Determine coincident coordinates

Description

coincident takes the coordinate matrices coords1 and coords2 and returns the indices of the coincident coordinates of the two matrices.

Usage

coincident(coords1, coords2)

Arguments

Details

This function calls a compiled C++ program created using the Rcpp package. This (may) result in a significant speedup over pure R code since the search algorithm involves loops. We assume that there are no duplicate coordinates in coords1 and coords2, respectively. In other words, each row of coords1 is unique and each row of coords2 is unique. There is at most 1 row of coords1 that will match with a row in coords2.

Value

Returns a matrix with the indices of the coincident locations. Specifically, an r \times 2 matrix will be returned, with r being the number of coordinates in coords1 coinciding with coordinates in coords2. If row i of the matrix is c(2, 5), then the ith set of coincident locations is between the 2nd row of coords1 and the 5th row of coords2. If there are no coincident locations, then a matrix of size 0 \times 2 is returned.

Author(s)

Joshua French

Examples

#Generate two sets of coordinates
loc1 <- as.matrix(expand.grid(seq(0, 1, len = 25), seq(0, 1, len = 25)))
loc2 <- as.matrix(expand.grid(seq(0, 1, len = 101), seq(0, 1, len = 101)))
coincident(loc1, loc2)

Calculates spatial covariance

Description

Calculates spatial covariance matrix of the observed responses, and possibly, the responses to be predicted. If pcoords is not provided, then only V, the covariance matrix of the observed responses will be returned. If pcoords is provided, then Vp and Vop (the covariance matrix for predicted responses and between observed and predicted responses, respectively) will also be returned.

Usage

	cov.sp(coords, sp.type = "exponential", 
		sp.par = stop("specify sp.par argument"), 
		error.var = 0, smoothness = 0.5, finescale.var = 0, 
		pcoords = NULL, D = NULL, Dp = NULL, Dop = NULL)

Arguments

Details

The spatial covariance functions are parameterized in a manner consistent with the cov.spatial function in the geoR package. The matern2 covariance function is an alternative covariance function suggested by Handcock and Wallis (1994). The benefit of this parameterization is that the range parameter is that it allows the effective range to be less dependent on the smoothness parameter.

The D, Dp, and Dop arguments are supplied to decrease the number of necessary computations needed when performing repetitive analysis or simulations. It is probably in the user's interest to not supply these arguments unless the duration of analysis is an important consideration. Note that these arguments override the information given in coords and pcoords, i.e., if dist1(coords) != D, then D is used in subsequent calculations, etc. This could create problems.

Value

Returns a list with the following elements:

Author(s)

Joshua French

References

M.S. Handcock, J.R. Wallis. An approach to statistical spatial-temporal modeling of meteorological fields (with discussion). Journal of the American Statistical Association, 89 (1994), pp. 368–390.

See Also

simple.cov.sp

Examples

    coords <- matrix(rnorm(30), ncol = 3)
    cov.sp(coords = coords, sp.type = "exponential", sp.par = c(2, 1),
        error.var = 1)

Calculates spatio-temporal covariance

Description

Calculates spatial covariance matrix of the observed responses, and possibly, the responses to be predicted. If pcoords is not provided, then only V, the covariance matrix of the observed responses will be returned. If pcoords is provided, then Vp and Vop (the covariance matrix for predicted responses and between observed and predicted responses, respectively) will also be returned.

Usage

	cov.st(coords, time, sp.type = "exponential", 
	    sp.par = stop("specify sp.par argument"),
 	    error.var = 0, smoothness = 0.5, finescale.var = 0, 
	    t.type = "ar1", t.par = .5, 
	    pcoords = NULL, ptime = NULL, 
	    D = NULL, Dp = NULL, Dop = NULL, 
	    T = NULL, Tp = NULL, Top = NULL)

Arguments

Details

At this point, this function only implements a separable spatio-temporal covariance funcation. If h is the distance between two sites, and t is the temporal lag between the times when the associated responses were observed, then the covariance function C(h,t) = Cs(h) \times Ct(t) where Cs is a spatial covariance function corresponding to the exponential, matern, gaussian, or spherical and Ct is the temporal covariance function corresponding to an ar1 process with Ct(t) = \phi ^ t.

