Bayesian Estimation of Change-Points in the Slope of Multivariate Time-Series

Description

Assume that a temporal process is composed of contiguous segments with differing slopes and replicated noise-corrupted time series measurements are observed. The unknown mean of the data generating process is modelled as a piecewise linear function of time with an unknown number of change-points. The package infers the joint posterior distribution of the number and position of change-points as well as the unknown mean parameters per time-series by MCMC sampling. A-priori, the proposed model uses an overfitting number of mean parameters but, conditionally on a set of change-points, only a subset of them influences the likelihood. An exponentially decreasing prior distribution on the number of change-points gives rise to a posterior distribution concentrating on sparse representations of the underlying sequence, but also available is the Poisson distribution. See Papastamoulis et al (2017) <arXiv:1709.06111> for a detailed presentation of the method.

Details

The beast package deals with Bayesian estimation of change-points in the slope of multivariate time-series, introduced by Papastamoulis et al (2017). For a given period t = 1,\ldots,T we observe multiple time series which are assumed independent, each one consisting of multiple measurements (replicates). Each time series is assumed to have its own segmentation, which is common among its replicates. Thus, different time series have distinct mean parameters in the underlying normal distribution. The variance, which is assumed known, can be either shared between different time series or not and in practice it is estimated at a pre-processing stage.

Our model infers the joint posterior distribution of the number and location of change-points by MCMC sampling. For this purpose a Metropolis-Hastings MCMC sampler is used. The main function of the package is beast.

Assume that the observed data consists of N time-series, each one consisting of R variables (or replicates) measured at T consecutive time-points. The input of the main function beast should be given in the form of a list myDataList with the following attributes:

• length(myDataList) should be equal to R, that is, the number of variables (or replicates)

• dim(myDataList)[[1]], \ldots, dim(myDataList)[[R]] should be all T\times N, that is, rows and columns should correspond to time-points and different series, respectively.

Then, a basic usage of the package consists of the following commands:

• beastRun <- beast( myDataList = myDataList )

• print(beastRun)

• plot(beastRun)

which correspond to running the MCMC sampler, printing and plotting output summaries, respectively.

Author(s)

Panagiotis Papastamoulis

Maintainer: Panagiotis Papastamoulis <papapast@yahoo.gr>

References

Papastamoulis P., Furukawa T., van Rhijn N., Bromley M., Bignell E. and Rattray M. (2017). Bayesian detection of piecewise linear trends in replicated time-series with application to growth data modelling. arXiv:1709.06111 [stat.AP]

See Also

beast, print.beast.object, plot.beast.object

Examples

# toy-example (MCMC iterations not enough)
library('beast')	# load package
data("FungalGrowthDataset")	# load dataset
myIndex <- c(392, 62, 3, 117)	# run the sampler only for the 
#                                 specific subset of time-series
set.seed(1)	
# Run MCMC sampler with very small number of iterations (nIter):
run_mcmc <- beast(myDataList = FungalGrowthDataset, subsetIndex = myIndex, 
			zeroNormalization = TRUE, nIter = 40, burn = 20) 
# Print output:
print(run_mcmc)
# Plot output to file: "beast_plot.pdf"
plot(run_mcmc, fileName = "beast_plot_toy.pdf", timeScale=1/6, xlab = "hours", ylab = "growth")
# Run the following commands to obtain convergence:

## Not run: 
# This example illustrates the package using a subset of four 
#      time-series of the fungal dataset. 
library('beast')	# load package
data("FungalGrowthDataset")	# load dataset
myIndex <- c(392, 62, 3, 117)	# run the sampler only for the 
#                                 specific subset of time-series
set.seed(1)		# optional
# Run MCMC sampler with the default number of iterations (nIter =70000):
run_mcmc <- beast(myDataList = FungalGrowthDataset, subsetIndex = myIndex, 
			zeroNormalization = TRUE) 
# Print output:
print(run_mcmc)
# Plot output to file: "beast_plot.pdf"
plot(run_mcmc, fileName = "beast_plot.pdf", timeScale=1/6, xlab = "hours", ylab = "growth")
# NOTE 1: for a complete analysis remove the `subsetIndex = myIndex` argument.
# NOTE 2: `zeroNormalization = TRUE` is an optional argument that forces all 
#	   time-series to start from zero. It is not supposed to be used 
#	   for other applications.

## End(Not run)

Fungal Growth Dataset

Description

Time-series dataset with N\times R \times T growth levels for R = 3 replicates of N = 411 objects (mutants) measured every 10 minutes for T = 289 time-points. See Papastamoulis et al (2017) for a detailed description.

