Help for package bbl

Title:

Boltzmann Bayes Learner

Version:

1.0.0

Maintainer:

Jun Woo <junwoo035@gmail.com>

Depends:

R (≥ 3.6.0)

Imports:

methods, stats, utils, Rcpp (≥ 0.12.16), pROC, RColorBrewer

Description:

Supervised learning using Boltzmann Bayes model inference, which extends naive Bayes model to include interactions. Enables classification of data into multiple response groups based on a large number of discrete predictors that can take factor values of heterogeneous levels. Either pseudo-likelihood or mean field inference can be used with L2 regularization, cross-validation, and prediction on new data. <doi:10.18637/jss.v101.i05>.

License:

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

LinkingTo:

Rcpp

Encoding:

UTF-8

RoxygenNote:

7.1.2

NeedsCompilation:

yes

Suggests:

glmnet, BiocManager, Biostrings

Author:

Jun Woo

[aut, cre]

Repository:

CRAN

Packaged:

2022-01-27 16:11:27 UTC; junwoo

Date/Publication:

2022-01-27 23:20:12 UTC

Boltzmann Bayes Learning Inference

Description

Main driver for bbl inference

Usage

bbl(
  formula,
  data,
  weights,
  xlevels = NULL,
  verbose = 1,
  method = "pseudo",
  novarOk = FALSE,
  testNull = TRUE,
  prior.count = 1,
  ...
)

Arguments

formula

Formula for modeling

data

Data for fitting

weights

Vector of weights for each instance in data. Restricted to non-negative integer frequencies, recoding the number of times each row of data must be repeated. If NULL, assumed to be all 1. Fractional weights are not supported. Can be a named column in data

xlevels

List of factor levels for predictors. If NULL, will be inferred from data with factor levels ordered alphanumerically.

verbose

Output verbosity level. Will be send to down-stream function calls with one level lower

method

BB inference algorithm; pseudo-likelihood inference ('pseudo') or mean field ('mf')

novarOk

If TRUE, will proceed with predictors having only one level

testNull

Repeat the inference for the ‘pooled’ sample; i.e., under the null hypothesis of all rows in data belonging to a single group

prior.count

Prior count for computing single predictor and pairwise frequencies

...

Other parameters to mlestimate.

Details

Formula argument and data are used to tabulate xlevels unless explicitly given as list. Data are expected to be factors or integers. This function is a driver interepreting formula and calls bbi.fit. Will stop with error if any predictor has only one level unless novarOk='TRUE'. Use removeConst to remove the non-varying predictors before calling if this happens.

Value

A list of class bbl with the following elements:

coefficients

List of inferred coefficients with elements h, J, h0, and J0. The bias parameter h is a list of length equal to no. of response groups, each of which is a list of the same struture as xlevels: length equal to no. of predictors, containing vectors of length equal to each predictor factor levels: h_i^{(y)}(x) represented by h[[y]][[i]][x]. The interaction parameter J is a list of lists of dimension m \times m, where m is the number of predictors. Each element is a matrix of dimension L_i \times L_j, where L_i and L_j are numbers of factor levels in predictor i and j: J_{ij}^{(y)}(x_1,x_2) represented by J[[y]][[i]][[j]][x1,x2]. All elements of lists are named. The pooled parameters h0 and J0, if computed, are of one less dimension, omitting response group argument.

xlevels

List of vectors containing predictor levels.

terms

The terms of formula input.

groups

Vector of response groups.

groupname

Name of the response variable.

qJ

Matrix of logicals whose elements record whether formula includes interaction between the two predictors.

model

Model data frame derived from formula and data.

lkh

Log likelihood.

lz

Vector log partition function. Used in predict.

weights

Vector of integral weights (frequencies).

call

Function call.

df

Degrees of freedom.

