# Statistics of the True ARMA Process

### Description

A collection and description of functions
to compute statistics of a true ARMA time
series process.

The functions are:

`armaRoots` | Roots of the characteristic ARMA polynomial, |

`armaTrueacf` | True autocorrelation function of an ARMA process. |

### Usage

1 2 3 | ```
armaRoots(coefficients, n.plot = 400, digits = 4, ...)
armaTrueacf(model, lag.max = 20, type = c("correlation", "partial", "both"),
doplot = TRUE)
``` |

### Arguments

`coefficients` |
[armaRoots] - |

`digits` |
[armaRoots] - |

`doplot` |
[armaRoots] - |

`lag.max` |
[armaTrueacf] - |

`model` |
[armaTrueacf] - |

`n` |
[armaSim] - |

`n.plot` |
[armaRoots] - |

`type` |
[armaTrueacf] - |

`...` |
additional arguments to be passed. |

### Value

`armaRoots`

returns a three column data frame with the real, the imaginary part
and the radius of the roots. The number of rows corresponds
to the coefficients.

`armaTrueacf`

returns a two column data frame with the lag and the correlation
function.

### Author(s)

M. Plummer and B.D. Ripley for `ar`

functions and code,

B.D. Ripley for `arima`

and `ARMAacf`

functions and code,

C. Fraley and F. Leisch for `fracdiff`

functions and code, and

Diethelm Wuertz for the Rmetrics **R**-port.

### References

Brockwell, P.J. and Davis, R.A. (1996);
*Introduction to Time Series and Forecasting*,
Second Edition, Springer, New York.

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

Gardner, G, Harvey, A.C., 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.

Hannan E.J. and Rissanen J. (1982);
*Recursive Estimation of Mixed Autoregressive-Moving
Average Order.*
Biometrika 69, 81–94.

Harvey, A.C. (1993);
*Time Series Models*,
2nd Edition, Harvester Wheatsheaf, Sections 3.3 and 4.4.

Jones, R.H. (1980);
*Maximum likelihood fitting of ARMA models to time
series with missing observations*,
Technometrics, 20, 389–395.

Percival, D.P. and Walden, A.T. (1998);
*Spectral Analysis for Physical Applications.*
Cambridge University Press.

Whittle, P. (1963);
*On the fitting of multivariate autoregressions
and the approximate canonical factorization of a spectral
matrix.*
Biometrika 40, 129–134.

Haslett J. and Raftery A.E. (1989);
*Space-time Modelling with Long-memory Dependence: Assessing
Ireland's Wind Power Resource (with Discussion)*,
Applied Statistics 38, 1–50.

### Examples

1 2 3 4 5 6 7 8 9 10 11 | ```
## armaRoots -
# Calculate and plot the roots of an ARMA process:
par(mfrow = c(2, 2), cex = 0.7)
coefficients = c(-0.5, 0.9, -0.1, -0.5)
armaRoots(coefficients)
## armaTrueacf -
model = list(ar = c(0.3, +0.3), ma = 0.1)
armaTrueacf(model)
model = list(ar = c(0.3, -0.3), ma = 0.1)
armaTrueacf(model)
``` |