| Title: | Semiparametric Spatial Filtering with Eigenvectors in (Generalized) Linear Models |
|---|---|
| Description: | Tools to decompose (transformed) spatial connectivity matrices and perform supervised or unsupervised semiparametric spatial filtering in a regression framework. The package supports unsupervised spatial filtering in standard linear as well as some generalized linear regression models. |
| Authors: | Sebastian Juhl [aut, cre] |
| Maintainer: | Sebastian Juhl <[email protected]> |
| License: | GPL-3 |
| Version: | 2.1.0 |
| Built: | 2025-02-21 11:41:33 UTC |
| Source: | https://github.com/sjuhl/spfilter |
An artificially generated cross-sectional dataset together with
an accompanying binary connectivity matrix W. The
units are located on a regular grid and W
is defined according to rook's adjacency definition of contiguity.
The synthetic data can be used to illustrate the functionality
of this package.
data(fakedata) Wdata(fakedata) W
An object of class data.frame with 100 rows and 8 columns.
An object of class matrix (inherits from array) with 100 rows and 100 columns.
The file contains two objects:
fakedataset |
a synthetic dataset | |
W |
an artificial spatial connectivity matrix | |
data(fakedata) head(fakedataset) dim(W)data(fakedata) head(fakedataset) dim(W)
Extract eigenvectors and corresponding eigenvalues from the matrix MWM, where M denotes a symmetric and idempotent projection matrix and W is the spatial connectivity matrix. This function also reports the Moran coefficient associated with each of the eigenvectors.
getEVs(W, covars = NULL)getEVs(W, covars = NULL)
W |
spatial connectivity matrix |
covars |
vector/ matrix of regressors included in the construction of the projection matrix M - see Details |
The eigenfunctions obtained by getEVs
can be used to perform supervised eigenvector selection and to
manually create a spatial filter. To this end, a candidate set
may be determined by 1) the sign of the spatial autocorrelation
in model residuals and 2) the strength of spatial association
found in each eigenvector as indicated by moran.
Prior to the spectral decomposition, getEVs symmetrizes the
spatial connectivity matrix by: 1/2 * (W + W').
If covars are supplied, the function uses the covariates to construct
projection matrix: M = I - X (X'
X)^-1X'. Using this matrix results in a set of
eigenvectors that are uncorrelated to each other as well as to the
covariates. If covars = NULL, only the intercept term is used
to construct M. See e.g., Griffith and Tiefelsdorf (2007)
for more details on the appropriate choice of M.
A list containing the following objects:
vectorsmatrix of all eigenvectors
valuesvector of the corresponding eigenvalues
moranvector of the Moran coefficients associated with the eigenvectors
Sebastian Juhl
Tiefelsdorf, Michael and Daniel A. Griffith (2007): Semiparametric filtering of spatial autocorrelation: the eigenvector approach. Environment and Planning A: Economy and Space, 39 (5): pp. 1193 - 1221.
lmFilter, glmFilter, MI.ev,
MI.sf, vif.ev, partialR2
data(fakedata) E <- getEVs(W = W, covars = NULL)data(fakedata) E <- getEVs(W = W, covars = NULL)
This function implements the eigenvector-based semiparametric spatial filtering approach in a generalized linear regression framework using maximum likelihood estimation (MLE). Eigenvectors are selected by an unsupervised stepwise regression technique. Supported selection criteria are the minimization of residual autocorrelation, maximization of model fit, significance of residual autocorrelation, and the statistical significance of eigenvectors. Alternatively, all eigenvectors in the candidate set can be included as well.
