Help for package gjam

Type:

Package

Title:

Generalized Joint Attribute Modeling

Version:

2.6.2

Date:

2022-5-23

Author:

James S. Clark, Daniel Taylor-Rodriquez

Maintainer:

James S. Clark <jimclark@duke.edu>

Description:

Analyzes joint attribute data (e.g., species abundance) that are combinations of continuous and discrete data with Gibbs sampling. Full model and computation details are described in Clark et al. (2018) <doi:10.1002/ecm.1241>.

License:

GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]

Imports:

Rcpp (≥ 0.11.5), MASS, RANN

Depends:

R (≥ 2.10)

LinkingTo:

Rcpp, RcppArmadillo

Suggests:

knitr

VignetteBuilder:

knitr

NeedsCompilation:

yes

Packaged:

2022-05-23 09:43:25 UTC; jimclark

Repository:

CRAN

Date/Publication:

2022-05-23 18:20:13 UTC

Generalized Joint Attribute Modeling

Description

Inference and prediction for jointly distributed responses that are combinations of continous and discrete data. Functions begin with 'gjam' to avoid conflicts with other packages.

Details

Package:	gjam
Type:	Package
Version:	2.6.2
Date:	2022-5-23
License:	GPL (>= 2)
URL:	http://sites.nicholas.duke.edu/clarklab/code/

The generalized joint attribute model (gjam) analyzes multivariate data that are combinations of presence-absence, ordinal, continuous, discrete, composition, zero-inflated, and censored. It does so as a joint distribution over response variables. gjam provides inference on sensitivity to input variables, correlations between responses on the data scale, model selection, and prediction.

Importantly, analysis is done on the observation scale. That is, coefficients and covariances are interpreted on the same scale as the data. Contrast this approach with standard Generalized Linear Models, where coefficients and covariances are difficult to interpret and cannot be compared across responses that are modeled on different scales.

gjam was motivated by species distribution and abundance data in ecology, but can provide an attractive alternative to traditional methods wherever observations are multivariate and combine multiple scales and mixtures of continuous and discrete data.

gjam can be used to model ecological trait data, where species traits are translated to locations as community-weighted means and modes.

Posterior simulation is done by Gibbs sampling. Analysis is done by these functions, roughly in order of how frequently they might be used:

gjam fits model with Gibbs sampling.

gjamSimData simulates data for analysis by gjam.

gjamPriorTemplate sets up prior distribution for coefficients.

gjamSensitivity evaluates sensitivity to predictors from gjam.

gjamCensorY defines censored values and intervals.

gjamTrimY trims the response matrix and aggregates rare types.

gjamPlot plots output from gjam.

gjamSpec2Trait constructs plot by trait matrix.

gjamPredict does conditional prediction.

gjamConditionalParameters obtains the conditional coefficient matrices.

gjamOrdination ordinates the response matrix.

gjamDeZero de-zeros response matrix for storage.

gjamReZero recovers response matrix from de-zeroed format.

gjamIIE evaluates indirect effects and interactions.

gjamIIEplot plots indirect effects and interactions.

gjamSpec2Trait generates trait values.

gjamPoints2Grid aggregates incidence data to counts on a lattice.

Author(s)

Author: James S Clark, jimclark@duke.edu, Daniel Taylor-Rodriquez

References

Clark, J.S., D. Nemergut, B. Seyednasrollah, P. Turner, and S. Zhang. 2017. Generalized joint attribute modeling for biodiversity analysis: Median-zero, multivariate, multifarious data. Ecological Monographs 87, 34-56.

Clark, J.S. 2016. Why species tell more about traits than traits tell us about species: Predictive models. Ecology 97, 1979-1993.

Taylor-Rodriguez, D., K. Kaufeld, E. M. Schliep, J. S. Clark, and A. E. Gelfand. 2016. Joint species distribution modeling: dimension eduction using Dirichlet processes. Bayesian Analysis, 12, 939-967. doi: 10.1214/16-BA1031.

Gibbs sampler for gjam data

Description

Analyzes joint attribute data (e.g., species abundance) with Gibbs sampling. Input can be output from gjamSimData. Returns a list of objects from Gibbs sampling that can be plotted by gjamPlot.

Usage

  gjam(formula, xdata, ydata, modelList)
  
  ## S3 method for class 'gjam'
print(x, ...)
  
  ## S3 method for class 'gjam'
summary(object, ...)

Arguments

formula

R formula for model, e.g., ~ x1 + x2.

xdata

data.frame containing predictors in formula. If not found in xdata variables, they must be available from the user's workspace.

ydata

n by S response matrix or data.frame. Column names are unique labels, e.g., species names. All columns will be included in the analysis.

modelList

list specifying inputs, including ng (number of Gibbs steps), burnin, and typeNames. Can include the number of holdouts for out-of-sample prediction, holdoutN. See Details.

x

object of class gjam.

object

currently, also an object of class gjam.

...

further arguments not used here.

Details

Note that formula begins with ~, not y ~. The response matrix is passed in the form of a n by S matrix or data.frame ydata.

Both predictors in xdata and responses in ydata can include missing values as NA. Factors in xdata should be declared using factor. For computational stability variables that are not factors are standardized by mean and variance, then transformed back to original scales in output. To retain a variable in its original scale during computation include it in the character string notStandard as part of the list modelList. (example shown in the vignette on traits).

modelList has these defaults and provides these options:

ng = 2000, number of Gibbs steps.

burnin = 500, no. initial steps, must be less than ng.

typeNames can be 'PA' (presenceAbsence), 'CON' (continuous on (-Inf, Inf)), 'CA' (continuous abundance, zero censoring), 'DA' (discrete abundance), 'FC' (fractional composition), 'CC' (count composition), 'OC' (ordinal counts), 'CAT' (categorical classes). typeNames can be a single value that applies to all columns in ydata, or there can be one value for each column.

holdoutN = 0, number of observations to hold out for out-of-sample prediction.

holdoutIndex = numeric(0), numeric vector of observations (row numbers) to holdout for out-of-sample prediction.

censor = NULL, list specifying columns, values, and intervals for censoring, see gjamCensorY.

effort = NULL, list containing 'columns', a vector of length <= S giving the names of columns in in y, and 'values', a length-n vector of effort or a n by S matrix (see Examples). effort can be plot area, search time, etc. for discrete count data 'DA'.

