ArmaModelling            package:fSeries            R Documentation

_I_n_t_e_g_r_a_t_e_d _A_R_M_A _T_i_m_e _S_e_r_i_e_s _M_o_d_e_l_l_i_n_g

_D_e_s_c_r_i_p_t_i_o_n:

     A collection and description of simple to  use functions to model
     univariate autoregressive  moving average time series processes,
     including  time series simulation, parameter estimation, 
     diagnostic analysis of the fit, and predictions  of future values. 

     The functions are:

       'armaSim'    Simulates an artificial ARMA time series process,
       'armaFit'    Fits the parameters of an ARMA time series process,
       'predict'    Forecasts an ARMA time series process,
       'print'      S3 Print Method,
       'plot'       S3 Plot Method,
       'summary'    S3 Summary Method,
       'fitted'     S3 Method, returns fitted values,
       'residuals'  S3 Method, returns residuals.

     The function 'armaFit' allows for the following formula arguments:

       'x ~ ar()'        autoregressive time series processes,
       'x ~ arma()'      autoregressive moving average processes,
       'x ~ arima()'     autoregressive integrated moving average processes, and
       'x ~ fracdiff()'  fractionally integrated ARMA processes.

     For the first and third selection 'x ~ ar()' or 'x ~ arima()' the
     function uses the AR and ARIMA modelling as implemented in  R's
     'ts' package, for the second selection 'x~arma()'  the function
     uses the ARMA modelling implemented in R's  contributed 'tseries'
     package, and for the last selection  'x~fracdiff()' the function
     uses fractional ARIMA modelling  from R's contributed 'fracdiff'
     package. Note that AR, MA, and ARMA processes can all be modelled
     by the same algorithm specifying the formula 'x~arima(p,d,q)' in
     the proper way,  i.e. setting 'd=0' and choosing the orders of 'p'
     and  'q' as zero in agreement with the desired model
     specification.

     Alternatively, one can use the functions from R's '"stats"'
     package:

       'arima.sim'          Simulates from an ARIMA time series model,
       'ar, arima, arima0'  Fits an AR, ARIMA model to a univariate time series,
       'predict.ARIMA'      Forecast from models fitted by "arima",
       'tsdiag'             Plots time-series diagnostics.

     In addition there are two functions for the analysis of the true  
     ARMA process. The first investigates the characteristic polynomial
     and the second the true autocorrelation function:

       'armaRoots'     Roots of the characteristic ARMA polynomial,
       'armaTrueacf'   True autocorrelation function of an ARMA process,
       'armaToeplitz'  True Toeplitz matrix of a vector,
       'armaFischer'   Fischer Information Matrix of an ARMA Process.

_U_s_a_g_e:

     armaSim(model = list(ar = c(0.5, -0.5), d = 0, ma = 0.1), n = 100, 
         innov = NULL, n.start = 100, start.innov = NULL, rand.gen = rnorm, 
         rseed = NULL, ...) 

     armaFit(formula = x ~ arima(2, 0, 1), 
         method = c("CSS-ML", "ML", "CSS", "yw", "burg1", "burg2", "ols", "mle"), 
         include.mean = TRUE, fixed = NULL, fracdiff.M = 100, fracdiff.h = -1, 
         title = NULL, description = NULL, ...)

     ## S3 method for class 'fARMA':
     predict(object, n.ahead = 10, n.back = 50, conf = c(80, 95), 
         doplot = TRUE, doprint = TRUE, ...)
         
     ## S3 method for class 'fARMA':
     print(x, ...)
     ## S3 method for class 'fARMA':
     plot(x, gof.lag = 10, reference.grid = TRUE, 
         col = "steelblue", ...)
     ## S3 method for class 'fARMA':
     summary(object, doplot = TRUE, ...)