The D, Dp, Dop, T, Tp, Top arguments are supplied to decrease the number of necessary computations needed when performing repetitive analysis or simulations. It is probably in the user's interest to not supply these arguments unless the duration of analysis is an important consideration. Note that these arguments override the information given in coords, pcoords, time, and prime, i.e., if dist1(coords) != D, then D is used in subsequent calculations, etc. This could create problems.

Value

Returns a list with the following elements:

Author(s)

Joshua French

See Also

simple.cov.sp

Examples

	coords <- matrix(rnorm(30), ncol = 3)
	pcoords <- matrix(rnorm(90), ncol = 3)
	time <- 1:10
	ptime <- 1:30
	cov.st(coords = coords, time = time, sp.type = "exponential", 
	    sp.par = c(2, 1), error.var = 1, t.type = "ar1", t.par = .5, 
	    pcoords = pcoords, ptime = ptime)

Calculates decomposition of covariance matrix

Description

Calculates a decomposition of the provided covariance matrix, V, using the chosen method.

Usage

	decomp.cov(V, method = "eigen")

Arguments

Details

The matrix V is assumed to be symmetric and positive definite. Symmetry is checked, but the positive definiteness of the matrix is not. Returns a decomposition matrix U such that V = U %*% t(U).

Value

Returns a decomposition matrix U such that V = U %*% t(U).

Author(s)

Joshua French

See Also

cov.sp

Examples

	data(toydata)
	U <- decomp.cov(toydata$V, method = "chol")
	#range(toydata$V - U %*% t(U))

Calculate Euclidean distance matrix for a matrix of coordinates

Description

dist1 takes a matrix of coordinates and returns the Euclidean distance matrix of the coordinates. It does this using a compiled C program, so it is faster than the builtin R dist function.

Usage

dist1(coords)

Arguments

Value

An nr \times nr matrix of Euclidean distances.

Author(s)

Joshua French

See Also

dist, dist2

Examples

x <- matrix(rnorm(30), ncol = 3)
dist1(x)

Calculate Euclidean distance matrix between coordinates of two matrices

Description

dist2 takes the matrices of coordinates coords1 and coords2 and returns the inter-Euclidean distances between coordinates.

Usage

dist2(coords1, coords2)

Arguments

Value

An nr1 \times nr2 matrix of Euclidean distances.

Author(s)

Joshua French

See Also

dist, dist1

Examples

x1 <- matrix(rnorm(30), ncol = 3)
x2 <- matrix(rnorm(60), ncol = 3)
dist2(x1, x2)

Extracts coordinates from contourLines function

Description

Takes contours of contourLines function and extracts the associated coordinates.

Usage

	get.contours(x)

Arguments

Value

Returns a 2-column matrix containing the coordinates making up the contours in contours.list.

Author(s)

Joshua French

See Also

contourLines, contour

Examples

data(volcano)
x <- 10*1:nrow(volcano)
y <- 10*1:ncol(volcano)
cL <- contourLines(x, y, volcano)
out <- get.contours(cL)

Performs Ordinary Kriging

Description

Performs Ordinary Kriging using y, the n \times 1 matrix of observed responses, V, the (positive definite) covariance matrix of the observed responses, Vp, the np \times np covariance matrix of the responses to be predicted, and Vop, the n \times np matrix of covariances between the observed responses and the responses to be predicted.

Usage

	krige.ok(y, V, Vp, Vop, nsim = 0, Ve.diag = NULL, method = "eigen")

Arguments

Details

It is assumed that there are n observed data values and that we wish to make predictions at np locations.

If doing conditional simulation, the Cholesky decomposition should not work when there are coincident locations between the observed data locations and the predicted data locations. Both the Eigen and Singular Value Decompositions should work.

If user specifies nsim to be a positive integer, then nsim conditional realizations of the predictive distribution will be generated. If this is less than 1, then no conditional simulation is done. If nsim is a positive integer, then Ve.diag must also be supplied. Ve.diag is should be a vector of length n specifying the measurement error variances of the observed data. This information is only used for conditional simulation, so this argument is only needed when nsim > 0. When conditional simulation is desired, then the argument method can be to specify the method used to decompose V. Options are "eigen", "chol", or "svd" (Eigen decomposition, Cholesky decomposition, or Singular value decomposition, respectively). This information is only used for conditional simulation, so this argument is only applicable when nsim > 0.

