Methods for Computing Spatial, Temporal, and Spatiotemporal Statistics

Description

Methods for computing spatial, temporal, and spatiotemporal statistics as described in Gouhier and Guichard (2014) <doi:10.1111/2041-210X.12188>. These methods include empirical univariate, bivariate and multivariate variograms; fitting variogram models; phase locking and synchrony analysis; generating autocorrelated and cross-correlated matrices.

Details

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

Maintainer: Tarik C. Gouhier (tarik.gouhier@gmail.com)

References

Bjornstad, O. N., and W. Falck. 2001. Nonparametric spatial covariance functions: Estimation and testing. Environmental and Ecological Statistics 8:53-70.

Bjornstad, O. N., R. A. Ims, and X. Lambin. 1999. Spatial population dynamics: analyzing patterns and processes of population synchrony. Trends in Ecology & Evolution 14:427-432.

Buonaccorsi, J. P., J. S. Elkinton, S. R. Evans, and A. M. Liebhold. 2001. Measuring and testing for spatial synchrony. Ecology 82:1668-1679.

Cazelles, B., and L. Stone. 2003. Detection of imperfect population synchrony in an uncertain world. Journal of Animal Ecology 72:953-968.

Fortin, M. J., and M. R. T. Dale. 2005. Spatial Analysis: A Guide for Ecologists. Cambridge University Press.

Gouhier, T. C., and F. Guichard. 2007. Local disturbance cycles and the maintenance of spatial heterogeneity across scales in marine metapopulations. Ecology 88:647-657.

Gouhier, T. C., F. Guichard, and A. Gonzalez. 2010. Synchrony and stability of food webs in metacommunities. The American Naturalist 175:E16-E34.

Gouhier, T. C., F. Guichard, and B. A. Menge. 2010. Ecological processes can synchronize marine population dynamics over continental scales. Proceedings of the National Academy of Sciences 107:8281-8286.

Loreau, M., and C. de Mazancourt. 2008. Species synchrony and its drivers: Neutral and nonneutral community dynamics in fluctuating environments. The American Naturalist 172:E48-E66.

Purves, D. W., and R. Law. 2002. Fine-scale spatial structure in a grassland community: quantifying the plant's eye view. Journal of Ecology 90:121-129.

Vasseur, D. A. 2007. Environmental colour intensifies the Moran effect when population dynamics are spatially heterogeneous. Oikos 116:1726-1736.

Zar, J. H. 1999. Biostatistical Analysis, Fourth edition. Prentice-Hall, Inc., Upper Saddle River, NJ.

Examples

# Compute phase synchrony
t1=runif(100)
t2=runif(100)
sync=phase.sync(t1, t2)
# Distribution of phase difference
hist(sync$deltaphase$mod_phase_diff_2pi)

# Compute concordant peaks
p=peaks(t1, t2, nrands=100)
# Find proportion of time steps where both time series peak together
p$peaks
# Plot (null) distribution of proportion of time steps where both time
# series peak together
hist(p$rand)
# p-value of observed value
p$pval

# Compute Kendall's W 
data(bird.traits)
(w=kendall.w(bird.traits))

# Community matrix for 20 species undergoing random fluctuations 
comm.rand=matrix(runif(100), nrow=5, ncol=20)
community.sync(comm.rand, nrands=10)
# Community matrix for 20 species undergoing synchronized fluctuations 
comm.corr=matrix(rep(comm.rand[,1], 20), nrow=5, ncol=20)
community.sync(comm.corr, nrands=10)

bird trait dataset

Description

Contains the wing length, tail length, and bill length from 12 birds

Usage

data(bird.traits)

Format

A data frame with 12 observations (birds) on the following 3 variables.

wing.length

a numeric vector containing wing length in cm

tail.length

a numeric vector containing tail length in cm

bill.length

a numeric vector containing bill length in cm

Details

Dataset from Zar (1999; page 444)

Source

Zar, J. H. 1999. Biostatistical Analysis, Fourth edition. Prentice-Hall, Inc., Upper Saddle River, NJ.

Examples

data(bird.traits)
(w=kendall.w(bird.traits))

Compute community-wide synchrony and its significance via Monte Carlo randomizations

Description

Compute community-wide synchrony and its the significance via Monte Carlo randomizations. If all species fluctuate in perfect unison, the community-wide synchrony will be 1. If species undergo uncorrelated fluctuations, the community-wide synchrony will be 1/S. The Monte Carlo randomizations are performed by shuffling the columns of the community matrix independently. This function also returns the mean correlation between the columns of the matrix.