glmFilter( y, x = NULL, W, objfn = "AIC", MX = NULL, model, optim.method = "BFGS", sig = 0.05, bonferroni = TRUE, positive = TRUE, ideal.setsize = FALSE, min.reduction = 0.05, boot.MI = 100, resid.type = "pearson", alpha = 0.25, tol = 0.1, na.rm = TRUE )glmFilter( y, x = NULL, W, objfn = "AIC", MX = NULL, model, optim.method = "BFGS", sig = 0.05, bonferroni = TRUE, positive = TRUE, ideal.setsize = FALSE, min.reduction = 0.05, boot.MI = 100, resid.type = "pearson", alpha = 0.25, tol = 0.1, na.rm = TRUE )
y |
response variable |
x |
vector/ matrix of regressors (default = NULL) |
W |
spatial connectivity matrix |
objfn |
the objective function to be used for eigenvector selection. Possible criteria are: the maximization of model fit ('AIC' or 'BIC'), minimization of residual autocorrelation ('MI'), significance level of candidate eigenvectors ('p'), significance of residual spatial autocorrelation ('pMI'), or all eigenvectors in the candidate set ('all') |
MX |
covariates used to construct the projection matrix (default = NULL) - see Details |
model |
a character string indicating the type of model to be estimated. Currently, 'probit', 'logit', 'poisson', and 'nb' (for negative binomial model) are valid inputs |
optim.method |
a character specifying the optimization method used by
the |
sig |
significance level to be used for eigenvector selection
if |
bonferroni |
Bonferroni adjustment for the significance level
(TRUE/ FALSE) if |
positive |
restrict search to eigenvectors associated with positive levels of spatial autocorrelation (TRUE/ FALSE) |
ideal.setsize |
if |
min.reduction |
if |
boot.MI |
number of iterations used to estimate the variance of Moran's I
(default is 100). Alternatively, if |
resid.type |
character string specifying the residual type to be used. Options are 'raw', 'deviance', and 'pearson' (default) |
alpha |
a value in (0,1] indicating the range of candidate eigenvectors according to their associated level of spatial autocorrelation, see e.g., Griffith (2003) |
tol |
if |
na.rm |
remove observations with missing values (TRUE/ FALSE) |
If W is not symmetric, it gets symmetrized by 1/2 * (W + W') before the decomposition.
If covariates are supplied to MX, the function uses these regressors
to construct the following projection matrix:
M = I - X (X'X)^-1X'
Eigenvectors from MWM using this specification of
M are not only mutually uncorrelated but also orthogonal
to the regressors specified in MX. Alternatively, if MX = NULL, the
projection matrix becomes M = I - 11'/ *n*,
where 1 is a vector of ones and *n* represents the number of
observations. Griffith and Tiefelsdorf (2007) show how the choice of the appropriate
M depends on the underlying process that generates the spatial
dependence.
The Bonferroni correction is only possible if eigenvector selection is based on
the significance level of the eigenvectors (objfn = 'p'). It is set to
FALSE if eigenvectors are added to the model until the residuals exhibit no
significant level of spatial autocorrelation (objfn = 'pMI').
For the negative binomial model, deviance residuals are currently not computed.
The function sets resid.type = 'pearson' and prints a message to the console.
An object of class spfilter containing the following
information:
estimatessummary statistics of the parameter estimates
varcovarestimated variance-covariance matrix
EVa matrix containing the summary statistics of selected eigenvectors
selvecsvector/ matrix of selected eigenvectors
evMIMoran coefficient of eigenvectors
moranresidual autocorrelation in the initial and the filtered model
fitadjusted R-squared of the initial and the filtered model
residualsinitial and filtered model residuals
othera list providing supplementary information:
ncandidatesnumber of candidate eigenvectors considered
nevnumber of selected eigenvectors
condnumcondition number to assess the degree of multicollinearity among the eigenvectors induced by the link function, see e.g., Griffith/ Amrhein (1997)
sel_idID of selected eigenvectors
sfvector representing the spatial filter
sfMIMoran coefficient of the spatial filter
modeltype of the regression model
dependencefiltered for positive or negative spatial dependence
objfnselection criterion specified in the objective function of the stepwise regression procedure
bonferroniTRUE/ FALSE: Bonferroni-adjusted significance level
(if objfn='p')
siglevelif objfn = 'p' or objfn = 'pMI': actual
(unadjusted/ adjusted) significance level
resid.typeresidual type ('raw', 'deviance', or 'pearson')
pseudoR2McFadden's (adjusted) pseudo R-squared (filtered vs. unfiltered model) based on the models' likelihood functions
If the condition number (condnum) suggests high levels of
multicollinearity, eigenvectors can be sequentially removed from selvecs
and the model can be re-estimated using the glm function in order to
identify and manually remove the problematic eigenvectors. Moreover, if other
models that are currently not implemented here need to be estimated
(e.g., quasi-binomial models), users can extract eigenvectors using the function
getEVs and perform a supervised eigenvector search using the glm
function.