Author(s)

Jun Woo, junwoo035@gmail.com

References

doi: 10.18637/jss.v101.i05

Examples

titanic <- as.data.frame(Titanic)
b <- bbl(Survived ~ (Class + Sex + Age)^2, data = titanic, weights = Freq)
b

bbl Inference with model matrix

Description

Performs bbl inference using response vector and predictor matrix

Usage

bbl.fit(
  x,
  y,
  qJ = NULL,
  weights = NULL,
  xlevels = NULL,
  verbose = 1,
  method = "pseudo",
  prior.count = 1,
  ...
)

Arguments

x

Data frame of factors with each predictor in columns.

y

Vector of response variables.

qJ

Matrix of logicals indicating which predictor combinations are interacting.

weights

Vector of non-negative integer frequencies, recoding the number of times each row of data must be repeated. If NULL, assumed to be all 1. Fractional weights are not supported.

xlevels

List of factor levels for predictors. If NULL, will be inferred from data with factor levels ordered alphanumerically.

verbose

Verbosity level of output. Will be propagated to mlestimate with one level down.

method

c('pseudo','mf'); inference method.

prior.count

Prior count for computing single predictor and pairwise frequencies

...

Other arguments to mlestimate.

Details

This function would normally be called by bbl rather than directly. Expects the predictor data x and response vector y instead of formula input to bbl.

Value

List of named components h, J, lkh, and lz; see bbl for information regarding these components.

Examples

titanic <- as.data.frame(Titanic)
freq <- titanic$Freq
x <- titanic[,1:3]
y <- titanic$Survived
b <- bbl.fit(x=x,y=y, weights=freq)
b

Cross-Validation of BB Learning

Description

Run multiple fittings of bbl model with training/validation division of data

Usage

crossVal(
  formula,
  data,
  weights,
  novarOk = FALSE,
  lambda = 1e-05,
  lambdah = 0,
  eps = 0.9,
  nfold = 5,
  method = "pseudo",
  use.auc = TRUE,
  verbose = 1,
  progress.bar = FALSE,
  storeOpt = TRUE,
  ...
)

Arguments

formula

Formula for model. Note that intercept has no effect.

data

Data frame of data. Column names must match formula.

weights

Frequency vector of how many times each row of data must be repeated. If NULL, defaults to vector of 1s. Fractional values are not supported.

novarOk

Proceed even when there are predictors with only one factor level.

lambda

Vector of L2 penalizer values for method = 'pseudo'. Inferences will be repeated for each value. Restricited to non-negative values.

lambdah

L2 penalizer in method = 'pseudo' applied to parameter h. In contrast to lambda, only a single value is allowed.

eps

Vector of regularization parameters, \epsilon\in[0,1], for method = 'mf'. Inference will be repeated for each value.

nfold

Number of folds for training/validation split.

method

c('pseudo','mf') for pseudo-likelihood maximization or mean field.

use.auc

Use AUC as the measure of prediction accuracy. Only works if response groups are binary. If FALSE, mean prediction group accuracy will be used as score.

verbose

Verbosity level. Downgraded when relayed into bbl.

progress.bar

Display progress bar in predict.

storeOpt

Store the optimal fitted object of class bbl.

...

Other parameters to mlestimate.

Details

The data slot of object is split into training and validation subsets of (nfold-1):1 ratio. The model is trained with the former and validated on the latter. Individual division/fold results are combined into validation result for all instances in the data set and prediction score is evaluated using the known response group identity.

Value

Object of class cv.bbl extending bbl, a list with extra components: regstar, Value of regularization parameter, lambda and eps for method='pseudo' and method='mf',respectively, at which the accuracy score is maximized; maxscore, Value of maximum accuracy; cvframe, Data frame of regularization parameters and scores scanned. If use.auc=TRUE, also contains 95

Examples

set.seed(513)
m <- 5
n <- 100
predictors <- list()
for(i in 1:m) predictors[[i]] <- c('a','c','g','t')
names(predictors) <- paste0('v',1:m)
par <- list(randompar(predictors), randompar(predictors, h0=0.1, J0=0.1))
dat <- randomsamp(predictors, response=c('ctrl','case'), par=par, nsample=n)
cv <- crossVal(y ~ .^2, data=dat, method='mf', eps=seq(0.1,0.9,0.1))
cv

Fitted Response Group Probabilities

Description

Response group probabilities from BBL fit

Usage

## S3 method for class 'bbl'
fitted(object, ...)

Arguments

object

Object of class bbl.

...

Other arguments

Details

This method returns predicted response group probabilities of trainig data

Value

Matrix of response group probabities with data points in rows and response groups in columns

Examples

titanic <- as.data.frame(Titanic)
fit <- bbl(Survived ~ Class + Sex + Age, data=titanic, weights=titanic$Freq)

Formula in BBL Fitting

Description

Returns the formula used in BBL fit

Usage

## S3 method for class 'bbl'
formula(x, ...)

Arguments

x

Object of class bbl

...