     ## S3 method for class 'fARMA':
     fitted(object, ...)
     ## S3 method for class 'fARMA':
     residuals(object, ...)

     armaRoots(coefficients, n.plot = 400, digits = 4, ...) 
     armaTrueacf(model, lag.max = 20, type = "correlation", doplot = TRUE) 
     armaToeplitz(x)
     armaFischer(model = list(ar = c(0.5, -0.5), ma = 0.1))

_A_r_g_u_m_e_n_t_s:

coefficients: [armaRoots] - 
           a numeric vector with the coefficients of the
          characterisitic  polynomial. 

col, reference.grid: [plot] - 
           a character string to set the plot color, by default 
          '"steelblue4"'; and a logical flag, by default 'TRUE', to
          draw a reference grid on the plot or not. 

description: a character string which allows for a brief description. 

  digits: [armaRoots] - 
           output precision, an integer value. 

  doplot: [armaRoots] - 
           a logical. Should a plot be displayed? 
           [predict][summary] - 
           is used by the 'predict' and 'summary' methods. By default,
          this value is set to 'TRUE' and thus the function calls
          generate  beside written also graphical printout. Additional
          arguments required by underlying functions have  to be passed
          through the 'dots' argument. 

 doprint: [predict] - 
           should intermediate results be printed? Arguments used by
          the  'predict' method. 

   fixed: [armaFit] - 
           is an optional numeric vector of the same length as the
          total number of parameters. If supplied, only 'NA' entries in
          'fixed' will be varied. In this way subset ARMA processes can
          be modeled. ARIMA modelling supports this option. Thus for
          estimating parameters of subset ARMA and AR models the  most
          easiest way is to specify them by the formulas  'x~ARIMA(p,
          0, q)' and 'x~ARIMA(p, 0, 0)', respectively. 

 formula: [armaFit] - 
           a formula specifying the general structure of the ARMA form.
           Can have one of the forms   'x ~ ar(q)',  'x ~ arma(p, q)', 
          'x ~ arima(p, d, q)', or 'x ~ fracdiff(p, q)'. 'x' is the
          response variable and must appear in the  formula expression.
          In the first case R's function 'ar' from the 'ts' package
          will be used to estimate the parameters, in the second case
          R's function 'arma' from the 'tseries' package will be used,
          in the third case R's function 'arima' from  the 'ts' package
          will be used, and in the last case R's function 'fracdiff'
          from the 'fracdiff' package will be used. The state space
          modelling based 'arima' function allows also to fit ARMA
          models using 'arima(p, d=0, q)', and AR models using
          'arima(q, d=0, q=0)', or pure MA models using 'arima(q=0,
          d=0, p)'. (Exogenous variables are also  allowed and can be
          passed through the '...' argument.) 

fracdiff.M, fracdiff.h: two numeric parameter values for the
          '"fracdiff"'  estimator. 'M' is the number of terms in the
          likelihood  approximation. 'h' is the size of the finite
          difference  interval for numerical derivatives. By default,
          or if negative,  'h=min(0.1,eps.5*(1+ abs(cllf)))', where 
          'clff:=log.max.likelihood', as returned, and 
          'eps.5:=sqrt(.Machine$double.neg.eps)', typically  '1.05e-8'. 

 gof.lag: [print][plot][summary][predict] - 
           the maximum number of lags for a goodness-of-fit test. 

include.mean: [armaFit] - 
           Should the ARIMA model include a mean term? The default is 
          'TRUE', note that for differenced series a mean would  not
          affect the fit nor predictions. 

   innov: [armaSim] - 
           is a univariate time series or vector of innovations to
          produce  the series. If not provided, 'innov' will be
          generated using  the random number generator specified by
          'rand.gen'.  Missing values are not allowed. By default the
          normal random number generator will be used. 

 lag.max: [armaTrueacf] - 
           maximum number of lags at which to calculate the acf or 
          pacf, an integer value by default 20. 