FULL = F in modelList will save full prediction chains in $chains$ygibbs.

notStandard = NULL, character vector of column names in xdata that should not be standardized.

reductList = list(N = 20, r = 3), list of dimension reduction parameters, invoked when reductList is included in modelList or automatically when ydata has too many columns. See vignette on Dimension Reduction.

random, character string giving the name of a column in xdata that will be used to specify random effects. The random group column should be declared as a factor. There should be replication, i.e., each group level occurs multiple times.

REDUCT = F in modelList overrides automatic dimension reduction.

FCgroups, CCgroups, are length-S vectors assigned to columns in ydata indicating composition 'FC' or 'CC' group membership. For example, if there are two 'CA' columns in ydata followed by two groups of fractional composition data, each having three columns, then typeNames = c('CA','CA','FC','FC','FC','FC','FC','FC') and FCgroups = c(0,0,1,1,1,2,2,2). note: gjamSimData is not currently set up to simulate multiple composition groups, but gjam will model it.

PREDICTX = T executes inverse prediction of x. Speed-up by setting PREDICTX = F.

ematAlpha = .5 is the probability assigned for conditional and marginal independence in the ematrix.

traitList = list(plotByTrait, traitTypes, specByTrait), list of trait objects. See vignette on Trait analysis.

More detailed vignettes can be obtained with:

browseVignettes('gjam')

Value

Returns an object of class "gjam", which is a list containing the following components:

call

function call

chains

list of MCMC matrices, each with ng rows; includes coefficients bgibbs(Q*S columns), bgibbsUn (unstandardized for x), sensitivity fgibbs (Q1 columns), and fbgibbs (Q1 columns, where Q1 = Q - 1, unless there are multilevel factors); covariance sgibbs has S*(S + 1)/2 columns (REDUCT == F) or N*r columns (REDUCT == T).

fit

list of diagnostics (DIC, rmspeAll, rmspeBySpec, xscore, yscore).

inputs

list of input summaries, including breakMat (partition matrix), classBySpec (interval assignment), designTable (summary of design matrix), [factorBeta, interBeta, intMat, linFactor] (factor and interaction information), other, notOther (response columns to exclude and not), [standMat, standRows, standX] means and variances to standardize x, [x, xdata, y] cleaned versions of data.

missing

list of missing objects, including locations for predictors xmiss and responses ymiss in xdata and ydata, respectively, predictor means xmissMu and standard errors xmissSe, response means ymissMu and standard errors ymissSe .

modelList

list of model specifications from input modelList.

parameters

list of parameter estimates, including coefficient matrices on standardized (betaMu, betaSe), unstandardized (betaMuUn, betaSeUn), and dimensionless (fBetaMu, fBetaSd) scales; correlation (corMu, corSe) and covariance (sigMu, sigSe) matrices; sensitivities to predictors (fmatrix, fMu, fSe); environmental response matrix (ematrix), with locations of zero elements, conditionally (whConZero) and marginally (whichZero), set at probability level modelList$ematAlpha); and latent variables (wMu, wSd).

prediction

list of predicted values, including species richness (responses predicted > 0); inverse predicted x (xpredMu, xpredSd) and predicted y (ypredMu, ypredSd) matrices.

If traits are modeled, then parameters will additionally include betaTraitMu, betaTraitSe (coefficients), sigmaTraitMu, sigmaTraitSe (covariance). prediction will additionally include tMuOrd (ordinal trait means), tMu, tSe (trait predictions).

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
## combinations of scales
types <- c('DA','DA','OC','OC','OC','OC','CON','CON','CON','CON','CON','CA','CA','PA','PA')         
f    <- gjamSimData(S = length(types), typeNames = types)
ml   <- list(ng = 500, burnin = 50, typeNames = f$typeNames)
out  <- gjam(f$formula, f$xdata, f$ydata, modelList = ml)
summary(out)

# repeat with ng = 5000, burnin = 500, then plot data:
pl  <- list(trueValues = f$trueValues)
gjamPlot(out, plotPars = pl)

## discrete abundance with heterogeneous effort 
S   <- 5                             
n   <- 1000
eff <- list( columns = 1:S, values = round(runif(n,.5,5),1) )
f   <- gjamSimData(n, S, typeNames='DA', effort=eff)
ml  <- list(ng = 2000, burnin = 500, typeNames = f$typeNames, effort = eff)
out <- gjam(f$formula, f$xdata, f$ydata, modelList = ml)
summary(out)

# repeat with ng = 2000, burnin = 500, then plot data:
pl  <- list(trueValues = f$trueValues)
gjamPlot(out, plotPars = pl)

## End(Not run)

Censor gjam response data

Description

Returns a list with censored values, intervals, and censored response matrix y.

Usage

  gjamCensorY(values, intervals, y, type='CA', whichcol = c(1:ncol(y)))

Arguments

values

Values in y that are censored, specified by intervals

intervals

matrix having two rows and one column for each value in values. The first row holds lower bounds. The second row holds upper bounds. See Examples.

y

Response matrix, n rows by S columns. All values within intervals will be replaced with values

type

Response type, see typeNames in gjam

whichcol

Columns in y that are censored (often not all responses are censored)

Details

Any values in y that fall within censored intervals are replaced with censored values. The example below simulates data collected on an 'octave scale': 0, 1, 2, 4, 8, ..., an approach to accelerate data collection with approximate bins.

Value

Returns a list containing two elements.

y

n by S matrix updated with censored values substituted for those falling within intervals.

censor

list containing $columns that are censored and $partition, a matrix with 3 rows used in gjam and gjamPlot, one column per censor interval. Rows are values, followed by lower and upper bounds.