Value

The function returns a list containing the following objects:

If nsim > 0, this list has class "krigeConditionalSample".

Author(s)

Joshua French

References

Statistical Methods for Spatial Data Analysis, Schabenberger and Gotway (2003). See p. 226-228.

Examples

# create observed and predicted coordinates
ocoords <- matrix(runif(100), ncol = 2)
pcoords <- matrix(runif(200), ncol = 2)

# include some observed locations in the predicted coordinates
acoords <- rbind(ocoords, pcoords)

# create covariance matrix
C3 <- cov.sp(coords = ocoords, sp.type = "matern", sp.par = c(2, 1), smoothness = 1, 
	finescale = 0, error = 0.5, pcoords = acoords)

# generate data with error
y <- rmvnorm(nsim = 1, mu = rep(2, 50), V = C3$V) + rnorm(50, sd = sqrt(.5))

# use universal kriging to make predictions.  Do not do conditional simulation
krige.obj <- krige.ok(as.vector(y), V = C3$V, Vp = C3$Vp, Vop = C3$Vop, 
	nsim = 0)

#Do conditional simulation
krige.obj2 <- krige.ok(as.vector(y), V = C3$V, Vp = C3$Vp, Vop = C3$Vop, 
	nsim = 100, Ve.diag = rep(.5, 50), method = "eigen")

Performs simple kriging

Description

Performs simple kriging using y, a vector of length n, V, the (positive definite) covariance matrix of the observed responses, Vp, the np \times np covariance matrix of the responses to be predicted, Vop, the n \times np matrix of covariances between the observed responses and the responses to be predicted, and m, a numeric vector of length 1 identifying the value of the mean for each response.

Usage

	krige.sk(y, V, Vp, Vop, m = 0, nsim = 0, Ve.diag = NULL, method = "eigen")

Arguments

Details

It is assumed that there are n observed data values and that we wish to make predictions at np locations. The mean is subtracted from each value of y before determining the kriging weights, and then the mean is added onto the predicted response.

If doing conditional simulation, the Cholesky decomposition should not work when there are coincident locations between the observed data locations and the predicted data locations. Both the Eigen and Singular Value Decompositions should work.

If user specifies nsim to be a positive integer, then nsim conditional realizations of the predictive distribution will be generated. If this is less than 1, then no conditional simulation is done. If nsim is a positive integer, then Ve.diag must also be supplied. Ve.diag is should be a vector of length n specifying the measurement error variances of the observed data. This information is only used for conditional simulation, so this argument is only needed when nsim > 0. When conditional simulation is desired, then the argument method can be to specify the method used to decompose V. Options are "eigen", "chol", or "svd" (Eigen decomposition, Cholesky decomposition, or Singular value decomposition, respectively). This information is only used for conditional simulation, so this argument is only applicable when nsim > 0.

Value

The function returns a list containing the following objects:

. If nsim > 0, this list has class "krigeConditionalSample".

Author(s)

Joshua French

References

Statistical Methods for Spatial Data Analysis, Schabenberger and Gotway (2003). See p. 226-228.

Examples

	data(toydata)
	y <- as.vector(toydata$y)
	V <- toydata$V
	Vp <- toydata$Vp
	Vop <- toydata$Vop
	krige.sk(y, V, Vp, Vop, m = 2)

Performs universal kriging

Description

Performs universal kriging using X, the n \times k design matrix for the regression coefficients of the observed data, y, the n \times 1 matrix of observed responses, V, the (positive definite) covariance matrix of the observed responses, Xp, the np \times k design matrix of the responses to be predicted, Vp, the np \times np covariance matrix of the responses to be predicted, and Vop, the n \times np matrix of covariances between the observed responses and the responses to be predicted. If user specifies nsim to be a positive integer, then nsim conditional realizations of the predictive distribution will be generated.

Usage

	krige.uk(y, V, Vp, Vop, X, Xp, nsim = 0, Ve.diag = NULL, method = "eigen")

Arguments

.

Details

It is assumed that there are n observed data values and that we wish to make predictions at np locations. We assume that there are k regression coefficients (including the intercept). Both X and Xp should contain a column of 1's if an intercept is desired.

If doing conditional simulation, the Cholesky decomposition should not work when there are coincident locations between the observed data locations and the predicted data locations. Both the Eigen and Singular Value Decompositions should work.