In contrast to eigenvector-based spatial filtering in linear regression models, Chun (2014) notes that only a limited number of studies address the problem of measuring spatial autocorrelation in generalized linear model residuals. Consequently, eigenvector selection may be based on an objective function that maximizes model fit rather than a function that minimizes residual spatial autocorrelation.
Chun, Yongwan (2014): Analyzing Space-Time Crime Incidents Using Eigenvector Spatial Filtering: An Application to Vehicle Burglary. Geographical Analysis 46 (2): pp. 165 - 184.
Tiefelsdorf, Michael and Daniel A. Griffith (2007): Semiparametric filtering of spatial autocorrelation: the eigenvector approach. Environment and Planning A: Economy and Space, 39 (5): pp. 1193 - 1221.
Griffith, Daniel A. (2003): Spatial Autocorrelation and Spatial Filtering: Gaining Understanding Through Theory and Scientific Visualization. Berlin/ Heidelberg, Springer.
Griffith, Daniel A. and Carl G. Amrhein (1997): Multivariate Statistical Analysis for Geographers. Englewood Cliffs, Prentice Hall.
lmFilter, getEVs, MI.resid,
optim
data(fakedata) # poisson model y_pois <- fakedataset$count poisson <- glmFilter(y = y_pois, x = NULL, W = W, objfn = "MI", positive = FALSE, model = "poisson", boot.MI = 100) print(poisson) summary(poisson, EV = FALSE) # probit model - summarize EVs y_prob <- fakedataset$indicator probit <- glmFilter(y = y_prob, x = NULL, W = W, objfn = "p", positive = FALSE, model = "probit", boot.MI = 100) print(probit) summary(probit, EV = TRUE) # logit model - AIC objective function y_logit <- fakedataset$indicator logit <- glmFilter(y = y_logit, x = NULL, W = W, objfn = "AIC", positive = FALSE, model = "logit", min.reduction = .05) print(logit) summary(logit, EV = FALSE)data(fakedata) # poisson model y_pois <- fakedataset$count poisson <- glmFilter(y = y_pois, x = NULL, W = W, objfn = "MI", positive = FALSE, model = "poisson", boot.MI = 100) print(poisson) summary(poisson, EV = FALSE) # probit model - summarize EVs y_prob <- fakedataset$indicator probit <- glmFilter(y = y_prob, x = NULL, W = W, objfn = "p", positive = FALSE, model = "probit", boot.MI = 100) print(probit) summary(probit, EV = TRUE) # logit model - AIC objective function y_logit <- fakedataset$indicator logit <- glmFilter(y = y_logit, x = NULL, W = W, objfn = "AIC", positive = FALSE, model = "logit", min.reduction = .05) print(logit) summary(logit, EV = FALSE)
This function implements the eigenvector-based semiparametric spatial filtering approach in a linear regression framework using ordinary least squares (OLS). Eigenvectors are selected by an unsupervised stepwise regression technique. Supported selection criteria are the minimization of residual autocorrelation, maximization of model fit, significance of residual autocorrelation, and the statistical significance of eigenvectors. Alternatively, all eigenvectors in the candidate set can be included as well.
