risk.index {SoPhy} | R Documentation |
Estimation of the index of potential risk to groundwater
risk.index(data, selected.dist=0.95, selected.rate=cbind(c(0.5, 0.8), c(0.4, 0.9), c(0.3, 1.0)), weights=1, measure=function(x) x^2, method=c('fix.m', 'optim.m', 'ml'), min.neg.xi = -10, max.neg.xi = -0.1, max.pos.xi = 10, endpoint.tolerance = 0, front.factor = 2, # min.no.paths=max(data[, 2]), max.no.paths=10 * max(data[, 2]), PrintLevel=RFparameters()$Print, max.rate=TRUE)
data |
matrix of two columns. First column gives the distances (depths in the profile measured from the surface) and the second column the number of observed blue pixels. |
selected.dist |
scale or vector with values in (0,1)
or a vector of integers.
Distances for which the form parameter of the pareto distribution
is estimated; see Details.
If selected.dist is a number in (0,1) the distances are
1:(1 + round(max(data[,1]-1) * selected.dist)) . If it is a
vector with values in (0,1), then the vector must have an even
number of elements and pairs of elements are interpreted as intervals.
Otherwise the integers are interpreted as indices for data .
|
selected.rate |
vector of matrix of nrows.
additionally to the indices given by
selected.dist , the form parameter is estimated also for
those distances where the corresponding number of observed relative stained
pixels (w.r.t. to the maximum number of observed pixels)
is within the interval given by the first column of
selected.rate .
The risk index is calculated as the median of the estimated form
parameters.
In case no values are in the given interval of the first column the second column is considered etc, i.e. the first row should contain decreasing values and the second row increasing values. |
weights |
the estimation algorithm is based on a weighted least
square algorithm; weights is usually either 1 or a vector
of length nrow(data) . |
measure |
instead of the default least squares another distance function can be given. |
method |
the number of observed paths is a free parameter when
fitting the Pareto distribution. It can either be set as the maximum
number of stained pixels for the currently considered distances or
depths ('fix.m' ) or fitted within the optimisation algorithm
('optim.m' ). Usually, it is not worth using the slower
'optim.m' option.
See also the Details.
|
max.neg.xi |
optimisation parameter : largest negative value that is allowed as shape parameter of the Pareto distribution, i.e. a negative value close to 0. |
min.neg.xi |
optimisation parameter : smallest negative value that is allowed as shape parameter of the Pareto distribution |
max.pos.xi |
optimisation parameter : largest allowed shape parameter of the Pareto distribution |
endpoint.tolerance |
optimisation parameter. If the shape
parameter is negative then the distribution has a finite upper
endpoint. Hence, mathematically, the lowest upper end point of the
Pareto distribution is given as the largest distance for which at
least one stained pixel is observed.
For stability reasons and because the observed data might be a
scale mixure of Pareto distribution it is advantageous to allow for
some tolerance of the minimal upper end point.
If endpoint.tolerance is positive then the lower threshold
for the upper end point is the largest distance for which the
number of observed stained pixels is larger than
endpoint.tolerance .
If endpoint.tolerance is negative then the lower threshold
equals largest distance for which at least one stained pixel is
observed minus the modulus of endpoint.tolerance .
|
front.factor |
optimisation parameter . The upper bound
for the upper endpoint equals the front.factor times the
largest distance for which at least one stained pixel is observed.
The value should best not be changed. |
max.no.paths |
the number of paths is estimated as nuisance
parameter when estimating the risk index;
max.no.paths give the upper bound for the nuisance
parameter in the optimisation. |
PrintLevel |
The higher the value of PrintLevel the more tracing
information is given. Up to value 1, no information is given.
Note that if PrintLevel>=2 a running counter is shown that
includes the printing of backspaces
(^H).
The backspaces may have undesirable interactions with some few other
R functions, e.g. Sweave .
See package RandomFields for the default option
RFparameters ()$Print .
|
max.rate |
logical. If TRUE then
the lines for which m(D) / m(0) is in selected.rate
are used to calculate the final risk index.
Here m(D) gives the maximum of p(d), d=D, D+1, ... where p(d) the number of stained
pixels in depth d.
If FALSE then the criterion m(D) / m(0) is replaced
by p(D) / p(0).
|
Denote by f(d) the number of blue pixels registered at depth
d (or distances from the soil surface).
Then, the risk index is by definition a shape parameter of
f(d) for large distances d. Since the term large cannot be
defined precisely, the shape parameter is calculated for the
function values f(d) for distances d>=d_i
and several fixed starting distances d_i.
The distances d_i are given by selected.rate
.
(The approach is similar to that for analyzing extremal events.)
The selection criterion m(D) / m(0) is always based
on method='fix.m'
, whatever method
is chosen to estimate xi.
list of the following components
par |
matrix of estimated parameters; first row: risk index;
second row: scale parameter; third row: estimated maximum number of
paths m(D) except m(0) that is given by max.freq
and is always set to the maximum number of pixels.
forth row: D (sel.dist ).
|
data |
the input data except for some reordering |
weights |
the input weights expcept for some reordering |
selected.dist |
the selected distances in form of indices (in clear text, in case they were given in form of a real value in (0,1)). |
selected.rate |
range of the selected number of stained pixels |
sel.rate |
index set for the data where the observed
number of stained pixels are within selected.rate |
sel.dist |
the index set containing selected.dist and
sel.rate |
max.freq |
maximum number of observed stained pixels |
values |
the minimal least squares values |
method |
the input parameter method |
measure |
the input parameter measure |
raw.risk.index |
risk index calculated as median of the
estimated form parameters for selected.rate
|
risk.index |
the median is calculated only for values greater than 0.999
min.neg.xi and less than 0.999
max.pos.xi
|
Martin Schlather, martin.schlather@math.uni-goettingen.de http://www.stochastik.math.uni-goettingen.de/institute
Embrechts, P., Klueppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events. Berlin: Springer.
Schlather, M. and Huwe, B. (2005) A risk index for characterising flow pattern in soils using dye tracer distributions. J. Contam. Hydrol. 79, to appear.
SoPhy
,
analyse.profile
,
xswms2d
sample.depth <- 1 : 100 d <- rexp(1000, 1/25) freq <- numeric(length(sample.depth)) for (i in 1:length(sample.depth)) freq[i] <- sum(d>=sample.depth[i]) cr <- risk.index(cbind(sample.depth, freq), selected.rate=c(0.95, 0.9), endpoint.tolerance=20, method="fix.m") cr$risk.index ## the true value is 0