ArmaStatistics: Statistics of the True ARMA Process
In fArma: Rmetrics - Modelling ARMA Time Series Processes

Description Usage Arguments Value Author(s) References Examples

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.

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

`coefficients`	[armaRoots] - a numeric vector with the coefficients of the characterisitic polynomial.
`digits`	[armaRoots] - output precision, an integer value.
`doplot`	[armaRoots] - a logical. Should a plot be displayed?
`lag.max`	[armaTrueacf] - maximum number of lags at which to calculate the acf or pacf, an integer value by default 20.
`model`	[armaTrueacf] - a specification of the ARMA model with two elements: `model$ar` is the vector of the AR coefficients, and `model$ma` is the vector of the MA coefficients.
`n`	[armaSim] - an integer value setting the length of the series to be simulated (optional if `innov` is provided). The default value is 100.
`n.plot`	[armaRoots] - the number of data points to plot the unit circle; an integer value.
`type`	[armaTrueacf] - a character string, "correlation" to compute the true autocorrelation function, "partial" to compute the true partial autocorrelation function, or "both" if both functions are desired. The start of one of the strings will suffice.
`...`	additional arguments to be passed.

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.

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.

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.

## 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)

Loading required package: timeDate
Loading required package: timeSeries
Loading required package: fBasics
       re      im   dist
1  1.0291  0.7284 1.2608
2 -1.0000  0.0000 1.0000
3 -1.2582  0.0000 1.2582
4  1.0291 -0.7284 1.2608
   lag         acf
0    0 1.000000000
1    1 0.525423729
2    2 0.457627119
3    3 0.294915254
4    4 0.225762712
5    5 0.156203390
6    6 0.114589831
7    7 0.081237966
8    8 0.058748339
9    9 0.041995892
10  10 0.030223269
11  11 0.021665748
12  12 0.015566705
13  13 0.011169736
14  14 0.008020932
15  15 0.005757201
16  16 0.004133440
17  17 0.002967192
18  18 0.002130190
19  19 0.001529215
20  20 0.001097821
   lag           acf
0    0  1.000000e+00
1    1  2.935178e-01
2    2 -2.119446e-01
3    3 -1.516387e-01
4    4  1.809177e-02
5    5  5.091916e-02
6    6  9.848216e-03
7    7 -1.232128e-02
8    8 -6.650849e-03
9    9  1.701130e-03
10  10  2.505594e-03
11  11  2.413392e-04
12  12 -6.792764e-04
13  13 -2.761847e-04
14  14  1.209275e-04
15  15  1.191337e-04
16  16 -5.381576e-07
17  17 -3.590154e-05
18  18 -1.060902e-05
19  19  7.587758e-06
20  20  5.459032e-06