R: Kalman Filtering

KalmanLike {stats} | R Documentation

## Kalman Filtering

### Description

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

### Usage

    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 AS 154: An algorithm for exact maximum likelihood estimation of autoregressive-moving average models by means of Kalman filtering. Applied Statistics, 29, 311–322. doi: 10.2307/2346910.

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

### See Also

`arima`, `StructTS`. `tsSmooth`.

### Examples

    ## 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")
    }

[Package stats version 4.0.5 Index]