Usage

community.sync (data, nrands = 0, method = c("pearson", "kendall", "spearman"), 
                alternative = c("greater", "less"), type = 1, quiet = FALSE, ...)

Arguments

Details

Loreau and de Mazancourt (2008) show that community-wide synchrony \varphi can be quantified by computing the temporal variance \sigma_{x_T}^2 of the community time series x_T(t)=\sum{x_i(t)} and the sum of the temporal standard deviation of the time series across all species \left(\sum{\sigma_{x_i}}\right)^2 such that: \varphi=\frac{\sigma_{x_T}^2}{\left(\sum{\sigma_{x_i}}\right)^2}

Value

Returns a named list containing:

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

References

Loreau, M., and C. de Mazancourt. 2008. Species synchrony and its drivers: Neutral and nonneutral community dynamics in fluctuating environments. The American Naturalist 172:E48-E66.

Purves, D. W., and R. Law. 2002. Fine-scale spatial structure in a grassland community: quantifying the plant's eye view. Journal of Ecology 90:121-129.

Examples

# Community matrix for 20 species undergoing random fluctuations 
comm.rand=matrix(runif(100), nrow=5, ncol=20)
community.sync(comm.rand, nrands=20)$pval
# Community matrix for 20 species undergoing synchronized fluctuations 
comm.corr=matrix(rep(comm.rand[,1], 20), nrow=5, ncol=20)
community.sync(comm.corr, nrands=20)$pval
# On "real" data
data(bird.traits)
community.sync(bird.traits, nrands=20)$pval

coord2dist

Description

Calculate distance between all pairs of sites

Usage

coord2dist (coords, is.latlon = TRUE, lower.tri = TRUE)

Arguments

Value

Returns the distance between all pairs of sites

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

Examples

coords=rbind(c(32, -125), c(43, -130))
# Compute great circle distance
coord2dist(coords)

correlated.matrix

Description

Create an ntimes x nspecies matrix with correlation rho, standard deviation sigma, and mean mu

Usage

correlated.matrix (rho = 0, sigma = 1, mu = 0, ntimes = 200, nspecies = 10)

Arguments

Details

This function is based on the Cholesky factorization method described by Legendre (2000).

Value

Returns a named list containing the following:

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

References

Gouhier, T. C., F. Guichard, and A. Gonzalez. 2010. Synchrony and stability of food webs in metacommunities. The American Naturalist 175:E16-E34.

Legendre, P. 2000. Comparison of permutation methods for the partial correlation and partial mantel tests. Journal of Statistical Computation and Simulation 67:37-73.

Examples

mat=correlated.matrix(rho=0.85, sigma=30, mu=10, nspecies=10)
# Check sd of each column
apply(mat$community, 2, sd)
# Check mean of each column
apply(mat$community, 2, mean)
# Check correlation of matrix
community.sync(mat$community)

Find min/max of a time series

Description

Find local minima and maxima of a time series

Usage

find.minmax (timeseries)

Arguments

Value

Returns a named list containing:

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

Examples

t1=runif(100)
min.max=find.minmax(t1)
min.max$maxs
plot (t1, t="l")
points (min.max$mins, col="blue", bg="blue", pch=19)
points (min.max$maxs, col="red", bg="red", pch=19)

Kendall's W

Description

Compute Kendall's coefficient of concordance (W)

Usage

kendall.w (data, nrands = 0, type = 1, quiet = FALSE)

Arguments

Details

Kendall's W is a non-parametric statistic that ranges from 0 to 1 and measures the level of agreement between multiple variables. When the number of observations n>10, its significance can be determined by using a \chi^2 distribution with df=n-1. Legendre (2005) shows that the \chi^2 test is always too conservative (low power) compared to the randomization test. Hence, both tests have been made available in this function. The Monte Carlo randomizations are performed by shuffling the columns of the community matrix independently (Legendre 2005).

Value

Returns a named list containing:

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

References

Buonaccorsi, J. P., J. S. Elkinton, S. R. Evans, and A. M. Liebhold. 2001. Measuring and testing for spatial synchrony. Ecology 82:1668-1679.

Gouhier, T. C., and F. Guichard. 2007. Local disturbance cycles and the maintenance of spatial heterogeneity across scales in marine metapopulations. Ecology 88:647-657.

