Kalman Filtering

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Description

Use Kalman Filtering to find the (Gaussian) log-likelihood, or for forecasting or smoothing.

Usage

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KalmanLike(y, mod, nit = 0L, update = FALSE)
KalmanRun(y, mod, nit = 0L, update = FALSE)
KalmanSmooth(y, mod, nit = 0L)
KalmanForecast(n.ahead = 10L, mod, update = FALSE)

makeARIMA(phi, theta, Delta, kappa = 1e6,
          SSinit = c("Gardner1980", "Rossignol2011"),
          tol = .Machine$double.eps)

Arguments

y

a univariate time series.

mod

a list describing the state-space model: see ‘Details’.

nit

the time at which the initialization is computed. nit = 0L implies that the initialization is for a one-step prediction, so Pn should not be computed at the first step.

update

if TRUE the update mod object will be returned as attribute "mod" of the result.

n.ahead

the number of steps ahead for which prediction is required.

phi, theta

numeric vectors of length ≥ 0 giving AR and MA parameters.

Delta

vector of differencing coefficients, so an ARMA model is fitted to y[t] - Delta[1]*y[t-1] - ....

kappa

the prior variance (as a multiple of the innovations variance) for the past observations in a differenced model.

SSinit

a string specifying the algorithm to compute the Pn part of the state-space initialization; see ‘Details’.

tol

tolerance eventually passed to solve.default when SSinit = "Rossignol2011".

Details

These functions work with a general univariate state-space model with state vector a, transitions a <- T a + R e, e ~ N(0, kappa Q) and observation equation y = Z'a + eta, eta ~ N(0, kappa h). The likelihood is a profile likelihood after estimation of kappa.

The model is specified as a list with at least components

T

the transition matrix

Z

the observation coefficients

h

the observation variance

V

RQR'

a

the current state estimate

P

the current estimate of the state uncertainty matrix Q

Pn

the estimate at time t-1 of the state uncertainty matrix Q (not updated by KalmanForecast.

KalmanSmooth is the workhorse function for tsSmooth.

makeARIMA constructs the state-space model for an ARIMA model, see also arima.

The state-space initialization has used Gardner et al's method (SSinit = "Gardner1980"), as only method for years. However, that suffers sometimes from deficiencies when close to non-stationarity. For this reason, it may be replaced as default in the future and only kept for reproducibility reasons. Explicit specification of SSinit is therefore recommended, notably also in arima(). The "Rossignol2011" method has been proposed and partly documented by Raphael Rossignol, Univ. Grenoble, on 2011-09-20 (see PR#14682, below), and later been ported to C by Matwey V. Kornilov. It computes the covariance matrix of (X_{t-1},...,X_{t-p},Z_t,...,Z_{t-q}) by the method of difference equations (page 93 of Brockwell and Davis), apparently suggested by a referee of Gardner et al (see p.314 of their paper).

Value

For KalmanLike, a list with components Lik (the log-likelihood less some constants) and s2, the estimate of kappa.

For KalmanRun, a list with components values, a vector of length 2 giving the output of KalmanLike, resid (the residuals) and states, the contemporaneous state estimates, a matrix with one row for each observation time.

For KalmanSmooth, a list with two components. Component smooth is a n by p matrix of state estimates based on all the observations, with one row for each time. Component var is a n by p by p array of variance matrices.

For KalmanForecast, a list with components pred, the predictions, and var, the unscaled variances of the prediction errors (to be multiplied by s2).

For makeARIMA, a model list including components for its arguments.

Warning

These functions are designed to be called from other functions which check the validity of the arguments passed, so very little checking is done.

References

Durbin, J. and Koopman, S. J. (2001) Time Series Analysis by State Space Methods. Oxford University Press.

Gardner, G, Harvey, A. C. and Phillips, G. D. A. (1980) Algorithm AS154. An algorithm for exact maximum likelihood estimation of autoregressive-moving average models by means of Kalman filtering. Applied Statistics 29, 311–322.

R bug report PR#14682 (2011-2013) https://bugs.r-project.org/bugzilla3/show_bug.cgi?id=14682.

See Also

arima, StructTS. tsSmooth.

Examples

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## an ARIMA fit
fit3 <- arima(presidents, c(3, 0, 0))
predict(fit3, 12)
## reconstruct this
pr <- KalmanForecast(12, fit3$model)
pr$pred + fit3$coef[4]
sqrt(pr$var * fit3$sigma2)
## and now do it year by year
mod <- fit3$model
for(y in 1:3) {
  pr <- KalmanForecast(4, mod, TRUE)
  print(list(pred = pr$pred + fit3$coef["intercept"], 
             se = sqrt(pr$var * fit3$sigma2)))
  mod <- attr(pr, "mod")
}

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