lmFilter( y, x = NULL, W, objfn = "MI", MX = NULL, sig = 0.05, bonferroni = TRUE, positive = TRUE, ideal.setsize = FALSE, alpha = 0.25, tol = 0.1, boot.MI = NULL, na.rm = TRUE ) ## S3 method for class 'spfilter' summary(object, EV = FALSE, ...)lmFilter( y, x = NULL, W, objfn = "MI", MX = NULL, sig = 0.05, bonferroni = TRUE, positive = TRUE, ideal.setsize = FALSE, alpha = 0.25, tol = 0.1, boot.MI = NULL, na.rm = TRUE ) ## S3 method for class 'spfilter' summary(object, EV = FALSE, ...)
y |
response variable |
x |
vector/ matrix of regressors (default = NULL) |
W |
spatial connectivity matrix |
objfn |
the objective function to be used for eigenvector selection. Possible criteria are: the maximization of the adjusted R-squared ('R2'), minimization of residual autocorrelation ('MI'), significance level of candidate eigenvectors ('p'), significance of residual spatial autocorrelation ('pMI') or all eigenvectors in the candidate set ('all') |
MX |
covariates used to construct the projection matrix (default = NULL) - see Details |
sig |
significance level to be used for eigenvector selection
if |
bonferroni |
Bonferroni adjustment for the significance level
(TRUE/ FALSE) if |
positive |
restrict search to eigenvectors associated with positive levels of spatial autocorrelation (TRUE/ FALSE) |
ideal.setsize |
if |
alpha |
a value in (0,1] indicating the range of candidate eigenvectors according to their associated level of spatial autocorrelation, see e.g., Griffith (2003) |
tol |
if |
boot.MI |
number of iterations used to estimate the variance of Moran's I.
If |
na.rm |
listwise deletion of observations with missing values (TRUE/ FALSE) |
object |
an object of class |
EV |
display summary statistics for selected eigenvectors (TRUE/ FALSE) |
... |
additional arguments |
If W is not symmetric, it gets symmetrized by 1/2 * (W + W') before the decomposition.
If covariates are supplied to MX, the function uses these regressors
to construct the following projection matrix:
M = I - X (X'X)^-1X'
Eigenvectors from MWM using this specification of
M are not only mutually uncorrelated but also orthogonal
to the regressors specified in MX. Alternatively, if MX = NULL, the
projection matrix becomes M = I - 11'/*n*,
where 1 is a vector of ones and *n* represents the number of
observations. Griffith and Tiefelsdorf (2007) show how the choice of the appropriate
M depends on the underlying process that generates the spatial
dependence.
The Bonferroni correction is only possible if eigenvector selection is based on
the significance level of the eigenvectors (objfn = 'p'). It is set to
FALSE if eigenvectors are added to the model until the residuals exhibit no
significant level of spatial autocorrelation (objfn = 'pMI').
An object of class spfilter containing the following
information:
estimatessummary statistics of the parameter estimates
varcovarestimated variance-covariance matrix
EVa matrix containing the summary statistics of selected eigenvectors
selvecsvector/ matrix of selected eigenvectors
evMIMoran coefficient of eigenvectors
moranresidual autocorrelation in the initial and the filtered model
fitadjusted R-squared of the initial and the filtered model
residualsinitial and filtered model residuals
othera list providing supplementary information:
ncandidatesnumber of candidate eigenvectors considered
nevnumber of selected eigenvectors
sel_idID of selected eigenvectors
sfvector representing the spatial filter
sfMIMoran coefficient of the spatial filter
modeltype of the fitted regression model
dependencefiltered for positive or negative spatial dependence
objfnselection criterion specified in the objective function of the stepwise regression procedure
bonferroniTRUE/ FALSE: Bonferroni-adjusted significance level
(if objfn = 'p')
siglevelif objfn = 'p' or objfn = 'pMI': actual
(unadjusted/ adjusted) significance level
Tiefelsdorf, Michael and Daniel A. Griffith (2007): Semiparametric filtering of spatial autocorrelation: the eigenvector approach. Environment and Planning A: Economy and Space, 39 (5): pp. 1193 - 1221.