Gouhier, T. C., F. Guichard, and A. Gonzalez. 2010. Synchrony and stability of food webs in metacommunities. The American Naturalist 175:E16-E34.

Legendre, P. 2005. Species associations: the Kendall coefficient of concordance revisited. Journal of Agricultural, Biological, and Environmental Statistics 10:226-245.

Purves, D. W., and R. Law. 2002. Fine-scale spatial structure in a grassland community: quantifying the plant's eye view. Journal of Ecology 90:121-129.

Zar, J. H. 1999. Biostatistical Analysis, Fourth edition. Prentice-Hall, Inc., Upper Saddle River, NJ.

Examples

data(bird.traits)
(w=kendall.w(bird.traits))

latlon2dist

Description

Calculate distance between a pair of coordinates

Usage

latlon2dist (coords)

Arguments

Value

Returns the great circle distance distance between the pair of coordinates

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

See Also

coord2dist

Examples

coords=c(32, -125, 43, -130)
# Compute great circle distance
latlon2dist(coords)

Compute mean column-wise correlation and determine its significance via Monte Carlo randomizations

Description

Compute mean column-wise correlation and determine its significance via Monte Carlo randomizations. The Monte Carlo randomizations are performed by shuffling the columns of the community matrix independently.

Usage

meancorr (data, nrands = 0, alternative = c("two.tailed", "greater", "less"), 
                            method = c("pearson", "kendall", "spearman"), 
                            type = 1, quiet = FALSE, ...)

Arguments

Value

Returns a named list containing:

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

References

Purves, D. W., and R. Law. 2002. Fine-scale spatial structure in a grassland community: quantifying the plant's eye view. Journal of Ecology 90:121-129.

Examples

# Community matrix for 20 species undergoing random fluctuations 
comm.rand=matrix(runif(100), nrow=5, ncol=20)
meancorr(comm.rand, nrands=20)$pval
# Community matrix for 20 species undergoing synchronized fluctuations 
comm.corr=matrix(rep(comm.rand[,1], 20), nrow=5, ncol=20)
meancorr(comm.corr, nrands=20)$pval
# On "real" data
data(bird.traits)
meancorr(bird.traits, nrands=20)$pval

Find the proportion of local minima/maxima common to both time series and compute its significance via Monte Carlo randomizations

Description

Find the proportion of local minima/maxima common to both time series and compute its significance via Monte Carlo randomizations

Usage

peaks (t1, t2, nrands = 0, type = 1, quiet = FALSE)

Arguments

Value

Returns a named list containing:

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

References

Buonaccorsi, J. P., J. S. Elkinton, S. R. Evans, and A. M. Liebhold. 2001. Measuring and testing for spatial synchrony. Ecology 82:1668-1679.

Purves, D. W., and R. Law. 2002. Fine-scale spatial structure in a grassland community: quantifying the plant's eye view. Journal of Ecology 90:121-129.

Examples

t1=runif(100)
t2=runif(100)
(p=peaks(t1, t2))

Phase partnered time series

Description

Create two time series with specific autocorrelation \gamma, cross-correlation \rho, mean ts.mean, and standard deviation ts.sd using the phase partnered algorithm described by Vasseur (2007)

Usage

phase.partnered (n = 2000, rho = 1, gamma = 1, sigma = 0.1, mu = 0)

Arguments

Value

Returns a named list containing the following:

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

References

Gouhier, T. C., F. Guichard, and A. Gonzalez. 2010. Synchrony and stability of food webs in metacommunities. The American Naturalist 175:E16-E34.

Vasseur, D. A. 2007. Environmental colour intensifies the Moran effect when population dynamics are spatially heterogeneous. Oikos 116:1726-1736.

Examples

# Positively cross-correlated white noise
pos.corr=phase.partnered(n = 100, rho = 0.7, gamma = 0)
# Negatively cross-correlated white noise
neg.corr=phase.partnered(n = 100, rho = -1, gamma = 0)
par(mfrow=c(2,1))
matplot (pos.corr$timeseries, t="l", lty=1)
matplot (neg.corr$timeseries, t="l", lty=1)

Phase synchrony of quasi-periodic time series

Description

Compute the phase synchrony between two quasi-periodic time series by quantifying their phase difference at each time step

Usage

phase.sync (t1, t2,  nrands = 0, mod = 1, method = c("markov", "fft"), 
            nbreaks = 10, mins = FALSE, quiet = FALSE)

Arguments

Details

Two time series are phase-locked if the relationship between their phases remains constant over time. This function computes the phase of successive local maxima or minima for each time series and then uses linear interpolation to find the phase at time steps that fall between local maxima/minima. A histogram can be used to determine if the distribution of the phase difference at each time step is uniform (indicating no phase locking) or has a clear peak (indicating phase locking).