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
# data in octaves
v  <- up <- c(0, 2^c(0:4), Inf)         
dn <- c(-Inf, v[-length(v)])
i  <- rbind( dn, up )  # intervals

f  <- gjamSimData(n = 2000, S = 15, Q = 3, typeNames='CA')
y  <- f$y
cc <- c(3:6)                                                # censored columns
g  <- gjamCensorY(values = v, intervals = i, y = y, whichcol = cc)
y[,cc] <- g$y                                               # replace columns
ml     <- list(ng = 500, burnin = 100, censor = g$censor, typeNames = f$typeNames)
output <- gjam(f$formula, xdata = f$xdata, ydata = y, modelList = ml)

#repeat with ng = 2000, burnin = 500, then:
pl  <- list(trueValues = f$trueValues, width = 3, height = 3)   
gjamPlot(output, pl)

# upper detection limit
up <- 5
v  <- up
i  <- matrix(c(up,Inf),2)
rownames(i) <- c('down','up')

f   <- gjamSimData(typeNames='CA')   
g   <- gjamCensorY(values = v, intervals = i, y = f$y)
ml  <- list(ng = 500, burnin = 100, censor = g$censor, typeNames = f$typeNames)
out <- gjam(f$formula, xdata = f$xdata, ydata = g$y, modelList = ml)

#repeat with ng = 2000, burnin = 500, then:
pl  <- list(trueValues = f$trueValues, width = 3, height = 3)           
gjamPlot(out, pl)

# lower detection limit
lo        <- .001
values    <- upper <- lo
intervals <- matrix(c(-Inf,lo),2)
rownames(intervals) <- c('lower','upper')

## End(Not run)

Parameters for gjam conditional prediction

Description

Conditional parameters quantify the direct effects of predictors including those that come through other species.

Usage

  gjamConditionalParameters(output, conditionOn, nsim = 2000)

Arguments

output

object of class "gjam".

conditionOn

a character vector of responses to condition on (see Details).

nsim

number of draws from the posterior distribution.

Details

Responses in ydata are random with a joint distribution that comes through the residual covariance having mean matrix parameters$sigMu and standard error matrix parameters$sigSe. Still, it can be desirable to use some responses, along with covariates, as predictors of others. The responses (columns) in ydata are partitioned into two groups, a group to condition on (the names included in character vector conditionOn) and the remaining columns. conditionOn gives the names of response variables (colnames for ydata). The conditional distribution is parameterized as the sum of effects that come directly from predictors in xdata, in a matrix C, and from the other responses, i.e., those in conditionOn, a matrix A. A third matrix P holds the conditional covariance. If dimension reduction is used in model fitting, then there will some redundancy in conditional coefficients.

See examples below.

Value

Amu

posterior mean for matrix A.

Ase

standard error for matrix A.

Atab

parameter summary for matrix A.

Cmu

posterior mean for matrix C.

Cse

standard error for matrix C.

Ctab

parameter summary for matrix C.

Pmu

posterior mean for matrix P.

Pse

standard error for matrix P.

Ptab

parameter summary for matrix P.

Author(s)

James S Clark, jimclark@duke.edu

References

Qiu, T., S. Shubhi, C. W. Woodall, and J.S. Clark. 2021. Niche shifts from trees to fecundity to recruitment that determine species response to climate change. Frontiers in Ecology and Evolution 9, 863. 'https://www.frontiersin.org/article/10.3389/fevo.2021.719141'.

Examples

## Not run: 
f   <- gjamSimData(n = 200, S = 10, Q = 3, typeNames = 'CA') 
ml  <- list(ng = 2000, burnin = 50, typeNames = f$typeNames, holdoutN = 10)
output <- gjam(f$formula, f$xdata, f$ydata, modelList = ml)

# condition on three species
gjamConditionalParameters( output, conditionOn = c('S1','S2','S3') )

## End(Not run)

Compress (de-zero) gjam data

Description

Returns a de-zeroed (sparse matrix) version of matrix ymat with objects needed to re-zero it.

Usage

  gjamDeZero(ymat)

Arguments

ymat

n by S response matrix

Details

Many abundance data sets are mostly zeros. gjamDeZero extacts non-zero elements for storage.

Value

Returns a list containing the de-zeroed ymat as a vector yvec.

yvec

non-zero elements of ymat

n

no. rows of ymat

S

no. cols of ymat

index

index for non-zeros

ynames

column names of ymat

Author(s)

James S Clark, jimclark@duke.edu

References

Clark, J.S., D. Nemergut, B. Seyednasrollah, P. Turner, and S. Zhang. 2016. Generalized joint attribute modeling for biodiversity analysis: Median-zero, multivariate, multifarious data. Ecological Monographs 87, 34-56.

Examples

## Not run: 
library(repmis)
source_data("https://github.com/jimclarkatduke/gjam/blob/master/fungEnd.RData?raw=True")

ymat <- gjamReZero(fungEnd$yDeZero)  # OTUs stored without zeros
length(fungEnd$yDeZero$yvec)         # size of stored version
length(ymat)                         # full size
yDeZero <- gjamDeZero(ymat)
length(yDeZero$yvec)                 # recover de-zeroed vector

## End(Not run)

Fill out data for time series (state-space) gjam

Description

Fills in predictor, response, and effort matrics for time series data where there are multiple multivariate time series. Time series gjam is here https://htmlpreview.github.io/?https://github.com/jimclarkatduke/gjam/blob/master/gjamTimeMsVignette.html

Usage

  gjamFillMissingTimes(xdata, ydata, edata, groupCol, timeCol, groupVars = groupCol,
                                 FILLMEANS = FALSE, typeNames = NULL, missingEffort = .1)

Arguments

xdata

n by Q data.frame holding predictor variables

ydata

n by S matrix holding response variables

edata

n by S matrix holding effort

groupCol

column name in xdata for group variable, i.e., observations part of the same time series

timeCol

column name in xdata for time index

groupVars

character vector of column names in xdata having values that are fixed for a value of groupCol, i.e., they do not change with time index in timeCol

FILLMEANS

fill new rows in ydata with mean for groupCol times missingEffort; otherwise NA

typeNames

typenames current limited to 'DA' for discrete counts

missingEffort

effort assigned to missing values of edata and ydata

Details

Missing times in the data occur where there are gaps in timeCol column of xdata and the initial time 0 for each sequence. New versions of the data have NA (xdata) or prior values with appropriate weight (ydata). Missing times are filled in xdata, ydata, edata, including a time 0 which serves as a prior mean for ydata for time code1. The group and time indices in columns groupCol and timeCol of xdata reference the time for a given time series. Missing values in the columns groupVars of xdata are filled automatically filled in. This assumes that values for these variables are fixed for the group. If FILLMEANS, the missing values in ydata are filled with means for the group and given a low weight specified in missingEffort.