Griffith, Daniel A. (2003): Spatial Autocorrelation and Spatial Filtering: Gaining Understanding Through Theory and Scientific Visualization. Berlin/ Heidelberg, Springer.
Chun, Yongwan, Daniel A. Griffith, Monghyeon Lee, Parmanand Sinha (2016): Eigenvector selection with stepwise regression techniques to construct eigenvector spatial filters. Journal of Geographical Systems, 18, pp. 67 – 85.
Le Gallo, Julie and Antonio Páez (2013): Using synthetic variables in instrumental variable estimation of spatial series models. Environment and Planning A: Economy and Space, 45 (9): pp. 2227 - 2242.
Tiefelsdorf, Michael and Barry Boots (1995): The Exact Distribution of Moran's I. Environment and Planning A: Economy and Space, 27 (6): pp. 985 - 999.
data(fakedata) y <- fakedataset$x1 X <- cbind(fakedataset$x2, fakedataset$x3, fakedataset$x4) res <- lmFilter(y = y, x = X, W = W, objfn = 'MI', positive = FALSE) print(res) summary(res, EV = TRUE) E <- res$selvecs (ols <- coef(lm(y ~ X + E))) coef(res)data(fakedata) y <- fakedataset$x1 X <- cbind(fakedataset$x2, fakedataset$x3, fakedataset$x4) res <- lmFilter(y = y, x = X, W = W, objfn = 'MI', positive = FALSE) print(res) summary(res, EV = TRUE) E <- res$selvecs (ols <- coef(lm(y ~ X + E))) coef(res)
A decomposition of the Moran coefficient in order to separately test for the simultaneous presence of positive and negative autocorrelation in a variable.
MI.decomp(x, W, nsim = 100, na.rm = TRUE)MI.decomp(x, W, nsim = 100, na.rm = TRUE)
x |
a vector or matrix |
W |
spatial connectivity matrix |
nsim |
number of iterations to simulate the null distribution |
na.rm |
listwise deletion of observations with missing values (TRUE/ FALSE) |
If x is a matrix, this function computes the Moran
test for spatial autocorrelation for each column.
The p-values calculated for I+ and I- assume
a directed alternative hypothesis. Statistical significance is assessed
using a permutation procedure to generate a simulated null distribution.
Returns a data.frame that contains the following information
for each variable:
I+observed value of Moran's I (positive part)
VarI+variance of Moran's I (positive part)
pI+simulated p-value of Moran's I (positive part)
I-observed value of Moran's I (negative part)
VarI-variance of Moran's I (negative part)
pI-simulated p-value of Moran's I (negative part)
pItwo.sidedsimulated p-value of the two-sided test
Sebastian Juhl
Dary, Stéphane (2011): A New Perspective about Moran’s Coefficient: Spatial Autocorrelation as a Linear Regression Problem. Geographical Analysis, 43 (2): pp. 127 - 141.
MI.vec, MI.ev, MI.sf,
MI.resid, MI.local, getEVs
data(fakedata) X <- cbind(fakedataset$x1, fakedataset$x2, fakedataset$x3, fakedataset$negative) (MI.dec <- MI.decomp(x = X, W = W, nsim = 100)) # the sum of I+ and I- equals the observed Moran coefficient: I <- MI.vec(x = X, W = W)[, "I"] cbind(MI.dec[, "I+"] + MI.dec[, "I-"], I)data(fakedata) X <- cbind(fakedataset$x1, fakedataset$x2, fakedataset$x3, fakedataset$negative) (MI.dec <- MI.decomp(x = X, W = W, nsim = 100)) # the sum of I+ and I- equals the observed Moran coefficient: I <- MI.vec(x = X, W = W)[, "I"] cbind(MI.dec[, "I+"] + MI.dec[, "I-"], I)
Calculates the Moran coefficient for each eigenvector.