Other arguments

Value

Formula object

Examples

titanic <- as.data.frame(Titanic)
fit <- bbl(Survived ~ Class + Sex + Age, data=titanic, weights=titanic$Freq)
formula(fit)

Convert Frequency Table into Raw Data

Description

Data with unique rows and a frequency column is converted into data with duplicate rows.

Usage

freq2raw(data, freq)

Arguments

data

Data frame with factors in columns

freq

Vector of frequency of each row in data; can be a named column in data; if missing, the column Freq is looked for in data

Details

The ouput data frame can be used as input to bbl.

Value

Data frame with one row per instances

Examples

Titanic
x <- as.data.frame(Titanic)
head(x)
titanic <- freq2raw(data=x[,1:3], freq=x$Freq)
head(titanic)

Log likelihood for bbl object

Description

Compute log likelihood from a fitted bbl object

Usage

## S3 method for class 'bbl'
logLik(object, ...)

Arguments

object

Object of class bbl

...

Other arguments to methods

Details

This method uses inferred parameters from calls to bbl and data to compute the log likelihood.

Value

An object of class logLik, the Log likelihood value and the attribute "df" (degrees of freedom), the number of parameters.

Sample Predictor Distributions

Description

Uses fitted BBL model to explore predictor distributions

Usage

mcSample(object, nsteps = 1000, verbose = 1, progress.bar = TRUE)

Arguments

object

Object of class bbl

nsteps

Total number of MC steps

verbose

Verbosity level of output

progress.bar

Display progress bar

Details

After bbl fit, the resulting model is used by this function to sample predictor distributions in each response group and find the most likely preditor set using MCMC.

Examples

titanic <- as.data.frame(Titanic)
b <- bbl(Survived~., data=titanic[,1:4], weights=titanic$Freq)
pxy <- mcSample(b)
pxy

Maximum likelihood estimate

Description

Perform inference of bias and interaction parameters for a single response group

Usage

mlestimate(
  xi,
  weights = NULL,
  qJ = NULL,
  method = "pseudo",
  L = NULL,
  lambda = 1e-05,
  lambdah = 0,
  symmetrize = TRUE,
  eps = 0.9,
  nprint = 100,
  itmax = 1e+05,
  tolerance = 1e-04,
  verbose = 1,
  prior.count = 1,
  naive = FALSE,
  lz.half = FALSE
)

Arguments

xi

Data matrix; expected to be numeric with elements ranging from zero to positive integral upper bound L-1.

weights

Frequency vector of number of times each row of xi is to be repeated. If NULL, defaults to 1. Expected to be non-negative integers.

qJ

Matrix of logicals indicating which predictor pairs are interacting. If NULL, all are allowed.

method

c('pseudo','mf') for pseudo-likelihood maximization or mean field inference.

L

Vector of number of factor levels in each predictor. If NULL, will be inferred from xi.

lambda

Vector of L2 regularization parameters for method = 'pseudo'. Applies to interaction parameters J.

lambdah

L2 parameters for h in 'pseudo'. If NULL, it is set equal to lambda. lambdah = 0 will free h from penalization.

symmetrize

Enforce the symmetry of interaction parameters by taking mean values of the matrix and its trace: J_{ij}^{(y)}(x_1,x_2)=J_{ji}^{(y)}(x_2,x_1).

eps

Vector of regularization parameters for mf. Must be within the range of \epsilon \in [0,1].

nprint

Frequency of printing iteration progress under 'pseudo'.

itmax

Maximum number of iterations for 'pseudo'.

tolerance

Upper bound for fractional changes in pseduo-likelihood values before termiating iteration in 'pseudo'.

verbose

Verbosity level.

prior.count

Prior count for method = 'mf' to reduce numerical instability.

naive

Naive Bayes inference. Equivalent to method = 'mf' together with eps = 0.

lz.half

Divide interaction term in approximation to \ln Z_{iy} in 'pseudo'.

Details

Given numeric data matrix, either pseudo-likelihood of mean-field theory is used to find the maximum likelihood estimate of bias h and interaction J parameters. Normally called by bbl rather than directly.

Value

List of inferred parameters h and J. See bbl for parameter structures.