If user specifies nsim to be a positive integer, then nsim conditional realizations of the predictive distribution will be generated. If this is less than 1, then no conditional simulation is done. If nsim is a positive integer, then Ve.diag must also be supplied. Ve.diag is should be a vector of length n specifying the measurement error variances of the observed data. This information is only used for conditional simulation, so this argument is only needed when nsim > 0. When conditional simulation is desired, then the argument method can be to specify the method used to decompose V. Options are "eigen", "chol", or "svd" (Eigen decomposition, Cholesky decomposition, or Singular value decomposition, respectively). This information is only used for conditional simulation, so this argument is only applicable when nsim > 0.

Value

The function returns a list containing the following objects:

If nsim > 0, this list has class "krigeConditionalSample".

Author(s)

Joshua French

References

Statistical Methods for Spatial Data Analysis, Schabenberger and Gotway (2003). See p. 241-243.

Examples

# create observed and predicted coordinates
ocoords <- matrix(runif(100), ncol = 2)
pcoords <- matrix(runif(200), ncol = 2)

# include some observed locations in the predicted coordinates
acoords <- rbind(ocoords, pcoords)

# create design matrices
X <- as.matrix(cbind(1, ocoords))
Xa <- as.matrix(cbind(1, acoords))

# create covariance matrix
C3 <- cov.sp(coords = ocoords, sp.type = "matern", sp.par = c(2, 1), smoothness = 1, 
	finescale = 0, error = 0.5, pcoords = acoords)

# set values of regression coefficients
coeff <- matrix(c(1, 2, 3), nrow = 1)

# generate data with error
y <- rmvnorm(nsim = 1, mu = tcrossprod(X, coeff), V = C3$V) + rnorm(50, sd = sqrt(.5))

# use universal kriging to make predictions.  Do not do 
# conditional simulation
krige.obj <- krige.uk(as.vector(y), V = C3$V, Vp = C3$Vp, Vop = C3$Vop, 
	X = X, Xp = Xa, nsim = 0)

#Do kriging with conditional simulation
krige.obj2 <- krige.uk(as.vector(y), V = C3$V, Vp = C3$Vp, Vop = C3$Vop, 
	X = X, Xp = Xa, nsim = 100, 
	Ve.diag = rep(.5, 50), method = "eigen")

Determines maximum likelihood estimates of covariance parameters

Description

Estimates covariance parameters of spatial covariance functions using maximum likelihood or restricted maximum likelihood. See cov.sp for more details of covariance functions to be estimated.

Usage

	maxlik.cov.sp(X, y, coords, sp.type = "exponential", 
	    range.par = stop("specify range.par argument"), 
	    error.ratio = stop("specify error.ratio argument"), 
	    smoothness = 0.5,  
	    D = NULL, reml = TRUE, lower = NULL, upper = NULL, 
	    control = list(trace = TRUE), optimizer="nlminb")

Arguments

Details

When doing the numerical optimizaiton, the covariance function is reparameterized slightly to speedup computation. Specifically, the variance parameter for the process of interest,sp.par[1], is profiled out, and the error.var parameter is parameterized as sp.par[1] * error.ratio, where error.ratio = error.var/sp.par[1].

Value

Returns a list with the following elements:

Author(s)

Joshua French

See Also

cov.st

Examples

#generate 20 random (x, y) coordinates
coords <- matrix(rnorm(20), ncol = 2)

#create design matrix
X <- cbind(1, coords)

#create mean for observed data to be generated
mu <- X %*% c(1, 2, 3)

#generate covariance matrix
V <- exp(-dist1(coords))

#generate observe data
y <- rmvnorm(mu = mu, V = V)

#find maximum likelihood estimates of covariance parameters
maxlik.cov.sp(X = X, y = y, coords = coords,
    sp.type = "exponential", range.par = 1, error.ratio = 0, 
    reml = TRUE)

Determines maximum likelihood estimates of covariance parameters

Description

Estimates covariance parameters of spatio-temporal covariance functions using maximum likelihood or restricted maximum likelihood. See cov.st for more details of covariance functions to be estimated. The covariance function is reparameterized slightly to speedup computation. Specifically, the variance parameter for the hidden process, sp.par[1], is profiled out and the error.var parameter is parameterized as sp.par[1] * error.ratio.