MI.ev(W, evals)MI.ev(W, evals)
W |
spatial connectivity matrix |
evals |
vector of eigenvalues |
Returns a vector containing the Moran coefficients of the eigenvectors associated with the supplied eigenvalues.
Sebastian Juhl
Le Gallo, Julie and Antonio Páez (2013): Using synthetic variables in instrumental variable estimation of spatial series models. Environment and Planning A, 45 (9): pp. 2227 - 2242.
Tiefelsdorf, Michael and Barry Boots (1995): The Exact Distribution of Moran's I. Environment and Planning A: Economy and Space, 27 (6): pp. 985 - 999.
lmFilter, glmFilter, getEVs,
MI.sf
Reports the local Moran Coefficient for each unit.
MI.local(x, W, alternative = "greater", na.rm = TRUE)MI.local(x, W, alternative = "greater", na.rm = TRUE)
x |
a vector |
W |
spatial connectivity matrix |
alternative |
specification of alternative hypothesis as 'greater' (default), 'lower', or 'two.sided' |
na.rm |
listwise deletion of observations with missing values (TRUE/ FALSE) |
Returns an object of class data.frame that contains the
following information for each variable:
Iiobserved value of local Moran's I
EIiexpected value of local Moran coefficients
VarIivariance of local Moran's I
zIistandardized local Moran coefficient
pIip-value of the test statistic
The calculation of the statistic and its moments follows Anselin (1995) and Sokal et al. (1998).
Sebastian Juhl
Anselin, Luc (1991): Local Indicators of Spatial Association-LISA. Geographical Analysis, 27 (2): pp. 93 - 115.
Bivand, Roger S. and David W. S. Wong (2018): Comparing Implementations of Global and Local Indicators of Spatial Association. TEST, 27: pp. 716 - 748.
Sokal, Robert R., Neal L. Oden, Barbara A. Thomson (1998): Local Spatial Autocorrelation in a Biological Model. Geographical Analysis, 30 (4): pp. 331 - 354.
MI.vec, MI.ev, MI.sf,
MI.resid, MI.decomp
data(fakedata) x <- fakedataset$x2 (MIi <- MI.local(x = x, W = W, alternative = "greater"))data(fakedata) x <- fakedataset$x2 (MIi <- MI.local(x = x, W = W, alternative = "greater"))
This function assesses the degree of spatial autocorrelation present in regression residuals by means of the Moran coefficient.
MI.resid( resid, x = NULL, W, alternative = "greater", boot = NULL, na.rm = TRUE )MI.resid( resid, x = NULL, W, alternative = "greater", boot = NULL, na.rm = TRUE )
resid |
residual vector |
x |
vector/ matrix of regressors (default = NULL) |
W |
spatial connectivity matrix |
alternative |
specification of alternative hypothesis as 'greater' (default), 'lower', or 'two.sided' |
boot |
optional integer specifying the number of simulation iterations to compute the variance. If NULL (default), variance calculated under assumed normality |
na.rm |
listwise deletion of observations with missing values (TRUE/ FALSE) |
The function assumes an intercept-only model if x = NULL.
Furthermore, MI.resid automatically symmetrizes the matrix
W by: 1/2 * (W + W').
A data.frame object with the following elements:
Iobserved value of the Moran coefficient
EIexpected value of Moran's I
VarIvariance of Moran's I
zIstandardized Moran coefficient
pIp-value of the test statistic
Calculations are based on the procedure proposed by Cliff and Ord (1981). See also Cliff and Ord (1972).
Sebastian Juhl
Cliff, Andrew D. and John K. Ord (1981): Spatial Processes: Models & Applications. Pion, London.