Examples

set.seed(535)
predictors <- list()
for(i in 1:5) predictors[[i]] <- c('a','c','g','t')
par <- randompar(predictors)
par
xi <- sample_xi(nsample=5000, predictors=predictors, h=par$h, J=par$J,
                code_out=TRUE)
head(xi)
ps <- mlestimate(xi=xi, method='pseudo', lambda=0)
ps$h
ps$J[[1]]
mf <- mlestimate(xi=xi, method='mf', eps=0.9)
plot(x=unlist(par$h), y=unlist(ps$h), xlab='True', ylab='Inferred')
segments(x0=-2, x1=2, y0=-2, y1=2, lty=2)
points(x=unlist(par$J), y=unlist(ps$J), col='red')
points(x=unlist(par$h), y=unlist(mf$h), col='blue')
points(x=unlist(par$J), y=unlist(mf$J), col='green')

Model Frame for BBL

Description

Returns the model frame used in BBL fit

Usage

## S3 method for class 'bbl'
model.frame(formula, ...)

Arguments

formula

Object of class bbl

...

Other arguments

Value

Data frame used for fitting

Examples

titanic <- as.data.frame(Titanic)
fit <- bbl(Survived ~ Class + Sex + Age, data=titanic[,1:4], weights=titanic$Freq)
head(model.frame(fit))

Number of Observations in BBL Fit

Description

Returns the number of observations from a BBL fit

Usage

## S3 method for class 'bbl'
nobs(object, ...)

Arguments

object

Object of class bbl

...

Other arguments

Value

An integer of number of observations

Examples

titanic <- as.data.frame(Titanic)
fit <- bbl(Survived ~ Class + Sex + Age, data=titanic[,1:4], weights=titanic$Freq)
nobs(fit)

Plot bbl object

Description

Visualize bias and interaction parameters

Usage

## S3 method for class 'bbl'
plot(x, layout = NULL, hcol = NULL, Jcol = NULL, npal = 100, ...)

Arguments

x

Object of class bbl

layout

Matrix of layouts for arrangment of linear and interaction parameters. If NULL, the top half will be used for linear parameter barplot and bottom half will be divided into interaction heatmaps for each response group.

hcol

Color for linear barplots. Grayscale if NULL.

Jcol

Color for interaction heatmaps. Default (NULL) is RdBu from RColorBrewer.

npal

Number of color scales.

...

Other graphical parameters for plot.

Details

This method displays a barplot of bias parameters and heatmaps (one per response group) of interaction parameters. All parameters are offset by the pooled values (single group inference) unless missing.

Plot Cross-validation Outcome

Description

Plot cross-validation score as a function of regularization parameter

Usage

## S3 method for class 'cv.bbl'
plot(
  x,
  type = "b",
  log = "x",
  pch = 21,
  bg = "white",
  xlab = NULL,
  ylab = NULL,
  las = 1,
  ...
)

Arguments

x

Object of class cv.bbl from a call to crossVal

type

Symbol type in plot, present here to set default.

log

Log scale argument to plot.

pch

Symbol type code in par.

bg

Symbol background color in par.

xlab

X axis label

ylab

Y axis label

las

Orientation of axis labels in par.

...

Other arguments to plot.

Details

This function will plot accuracy score as a function of regularization parameter from a call to crossVal.

Predict Response Group Using `bbl` Model

Description

Make prediction of response group identity based on trained model

Usage

## S3 method for class 'bbl'
predict(object, newdata, type = "link", verbose = 1, progress.bar = FALSE, ...)

Arguments

object

Object of class bbl containing trained model

newdata

Data frame of new data for which prediction is to be made. Columns must contain all of those in model@data. If column names are present, the columns will be matched based on them. Extra columns will be ignored. If column names are not provided, the columns should exactly match model@data predictor parts. If NULL, replaced by model@data (self-prediction).

type

Return value type. If 'link', the logit scale probabilities. If 'prob' the probability itself.

verbose

Verbosity level

progress.bar

Display progress of response group probability. Useful for large samples.

...

Other arguments to methods

Details

This method uses a new data set for predictors and trained bbl model parameters to compute posterior probabilities of response group identity.

Value

Data frame of predicted posterior probabilities with samples in rows and response groups in columns. The last column is the predicted response group with maximum probability.