Usage

FungalGrowthDataset

Format

Time-series data.

References

Papastamoulis P., Furukawa T., van Rhijn N., Bromley M., Bignell E. and Rattray M. (2017). Bayesian detection of piecewise linear trends in replicated time-series with application to growth data modelling. arXiv:1709.06111 [stat.AP]

Main function

Description

This is the main function of the package, implementing the MCMC sampler described in Papastamoulis et al (2017).

Usage

beast(myDataList, burn, nIter, mhPropRange, mhSinglePropRange, startPoint,
    timeScale, savePlots, zeroNormalization, LRange, tau,
    gammaParameter, nu0, alpha0, beta0, subsetIndex, saveTheta, sameVariance,
    Prior
)

Arguments

Value

The output of the sampler is returned as a list, with the following features:

Note

The complexity prior distribution with parameter gammaParameter = 2 is the default prior assumption imposed on the number of change-points. Smaller (larger) values of gammaParameter will a-priori support larger (respectively: smaller) number of change-points.

For completeness purposes, the Poisson distribution is also allowed in the Prior. In this latter case, the gammaParameter denotes the rate parameter of the Poisson distribution. Note that in this case the interpretation of gammaParameter is reversed: Smaller (larger) values of gammaParameter will a-priori support smaller (respectively: larger) number of change-points.

Author(s)

Panagiotis Papastamoulis

References

Papastamoulis P., Furukawa T., van Rhijn N., Bromley M., Bignell E. and Rattray M. (2017). Bayesian detection of piecewise linear trends in replicated time-series with application to growth data modelling. arXiv:1709.06111 [stat.AP]

Examples

# toy-example (MCMC iterations not enough)
library('beast')	# load package
data("FungalGrowthDataset")	# load dataset
myIndex <- c(392, 62, 3, 117)	# run the sampler only for the 
#                                 specific subset of time-series
set.seed(1)	
# Run MCMC sampler with very small number of iterations (nIter):
run_mcmc <- beast(myDataList = FungalGrowthDataset, subsetIndex = myIndex, 
			zeroNormalization = TRUE, nIter = 40, burn = 20) 
# Print output:
print(run_mcmc)
# Plot output to file: "beast_plot.pdf"
plot(run_mcmc, fileName = "beast_plot_toy.pdf", timeScale=1/6, xlab = "hours", ylab = "growth")
# Run the following commands to obtain convergence:

## Not run: 
# This example illustrates the package using a subset of four 
#      time-series of the fungal dataset. 
library('beast')	# load package
data("FungalGrowthDataset")	# load dataset
myIndex <- c(392, 62, 3, 117)	# run the sampler only for the 
#                                 specific subset of time-series
set.seed(1)		# optional
# Run MCMC sampler with the default number of iterations (nIter =70000):
run_mcmc <- beast(myDataList = FungalGrowthDataset, subsetIndex = myIndex, 
			zeroNormalization = TRUE) 
# Print output:
print(run_mcmc)
# Plot output to file: "beast_plot.pdf"
plot(run_mcmc, fileName = "beast_plot.pdf", timeScale=1/6, xlab = "hours", ylab = "growth")
# NOTE 1: for a complete analysis remove the `subsetIndex = myIndex` argument.
# NOTE 2: `zeroNormalization = TRUE` is an optional argument that forces all 
#	   time-series to start from zero. It is not supposed to be used 
#	   for other applications.

## End(Not run)

Birth Probabilities

Description

This function defines the probability of proposing an addition of a change-point.

Usage

birthProbs(LRange)

Arguments

Value

Author(s)

Panagiotis Papastamoulis

Complexity prior distribution

Description

This function computes the complexity prior distribution on the number of change-points, defined as f(\ell) = P(\ell_n = \ell)\propto e^{-\alpha\ell\log(bT/\ell)}, a, b > 0; \ell = 0,1,2,\ldots. Note that this distribution has exponential decrease (Castillo and van der Vaart, 2012) when b>1+e, so we set b=3.72.

Usage

complexityPrior(Lmax = 20, gammaParameter, nTime)

Arguments

Value

Author(s)

Panagiotis Papastamoulis

References

Castillo I. and van der Vaart A (2012). Needles and Straw in a Haystack: Posterior concentration for possibly sparse sequences. The Annals of Statistics, 40(4), 2069–2101.

Compute the empirical mean.

Description

This function computes the empirical mean of the time-series.

