernan, J. and Tawn, J. A. (1999) Dependence
     measures for extreme value analyses. _Extremes_, *2*, 339-365.

     Coles, S. G. (2001) _An Introduction to Statistical Modelling of
     Extreme Values_, London: Springer-Verlag.

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

     'fbvevd', 'fbvpot', 'matplot'

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

     par(mfrow = c(1,2))
     smdat1 <- rbvevd(1000, dep = 0.6, model = "log")
     smdat2 <- rbvevd(1000, dep = 1, model = "log")
     chiplot(smdat1)
     chiplot(smdat2)