Examples

set.seed(154)

m <- 5
L <- 3
n <- 1000

predictors <- list()
for(i in 1:m) predictors[[i]] <- seq(0,L-1)
names(predictors) <- paste0('v',1:m)
par <- list(randompar(predictors=predictors, dJ=0.5),
            randompar(predictors=predictors, h0=0.1, J0=0.1, dJ=0.5))
dat <- randomsamp(predictors=predictors, response=c('ctrl','case'), par=par, 
                 nsample=n)
dat <- dat[sample(n),]
dtrain <- dat[seq(n/2),]
dtest <- dat[seq(n/2+1,n),]

model <- bbl(y ~ .^2, data=dtrain)
pred <- predict(model, newdata=dtest)
score <- mean(dtest$y==pred$yhat)
score

auc <- pROC::roc(response=dtest$y, predictor=pred$case, direction='<')$auc
auc

Predict using Cross-validation Object

Description

Use the optimal fitted model from cross-validation run to make prediction

Usage

## S3 method for class 'cv.bbl'
predict(object, ...)

Arguments

object

Object of class cv.bbl.

...

Other parameters to predict.bbl.

Details

This method will use the fitted model with maximum accuracy score returned by a call to crossVal to make prediction on new data

Value

Data frame of prediction; see predict.bbl.

Print Boltzmann Bayes Learning Fits

Description

This method displays model structure and first elements of coefficients

Usage

## S3 method for class 'bbl'
print(x, showcoeff = TRUE, maxcoeff = 3L, ...)

Arguments

x

An object of class bbl, usually dervied from a call to bbl.

showcoeff

Display first few fit coefficients

maxcoeff

Maximum number of coefficients to display

...

Further arguments passed to or from other methods

Details

Displays the call to bbl, response variable and its levels, predictors and their levels, and the first few fit coefficients.

Display Cross-validation Result

Description

Print cross-validation optimal result and data frame

Usage

## S3 method for class 'cv.bbl'
print(x, ...)

Arguments

x

Object of class cv.bbl

...

Other arguments to methods

Details

This method prints crossVal object with the optimal regularization condition and maximum accuracy score on top and the entire score profile as a data frame below.

Print Summary of Boltzmann Bayes Learning

Description

This method prints the summary of bbl object

Usage

## S3 method for class 'summary.bbl'
print(x, ...)

Arguments

x

Object of class summary.bbl

...

Other arguments to methods

Details

The naive Bayes summary of summary.bbl object is displayed.

Generate Random Parameters

Description

Random values of bias and interaction parameters are generated using either uniform or normal distributions.

Usage

randompar(predictors, distr = "unif", h0 = 0, dh = 1, J0 = 0, dJ = 1)

Arguments

predictors

List of predictor factor levels. See bbl.

distr

c('unif','norm') for uniform or normal distributions.

h0

Mean of bias parameters

dh

sd of bias if distr = 'unif'. If distr = 'norm', h = [h_0-dh, h_0+dh].

J0

Mean of interaction parameters.

dJ

sd of interactions if distr = 'unif'. If distr = 'norm', J = [J_0-dJ, J_0+dJ].

Details

Input argument predictors is used to set up proper list structures of parameters.

Value

List of parameters, h and J.

Examples

set.seed(311)
predictors <- list()
for(i in 1:5) predictors[[i]] <- c('a','c')
par <- randompar(predictors=predictors)
par

Generate Random Boltzmann Bayes Model Data

Description

Predictor-response paired data are generated

Usage

randomsamp(predictors, response, prob = NULL, par, nsample = 100)

Arguments

predictors

List of vectors of predictor levels

response

Vector of response variables

prob

Vector of probabilities for sampling each response group

par

List of bbl parameters for each response group; e.g., generated from calls to randompar.

nsample

Sample size

Details

The argument response is used to set up all possible levels of response groups and likewise for predictors. The parameter argument par must have the appropriate structure consistent with response and predictors. This function is a wrapper calling sample_xi multiple times.

Value

Data frame of response and predictor variables.

Read FASTA File

Description

Read nucleotide sequence files in FASTA format

Usage

readFasta(file, rownames = FALSE)

Arguments

file

File name of FASTA input.

rownames

Use the sequence annotation line in file (starts with '>') as the row names. Will fail if there are duplicate items.

