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

```
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)

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 \(\ge 0\) 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"`

.

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.

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

These functions work with a general univariate state-space model
with state vector `a`, transitions `a <- T a + R e`,
\(e \sim {\cal N}(0, \kappa Q)\) and observation
equation `y = Z'a + eta`,
\((eta\equiv\eta), \eta \sim {\cal N}(0, \kappa h)\).
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).

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.
10.2307/2346910.

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

```
# NOT RUN {
## 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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