Cliff, Andrew D. and John K. Ord (1972): Testing for Spatial Autocorrelation Among Regression Residuals. Geographical Analysis, 4 (3): pp. 267 - 284
lmFilter, glmFilter, MI.vec,
MI.local
data(fakedata) y <- fakedataset$x1 x <- fakedataset$x2 resid <- y - x %*% solve(crossprod(x)) %*% crossprod(x,y) (Moran <- MI.resid(resid = resid, x = x, W = W, alternative = "greater")) # intercept-only model x <- rep(1, length(y)) resid2 <- y - x %*% solve(crossprod(x)) %*% crossprod(x,y) intercept <- MI.resid(resid = resid2, W = W, alternative = "greater") # same result with MI.vec for the intercept-only model vec <- MI.vec(x = resid2, W = W, alternative = "greater") rbind(intercept, vec)data(fakedata) y <- fakedataset$x1 x <- fakedataset$x2 resid <- y - x %*% solve(crossprod(x)) %*% crossprod(x,y) (Moran <- MI.resid(resid = resid, x = x, W = W, alternative = "greater")) # intercept-only model x <- rep(1, length(y)) resid2 <- y - x %*% solve(crossprod(x)) %*% crossprod(x,y) intercept <- MI.resid(resid = resid2, W = W, alternative = "greater") # same result with MI.vec for the intercept-only model vec <- MI.vec(x = resid2, W = W, alternative = "greater") rbind(intercept, vec)
Computes the Moran coefficient of the spatial filter.
MI.sf(gamma, evMI)MI.sf(gamma, evMI)
gamma |
vector of regression coefficients associated with the eigenvectors |
evMI |
Moran coefficient of eigenvectors |
Moran coefficient of the spatial filter.
Sebastian Juhl
Le Gallo, Julie and Antonio Páez (2013): Using synthetic variables in instrumental variable estimation of spatial series models. Environment and Planning A: Economy and Space, 45 (9): pp. 2227 - 2242.
lmFilter, glmFilter, getEVs,
MI.ev
Tests for the presence of spatial autocorrelation in variables as indicated by the Moran coefficient. The variance is calculated under the normality assumption.
MI.vec(x, W, alternative = "greater", symmetrize = TRUE, na.rm = TRUE)MI.vec(x, W, alternative = "greater", symmetrize = TRUE, na.rm = TRUE)
x |
a vector or matrix |
W |
spatial connectivity matrix |
alternative |
specification of alternative hypothesis as 'greater' (default), 'lower', or 'two.sided' |
symmetrize |
symmetrizes the connectivity matrix W by: 1/2 * (W + W') (TRUE/ FALSE) |
na.rm |
listwise deletion of observations with missing values (TRUE/ FALSE) |
If x is a matrix, this function computes the Moran
test for spatial autocorrelation for each column.
Returns an object of class data.frame that contains the
following information for each variable:
Iobserved value of the Moran coefficient
EIexpected value of Moran's I
VarIvariance of Moran's I (under normality)
zIstandardized Moran coefficient
pIp-value of the test statistic
Estimation of the variance (under the normality assumption)
follows Cliff and Ord (1981), see also Upton and Fingleton (1985).
It assumes the connectivity matrix W to be symmetric.
For inherently non-symmetric matrices, it is recommended to specify
symmetrize = TRUE.
Sebastian Juhl
Cliff, Andrew D. and John K. Ord (1981): Spatial Processes: Models & Applications. Pion, London.
Upton, Graham J. G. and Bernard Fingleton (1985): Spatial Data Analysis by Example, Volume 1. New York, Wiley.
Bivand, Roger S. and David W. S. Wong (2018): Comparing Implementations of Global and Local Indicators of Spatial Association. TEST 27: pp. 716 - 748.
data(fakedata) X <- cbind(fakedataset$x1, fakedataset$x2, fakedataset$x3) (MI <- MI.vec(x = X, W = W, alternative = "greater", symmetrize = TRUE))data(fakedata) X <- cbind(fakedataset$x1, fakedataset$x2, fakedataset$x3) (MI <- MI.vec(x = X, W = W, alternative = "greater", symmetrize = TRUE))
This function computes the partial R-squared of all selected eigenvectors in a spatially filtered linear regression model.
partialR2(y, x = NULL, evecs)partialR2(y, x = NULL, evecs)
y |
response variable |
x |
vector/ matrix of regressors |
evecs |
(selected) eigenvectors |
Vector of partial R-squared values of the eigenvectors.