Value

Returns a list containing Q.obs, pval, rands, phases1, phases2, deltaphase, and icdf:

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

References

Cazelles, B., and L. Stone. 2003. Detection of imperfect population synchrony in an uncertain world. Journal of Animal Ecology 72:953–968.

Theiler, J., S. Eubank, A. Longtin, B. Galdrikian, and J. Doyne Farmer. 1992. Testing for nonlinearity in time series: the method of surrogate data. Physica D: Nonlinear Phenomena 58:77–94.

Examples

t1=runif(100)
t2=runif(100)
# Compute and interpolate phases using successive local minima
sync.mins=phase.sync(t1, t2, mins=TRUE)
# Compute and interpolate phases using successive local maxima
sync.maxs=phase.sync(t1, t2)
# Plot distribution of phase difference
hist(sync.mins$deltaphase$mod_phase_diff_2pi)

PISCO multi-year and spatially-explicit mussel and environmental dataset

Description

Contains the mean annual chl-a concentration, sea surface temperature, upwelling currents, and mussel abundance at 48 intertidal sites along the West Coast of the United States from 2000-2003.

Usage

data(pisco.data)

Format

A data frame with 192 observations on the following 7 variables.

latitude

latitude (degrees North)

longitude

longitude (degrees West)

chl

mean annual remote sensed chlorophyll-a concentration

sst

mean annual remote sensed sea surface temperature

upwelling

mean annual remote sensed upwelling currents

mussel_abund

mean annual mussel cover (Mytilus californianus)

year

sampling year

References

Gouhier, T. C., F. Guichard, and B. A. Menge. 2010. Ecological processes can synchronize marine population dynamics over continental scales. Proceedings of the National Academy of Sciences 107:8281-8286.

Examples

data(pisco.data)

Plot synchrony objects

Description

Plot synchrony objects

Usage

  ## S3 method for class 'synchrony'
 plot(x, main = "", xlab = "Values from randomizations", 
                            ylab = "Frequency", line.col = "red", lty = 2, 
                            lwd = 1, col = "grey", ...)

Arguments

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

Examples

comm.rand=matrix(runif(100), nrow=5, ncol=20)
comm.rand.sync=community.sync(comm.rand, nrands=20)
plot(comm.rand.sync)

Plot vario objects

Description

Plot vario objects

Usage

## S3 method for class 'vario'
 plot(x, xlab = "Lag distance", ylab = NULL, ylim = NULL,
                      xtype = c("mean.bin.dist", "bins"), rug = FALSE, ci = FALSE,
                      pch = 21, col.sig="black", col.nonsig="black", bg.sig="black", 
                      bg.nonsig = "white", alpha = 0.05, ...)

Arguments

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

Examples

data(pisco.data)
d=subset(pisco.data, subset=year==2000, select=c("latitude", "longitude", "sst"))
semiv=vario(data=d)
moran=vario(data=d, type="moran", nrand=100)
geary=vario(data=d, type="geary", nrand=100)

par(mfrow=c(3,1))
plot(semiv)
plot(moran, bg.sig="blue")
plot(geary, bg.sig="red")

Plot variofit objects

Description

Plot variofit objects

Usage

## S3 method for class 'variofit'
 plot(x, xlab = "Lag distance", ylab = "Variogram",
                         col.pts = "black", col.line = "red", 
                         pch = 21, ...)                      

Arguments

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

Examples

# Environmental variogram
data(pisco.data)
d=subset(pisco.data, subset=year==2000, select=c("latitude", "longitude", "upwelling"))
semiv=vario(data=d)
mod.sph=vario.fit(semiv$vario, semiv$mean.bin.dist)
plot(mod.sph)

Create surrogate time series via Markov process

Description

Create surrogate time series with the same short-term time correlation and overall temporal pattern as the original time series using the Markov process described by Cazelles and Stones (2003)

Usage

surrogate.ts (ts, distr.ts = NULL, trans.ts = NULL, nbreaks = 10)

Arguments

Details

The values of the time series x_n are grouped into nbreak equally-sized bins. The transition matrix M_{ij} describing the probability of x_{n+1} belonging to bin j based on x_n belonging to bin i is defined using the relative frequencies of the data such that: M_{ij}=Pr(x_{n+1} \in b_{j} | x_{n} \in b_{i}). The surrogate time series is then constructed by randomly selecting a starting value and randomly selecting the next value from the proper bin based on the transition matrix. This process is repeated until the surrogate time series has the same length as the original time series.