  method: [armaFit] - 
           denotes the method used to fit the model by a character
          string. 'method' must be one of the strings in the default
          argument. For ARIMA models a valid selection is one of
          '"CSS-ML"',  '"ML"', '"CSS"',  for ARMA models only one valid
          selection is possible '"CSS"', for AR models a valid
          selection is one of '"yw"', '"burg1"',  '"burg2"', '"ols"',
          '"mle"', and  for FRACDIFF models only one valid selection is
          possible  '"FRACDIFF"'. If no 'method' argument specified the
          following default settings are used: '"CSS-ML"' for ARIMA,
          '"FRACDIFF"' for FRACDIFF, '"CSS"' for ARMA, and '"yw"' for
          AR modelling.  
           Additional arguments required by the optimization algorithm
          have  to be passed through the 'dots' argument. 

   model: [armaSim] - 
           a list with one (AR), two (ARMA) or three (ARIMA, FRACDIFF) 
          elements . 'ar' is a numeric vector giving the AR
          coefficients,  'd' is an integer value giving the degree of
          differencing, and 'ma' is a numeric vector giving the MA
          coefficients. Thus the order of the time series process is
          (F)ARIMA(p, d, q) with 'p=length(ar)' and 'q=length(ma)'. 'd'
          is a positive integer for ARIMA models and a numeric value
          for FRACDIFF models. By default an ARIMA(2, 0, 1) model with 
          coefficients 'ar=c(0.5, -0.5)' and 'ma=0.1' will be 
          generated. 
           [armaTrueacf][armaFischer] - 
           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] - 
           is the length of the series to be simulated (optional  if
          'innov' is provided). The default value is 100. 

n.ahead, n.back, conf: [print][plot][summary][predict] - 
           are presetted arguments for the 'predict' method. 'n.ahead'
          determines how far ahead forecasts should be evaluated
          together with errors on the confidence intervals given by the
          argument 'conf'. If a forecast plot is desired, which is the
          default and expressed by 'doplot=TRUE', then 'n.back'  sets
          the number of time steps back displayed in the graph. 

  n.plot: [armaRoots] - 
           the number of data point to plot the unit circle; an integer
          value. 

 n.start: [armaSim] - 
           gives the number of start-up values discarded when
          simulating  non-stationary models. The start-up innovations
          will be generated  by 'rand.gen' if 'start.innov' is not
          provided. 

  object: [summary][predict] - 
           is an object of class 'fARMA' returned by the fitting
          function  'armaFit' and serves as input for the 'summary',
          and  'predict' methods. Some methods allow for additional 
          arguments. 

rand.gen: [armaSim] - 
           is the function which is called to generate the innovations.
           Usually, 'rand.gen' will be a random number generator. 
          Additional arguments required by the random number generator 
          'rand.gen', usually the location, scale and/or shape 
          parameter of the underlying distribution function, have to be
           passed through the 'dots' argument. 

   rseed: [armaSim] - 
           the random number seed, by default NULL. If this argument is
          set to an integervalue, then the function 'set.seed(rseed)'
          will be called. 

start.innov: [armaSim] - 
           is a univariate time series or vector of innovations to be
          used  as start up values. Missing values are not allowed. 

   title: a character string which allows for a project title. 

    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. 

       x: [print][plot] - 
           is an object of class 'fARMA' returned by the fitting
          function 'armaFit' and serves as input for the  'predict',
          'print', 'print.summary', and 'plot' methods. Some methods
          allow for additional  arguments. 
           [armaToeplitz] - 
           a vector of autocovariances. The returned Toeplitz matrix is
            the corresponding covariance matrix of the observatons. 

     ...: additional arguments to be passed. 

_D_e_t_a_i_l_s:

     *AR - Auto-Regressive Modelling:* 

      The argument 'x ~ar()' calls the underlying functions  'ar.yw',
     'ar.burg1', 'ar.burg2', code{ar.ols}  or 'ar.mle'.  For
     definiteness, the AR models are defined through


 '(x[t] - m) = a[1]*(x[t-1] - m) + ... +  a[p]*(x[t-p] - m) + e[t]'