The function assumes a linear regression model. Since the
eigenvectors are mutually uncorrelated, partialR2 evaluates
them sequentially. In generalized linear models, the presence of a link
function can corrupt the uncorrelatedness of the eigenvectors.
Sebastian Juhl
data(fakedata) y <- fakedataset$x1 x <- fakedataset$x2 # get eigenvectors E <-getEVs(W = W, covars = NULL)$vectors (out <- partialR2(y = y, x = x, evecs = E[, 1:5]))data(fakedata) y <- fakedataset$x1 x <- fakedataset$x2 # get eigenvectors E <-getEVs(W = W, covars = NULL)$vectors (out <- partialR2(y = y, x = x, evecs = E[, 1:5]))
Calculate the variance inflation factor (VIF) of the eigenvectors in the spatial filter.
vif.ev(x = NULL, evecs, na.rm = TRUE)vif.ev(x = NULL, evecs, na.rm = TRUE)
x |
vector/ matrix of regressors (default = NULL) |
evecs |
(selected) eigenvectors |
na.rm |
remove missing values in covariates (TRUE/ FALSE) |
Returns a vector containing the VIF for each eigenvector.
This function assumes a linear model which ensures the uncorrelatedness of the eigenvectors. Note that regression weights or the link function used in generalized linear models can corrupt this property.
Sebastian Juhl
data(fakedata) E <- getEVs(W = W, covars = NULL)$vectors (VIF <- vif.ev(x = fakedataset$x1, evecs = E[, 1:10]))data(fakedata) E <- getEVs(W = W, covars = NULL)$vectors (VIF <- vif.ev(x = fakedataset$x1, evecs = E[, 1:10]))
This function decomposes the variation in an outcome variable into four fractions: a) the influence of covariates, b) joint influence of covariates and space, c) the influence of space, and d) unexplained residual variation. Moran spectral randomization is applied to obtain the expected value of the coefficient of determination adjusted for spurious correlations.
vp(y, x = NULL, evecs = NULL, msr = 100)vp(y, x = NULL, evecs = NULL, msr = 100)
y |
outcome vector |
x |
vector/ matrix of covariates |
evecs |
selected eigenvectors |
msr |
number of permutations to compute the expected value under H0 |
Returns an object of class vpart which provides the following
information:
R2unadjusted fractions of explained variation
adjR2adjusted fractions (based on Moran spectral randomization)
msrnumber of permutations to obtain the expected value under H0
The adjusted R-squared values are obtained by: 1 - (1 - R2) / (1 - E(R2|H0)). For fractions [ab] and [a], Moran spectral randomization is used to derive E(R2|H0). To this end, the rows in matrix (or column vector) x are randomly permuted in order to preserve the correlation structure (see e.g., Clappe et al. 2018).
Sebastian Juhl
Clappe, Sylvie, Dray Stéphane. and Pedro R. Peres-Neto (2018): Beyond neutrality: disentangling the effects of species sorting and spurious correlations in community analysis. Ecology 99 (8): pp. 1737 - 1747.
Wagner, Helene H., and Stéphane Dray (2015): Generating spatially constrained null models for irregularly spaced data using Moran spectral randomization methods. Methods in Ecology and Evolution 6 (10): pp. 1169 - 1178.
data(fakedata) E <- getEVs(W = W, covars = NULL)$vectors (partition <- vp(y = fakedataset$x1, evecs = E[, 1:10], msr = 100))data(fakedata) E <- getEVs(W = W, covars = NULL)$vectors (partition <- vp(y = fakedataset$x1, evecs = E[, 1:10], msr = 100))