Value

Returns a named list containing:

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

References

Cazelles, B., and L. Stone. 2003. Detection of imperfect population synchrony in an uncertain world. Journal of Animal Ecology 72:953-968.

See Also

phase.sync

Examples

t1=runif(100)
surr.t1=surrogate.ts(ts=t1)
plot(t1, t="l")
lines(surr.t1$surr.ts, col="red")

vario

Description

Compute the empirical variogram and determine its significance via Monte Carlo randomizations

Usage

vario (n.bins = 20, size.bins = NULL, extent = 0.5, data, data2 = NULL, 
                  is.latlon = TRUE, is.centered = FALSE, nrands = 0, 
                  type = c("semivar", "cov", "pearson", 
                  "spearman", "kendall", "moran", "geary"),
                  alternative = c("one.tailed", "two.tailed"),
                  mult.test.corr = c("none", "holm", "hochberg", "bonferroni"),
                  regional = c("all", "extent"),
                  quiet = FALSE)

Arguments

Details

This function can be used to compute univariate correlograms using Moran's I, Geary's C, and the covariance function or variograms using the semivariance function. Multivariate (Mantel) correlograms can also be computed using the covariance function, Pearson's, Spearman's or Kendall's correlation coefficients. Cross-correlograms/variograms between data1 and data2 can be computed with the covariance function, Pearson's, Spearman's or Kendall's correlation coefficients for multivariate variograms and Moran's I, Geary's C, the covariance function, or semivariance for univariate variograms.

Value

Returns a named list containing the following variables:

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

References

Bjornstad, O. N., and W. Falck. 2001. Nonparametric spatial covariance functions: Estimation and testing. Environmental and Ecological Statistics 8:53-70.

Bjornstad, O. N., R. A. Ims, and X. Lambin. 1999. Spatial population dynamics: analyzing patterns and processes of population synchrony. Trends in Ecology & Evolution 14:427-432.

Fortin, M. J., and M. R. T. Dale. 2005. Spatial Analysis: A Guide for Ecologists. Cambridge University Press.

See Also

vario.func

Examples

data(pisco.data)
d=subset(pisco.data, subset=year==2000, select=c("latitude", "longitude", "sst"))
semiv=vario(data=d)
moran=vario(data=d, type="moran", nrands=100)
par(mfrow=c(2,1), mar=c(4.2, 4, 1, 1))
plot(semiv$mean.bin.dist, semiv$vario, xlab="Lag distance (km)", ylab="Semivariance")
plot(moran$mean.bin.dist, moran$vario, xlab="Lag distance (km)", ylab="Moran's I", t="l")
points(moran$mean.bin.dist[moran$pvals >= 0.05], moran$vario[moran$pvals >= 0.05], 
       bg="white", pch=21)
points(moran$mean.bin.dist[moran$pvals < 0.05], moran$vario[moran$pvals < 0.05], 
       bg="black", pch=21)
abline(h=0, lty=2)

# Compute spatial synchrony
d.upw=subset(pisco.data, select=c("latitude", "longitude", "year", "upwelling"))
d.cov=subset(pisco.data, select=c("latitude", "longitude", "year", "mussel_abund"))
# Reshape the data
d.upw.wide=reshape(data=d.upw, idvar=c("latitude", "longitude"), timevar=c("year"), 
                   direction="wide")
d.cov.wide=reshape(data=d.cov, idvar=c("latitude", "longitude"), timevar=c("year"), 
                   direction="wide")
# Generate variograms
v.upw=vario(n.bins=12, data=d.upw.wide, type="pearson", extent=1, nrands=999)
v.cov=vario(n.bins=12, data=d.cov.wide, type="pearson", extent=1, nrands=999)
## Fit variograms
v.cov.per=vario.fit(v.cov$vario, v.cov$mean.bin.dist, type="period", 
                    start.vals=list(a=1, b=3, c=0))
v.upw.lin=vario.fit(v.upw$vario, v.upw$mean.bin.dist, type="linear")

