bsts  R Documentation 
Uses MCMC to sample from the posterior distribution of a Bayesian structural time series model. This function can be used either with or without contemporaneous predictor variables (in a time series regression).
If predictor variables are present, the regression coefficients are fixed (as opposed to time varying, though time varying coefficients might be added as state component). The predictors and response in the formula are contemporaneous, so if you want lags and differences you need to put them in the predictor matrix yourself.
If no predictor variables are used, then the model is an ordinary state space time series model.
The model allows for several useful extensions beyond standard Bayesian dynamic linear models.
A spikeandslab prior is used for the (static) regression component of models that include predictor variables. This is especially useful with large numbers of regressor series.
Both the spikeandslab component (for static regressors) and
the Kalman filter (for components of time series state) require
observations and state variables to be Gaussian. The bsts
package allows for nonGaussian error families in the observation
equation (as well as some state components) by using data
augmentation to express these families as conditionally
Gaussian.
As of version 0.7.0, bsts
supports having multiple
observations at the same time point. In this case the basic model
is taken to be
y_{t,j} = Z_t^T \alpha_t + \beta^Tx_{t, j} + \epsilon_{t,j}.
This is a regression model where all observations with the same time point share a common time series effect.
bsts(formula,
state.specification,
family = c("gaussian", "logit", "poisson", "student"),
data,
prior,
contrasts = NULL,
na.action = na.pass,
niter,
ping = niter / 10,
model.options = BstsOptions(),
timestamps = NULL,
seed = NULL,
...)
formula 
A formula describing the regression portion of the relationship between y and X. If no regressors are desired then the formula can be replaced by a numeric vector giving the time series to be modeled. Missing values are not allowed in predictors, but they are allowed in the response variable. If the response variable is of class 
state.specification 
A list with elements created by

family 
The model family for the observation equation. NonGaussian model families use data augmentation to recover a conditionally Gaussian model. 
data 
An optional data frame, list or environment (or object
coercible by 
prior 
If regressors are supplied in the model formula, then
this is a prior distribution for the regression component of the
model, as created by If the model contains no regressors, then this is simply the prior
on the residual standard deviation, expressed as an object created
by 
contrasts 
An optional list containing the names of contrast
functions to use when converting factors numeric variables in a
regression formula. This argument works exactly as it does in

na.action 
What to do about missing values. The default is to allow missing responses, but no missing predictors. Set this to na.omit or na.exclude if you want to omit missing responses altogether. 
niter 
A positive integer giving the desired number of MCMC draws. 
ping 
A scalar giving the desired frequency of status messages.
If ping > 0 then the program will print a status message to the
screen every 
model.options 
An object (list) returned by

timestamps 
The timestamp associated with each value of the
response. This argument is primarily useful in cases where the
response has missing gaps, or where there are multiple observations
per time point. If the response is a "regular" time series with a
single observation per time point then you can leave this argument
as 
seed 
An integer to use as the random seed for the underlying
C++ code. If 
... 
Extra arguments to be passed to

