Compute empirical survivals (S) and return periods (T)

Compute the empirical survival values and the empirical return periods at the observations of an object. These are used as plotting positions in several plots.

Usage

SandT(object, points = c("p", "H"), a = 0, naive = FALSE)

Arguments

object

The object containing the data, with class "Renouv" or "Rendata".

points

Option for the computation of the plotting positions. When points is set to "p", the \(p\)-points formula is used with the selected value of a. This formula is used to compute the survival from which the return period is computed. When instead points is set to "H", Nelson's formula is used to compute the return periods, the survival value still being given by the \(p\)-points formula. When the data is heterogeneous, i.e. when object contains MAX and/or OTS data, Nelson's formula is used only to compute the return periods of the upper slice of levels.

a

Parameter used in the interpolation formula for the inverse return periods as in Hirsch and Stedinger (1987).

naive

Logical. When TRUE, naive plotting positions are used to display MAX or OTS data. These can be defined only when a main sample exists in object as a x.OT element. For each MAX or OTS block, the positions use the number of events predicted using the rate of events as estimated from the main sample. When the main sample has a small durations, such predictions are likely to be misleading.

Value

A list with the following elements

x

Numeric vector containing the ordered values from all the available sources in the object: main sample, historical periods either 'MAX' or 'OTS'.

group, groupNames

Integer and character vectors giving the source of the values in x, in the same order of the values. For instance, group[10] gives the group form which x[10] was extracted, and the name of this group is groupNames[group[10]].

S, T

Numeric vectors of the same length as x and containing the corresponding estimation of the survival value and of the return period.

thresh, lambda.thresh, S.thresh, T.thresh

Vector of thresholds and the corresponding estimation for the event rate, survival and return period. All the estimations are for the threshold values. The value of T.thresh[i] for a threshold thresh[i] results from a simple computation: divide the sum of the durations for blocks with thresholds >= thresh[i] by the number of events for these blocks.

Details

When the object contains historical information (MAX or OTS), the computation is an adaptation Hirsch and Stedinger (1987) for the Marked Process (MP) context. The original method is devoted to block maxima and interpolates the survival values at the thresholds which are computed first. For MP, the interpolation is done for the inverse return periods, and the survival values are deduced from those of the inverse return periods.

Nelson's formula provides unbiased estimates for the values of the cumulative hazard \(H(x)\) at the order statistics, and thus can be used to estimate the log-return periods as required on the return level plot.

Warning

When using points = "H" the estimated values of the survival returned in S and those for the return period T no longer verify T=1/S/lambda, where lambda is the estimated rate. In this case, the values in T should be used in the return level plot, while the values in S should be used in the PP-plot.

Author

Yves Deville

References

The original method for block maxima is described in

Hirsch R.M. and Stedinger J.R.(1887) Plotting Positions for Historical Floods and their precision. Water Ressources Research, vol. 23, N. 4 pp. 715-727.
Millard S. and Neerchal N. (2001). Environmental Statistics with S-Plus. CRC Press

The adaptation for the Marked Process context is described in the Renext Computing Details document.

Examples

## use an object with class "Rendata"
ST1 <- SandT(object = Garonne)
## basic return level plot
plot(ST1$T, ST1$x, col = ST1$group, log = "x")

## use an object with class "Renouv"
fit <- Renouv(x = Garonne, plot = FALSE)
ST2 <- SandT(object = fit)
plot(ST2$T, ST2$x, col = ST2$group, log = "x")