Usage

computeEmpiricalPriorParameters(myDataList, nu0 = 1, alpha0 = 1, beta0 = 1)

Arguments

Value

Author(s)

Panagiotis Papastamoulis

Compute empirical posterior parameters

Description

Compute empirical posterior parameters

Usage

computePosteriorParameters(myDataList, priorParameters)

Arguments

Value

list of posterior parameters

Author(s)

Panagiotis Papastamoulis

Posterior parameters

Description

Posterior parameters

Usage

computePosteriorParametersFree(myDataList, priorParameters)

Arguments

Value

list of posterior parameters.

Author(s)

Panagiotis Papastamoulis

Move 3.b

Description

Implements Move 3.b of the Metropolis-Hastings MCMC sampler.

Usage

localProposal(cutPoints, nTime, mhPropRange, startPoint)

Arguments

Value

Author(s)

Panagiotis Papastamoulis

Log-likelihood function.

Description

Log-likelihood function.

Usage

logLikelihoodFullModel(myData, cutPoints, theta, startPoint)

Arguments

Value

log-likelihood value.

Author(s)

Panagiotis Papastamoulis

Log-prior.

Description

Logarithm of the prior distribution on the number of change-points.

Usage

logPrior(cutPoints, nTime, startPoint)

Arguments

Value

logarithm of the prior distribution.

Author(s)

Panagiotis Papastamoulis

MCMC sampler

Description

This function implements the Metropolis-Hastings MCMC sampler for individual time-series.

Usage

mcmcSampler(myData, nIter, finalIterationPdf, modelVariance, mhPropRange, 
mhSinglePropRange, movesRange, startPoint, postPar, dName, timeScale, 
burn, iterPerPlotPrefix, priorParameters, L = 3, LRange, tau, 
gammaParameter, saveTheta, Prior = "complexity")

Arguments

Value

MCMC output.

Author(s)

Panagiotis Papastamoulis

Printing

Description

Printing various unicode symbols.

Usage

myUnicodeCharacters()

Value

printed symbol

Zero normalization

Description

Zero normalization at 1st time-point

Usage

normalizeTime0(myDataList)

Arguments

Value

null

Author(s)

Panagiotis Papastamoulis

Plot function

Description

This function plots objects returned by the beast function. All output is diverted to a pdf file provided in the fileName argument.

Usage

## S3 method for class 'beast.object'
plot(x, fileName, width, height, pointsize, ylab, xlab, timeScale, myPal, boxwex, ...)

Arguments

Details

The function will produce a plot with all time-series coloured according to the corresponding number of change-points. In addition, it will generate individual plots per time-series displaying the observed data with boxplots which summarize the posterior distribution of change-points locations, conditionally on the most probable number of change-points.

Author(s)

Panagiotis Papastamoulis

Print function

Description

This function prints a summary of objects returned by the beast function.

Usage

## S3 method for class 'beast.object'
print(x, ...)

Arguments

Details

The function prints a summary of the most probable number (MAP) of change-points per time-series in the form of a table, as well as a list containing the MAP number of change-points and the corresponding locations (posterior medians) per time-series.

Author(s)

Panagiotis Papastamoulis

Move 2

Description

Proposes an update of \theta according to Metropolis-Hastings move 2.

Usage

proposeTheta(thetaOld, tau, alpha, beta)

Arguments

Value

Author(s)

Panagiotis Papastamoulis

Prior random numbers

Description

Generation of mean values according to the prior

Usage

simMultiIndNormInvGamma(mu, nu, alpha, beta)

Arguments

Value

null

Author(s)

Panagiotis Papastamoulis

Generate change-points according to the prior

Description

Generate change-points according to the prior distribution conditionally on a given number of change-points.

Usage

simulateFromPrior(nTime, startPoint, L = 3)

Arguments

Value

Author(s)

Panagiotis Papastamoulis

Move 3.b

Description

Implement Metropolis-Hastings Move 3.b.

Usage

singleLocalProposal(cutPoints, nTime, mhSinglePropRange, startPoint)

Arguments

Value

Author(s)

Panagiotis Papastamoulis

Truncated Poisson pdf

Description

Probability density function of the truncated Poisson distribution.

Usage

truncatedPoisson(Lmax = 20, gammaParameter = 1)

Arguments

Value

Author(s)

Panagiotis Papastamoulis

Move 1

Description

Update the number of change-points according to Metropolis-Hastings move 1.

Usage

updateNumberOfCutpoints(cutPoints, nTime, startPoint, LRange, birthProbs)

Arguments

Value

Author(s)

Panagiotis Papastamoulis