Usage

	maxlik.cov.st(X, y, coords, time, sp.type = "exponential", 
	    range.par = stop("specify range.par argument"), 
	    error.ratio = stop("specify error.ratio argument"), 
	    smoothness = 0.5, t.type = "ar1", t.par = .5, D = NULL, T = NULL, 
	    reml = TRUE, lower = NULL, upper = NULL, control = list(trace = TRUE), 
	    optimizer="nlminb")

Arguments

Details

When doing the numerical optimization, the covariance function is reparameterized slightly to speedup computation. Specifically, the variance parameter for the process of interest,sp.par[1], is profiled out, and the error.var parameter is parameterized as sp.par[1] * error.ratio, where error.ratio = error.var/sp.par[1].

Value

Returns a list with the following elements:

Author(s)

Joshua French

See Also

cov.st

Examples

#Generate locations and observed times
coords <- matrix(rnorm(40), ncol = 2)
time <- rep(1:2, each = 10)

#Calculate distance matrix for time vector
T <- dist1(matrix(time))

#create design matrix
X <- cbind(1, coords)

#create mean for observed data to be generated
mu <- X %*% c(1, 2, 3)

#generate covariance matrix for spatio-temporal data
V <- exp(-dist1(coords)) * .25^T

#generate observe data
y <- rmvnorm(mu = mu, V = V)

maxlik.cov.st(X = X, y = y, coords = coords, time = time,
    sp.type = "exponential", range.par = 1, error.ratio = 0, 
    t.type = "ar1", t.par = .5, reml = TRUE)

Plot contour lines

Description

Plot contour lines from list produced by contourLines function.

Usage

## S3 method for class 'contourLines'
plot(x, begin=1, end = length(x), add = FALSE, ...)

Arguments

Value

This function does not return anything; it only creates a new plot or modifies an existing plot.

Author(s)

Joshua French

Examples

data(volcano)
x <- 10*1:nrow(volcano)
y <- 10*1:ncol(volcano)
cL <- contourLines(x, y, volcano)
plot.contourLines(cL)

Generate from conditional normal distribution

Description

Generates realizations from a multivariate normal distribution conditional on observed data vector

Usage

	rcondnorm(nsim = 1, y, mu, mup, V, Vp, Vop, method = "eigen")

Arguments

Value

An np \times nsim matrix containing the nsim realizations of the conditional normal distribution. Each column of the matrix represents a realization of the multivariate normal distribution.

Author(s)

Joshua French

See Also

rmvnorm

Examples

n <- 100
np <- 100

mu <- rep(1, 100)
mup <- rep(2, 100)

coords <- matrix(runif(2 * n), ncol = 2)
pcoords <- matrix(runif(2 * np), ncol = 2)

myV <- cov.sp(coords, sp.type = "exponential", c(1, 2), error.var = 1, pcoords = pcoords)

y <- rmvnorm(1, mu = mu, V = myV$V)

rcondnorm(3, y = y, mu = mu, mup = mup, V = myV$V, Vp = myV$Vp, Vop = myV$Vop, method = "chol")

Generates realizations from a multivariate normal distribution

Description

Generates realizations from a multivariate normal distribution.

Usage

	rmvnorm(nsim = 1, mu, V, method = "eigen")

Arguments

Value

An n \times nsim matrix containing the nsim realizations of the multivariate normal distribution. Each column of the matrix represents a realization of the multivariate normal distribution.

Author(s)

Joshua French

See Also

rmvnorm

Examples

n <- 20
mu <- 1:n
V <- exp(-dist1(matrix(rnorm(n))))
rmvnorm(nsim = 100, mu = mu, V = V, method = "eigen")

Calculates spatial covariance based on distance matrix

Description

Calculates a spatial covariance using a (Euclidean) distance matrix D. Not intended to be used directly by user (though it may be helpful to some). It is strongly recommended that you use the cov.sp function. No argument or error checking is provided for this function.

Usage

	simple.cov.sp(D, sp.type, sp.par, error.var, smoothness, finescale.var)

Arguments

Value

Returns a covariance matrix.