Value

A list containing the following:

xdata

filled version of xdata

ydata

filled version of ydata

edata

filled version of edata

timeList

time indices used for computation, including, timeZero (row numbers in new data where each time series begins, with times = 0), timeLast (row numbers in new data where each time series ends), rowInserts (row numbers for all inserted rows), noEffort (rows for which effort in edata is filled with missingEffort)

Author(s)

James S Clark, jimclark@duke.edu

References

Clark, J. S., C. L. Scher, and M. Swift. 2020. The emergent interactions that govern biodiversity change. Proceedings of the National Academy of Sciences, 117, 17074-17083.

Indirect effects and interactions for gjam data

Description

Evaluates direct, indirect, and interactions from a gjam object. Returns a list of objects that can be plotted by gjamIIEplot.

Usage

  gjamIIE(output, xvector, MEAN = T, keepNames = NULL, omitY = NULL, 
          sdScaleX = T, sdScaleY = F)

Arguments

output

object of class inheriting from "gjam".

xvector

vector of predictor values, with names, corresponding to columns in output$x.

MEAN

logical, if false, then median used.

omitY

character vector of columns in output$y to omit from calculations.

keepNames

character vector of columns in output$y. If omitted, all columns used.

sdScaleX

standardize coefficients to X scale.

sdScaleY

standardize coefficients to correlation scale.

Details

For plotting or recovering effects. The list fit$IIE has matrices for main effects (mainEffect), interactions (intEffect), direct effects (dirEffect), indirect effects (indEffectTo), and standard deviations for each. The direct effects are the sum of main effects and interactions. The indirect effects include main effects and interactions that come through other species, determined by covariance matrix sigma.

If sdScaleX = T effects are standandardized from the Y/X to Y scale. This is the typical standardization for predictor variables. If sdScaleY = T effects are given on the correlation scale. If both are true effects are dimensionless. See the gjam vignette on dimension reduction.

Value

A list of objects for plotting by gjamIIEplot.

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
sim <- gjamSimData(S = 12, Q = 5, typeNames = 'CA')
ml  <- list(ng = 50, burnin = 5, typeNames = sim$typeNames)
out <- gjam(sim$formula, sim$xdata, sim$ydata, modelList = ml)

xvector <- colMeans(out$inputs$xStand)  #predict at mean values for data
xvector[1] <- 1

fit <- gjamIIE(output = out, xvector)

gjamIIEplot(fit, response = 'S1', effectMu = c('main','ind'), 
            effectSd = c('main','ind'), legLoc = 'topleft')

## End(Not run)

Plots indirect effects and interactions for gjam data

Description

Using the object returned by gjamIIEplot generates a plot for a response variable.

Usage

  gjamIIEplot(fit, response, effectMu, effectSd = NULL, 
              ylim = NULL, col='black', legLoc = 'topleft', cex = 1)

Arguments

fit

object from gjamIIE.

response

name of a column in fit$y to plot.

effectMu

character vector of mean effects to plot, can include 'main','int','direct','ind'.

effectSd

character vector can include all or some of effectMu.

ylim

vector of two values defines vertical axis range.

col

vector of colors for barplot.

legLoc

character for legend location.

cex

font size.

Details

For plotting direct effects, interactions, and indirect effects from an object fit generated by gjamIIE. The character vector supplied as effectMu can include main effects ('main'), interactions ('int'), main effects plus interactions ('direct'), and/or indirect effects ('ind'). The list effectSd draws 0.95 predictive intervals for all or some of the effects listed in effectMu. Bars are contributions of each effect to the response.

For factors, effects are plotted relative to the mean over all factor levels.

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
f   <- gjamSimData(S = 10, Q = 6, typeNames = 'OC')
ml  <- list(ng = 50, burnin = 5, typeNames = f$typeNames)
out <- gjam(f$formula, f$xdata, f$ydata, modelList = ml)

xvector <- colMeans(out$inputs$xStand)  #predict at mean values for data, standardized x
xvector[1] <- 1

fit <- gjamIIE(out, xvector)

gjamIIEplot(fit, response = 'S1', effectMu = c('main','ind'), 
            effectSd = c('main','ind'), legLoc = 'topleft')

## End(Not run)

Ordinate gjam data

Description

Ordinate data from a gjam object using correlation corresponding to reponse matrix E.

Usage

  gjamOrdination(output, specLabs = NULL, col = NULL, cex = 1, 
                 PLOT=T, method = 'PCA')

Arguments

output

object of class "gjam".

specLabs

character vector of variable names in colnames(output$y).

col

character vector of columns in output$y to label in plots.

cex

text size in plot.

PLOT

logical, if true, draw plots.

method

character variable can specify 'NMDS'.

Details

Ordinates the response correlation ematrix contained in output$parameterTables. If method = 'PCA' returns eigenvalues and eigenvectors. If method = 'PCA' returns three NMDS dimensions. If PLOT, then plots will be generated. Uses principle components analysis or non-metric multidimensional scale (NMDS).

Value

eVecs

S x S or, if there is an other response variable to be excluded, S-1 x S-1 matrix of eigenvectors for species (rows) by eigenvectors (columns).

eValues

If method = 'PCA' returns length-S or, there is an other response variable to be excluded, length-S-1 vector of eigenvalues. If method = 'NMDS' this variable is NULL.

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
f      <- gjamSimData(S = 30, typeNames = 'CA') 
ml     <- list(ng = 1000, burnin = 200, typeNames = f$typeNames, holdoutN = 10)
output <- gjam(f$formula, f$xdata, f$ydata, modelList = ml)
ePCA   <- gjamOrdination(output, PLOT=TRUE)
eNMDS  <- gjamOrdination(output, PLOT=TRUE, method='NMDS')

## End(Not run)

Plot gjam analysis

Description

Constructs plots of posterior distributions, predictive distributions, and additional analysis from output of gjam.