par(mfrow=c(2,1))
plot(v.cov, xlab="Lag distance (km)", bg.sig="red", col.nonsig="red", 
     main="Mussel cover", 
     rug=TRUE, ylim=c(-0.3, 0.3))
lines(v.cov$mean.bin.dist, v.cov.per$fit, col="red")
plot(v.upw, xlab="Lag distance (km)", bg.sig="blue", col.nonsig="blue", 
     main="Upwelling", rug=TRUE)
lines(v.upw$mean.bin.dist, v.upw.lin$fit, col="blue")

vario.fit

Description

Fit model to the empirical variogram

Usage

vario.fit (vario, bins, weights = rep(1, length(vario)),
                       type = c("spherical", "gaussian", "nugget", "linear", 
                       "exponential", "sill", "periodic", "hole"),
                       start.vals = list(c0 = 0, c1 = max(vario), 
                                          a = max(bins)/4, b=0.1, c=0.1),
                       control = list(maxit=10000)) 

Arguments

Value

Return a named list containing the following variables:

Note

Selecting proper initial values is critical for fitting a reasonable model to the empirical variogram. If these values are off, nls will fail and fall-back functions will be used to determine the best parameter values that minimize the Root Mean Square Error (RMSE).

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

See Also

vario, vario.func

Examples

# Load data
data(pisco.data)
# Environmental variogram
d=subset(pisco.data, subset=year==2000, select=c("latitude", "longitude", "upwelling"))
semiv=vario(data=d)
plot(semiv, xlab="Lag distance (km)")
mod.sph=vario.fit(semiv$vario, semiv$mean.bin.dist)
# Weighted least squares fit based on the number of points
mod.exp=vario.fit(semiv$vario, semiv$mean.bin.dist, 
                  weights=semiv$npoints/sum(semiv$npoints), 
                  type="expo")
mod.gau=vario.fit(semiv$vario, semiv$mean.bin.dist, type="gauss")
mod.lin=vario.fit(semiv$vario, semiv$mean.bin.dist, type="lin")
lines(semiv$mean.bin.dist, mod.sph$fit, col="red")
lines(semiv$mean.bin.dist, mod.exp$fit, col="black")
lines(semiv$mean.bin.dist, mod.gau$fit, col="blue")
lines(semiv$mean.bin.dist, mod.lin$fit, col="green")
legend(x="topleft", legend=paste(c("Spherical AIC:", "Exponential AIC:", 
                                   "Gaussian AIC:", "Linear AIC:"), 
                                   c(format(mod.sph$AIC, dig=2), 
                                   format(mod.exp$AIC, dig=2), 
                                   format(mod.gau$AIC, dig=2), 
       format(mod.lin$AIC, dig=2))), lty=1, col=c("red", "black", "blue", "green"), 
       bty="n")

# Correlogram
cover=subset(pisco.data, subset=year==2000, 
             select=c("latitude", "longitude", "mussel_abund"))
moran=vario(data=cover, type="moran")
mod.hol=vario.fit(moran$vario, moran$mean.bin.dist, 
                  type="hole", start.vals=list(c0=0.6, a=25, c1=0.01))
mod.per=vario.fit(moran$vario, moran$mean.bin.dist, type="period",
                  start.vals=list(a=1, b=3, c=0))
mod.lin=vario.fit(moran$vario, moran$mean.bin.dist, type="linear")
plot(moran, xlab="Lag distance (km)", ylim=c(-0.6, 0.8))
lines(moran$mean.bin.dist, mod.per$fit, col="red")
lines(moran$mean.bin.dist, mod.hol$fit, col="black")
lines(moran$mean.bin.dist, mod.lin$fit, col="blue")
legend(x="topleft", legend=paste(c("Periodic AIC:", "Hole AIC:", 
                                   "Linear AIC:"), 
                                   c(format(mod.per$AIC, dig=2), 
                                   format(mod.hol$AIC, dig=2), 
                                   format(mod.lin$AIC, dig=2))), 
                                   lty=1, col=c("red", "black", "blue"), bty="n")

vario.func

Description

Compute the empirical variogram values for each bin

Usage

vario.func (x, y, glob.mean, glob.sd, glob.N, is.multivar = FALSE,
                        type = c("semivar", "cov", "pearson", 
                                 "spearman", "kendall", "moran", "geary"))

Arguments

Value

Return the value.

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com)

See Also

vario

Examples

  # Internal function used by vario