Details

Sequence data in FASTA files are converted into data frame suitable as input to bbl. If sequence lengths are different, instances longer than those already read will be truncated. Empty sequences are skipped.

Value

Data frame of each sequence in rows.

Examples

file <- tempfile('data')
write('>seq1', file)
write('atgcc', file, append=TRUE)
write('>seq2', file, append=TRUE)
write('gccaa', file, append=TRUE)
system(paste0('cat ',file))
x <- readFasta(file)
x

Remove Non-varying Predictors

Description

Constant predictor is identified and removed

Usage

removeConst(x)

Arguments

x

Data frame containing discrete factor variables in each column

Details

Variables with only one factor level is removed from data. Intended for use before calling bbl.

Value

Data frame omitting non-varying variables from x.

Examples

set.seed(351)
nt <- c('a','c','g','t')
x <- data.frame(v1=sample(nt,size=50,replace=TRUE),
                v2=rep('a',50),v3=sample(nt,size=50,replace=TRUE))
y <- sample(c('case','ctrl'),size=50,replace=TRUE)
dat <- cbind(data.frame(y=y), x)
summary(dat)
dat <- removeConst(dat)
summary(dat)

Residuals of BBL fit

Description

Binary-valued vector of fitted vs. true response group

Usage

## S3 method for class 'bbl'
residuals(object, ...)

Arguments

object

Object of class bbl

...

Other arguments

Details

Discrete response group identity for each data point is compared with the fitted group and 0 (discordant) or 1 (concordant) is returned

Value

Vector binary values for each data point

Examples

titanic <- as.data.frame(Titanic)
dat <- freq2raw(titanic[,1:4], freq=titanic$Freq)
fit <- bbl(Survived ~ .^2, data=dat)
x <- residuals(fit)
table(x)

Generate Random Samples from Boltzmann Distribution

Description

Random samples are drawn from Boltzmann distribution

Usage

sample_xi(nsample = 1, predictors = NULL, h, J, code_out = FALSE)

Arguments

nsample

Sample size

predictors

List of predictor factor levels.

h

Bias parameter; see bbl.

J

Interaction parameters; see bbl.

code_out

Ouput in integer codes; a_i = 0, \cdots, L_i-1. If FALSE, output in factors in predictors.

Details

All possible factor states are enumerated exhaustively using input argument predictors. If the number of predictors m or the number of factor levels L_i for each predictor i are even moderately large (m\ge 10 or L_i\ge 5), this function will likely hang because the number of all possible states grows exponentially.

Value

Data frame of samples in rows and predictors in columns.

Examples

set.seed(512)
m <- 5
n <- 1000
predictors <- list()
for(i in 1:m) predictors[[i]] <- c('a','c','g','t')
par <- randompar(predictors)
xi <- sample_xi(nsample=n, predictors=predictors, h=par$h, J=par$J)
head(xi)

Naive Bayes Summary

Description

Estimate significant of predictor-group association using naive Bayes model

Usage

## S3 method for class 'bbl'
summary(object, prior.count = 0, ...)

Arguments

object

Object of class bbl

prior.count

Prior count to be used for computing naive Bayes coefficients and test results. If 0, will produce NAs for factor levels without data points.

...

Other arguments to methods.

Details

This summary.bbl method gives a rough overview of associations within a bbl fit object via naive Bayes coefficients and test p-values. Note that naive Bayes results displayed ignore interactions even when interactions are present in the model being displayed. This feature is because simple analytic results exist for naive Bayes coefficients and test p-values. The likelihood ratio test is with respect to the null hypothesis that coefficients are identical for all response groups.

Value

Object of class summary.bbl extending bbl class; a list with extra components

h

List of bias coefficients of response groups under naive Bayes approximation

h0

Bias coefficients of pooled group under naive Bayes

chisqNaive

Vector of chi-square statistics for likelihood ratio test for each predictor

dfNaive

Vector of degrees of freedom for likelihood ratio test for each predictor

pvNaive

Vector p-values for each predictor

Weights in BBL Fit

Description

This method returns weights used in BBL fit.

Usage

## S3 method for class 'bbl'
weights(object, ...)

Arguments

object

Object of class bbl.

...

Other arguments

Details

Note that weithts are integral frequency values specifying repeat number of each instance in bbl. If no weights were used (default of 1s), NULL is returned.

Value

Vector of weights for each instance