Usage

  gjamPlot(output, plotPars)

Arguments

output

object of class "gjam"

plotPars

list having default values described in Details

Details

plotPars a list that can contain the following, listed with default values:

`PLOTY = T`	plot predicted `y`.
`PLOTX = T`	plot inverse predicted `x`.
`PREDICTX = T`	inverse prediction of `x`; does not work if `PREDICTX = F` in `link{gjam}`.
`ncluster`	number of clusters to highlight in cluster diagrams, default based on `S`.
`CORLINES = T`	draw grid lines on grid plots of R and E.
`cex = 1`	text size for grid plots, see `par`.
`BETAGRID = T`	draw grid of beta coefficients.
`PLOTALLY = F`	an individual plot for each column in `y`.
`SMALLPLOTS = T`	avoids plot margin error on some devices, better appearance if `FALSE`.
`GRIDPLOTS = F`	cluster and grid plots derived from parameters; matrices R and E are discussed in Clark et al. (2016).
`SAVEPLOTS = F`	plots saved in pdf format.
`outfolder = 'gjamOutput'`	folder for plot files if `SAVEPLOTS = T`.
`width, height = 4`	can be small values, in inches, to avoid plot margin error on some devices.
`specColor = 'black'`	color for posterior box-and-whisker plots.
`ematAlpha = .95`	prob threshold used to infer that a covariance value in `Emat` is not zero.
`ncluster = 4`	number of clusters to identify in `ematrix`.

The 'plot margin' errors mentioned above are device-dependent. They can be avoided by specifying small width, height (in inches) and by omitting the grid plots (GRIDPLOTS = F). If plotting does not produce a 'plot margin error', better appearance is obtained with SMALLPLOTS = F.

Names will not be legible for large numbers of species. Specify specLabs = F and use a character vector for specColor to identify species groups (see the gjam vignette on dimension reduction).

Box and whisker plots bound 0.68 and 0.95 credible and predictive intervals.

Value

Summary tables of parameter estimates are:

betaEstimates

Posterior summary of beta coefficients.

clusterIndex

cluster index for responses in grid/cluster plots.

clusterOrder

order for responses in grid/cluster plots.

eComs

groups based on clustering ematrix.

ematrix

S X S response correlation matrix for E.

eValues

eigenvalues of ematrix.

eVecs

eigenvectors of ematrix.

fit

list containing DIC, score, and rmspe.

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
## ordinal data
f   <- gjamSimData(S = 15, Q = 3, typeNames = 'OC') 
ml  <- list(ng = 1500, burnin = 500, typeNames = f$typeNames, holdoutN = 10)
out <- gjam(f$formula, f$xdata, f$ydata, modelList = ml)

# repeat with ng = 2000, burnin = 500, then plot data here:
pl  <- list(trueValues = f$trueValues, width=3, height=2)
fit <- gjamPlot(output = out, plotPars = pl)

## End(Not run)

Incidence point pattern to grid counts

Description

From point pattern data in (x, y) generates counts on a lattice supplied by the user or specified by lattice size or density. For analysis in gjam as counts (known effort) or count composition (unknown effort) data.

Usage

  gjamPoints2Grid(specs, xy, nxy = NULL, dxy = NULL, 
                  predGrid = NULL, effortOnly = TRUE)

Arguments

specs

character vector of species names or codes.

xy

matrix with rows = length(specs) and columns for (x, y).

nxy

length-2 numeric vector with numbers of points evenly spaced on (x, y).

dxy

length-2 numeric vector with distances for points evenly spaced on (x, y).

predGrid

matrix with 2 columns for (x, y).

effortOnly

logical to return only points where counts are positive (e.g., effort is unknown).

Details

For incidence data with species names specs and locations (x, y) constructs a lattice based a prediction grid predGrid, at a density of (dx, dy), or with numbers of lattice points (nx, ny). If effortOnly = T, returns only points with non-zero values.

A prediction grid predGrid would be passed when counts by locations of known effort are required or where multiple groups should be assign to the same lattice points.

The returned gridBySpec can be analyzed in gjam with known effort as count data "DA" or with unknown effort as count composition data "CC".

Value

gridBySpec

matrix with rows for grid locations, columns for counts by species.

predGrid

matrix with columns for (x, y) and rows matching gridBySpec.

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
## random data
n  <- 100
s  <- sample( letters[1:3], n, replace = TRUE)
xy <- cbind( rnorm(n,0,.2), rnorm(n,10,2) )

nx <- ny <- 5                                    # uniform 5 X 5 lattice
f  <- gjamPoints2Grid(s, xy, nxy = c(nx, ny))
plot(f$predGrid[,1], f$predGrid[,2], cex=.1, xlim=c(-1,1), ylim=c(0,20),
     xlab = 'x', ylab = 'y')
text(f$predGrid[,1], f$predGrid[,2], rowSums(f$gridBySpec))

dx <- .2                                          # uniform density
dy <- 1.5
g  <- gjamPoints2Grid(s, xy, dxy = c(dx, dy))
text(g$predGrid[,1], g$predGrid[,2], rowSums(g$gridBySpec), col='brown')

p  <- cbind( runif(30, -1, 1), runif(30, 0, 20) ) # irregular lattice
h  <- gjamPoints2Grid(s, xy, predGrid = p)
text(h$predGrid[,1], h$predGrid[,2], rowSums(h$gridBySpec), col='blue')

## End(Not run)

Predict gjam data

Description

Predicts data from a gjam object, including conditional and out-of-sample prediction.

Usage

  gjamPredict(output, newdata = NULL, y2plot = NULL, ylim = NULL, 
              FULL = FALSE)

Arguments

output

object of class "gjam".

newdata

a list of data for prediction, see Details.

y2plot

character vector of columns in output$y to plot.

ylim

vector of lower and upper bounds for prediction plot

FULL

will return full chains for predictions as output$ychains

Details

If newdata is not specified, the response is predicted from xdata as an in-sample prediction. If newdata is specified, prediction is either conditional or out-of-sample.

Conditional prediction on a new set of y values is done if newdata includes the matrix ycondData, which holds columns to condition on. ycondData must be a matrix and have column names matching those in y that it will replace. ycondData must have at least one column, but fewer than ncol(y) columns. Columns not included in ycondData will be predicted conditionally. Note that conditional prediction can be erratic if the observations on which the prediction is conditioned are unlikely given the model.

Alternatively, the list newdata can include a new version of xdata for out-of-sample prediction. The version of xdata passed in newdata has the columns with the same names and variable types as xdata passed to gjam. Note that factor levels must also match those included when fitting the model. All columns in y will be predicted out-of-sample.

For count composition data the effort (total count) is 1000.

Because there is no out-of-sample effort for 'CC' data, values are predicted on the [0, 1] scale.

