Fitting Linear Models for Spatial Stream Networks

This function works on spatial stream network objects to fit linear models with spatially autocorrelated errors using likelihood methods, allowing for non-spatial random effects, anisotropy, partition factors, big data methods, and more. The spatial formulation is described in Ver Hoef and Peterson (2010) and Peterson and Ver Hoef (2010).

Usage

ssn_lm(
  formula,
  ssn.object,
  tailup_type = "none",
  taildown_type = "none",
  euclid_type = "none",
  nugget_type = "nugget",
  tailup_initial,
  taildown_initial,
  euclid_initial,
  nugget_initial,
  additive,
  estmethod = "reml",
  anisotropy = FALSE,
  random,
  randcov_initial,
  partition_factor,
  ...
)

Arguments

formula: A two-sided linear formula describing the fixed effect structure of the model, with the response to the left of the ~ operator and the terms on the right, separated by + operators.
ssn.object: A spatial stream network object with class SSN.
tailup_type: The tailup covariance function type. Available options include "linear", "spherical", "exponential", "mariah", "epa", and "none". Parameterizations are described in Details.
taildown_type: The taildown covariance function type. Available options include "linear", "spherical", "exponential", "mariah", "epa", and "none". Parameterizations are described in Details.
euclid_type: The euclidean covariance function type. Available options include "spherical", "exponential", "gaussian", "cosine", "cubic", "pentaspherical", "wave", "jbessel", "gravity", "rquad", "magnetic", and "none". Parameterizations are described in Details.
nugget_type: The nugget covariance function type. Available options include "nugget" or "none". Parameterizations are described in Details.
tailup_initial: An object from tailup_initial() specifying initial and/or known values for the tailup covariance parameters.
taildown_initial: An object from taildown_initial() specifying initial and/or known values for the taildown covariance parameters.
euclid_initial: An object from euclid_initial() specifying initial and/or known values for the euclidean covariance parameters.
nugget_initial: An object from nugget_initial() specifying initial and/or known values for the nugget covariance parameters.
additive: The name of the variable in ssn.object that is used to define spatial weights. Can be quoted or unquoted. For the tailup covariance functions, these additive weights are used for branching. Technical details that describe the role of the additive variable in the tailup covariance function are available in Ver Hoef and Peterson (2010).
estmethod: The estimation method. Available options include "reml" for restricted maximum likelihood and "ml" for maximum likelihood. The default is "reml".
anisotropy: A logical indicating whether (geometric) anisotropy should be modeled. Not required if spcov_initial is provided with 1) rotate assumed unknown or assumed known and non-zero or 2) scale assumed unknown or assumed known and less than one. When anisotropy is TRUE, computational times can significantly increase. The default is FALSE.
random: A one-sided linear formula describing the random effect structure of the model. Terms are specified to the right of the ~ operator. Each term has the structure x1 + ... + xn | g1/.../gm, where x1 + ... + xn specifies the model for the random effects and g1/.../gm is the grouping structure. Separate terms are separated by + and must generally be wrapped in parentheses. Random intercepts are added to each model implicitly when at least one other variable is defined. If a random intercept is not desired, this must be explicitly defined (e.g., x1 + ... + xn - 1 | g1/.../gm). If only a random intercept is desired for a grouping structure, the random intercept must be specified as 1 | g1/.../gm. Note that g1/.../gm is shorthand for (1 | g1/.../gm). If only random intercepts are desired and the shorthand notation is used, parentheses can be omitted.
randcov_initial: An optional object specifying initial and/or known values for the random effect variances. See spmodel::randcov_initial().
partition_factor: A one-sided linear formula with a single term specifying the partition factor. The partition factor assumes observations from different levels of the partition factor are uncorrelated.
...: Other arguments to stats::optim().

Value

A list with many elements that store information about the fitted model object and has class ssn_lm. Many generic functions that summarize model fit are available for ssn_lm objects, including AIC, AICc, anova, augment, coef, cooks.distance, covmatrix, deviance, fitted, formula, glance, glances, hatvalues, influence, labels, logLik, loocv, model.frame, model.matrix, plot, predict, print, pseudoR2, summary, terms, tidy, update, varcomp, and vcov.