Author(s)

Joshua French

See Also

~ cov.sp

Examples

	coords <- matrix(rnorm(30), ncol = 3)
	D <- dist1(coords)
	simple.cov.sp(D = D, sp.type = "exponential", sp.par = c(2, 1),
		error.var = 1, smoothness = 0.5, finescale.var = 0)

Calculates temporal covariance based on distance matrix

Description

Calculates a temporal covariance using a (Euclidean) distance matrix T. Not intended to be used directly by user (though it may be helpful to some). It is used in the covets function. No argument or error checking is provided for this function.

Usage

	simple.cov.time(T, t.type, t.par)

Arguments

Value

Returns a covariance matrix.

Author(s)

Joshua French

See Also

~ cov.st

Examples

	T <- dist1(matrix(1:10))
	simple.cov.time(T = T, t.type = "ar1", t.par = .5)

Returns posterior predictive sample from spLM object

Description

The function spLMPredictJoint collects posterior predictive samples for a set of new locations given a spLM object from the spBayes package.

Usage

  spLMPredictJoint(sp.obj, pred.coords, pred.covars, start = 1, 
    end = nrow(sp.obj$p.theta.samples), thin = 1, verbose = TRUE, n.report = 100, 
    noisy = FALSE, method = "eigen")

Arguments

Details

This function samples from the joint posterior predictive distribution of a Bayesian spatial linear model. Specifically, it is intended to be similar to the spPredict function in the spBayes except that it samples from the joint distribution instead of the marginal distribution. However, it will only work for spLM objects and should have the same limitations as the spLM and spPredict functions. Note that the spRecover function is called internally to recover the posterior samples form the posterior distribution of the spatial model.

Value

The function returns a np \times B matrix of posterior predictive samples, where B is the number of posterior samples. The class is jointPredicitveSample.

Author(s)

Joshua French

See Also

spLM, spPredict, spRecover

Examples

# Set parameters
n <- 100
np <- 12
n.samples <- 10
burnin.start <- .5 * n.samples + 1
sigmasq <- 1
tausq <- 0.0
phi <- 1
cov.model <- "exponential"
n.report <- 5

# Generate coordinates
coords <- matrix(runif(2 * n), ncol = 2); 
pcoords <- as.matrix(expand.grid(seq(0, 1, len = 12), seq(0, 1, len = 12)))
  
# Construct design matrices
X <- as.matrix(cbind(1, coords))
Xp <- cbind(1, pcoords)

# Specify priors
starting <- list("phi" = phi, "sigma.sq"= sigmasq, "tau.sq" = tausq)
tuning <- list("phi"=0.1, "sigma.sq"=0.1, "tau.sq"=0.1)
priors.1 <- list("beta.Norm"=list(c(1, 2, 1), diag(100, 3)),
                     "phi.Unif"=c(0.00001, 10), "sigma.sq.IG"=c(1, 1))

# Generate data
B <- rnorm(3, c(1, 2, 1), sd = 10)
phi <- runif(1, 0, 10)
sigmasq <- 1/rgamma(1, 1, 1)
V <- simple.cov.sp(D = dist1(coords), cov.model, c(sigmasq, 1/phi), error.var = tausq, 
	smoothness = nu, finescale.var = 0)
y <- X %*% B + rmvnorm(1, rep(0, n), V) + rnorm(n, 0, sqrt(tausq))

# Create spLM object
library(spBayes)
m1 <- spBayes::spLM(y ~ X - 1, coords = coords, starting = starting,
	tuning = tuning, priors = priors.1, cov.model = cov.model,
	n.samples = n.samples, verbose = FALSE, n.report = n.report)

# Sample from joint posterior predictive distribution
y1 <- spLMPredictJoint(m1, pred.coords = pcoords, pred.covars = Xp, 
	start = burnin.start, verbose = FALSE, method = "chol")

A toy data set for use in examples.

Description

A list containing X, a 50 x 3 design matrix, y, a vector of length 50 of observed responses, V, a 50 x 50 covariance matrix for the observed data, Xp, a 121 x 3 design matrix for the predicted responses, Vp, the 121 x 121 covariance matrix of the predicted responses, Vop, the 50 x 121 covariance matrix between the observed responses and the predicted responses, coords, a 50 x 2 matrix containing the sites of the 50 observed responses, and pcoords, a 121 x 2 matrix containing the 121 sites for the predicted responses (a 11 x 11 regular grid over the domain [0, 1]x[0, 1]).

Usage

	data(toydata)

Author(s)

Joshua French

Examples

	data(toydata)