See examples below.

Value

x

design matrix.

sdList

list of predictive means and standard errors includes yMu, yPe (predictive mean, SE), wMu, wSe (mean latent states and SEs)

piList

predictive intervals, only generated if length(y) < 10000, includes yLo, yHi (0.025, 0.975) prediction interval, wLo, wHi (0.025, 0.975) for latent states

prPresent

n x S matrix of probabilities of presence

ematrix

effort

ychains

full prediction chains if FULL = T

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
S   <- 10
f   <- gjamSimData(n = 200, S = S, Q = 3, typeNames = 'CC') 
ml  <- list(ng = 1500, burnin = 50, typeNames = f$typeNames, holdoutN = 10)
out <- gjam(f$formula, f$xdata, f$ydata, modelList = ml)

# predict data
cols <- c('#018571', '#a6611a')
par(mfrow=c(1,3),bty='n')             
gjamPredict(out, y2plot = colnames(f$ydata)) # predict the data in-sample
title('full sample')

# out-of-sample prediction
xdata     <- f$xdata[1:20,]
xdata[,3] <- mean(f$xdata[,3])     # mean for x[,3]
xdata[,2] <- seq(-2,2,length=20)   # gradient x[,2]
newdata   <- list(xdata = xdata, nsim = 50 )
p1 <- gjamPredict( output = out, newdata = newdata)

# plus/minus 1 prediction SE, default effort = 1000
x2   <- p1$x[,2]
ylim <- c(0, max(p1$sdList$yMu[,1] + p1$sdList$yPe[,1]))
plot(x2, p1$sdList$yMu[,1],type='l',lwd=2, ylim=ylim, xlab='x2',
     ylab = 'Predicted', col = cols[1])
lines(x2, p1$sdList$yMu[,1] + p1$sdList$yPe[,1], lty=2, col = cols[1])
lines(x2, p1$sdList$yMu[,1] - p1$sdList$yPe[,1], lty=2, col = cols[1])

# upper 0.95 prediction error
lines(x2, p1$piList$yLo[,1], lty=3, col = cols[1])
lines(x2, p1$piList$yHi[,1], lty=3, col = cols[1])
title('SE and prediction, Sp 1')

# conditional prediction, assume first species is absent
ydataCond <- out$inputs$y[,1,drop=FALSE]*0                 # set first column to zero
newdata   <- list(ydataCond = ydataCond, nsim=50)
p0        <- gjamPredict(output = out, newdata = newdata)

ydataCond <- ydataCond + 10                                # first column is 10
newdata   <- list(ydataCond = ydataCond, nsim=50)
p1        <- gjamPredict(output = out, newdata = newdata)

s    <- 4         # species chosen at random to compare
ylim <- range( p0$sdList$yMu[,s], p1$sdList$yMu[,s] )
plot(out$inputs$y[,s],p0$sdList$yMu[,s], cex=.2, col=cols[1],
     xlab = 'Observed', ylab = 'Predicted', ylim = ylim)
abline(0,1,lty=2)
points(out$inputs$y[,s],p1$sdList$yMu[,s], cex=.2, col=cols[2])
title('Cond. on 1st Sp')
legend( 'topleft', c('first species absent', 'first species = 10'), 
        text.col = cols, bty = 'n')

# conditional, out-of-sample
n   <- 1000
S   <- 10
f   <- gjamSimData(n = n, S = S, Q = 3, typeNames = 'CA') 

holdOuts <- sort( sample(n, 50) )

xdata <- f$xdata[-holdOuts,] # fitted data
ydata <- f$ydata[-holdOuts,]

xx <- f$xdata[holdOuts,]     # use for prediction
yy <- f$ydata[holdOuts,]

ml  <- list(ng = 2000, burnin = 500, typeNames = f$typeNames)   # fit the non-holdouts
out <- gjam(f$formula, xdata, ydata, modelList = ml)

cdex <- sample(S, 4)        # condition on 4 species
ndex <- c(1:S)[-cdex]       # conditionally predict others

newdata <- list(xdata = xx, ydataCond = yy[,cdex], nsim = 200) # conditionally predict out-of-sample
p2      <- gjamPredict(output = out, newdata = newdata)

par(bty='n', mfrow=c(1,1))
plot( as.matrix(yy[,ndex]), p2$sdList$yMu[,ndex], 
      xlab = 'Observed', ylab = 'Predicted', cex=.3, col = cols[1])
abline(0,1,lty=2)
title('RMSPE')
mspeC <- sqrt( mean(  (as.matrix(yy[,ndex]) - p2$sdList$yMu[,ndex])^2 ) )

#predict unconditionally, out-of-sample
newdata   <- list(xdata = xx, nsim = 200 ) 
p1 <- gjamPredict(out, newdata = newdata)

points( as.matrix(yy[,ndex]), p1$sdList$yMu[,ndex], col=cols[2], cex = .3)
mspeU <- sqrt( mean(  (as.matrix(yy[,ndex]) - p1$sdList$yMu[,ndex])^2 ) )

e1 <- paste( 'cond, out-of-sample =', round(mspeC, 2) )
e2 <- paste( 'uncond, out-of-sample =', round(mspeU, 2) )

legend('topleft', c(e1, e2), text.col = cols, bty = 'n')

## End(Not run)

Prior coefficients for gjam analysis

Description

Constructs coefficient matrices for low and high limits on the uniform prior distribution for beta.

Usage

  gjamPriorTemplate(formula, xdata, ydata, lo = NULL, hi = NULL)

Arguments

formula

object of class formula, starting with ~, matches the formula passed to gjam

xdata

n x Q observation by predictor data.frame

ydata

n x Q observation by response data.frame

lo

list of lower limits

hi

list of upper limits

Details

The prior distribution for a coefficient beta[q,s] for predictor q and response s, is dunif(lo[q,s], hi[q,s]). gjamPriorTemplate generates these matrices. The default values are (-Inf, Inf), i.e., all values in lo equal to -Inf and hi equal to Inf. These templates can be modified by changing specific values in lo and/or hi.

Alternatively, desired lower limits can be passed as the list lo, assigned to names in xdata (same limit for all species in ydata), in ydata (same limit for all predictors in xdata), or both, separating names in xdata and ydata by "_". The same convention is used for upper limits in hi.

