Compute the empirical semivariogram for varying bin sizes and cutoff values.
Usage
Torgegram(
formula,
ssn.object,
type = c("flowcon", "flowuncon"),
bins = 15,
cutoff,
partition_factor
)
Arguments
- formula
A formula describing the fixed effect structure.
- ssn.object
A spatial stream network object with class
SSN
.- type
The Torgegram type. A vector with possible values
"flowcon"
for flow-connected distances,"flowuncon"
for flow-unconnected distances, and"euclid"
for Euclidean distances. The default is to show both flow-connected and flow-unconnected distances.- bins
The number of equally spaced bins. The default is 15.
- cutoff
The maximum distance considered. The default is half the diagonal of the bounding box from the coordinates.
- partition_factor
An optional formula specifying the partition factor. If specified, semivariances are only computed for observations sharing the same level of the partition factor.
Value
A list with elements correspond to type
. Each element
is data frame with distance bins (bins
), the average distance
(dist
), the semivariance (gamma
), and the
number of (unique) pairs (np
) for the respective type
.
Details
The Torgegram is an empirical semivariogram is a tool used to visualize and model
spatial dependence by estimating the semivariance of a process at varying distances
separately for flow-connected, flow-unconnected, and Euclidean distances.
For a constant-mean process, the
semivariance at distance \(h\) is denoted \(\gamma(h)\) and defined as
\(0.5 * Var(z1 - z2)\). Under second-order stationarity,
\(\gamma(h) = Cov(0) - Cov(h)\), where \(Cov(h)\) is the covariance function
at distance h
. Typically the residuals from an ordinary
least squares fit defined by formula
are second-order stationary with
mean zero. These residuals are used to compute the empirical semivariogram.
At a distance h
, the empirical semivariance is
\(1/N(h) \sum (r1 - r2)^2\), where \(N(h)\) is the number of (unique)
pairs in the set of observations whose distance separation is h
and
r1
and r2
are residuals corresponding to observations whose
distance separation is h
. In spmodel, these distance bins actually
contain observations whose distance separation is h +- c
,
where c
is a constant determined implicitly by bins
. Typically,
only observations whose distance separation is below some cutoff are used
to compute the empirical semivariogram (this cutoff is determined by cutoff
).
References
Zimmerman, D. L., & Ver Hoef, J. M. (2017). The Torgegram for fluvial variography: characterizing spatial dependence on stream networks. Journal of Computational and Graphical Statistics, 26(2), 253--264.
Examples
# Copy the mf04p .ssn data to a local directory and read it into R
# When modeling with your .ssn object, you will load it using the relevant
# path to the .ssn data on your machine
copy_lsn_to_temp()
temp_path <- paste0(tempdir(), "/MiddleFork04.ssn")
mf04p <- ssn_import(temp_path, overwrite = TRUE)
tg <- Torgegram(Summer_mn ~ 1, mf04p)
plot(tg)