     Order selection can be achieved through the comparison of AIC
     values for different model specifications. However this may be
     problematic, as of the methods here only 'ar.mle' performs true
     maximum likelihood estimation. The AIC is computed as if  the
     variance estimate were the MLE, omitting the determinant  term
     from the likelihood. Note that this is not the same as the 
     Gaussian likelihood evaluated at the estimated parameter values. 
     With 'method="yw"' the variance matrix of the innovations is
     computed from the fitted coefficients and the autocovariance of
     'x'. Burg's method allows for two alternatives  'method="burg1"'
     or 'method="burg2"' to estimate the innovations variance and hence
     AIC. Method 1 is to use the update  given by the Levinson-Durbin
     recursion (Brockwell and Davis, 1991),  and follows S-PLUS. Method
     2 is the mean of the sum of squares of  the forward and backward
     prediction errors (as in Brockwell and Davis,  1996). Percival and
     Walden (1998) discuss both.  
      '[ts:ar]' 

     *ARMA - Auto-Regressive Moving-Average Modelling:* 

         The argument 'x ~ arma(p,q)' maps the call to the argument 'x
     ~ arima(p, 0, q)'. 

     *ARIMA - Integrated ARMA Modelling:* 

      The argument 'x ~ arima()' calls the underlying function  'arima'
     from R's 'ts' package. For definiteness, the AR  models are
     defined through


 'x[t] = a[1]x[t-1] + ... + a[p]x[t-p] + e[t] + b[1]e[t-1] + ... + b[q]e[t-q]'


     and so the MA coefficients differ in sign from those of S-PLUS.
     Further, if 'include.mean' is 'TRUE', this formula applies to x-m
     rather than x.  For ARIMA models with differencing, the
     differenced series follows a zero-mean ARMA model. 
      The variance matrix of the estimates is found from the Hessian of
     the log-likelihood, and so may only be a rough guide. 
      Optimization is done by 'optim'. It will work best if the columns
     in 'xreg' are roughly scaled to zero mean and unit variance, but
     does attempt to estimate suitable scalings. The exact likelihood
     is computed via a state-space representation  of the ARIMA
     process, and the innovations and their variance found  by a Kalman
     filter. The initialization of the differenced ARMA  process uses
     stationarity. For a differenced process the  non-stationary
     components are given a diffuse prior (controlled  by 'kappa').
     Observations which are still controlled by the  diffuse prior
     (determined by having a Kalman gain of at least  '1e4') are
     excluded from the likelihood calculations. (This  gives comparable
     results to 'arima0' in the absence of missing values, when the
     observations excluded are precisely those dropped by the
     differencing.) 
      Missing values are allowed, and are handled exactly in method
     '"ML"'. 
      If 'transform.pars' is true, the optimization is done using an
     alternative parametrization which is a variation on that suggested
     by Jones (1980) and ensures that the model is stationary.  For an
     AR(p) model the parametrization is via the inverse tanh of the
     partial autocorrelations: the same procedure is applied
     (separately) to the AR and seasonal AR terms.  The MA terms are
     not constrained to be invertible during optimization, but they
     will be converted to invertible form after optimization if
     'transform.pars' is true. 
      Conditional sum-of-squares is provided mainly for expositional
     purposes.  This computes the sum of squares of the fitted
     innovations from observation 'n.cond' on, (where 'n.cond' is at
     least the maximum lag of an AR term), treating all earlier
     innovations to be zero.  Argument 'n.cond' can be used to allow
     comparability between different fits.  The ``part log-likelihood''
     is the first term, half the log of the estimated mean square. 
     Missing values are allowed, but will cause many of the innovations
     to be missing. 
      When regressors are specified, they are orthogonalized prior to
     fitting unless any of the coefficients is fixed. It can be helpful
     to roughly scale the regressors to zero mean and unit variance. 
      Note from 'arima': The functions parse their arguments to the 
     original time series functions available in R's time series
     library  'ts'.  
      The results are likely to be different from S-PLUS's 'arima.mle',
     which computes a conditional likelihood and does not include a
     mean in the model.  Further, the convention used by 'arima.mle'
     reverses the signs of the MA coefficients. 
      '[ts:arima]' 