These matrices are supplied in as list betaPrior, which is included in modelList passed to gjam. See examples and browseVignettes('gjam').

Note that the informative prior slows computation.

Value

A list containing two matrices. lo is a Q x S matrix of lower coefficient limits. hi is a Q x S matrix of upper coefficient limits. Unless specied in lo, all values in lo = -Inf. Likewise, unless specied in hi, all values in hiBeta = -Inf.

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
library(repmis)
source_data("https://github.com/jimclarkatduke/gjam/blob/master/forestTraits.RData?raw=True")

xdata       <- forestTraits$xdata
plotByTree  <- gjamReZero(forestTraits$treesDeZero) # re-zero
traitTypes  <- forestTraits$traitTypes
specByTrait <- forestTraits$specByTrait

tmp <- gjamSpec2Trait(pbys = plotByTree, sbyt = specByTrait, 
                      tTypes = traitTypes)
tTypes <- tmp$traitTypes
traity <- tmp$plotByCWM
censor <- tmp$censor

formula <- as.formula(~ temp + deficit)
lo <- list(temp_gmPerSeed = 0, temp_dioecious = 0 ) # positive effect on seed size, dioecy
b  <- gjamPriorTemplate(formula, xdata, ydata = traity, lo = lo)

ml <- list(ng=3000, burnin=1000, typeNames = tTypes, censor = censor, betaPrior = b)
out <- gjam(formula, xdata, ydata = traity, modelList = ml)

S   <- ncol(traity)
sc  <- rep('black',S)
sc[colnames(traity) 
pl  <- list(specColor = sc)           
gjamPlot(output = out, plotPars = pl)         

## End(Not run)

Expand (re-zero) gjam data

Description

Returns a re-zeroed matrix y from the de-zeroed vector, a sparse matrix.

Usage

  gjamReZero( yDeZero )

Arguments

yDeZero

list created by gjamReZero containing number of rows n, number of columns S, index for non-zeros index, the vector of non-zero values yvec, and the column names ynames.

Details

Many abundance data sets are mostly zeros. gjamReZero recovers the full matrix from de-zeroed list yDeZero written by gjamDeZero

Value

ymat

re-zeroed n by S matrix.

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
library(repmis)
source_data("https://github.com/jimclarkatduke/gjam/blob/master/fungEnd.RData?raw=True")
ymat <- gjamReZero(fungEnd$yDeZero)  # OTUs stored without zeros
length(fungEnd$yDeZero$yvec)         # size of stored version
length(ymat)                         # full size

## End(Not run)

Sensitivity coefficients for gjam

Description

Evaluates sensitivity coefficients for full response matrix or subsets of it. Uses output from gjam. Returns a matrix of samples by predictors.

Usage


  gjamSensitivity(output, group = NULL, nsim = 100, PERSPECIES = TRUE)

Arguments

output

object fitted with gjam.

group

character vector of response-variable names from output$inputs$y.

nsim

number of samples from posterior distribution.

PERSPECIES

divide variance by number of species in the group

Details

Sensitivity to predictors of entire reponse matrix or a subset of it, identified by the character string group. The equations for sensitivity are given here:

browseVignettes('gjam')

Value

Returns a nsim by predictor matrix of sensitivities to predictor variables, evaluated by draws from the posterior. Because sensitivities may be compared across groups represented by different numbers of species, PERSPECIES = TRUE returns sensitivity on a per-species basis.

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
## combinations of scales
types <- c('DA','DA','OC','OC','OC','OC','CC','CC','CC','CC','CC','CA','CA','PA','PA')         
f    <- gjamSimData(S = length(types), typeNames = types)
ml   <- list(ng = 50, burnin = 5, typeNames = f$typeNames)
out  <- gjam(f$formula, f$xdata, f$ydata, modelList = ml)

ynames <- colnames(f$y)
group  <- ynames[types == 'OC']

full <- gjamSensitivity(out)
cc   <- gjamSensitivity(out, group)

nt <- ncol(full)

ylim <- range(rbind(full, cc))

boxplot( full, boxwex = 0.25,  at = 1:nt - .21, col='blue', log='y',
         ylim = ylim, xaxt = 'n', xlab = 'Predictors', ylab='Sensitivity')
boxplot( cc, boxwex = 0.25, at = 1:nt + .2, col='forestgreen', add=T,
         xaxt = 'n')
axis(1,at=1:nt,labels=colnames(full))
legend('bottomleft',c('full response','CC data'),
       text.col=c('blue','forestgreen'))

## End(Not run)

Simulated data for gjam analysis

Description

Simulates data for analysis by gjam.

Usage

  gjamSimData(n = 1000, S = 10, Q = 5, x = NULL, nmiss = 0, typeNames, effort = NULL)

Arguments

n

Sample size

S

Number of response variables (columns) in y, typically less than n

Q

Number of predictors (columns) in design matrix x << n

x

design matrix, if supplied n and Q will be set to nrow(x) and ncol(x), respectively

nmiss

Number of missing values to in x << n

typeNames

Character vector of data types, see Details

effort

List containing 'columns' specifying columns to which effort applies, and 'values', a length-n vector of effort per observation.

Details

Generates simulated data and parameters for analysis by gjam. Because both parameters and data are stochastic, not all simulations will give good results.

typeNames can be 'PA' (presenceAbsence), 'CA' (continuous), 'DA' (discrete), 'FC' (fractional composition), 'CC' (count composition), 'OC' (ordinal counts), and 'CAT' (categorical levels). If more than one 'CAT' is included, each defines a multilevel categorical reponse. One additional type, 'CON' (continuous), is not censored at zero by default.

If defined as a single character value typeNames applies to all columns in y. If not, typeNames is length-S character vector, identifying each response by column in y. If a column 'CAT' is included, a random number of levels will be generated, a, b, c, ....

A more detailed vignette is can be obtained with:

browseVignettes('gjam')

website 'http://sites.nicholas.duke.edu/clarklab/code/'.

Value

formula

R formula for model, e.g., ~ x1 + x2

xdata

data.frame includes columns for predictors in the design matrix

ydata

data.frame for the simulated response

y

response as a n by S matrix as assembled in gjam.

w

n by S latent states

typeY

vector of data types corresponding to columns in y, see Details

typeNames

vector of data types corresponding to columns in ydata

trueValues

list containing true parameter values beta (regression coefficients), sigma (covariance matrix), corSpec (correlation matrix corresponding to sigma), and cuts (partition matrix for ordinal data).

effort

see Arguments.

