Description Usage Arguments Details Value Author(s) References See Also Examples

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

1 2 3 4 5 | ```
chiplot(data, nq = 100, qlim = NULL, which = 1:2, conf = 0.95, trunc =
TRUE, spcases = FALSE, lty = 1, cilty = 2, col = 1, cicol = 1,
xlim = c(0,1), ylim1 = c(-1,1), 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(), ...)
``` |

`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 |

`which` |
If only one plot is required, specify |

`conf` |
The confidence coefficient of the plotted confidence intervals. |

`trunc` |
Logical; truncate the estimates at their theoretical upper and lower bounds? |

`spcases` |
If |

`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 |

`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 |

`...` |
Other arguments to be passed to |

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 `abvevd`

. 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*.

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.

Jan Heffernan and Alec Stephenson

Coles, S. G., Heffernan, J. and Tawn, J. A. (1999)
Dependence measures for extreme value analyses.
*Extremes*, **2**, 339â€“365.

Coles, S. G. (2001)
*An Introduction to Statistical Modelling of Extreme Values*,
London: Springerâ€“Verlag.

1 2 3 4 5 |

```
```

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