This fitted model list contains the following elements:

additive: The name of the additive function value column.
anisotropy: Whether euclidean anisotropy was modeled.
call: The function call.
coefficients: Model coefficients.
contrasts: Any user-supplied contrasts.
cooks_distance: Cook's distance values.
crs: The geographic coordinate reference system.
deviance: The model deviance.
diagtol: A tolerance value that may be added to the diagonal of ovariance matrices to encourage decomposition stability.
estmethod: The estimation method.
euclid_max: The maximum euclidean distance.
fitted: Fitted values.
formula: The model formula.
hatvalues: The hat (leverage) values.
is_known: An object that identifies which parameters are known.
local_index: An index identifier used internally for sorting.
missing_index: Which rows in the "obs" object had missing responses.
n: The sample size.
npar: The number of estimated covariance parameters.
observed_index: Which rows in the "obs" object had observed responses.
optim: The optimization output.
p: The number of fixed effects.
partition_factor: The partition factor formula.
pseudoR2: The pseudo R-squared.
random: The random effect formula.
residuals: The residuals.
sf_column_name: The name of the geometry columns ssn.object
ssn.object: An updated ssn.object.
tail_max: The maximum stream distance.
terms: The model terms.
vcov: Variance-covariance matrices
xlevels: The levels of factors in the model matrix.

These list elements are meant to be used with various generic functions (e.g., residuals() that operate on the model object. While possible to access elements of the fitted model list directly, we strongly advise against doing so when there is a generic available to return the element of interest. For example, we strongly recommend using residuals() to obtain model residuals instead of accessing the fitted model list directly via object$residuals.

Details

The linear model for spatial stream networks can be written as $y = X \beta + zu + zd + ze + n$, where $X$ is the fixed effects design matrix, $\beta$ are the fixed effects, $zu$ is tailup random error, $zd$ is taildown random error, and $ze$ is Euclidean random error, and $n$ is nugget random error. The tailup random errors capture spatial covariance moving downstream (and depend on downstream distance), the taildown random errors capture spatial covariance moving upstream (and depend on upstream) distance, the Euclidean random errors capture spatial covariance that depends on Euclidean distance, and the nugget random errors captures variability independent of spatial locations. The response $y$ is modeled using a spatial covariance function expressed as $de(zu) * R(zu) + de(zd) * R(zd) + de(ze) * R(ze) + nugget * I$. $de(zu)$, $de(zu)$, and $de(zd)$ represent the tailup, taildown, and Euclidean variances, respectively. $R(zu)$, $R(zd)$, and $R(ze)$ represent the tailup, taildown, and Euclidean correlation matrices, respectively. Each correlation matrix depends on a range parameter that controls the distance-decay behavior of the correlation. $nugget$ represents the nugget variance and $I$ represents an identity matrix.

tailup_type Details: Let $D$ be a matrix of hydrologic distances, $W$ be a diagonal matrix of weights from additive, $r = D / range$, and $I$ be an identity matrix. Then parametric forms for flow-connected elements of $R(zu)$ are given below:

linear: $(1 - r) * (r <= 1) * W$
spherical: $(1 - 1.5r + 0.5r^3) * (r <= 1) * W$
exponential: $exp(-r) * W$
mariah: $log(90r + 1) / 90r * (D > 0) + 1 * (D = 0) * W$
epa: $(D - range)^2 * F * (r <= 1) * W / 16range^5$
none: $I$ * W

Details describing the F matrix in the epa covariance are given in Garreta et al. (2010). Flow-unconnected elements of $R(zu)$ are assumed uncorrelated. Observations on different networks are also assumed uncorrelated.

taildown_type Details: Let $D$ be a matrix of hydrologic distances, $r = D / range$, and $I$ be an identity matrix. Then parametric forms for flow-connected elements of $R(zd)$ are given below:

linear: $(1 - r) * (r <= 1)$
spherical: $(1 - 1.5r + 0.5r^3) * (r <= 1)$
exponential: $exp(-r)$
mariah: $log(90r + 1) / 90r * (D > 0) + 1 * (D = 0)$
epa: $(D - range)^2 * F1 * (r <= 1) / 16range^5$
none: $I$

Now let $A$ be a matrix that contains the shorter of the two distances between two sites and the common downstream junction, $r1 = A / range$, $B$ be a matrix that contains the longer of the two distances between two sites and the common downstream junction, $r2 = B / range$, and $I$ be an identity matrix. Then parametric forms for flow-unconnected elements of $R(zd)$ are given below:

linear: $(1 - r2) * (r2 <= 1)$
spherical: $(1 - 1.5r1 + 0.5r2) * (1 - r2)^2 * (r2 <= 1)$
exponential: $exp(-(r1 + r2))$
mariah: $(log(90r1 + 1) - log(90r2 + 1)) / (90r1 - 90r2) * (A =/ B) + (1 / (90r1 + 1)) * (A = B)$
epa: $(B - range)^2 * F2 * (r2 <= 1) / 16range^5$
none: $I$