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
## ordinal data, show true parameter values
sim <- gjamSimData(S = 5, typeNames = 'OC')  
sim$ydata[1:5,]                              # example data
sim$trueValues$cuts                          # simulated partition
sim$trueValues$beta                          # coefficient matrix

## continuous data censored at zero, note latent w for obs y = 0
sim <- gjamSimData(n = 5, S = 5, typeNames = 'CA')  
sim$w
sim$y

## continuous and discrete data
types <- c(rep('DA',5), rep('CA',4))
sim   <- gjamSimData(n = 10, S = length(types), Q = 4, typeNames = types)
sim$typeNames
sim$ydata
                             
## composition count data  
sim <- gjamSimData(n = 10, S = 8, typeNames = 'CC')
totalCount <- rowSums(sim$ydata)
cbind(sim$ydata, totalCount)  # data with sample effort

## multiple categorical responses - compare matrix y and data.frqme ydata
types <- rep('CAT',2)
sim   <- gjamSimData(S = length(types), typeNames = types)
head(sim$ydata)
head(sim$y)

## discrete abundance, heterogeneous effort 
S   <- 5
n   <- 1000
ef  <- list( columns = 1:S, values = round(runif(n,.5,5),1) )
sim <- gjamSimData(n, S, typeNames = 'DA', effort = ef)
sim$effort$values[1:20]

## combinations of scales, partition only for 'OC' columns
types <- c('OC','OC','OC','CC','CC','CC','CC','CC','CA','CA','PA','PA')
sim   <- gjamSimData(S = length(types), typeNames = types)
sim$typeNames                           
head(sim$ydata)
sim$trueValues$cuts

## End(Not run)

Ecological traits for gjam analysis

Description

Constructs community-weighted mean-mode (CWMM) trait matrix for analysis with gjam for n observations, S species, P traits, and M total trait levels.

Usage

  gjamSpec2Trait(pbys, sbyt, tTypes)

Arguments

pbys

n x S plot by species matrix (presence-absence, abundance)

sbyt

S x P species by trait matrix

tTypes

P data types for trait columns

Details

Generates the objects needed for a trait response model (TRM). As inputs the sbyt data.frame has P columns containing numeric values, ordinal scores, and categorical variables, identified by data type in tTypes. Additional trait columns can appear in the n x M output matrix plotByCWMM, because each level of a category becomes a new 'FC' column as a CWMM. Thus, M can exceed P, depending on the number of factors in sbyt. The exception is for categorical traits with only two levels, which can be treated as (0, 1) censored 'CA' data.

As output, the CWMM data types are given in traitTypes.

The list censor = NULL unless some data types are censored. In the example below there are two censored columns.

A detailed vignette on trait analysis is obtained with:

browseVignettes('gjam')

Value

plotByCWM

n x M matrix of community-weight means (numeric) or modes (ordinal)

traitTypes

character vector of data types for traits

specByTrait

S x M matrix translates species to traits

censor

list of censored columns, values, and intervals; see gjamCensorY

Author(s)

James S Clark, jimclark@duke.edu

References

Clark, J.S. 2016. Why species tell us more about traits than traits tell us about species: Predictive models. Ecology 97, 1979-1993.

Examples

## Not run: 
library(repmis)
source_data("https://github.com/jimclarkatduke/gjam/blob/master/forestTraits.RData?raw=True")

xdata       <- forestTraits$xdata
plotByTree  <- gjamReZero(forestTraits$treesDeZero) # re-zero
traitTypes  <- forestTraits$traitTypes
specByTrait <- forestTraits$specByTrait

tmp <- gjamSpec2Trait(pbys = plotByTree, sbyt = specByTrait, 
                      tTypes = traitTypes)
tTypes <- tmp$traitTypes
traity <- tmp$plotByCWM
censor <- tmp$censor

ml  <- list(ng=2000, burnin=500, typeNames = tTypes, censor = censor)
out <- gjam(~ temp + stdage + deficit, xdata, ydata = traity, modelList = ml)
gjamPlot( output = out )         

## End(Not run)

Trim gjam response data

Description

Returns a list that includes a subset of columns in y. Rare species can be aggregated into a single class.

Usage

  gjamTrimY(y, minObs = 2, maxCols = NULL, OTHER = TRUE)

Arguments

y

n by S numeric response matrix

minObs

minimum number of non-zero observations

maxCols

maximum number of response variables

OTHER

logical or character string. If OTHER = TRUE, rare species are aggregated in a new column 'other'. A character vector contains the names of columns in y to be aggregated with rare species in the new column 'other'.

Details

Data sets commonly have many responses that are mostly zeros, large numbers of rare species, even singletons. Response matrix y can be trimmed to include only taxa having > minObs non-zero observations or to <= maxCol total columns. The option OTHER is recommended for composition data ('CC', 'FC'), where the 'other' column is taken as the reference class. If there are unidentified species they might be included in this class. [See gjamSimData for typeName codes].

Value

Returns a list containing three elements.

y

trimmed version of y.

colIndex

length-S vector of indices for new columns in y.

nobs

number of non-zero observations by column in y.

Author(s)

James S Clark, jimclark@duke.edu

References

Examples

## Not run: 
library(repmis)
source_data("https://github.com/jimclarkatduke/gjam/blob/master/forestTraits.RData?raw=True")

y   <- gjamReZero(fungEnd$yDeZero)     # re-zero data
dim(y)
y   <- gjamTrimY(y, minObs = 200)$y    # species in >= 200 observations
dim(y)
tail(colnames(y))    # last column is 'other'

## End(Not run)

Generalized Joint Attribute Modeling

Description

Details

Author(s)

References

See Also

Gibbs sampler for gjam data

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Censor gjam response data

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Parameters for gjam conditional prediction

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Compress (de-zero) gjam data

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Fill out data for time series (state-space) gjam

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Indirect effects and interactions for gjam data

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Plots indirect effects and interactions for gjam data

Description

Usage

Arguments

Details

Author(s)

References

See Also

Examples

Ordinate gjam data

Description

Usage

Arguments

Details

Value