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)