If the model family is logit, then there are two ways one can format the response variable. If the response is 0/1, TRUE/FALSE, or 1/1, then the response variable can be passed as with any other model family. If the response is a set of counts out of a specified number of trials then it can be passed as a twocolumn matrix, where the first column contains the counts of successes and the second contains the count of failures.
Likewise, if the model family is Poisson, the response can be passed as a single vector of counts, under the assumption that each observation has unit exposure. If the exposures differ across observations, then the resopnse can be a two column matrix, with the first column containing the event counts and the second containing exposure times.
An object of class bsts
which is a list with the
following components
coefficients 
A 
sigma.obs 
A vector of length 
The returned object will also contain named elements holding the MCMC
draws of model parameters belonging to the state models. The names of
each component are supplied by the entries in
state.specification
. If a model parameter is a scalar, then
the list element is a vector with niter
elements. If the
parameter is a vector then the list element is a matrix with
niter
rows. If the parameter is a matrix then the list element
is a 3way array with first dimension niter
.
Finally, if a model formula was supplied, then the returned object will contain the information necessary for the predict method to build the design matrix when a new prediction is made.
Steven L. Scott steve.the.bayesian@gmail.com
Scott and Varian (2014) "Predicting the Present with Bayesian Structural Time Series", International Journal of Mathematical Modelling and Numerical Optimization. 4–23.
Scott and Varian (2015) "Bayesian Variable Selection for Nowcasting Economic Time Series", Economic Analysis of the Digital Economy, pp 119135.
Harvey (1990), "Forecasting, structural time series, and the Kalman filter", Cambridge University Press.
Durbin and Koopman (2001), "Time series analysis by state space methods", Oxford University Press.
George and McCulloch (1997) "Approaches for Bayesian variable selection", Statistica Sinica pp 339–374.
bsts
,
AddLocalLevel
,
AddLocalLinearTrend
,
AddSemilocalLinearTrend
,
AddSeasonal
AddDynamicRegression
SpikeSlabPrior
,
SdPrior
.
## Example 1: Time series (ts) data
data(AirPassengers)
y < log(AirPassengers)
ss < AddLocalLinearTrend(list(), y)
ss < AddSeasonal(ss, y, nseasons = 12)
model < bsts(y, state.specification = ss, niter = 500)
pred < predict(model, horizon = 12, burn = 100)
par(mfrow = c(1,2))
plot(model)
plot(pred)
## Not run:
MakePlots < function(model, ask = TRUE) {
## Make all the plots callable by plot.bsts.
opar < par(ask = ask)
on.exit(par(opar))
plot.types < c("state", "components", "residuals",
"prediction.errors", "forecast.distribution")
for (plot.type in plot.types) {
plot(model, plot.type)
}
if (model$has.regression) {
regression.plot.types < c("coefficients", "predictors", "size")
for (plot.type in regression.plot.types) {
plot(model, plot.type)
}
}
}
## Example 2: GOOG is the Google stock price, an xts series of daily
## data.
data(goog)
ss < AddSemilocalLinearTrend(list(), goog)
model < bsts(goog, state.specification = ss, niter = 500)
MakePlots(model)
## Example 3: Change GOOG to be zoo, and not xts.
goog < zoo(goog, index(goog))
ss < AddSemilocalLinearTrend(list(), goog)
model < bsts(goog, state.specification = ss, niter = 500)
MakePlots(model)
## Example 4: Naked numeric data works too
y < rnorm(100)
ss < AddLocalLinearTrend(list(), y)
model < bsts(y, state.specification = ss, niter = 500)
MakePlots(model)
## Example 5: zoo data with intraday measurements
y < zoo(rnorm(100),
seq(from = as.POSIXct("20120101 7:00 EST"), len = 100, by = 100))
ss < AddLocalLinearTrend(list(), y)
model < bsts(y, state.specification = ss, niter = 500)
MakePlots(model)
\dontrun{
## Example 6: Including regressors
data(iclaims)
ss < AddLocalLinearTrend(list(), initial.claims$iclaimsNSA)
ss < AddSeasonal(ss, initial.claims$iclaimsNSA, nseasons = 52)
model < bsts(iclaimsNSA ~ ., state.specification = ss, data =
initial.claims, niter = 1000)
plot(model)
plot(model, "components")
plot(model, "coefficients")
plot(model, "predictors")
}
## End(Not run)
## Not run:
## Example 7: Regressors with multiple time stamps.
number.of.time.points < 50
sample.size.per.time.point < 10
total.sample.size < number.of.time.points * sample.size.per.time.point
sigma.level < .1
sigma.obs < 1
## Simulate some fake data with a local level state component.
trend < cumsum(rnorm(number.of.time.points, 0, sigma.level))
predictors < matrix(rnorm(total.sample.size * 2), ncol = 2)
colnames(predictors) < c("X1", "X2")
coefficients < c(10, 10)
regression < as.numeric(predictors %*% coefficients)
y.hat < rep(trend, sample.size.per.time.point) + regression
y < rnorm(length(y.hat), y.hat, sigma.obs)
## Create some time stamps, with multiple observations per time stamp.
first < as.POSIXct("20130324")
dates < seq(from = first, length = number.of.time.points, by = "month")
timestamps < rep(dates, sample.size.per.time.point)
## Run the model with a local level trend, and an unnecessary seasonal component.
ss < AddLocalLevel(list(), y)
ss < AddSeasonal(ss, y, nseasons = 7)
model < bsts(y ~ predictors, ss, niter = 250, timestamps = timestamps,
seed = 8675309)
plot(model)
plot(model, "components")
## End(Not run)
## Example 8: NonGaussian data
## Poisson counts of shark attacks in Florida.
data(shark)
logshark < log1p(shark$Attacks)
ss.level < AddLocalLevel(list(), y = logshark)
model < bsts(shark$Attacks, ss.level, niter = 500,
ping = 250, family = "poisson", seed = 8675309)
## Poisson data can have an 'exposure' as the second column of a
## twocolumn matrix.
model < bsts(cbind(shark$Attacks, shark$Population / 1000),
state.specification = ss.level, niter = 500,
family = "poisson", ping = 250, seed = 8675309)
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