BiRS.impute: Brownian Bridge Data Imputation for Bivariate Diffusion...

In DiffusionRimp: Inference and Analysis for Diffusion Processes via Data Imputation and Method of Lines

Description

BiRS.impute performs inference on bivariate diffusion processes with quite arbitrary drift functionals by imputing missing sample paths with Brownian bridges. The procedure was developed by Roberts and Stramer (2001) and subsequently extended in later papers (). Currently, the diffusion is assumed to take on the form

dX_t = mux(X_t, Y_t, t, theta)dt+sigma[1]f(X_t)dW_t^1

dY_t = muy(X_t, Y_t, t, theta)dt+sigma[2]g(Y_t)dW_t^2,

where mux(X_t, Y_t, t, theta) and muy(X_t, Y_t, t, theta) may be defined with near-complete freedom as R-functions in the current workspace. f(.) and g(.) may take on predefined forms (see details).

Usage

BiRS.impute(X, time, M, theta, sds, diff.type = c(1, 1),
            burns = min(floor(updates/2),25000), updates, plot.chain = TRUE,
            imputation.plot = FALSE, palette = 'mono')

Arguments

X	N x 2 matrix of coordinates at which the diffusion process was observed, where N is the number of observations.
time	A vector of time nodes at which the process was observed.
M	The number of points to impute between successive observations. Note that as the transition horizon increases, more points may be required in order to get desirable acceptance rates for Brownian bridges. Note that some transition horizons may be removed from likelihood calculations - see exclude.
theta	Starting parameters for the model process.
sds	Proposal standard deviations for the drift parameter chain.
diff.type	2-Component vector of indicators, each assuming values 1, 2 or 3, for which of the predefined volatility structures to impose - see note [3].
burns	The number of updates to burn (only affects traceplots).
updates	The number of updates to perform.
imputation.plot	Display imputed paths for successive updates - see note [4]. Default = FALSE.
plot.chain	Display the resulting MCMC chain - see notes [5], [6] and [7].
palette	Colour palette for drawing trace plots. Default palette = 'mono', otherwise a qualitative palette will be used.

Details

DETAILS ABOUT THE METHODOLOGY

Value

per.matrix	Matrix containing the MCMC chain updates.
acceptence.rate	Acceptance rates for the two parameter chains.
bridge.rate	Average Acceptance rate for Brownian bridge updates.
run.time	The total run time of the algorithm.

Note

Note [1]: The functional form of the drift components may be defined by the user as R-functions. Although the function body may take on arbitrary forms both the name of the drift functions and the input structure must assume the form

mux = function(variable1, variable2, time, theta){...} and

muy = function(variable1, variable2, time, theta){...}.

Note that theta is a reserved variable for the parameters. It is left to the user to ensure that the functional forms are not nonsensical. See the examples below.

Note [2]: Both drift functions have to be defined in order for the algorithm to execute.

Note [3]: The functional form for the volatility of each dimension may take on one of the following: f(X_t)=1, corresponding to constant volatility , f(X_t)=√{X_t} corresponding to CIR type models and f(X_t)=X_t such as for geometric Brownian motion. Corresponding indicators are simply given buy 1, 2, and 3.

Note [4]: Paths are imputed on a unit -volatility process. For display purposes the back-transformed imputed trajectories may plotted along with vertical lines indicating which bridges remained unchanged over successive updates.

Note [5]: By default the MCMC chain is displayed in a panel plot. Standard MCMC diagnostics may be performed on this chain which is returned in the value list as per.matrix.

Note [6]: In addition to the MCMC chain, acceptance rates are given for both the drift vector parameter updates as well as the volatility parameter updates. A, target region for the acceptance rate trajectories is displayed in blue.

Note [7]: Average acceptance rates are indicated for bridges per the transition number by blue bars. A target region for the bridge acceptance rates is given in light blue. Average acceptance rates that are lower than 60% are indicated along with their respective transition number. Vertical grey lines indicate exclude-d transitions.

Author(s)

Etienne A.D. Pienaar [e-mail]

References

Dellaportas, P. 2006 Bayesian model selection for partially observed diffusion models. Biometrika, 93(4): 809.

Kalogeropoulos, K., Dellaportas, P., and Roberts, G. O. 2011 Likelihood based inference for correlated diffusions. Canadian Journal of Statistics, 39(1):52–72.

Roberts, G. and Stramer, O. 2001 On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm. Biometrika, 88:603–621.

See Also

RS.impute, BiMOL.density.

Examples

 

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DiffusionRimp documentation built on May 2, 2019, 2 p.m.