voila: Variational Inference for Langevin Equations

Description Usage Arguments Value Examples

Estimates the drift and diffusion functions of a Langevin equation from a single trajectory of the process. The estimation models the drift and diffusion terms as gaussian processes and makes use of Variational Inference (VI) to approximate the posteriors.

1 2	sde_vi(timeSeriesIndex = 1, timeSeries, samplingPeriod, xm, fKernel, sKernel, v, maxIterations = 20, hyperparamIterations = 5, relTol = 1e-06)

`timeSeriesIndex`	Integer indicating in which of the input dimensions the inference should be made.
`timeSeries`	Matrix of size number_observations x input_dimensionality representing the multivariate time series from which the estimation shall be made. Note that if the time series is univariate, a matrix with just one column should be used.
`samplingPeriod`	The sampling period of the time series
`xm`	Matrix with the same number of columns as the timeSeries representing the initial values of the pseudo-inputs.
`fKernel, sKernel`	An object representing a gaussian process' kernel used to model the drift/diffusion term, e.g. a sde_kernel object.
`v`	Numeric value representing the prior mean for the diffusion process.
`maxIterations`	Integer value specifying the maximum number of iterations of the method.
`hyperparamIterations`	Integer value specifying the maximum number of iterations to be used when optimizing the hyperparameters.
`relTol`	Relative tolerance used to assess the convergence of the method.

A list with two sgp_sde objects representing the drift and diffusion estimates (drift and diff fields) and other useful information related with the inference process.

likelihoodLowerBound: The likelihood lower bound in each of the iterations.
inducingPoints: The inducing points that result of the inference process.
driftHyperparams, diffHyperparams: the hyperparameters of the drift/diffusion kernels after the optimization.
driftKernelParameters, diffKernelParameters: Lists containing the mean and covariance matrix of the variational distributions of the drift/diffusion terms.

## Not run: 
data("ornstein")
# sde_vi takes as input a matrix representing the time series.
# The matrix should have as dimensions: number_samples x input_dimension
x = matrix(as.numeric(ornstein), ncol = 1)
# the sampling period
h = deltat(ornstein)
# we shall use only 10 pseudo-inputs
m = 10
uncertainty = 5
# There is only one dimension and thus, the target has to be 1
targetIndex = 1
inputDim = 1
# A small value to regularize the covariance matrices
epsilon = 1e-5
# Initialize the pseudo-inputs spreading them across the x support
xm = matrix(seq(min(x), max(x), len = m), ncol = 1)
# Create the kernels
diffParams = select_diffusion_parameters(x, h, priorOnSd = uncertainty)
v = diffParams$v
driftKer = sde_kernel("exp_kernel",
                      list('amplitude' = uncertainty,
                           'lengthScales' = 1),
                      inputDim, epsilon)
diffKer =  sde_kernel("exp_const_kernel",
                      list('maxAmplitude' = diffParams$kernelAmplitude,
                           'expAmplitude' = diffParams$kernelAmplitude * 1e-5,
                           'lengthScales' = 1),
                      inputDim, epsilon)

# perform the inference
inference = sde_vi(targetIndex, x, h, xm, driftKer, diffKer, v)

# get predictions in some new values of x
predictionSupport = matrix(seq(quantile(x,0.05), quantile(x,0.95), len = 100),
                           ncol = 1)
driftPred = predict(inference$drift, predictionSupport)
diffPred = predict(inference$diff, predictionSupport, log = TRUE)

plot(driftPred, main = "Drift", ylab = "f(x)", xlab="x")
# the true drift
lines(predictionSupport, -predictionSupport, col = 2)
plot(diffPred, main = "Diffusion", ylab = "g(x)", xlab="x")
# the true diffusion
lines(predictionSupport, rep(1.5, rep(length(predictionSupport))), col = 2)

## End(Not run)