     *FRACDIFF Modelling:* 

            The argument 'x ~ fracdiff()' calls the underlying
     functions from R's 'fracdiff' package. The estimator calculates
     the maximum  likelihood estimators of the parameters of a
     fractionally-differenced  ARIMA (p,d,q) model, together (if
     possible) with their estimated  covariance and correlation
     matrices and standard errors, as well  as the value of the
     maximized likelihood. The likelihood is  approximated using the
     fast and accurate method of Haslett and  Raftery (1989). Note, the
     number of AR and MA coefficients should  not be too large (say <
     10) to avoid degeneracy in the model.  
      The optimization is carried out in two levels: an outer
     univariate  unimodal optimization in d over the interval [0,.5],
     and an inner  nonlinear least-squares optimization in the AR and
     MA parameters to minimize white noise variance. 
      '[fracdiff:fracdiff]'

_V_a_l_u_e:

     'armaFit'  
      returns an S4 object of class '"fARMA"', with the folliwing 
     slots:

    call: the matched function call. 

    data: the input data in form of a data.frame. 

description: allows for a brief project description. 

     fit: the results as a list returned from the underlying time
          series model function.  

  method: the selected time series model naming the applied method. 

 formula: the formula expression describing the model. 

parameters: named parameters or coefficients of the fitted model. 

   title: a title string. 


     '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. 

     'armaToeplitz' 
      returns the Toeplitz matrix of a given vector 'x'. The  Toeplitz
     matrix is a square matrix of order length(x) and with  '[i,j]'
     entries given by 'x[abs(i-j)]'. Note, if  'x' is a vector of
     autocovariances, the Toeplitz matrix is   the corresponding
     covariance matrix of the observatons.

     'armaFischer' 
      returns the Fischer information matrix for an ARMA time series
     process.

_N_o_t_e:

     There is nothing really new in this package. The benefit you will 
     get with this collection is, that all functions have a common 
     argument list with a formula to specify the model and presetted 
     arguments for the specification of the algorithmic method. For 
     users who have already modeled GARCH processes with SPlus, this 
     approach will be quite natural.

     On the other hand, the user can still use the underlying functions
      from the 'ts', 'tseries', or 'fracdiff' packages. No  function
     from these packages is masked or overwritten.

     The output of the 'predict', 'print' and 'summary' methods have
     all the same style of format with some additional  algorithm
     specific printing. This makes it easier to interpret  the results
     obtained from different algorithms implemented in  different
     functions.

_A_u_t_h_o_r(_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.

_R_e_f_e_r_e_n_c_e_s:

     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.

_E_x_a_m_p_l_e_s:

     ## SOURCE("fSeries.31A-ArmaModelling")

     ## armaSim/armaFit -
        xmpSeries("\nStart: Simulate an ARMA(2, 1) process > ")
        x = armaSim(model = list(ar = c(0.5, -0.5), ma = 0.1), n = 1000)
        Continue = xmpSeries("Press any key > ")
        # Estimate the parameters:
        fit = armaFit(x ~ arma(2, 1))
        print(fit)
        Continue = xmpSeries("Press any key > ") 
        # Diagnostic Analysis:
        par(mfrow = c(3, 2), cex = 0.7)
        summary(fit)
        Continue = xmpSeries("Press any key > ")
        # 5 Steps ahead Forecasts:
        predict(fit, 5)
          
     ## armaFit -
        xmpSeries("\nNext: Estimate the parameters > ")
        # Now use the arima fitting method with d=0:
        fit = armaFit(x ~ arima(2, 0, 1))
        print(fit)
        Continue = xmpSeries("Press any key > ") 
        # Diagnostic Analysis:
        par(mfrow = c(3, 2), cex = 0.7)
        summary(fit)
        Continue = xmpSeries("Press any key > ") 
        # 5 Step ahead Forecasts:
        predict(fit, n.ahead = 5)
        
     ## armaRoots -
        xmpSeries("\nNext: Roots of the Characteristic Polynomial > ")
        # Calculate and plot the  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 -
        xmpSeries("\nNext: True ACF of an ARMA Process > ")
        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)