Details describing the F1 and F2 matrices in the epa covariance are given in Garreta et al. (2010). Observations on different networks are assumed uncorrelated.

euclid_type Details: Let $D$ be a matrix of Euclidean distances, $r = D / range$, and $I$ be an identity matrix. Then parametric forms for elements of $R(ze)$ are given below:

exponential: $exp(- r )$
spherical: $(1 - 1.5r + 0.5r^3) * (r <= 1)$
gaussian: $exp(- r^2 )$
cubic: $(1 - 7r^2 + 8.75r^3 - 3.5r^5 + 0.75r^7) * (r <= 1)$
pentaspherical: $(1 - 1.875r + 1.25r^3 - 0.375r^5) * (r <= 1)$
cosine: $cos(r)$
wave: $sin(r) * (h > 0) / r + (h = 0)$
jbessel: $Bj(h * range)$, Bj is Bessel-J function
gravity: $(1 + r^2)^{-0.5}$
rquad: $(1 + r^2)^{-1}$
magnetic: $(1 + r^2)^{-1.5}$
none: $I$

nugget_type Details: Let $I$ be an identity matrix and $0$ be the zero matrix. Then parametric forms for elements the nugget variance are given below:

nugget: $I$
none: $0$

In short, the nugget effect is modeled when nugget_type is "nugget" and omitted when nugget_type is "none".

estmethod Details: The various estimation methods are

reml: Maximize the restricted log-likelihood.
ml: Maximize the log-likelihood.

anisotropy Details: By default, all Euclidean covariance parameters except rotate and scale are assumed unknown, requiring estimation. If either rotate or scale are given initial values other than 0 and 1 (respectively) or are assumed unknown in euclid_initial(), anisotropy is implicitly set to TRUE. (Geometric) Anisotropy is modeled by transforming a Euclidean covariance function that decays differently in different directions to one that decays equally in all directions via rotation and scaling of the original Euclidean coordinates. The rotation is controlled by the rotate parameter in $[0, \pi]$ radians. The scaling is controlled by the scale parameter in $[0, 1]$. The anisotropy correction involves first a rotation of the coordinates clockwise by rotate and then a scaling of the coordinates' minor axis by the reciprocal of scale. The Euclidean covariance is then computed using these transformed coordinates.

random Details: If random effects are used, the model can be written as $y = X \beta + W1\gamma 1 + ... Wj\gamma j + zu + zd + ze + n$, where each Z is a random effects design matrix and each u is a random effect.

partition_factor Details: The partition factor can be represented in matrix form as $P$, where elements of $P$ equal one for observations in the same level of the partition factor and zero otherwise. The covariance matrix involving only the spatial and random effects components is then multiplied element-wise (Hadmard product) by $P$, yielding the final covariance matrix.

Other Details: Observations with NA response values are removed for model fitting, but their values can be predicted afterwards by running predict(object).

Note

This function does not perform any internal scaling. If optimization is not stable due to large extremely large variances, scale relevant variables so they have variance 1 before optimization.

References

Garreta, V., Monestiez, P. and Ver Hoef, J.M. (2010) Spatial modelling and prediction on river networks: up model, down model, or hybrid? Environmetrics 21(5), 439--456.

Peterson, E.E. and Ver Hoef, J.M. (2010) A mixed-model moving-average approach to geostatistical modeling in stream networks. Ecology 91(3), 644--651.

Ver Hoef, J.M. and Peterson, E.E. (2010) A moving average approach for spatial statistical models of stream networks (with discussion). Journal of the American Statistical Association 105, 6--18. DOI: 10.1198/jasa.2009.ap08248. Rejoinder pgs. 22--24.

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)

ssn_mod <- ssn_lm(
  formula = Summer_mn ~ ELEV_DEM,
  ssn.object = mf04p,
  tailup_type = "exponential",
  additive = "afvArea"
)
summary(ssn_mod)
#> 
#> Call:
#> ssn_lm(formula = Summer_mn ~ ELEV_DEM, ssn.object = mf04p, tailup_type = "exponential", 
#>     additive = "afvArea")
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -3.9487 -2.3775 -0.8840 -0.2776  0.9001 
#> 
#> Coefficients (fixed):
#>              Estimate Std. Error z value Pr(>|z|)    
#> (Intercept) 71.323162   6.193050  11.517   <2e-16 ***
#> ELEV_DEM    -0.028822   0.003075  -9.372   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Pseudo R-squared: 0.6688
#> 
#> Coefficients (covariance):
#>               Effect     Parameter   Estimate
#>   tailup exponential  de (parsill)  4.415e+00
#>   tailup exponential         range  1.175e+06
#>               nugget        nugget  2.562e-02
#>