ArmaInterface             package:fArma             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,
       'print'              Print Method,
       'plot'               Plot Method,
       'summary'            Summary Method,
       'predict'            Forecasts and optionally plots an ARMA process,
       'fitted'             Method, returns fitted values,
       'coef|coefficients'  Method, returns coefficients,
       'residuals'          Method, returns residuals.

_U_s_a_g_e:

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

     armaFit(formula, data, method = c("mle", "ols"), include.mean = TRUE, 
         fixed = NULL, title = NULL, description = NULL, ...)

     show.fARMA(object)

     ## S3 method for class 'fARMA':
     plot(x, which = "ask", gof.lag = 10, ...)
     ## S3 method for class 'fARMA':
     summary(object, doplot = TRUE, which = "all", ...)

     ## S3 method for class 'fARMA':
     predict(object, n.ahead = 10, n.back = 50, conf = c(80, 95), 
         doplot = TRUE, ...)
         
     ## S3 method for class 'fARMA':
     fitted(object, ...)
     ## S3 method for class 'fARMA':
     coef(object, ...)
     ## S3 method for class 'fARMA':
     residuals(object, ...) 

_A_r_g_u_m_e_n_t_s:

addControl: [armaSim] - 
           a logical value. Should control parameters added to the
          returned series as a control attribute? 

    data: an optional timeSeries or data frame object containing the
          variables  in the model. If not found in 'data', the
          variables are taken  from 'environment(formula)', typically
          the environment from which 'armaFit' is called. If 'data' is
          an univariate series, then the series is converted into a
          numeric vector and the name of the response in the formula
          will be neglected. 

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

  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. 

   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 ~ ma(p)',  'x ~
          arma(p, q)', 'x ~ arima(p, d, q)', or 'x ~ arfima(p, q)'. 'x'
          is the response variable optionally to 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.) 

 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. 

  method: [armaFit] - 
           a character string denoting the method used to fit the
          model. The default method for all models is the
          log-likelihood parameter  estimation approach,
          'method="mle"'. In the case of an AR model the parameter
          estimation can also be done by ordinary least square
          estimation, '"ols"'.  

   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. 

       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.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.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. 

positions: [armaSim] - 
           determines the class of the returned time series. If set to
          'NULL', the default value, then an object of class '"ts"'
          will be returned. If 'positions' is a 'timeDate' vector of
          length 'n' then an object of class '"timeSeries"' will be
          returned. 

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. 

   which: [plot][summary] - 
           if 'which' is set to '"ask"' the function will 
          interactively ask which plot should be displayed. This is the
          default value for the 'plot' method. If  'which="all"' is
          specified all plots will be displayed. This is the default
          setting for the 'summary' method. On the other hand, if a
          vector of logicals is specified, then those plots will be
          displayed for which the elements  of the vector are set to
          'TRUE'. 

       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. 

     ...: additional arguments to be passed. 

_D_e_t_a_i_l_s:

     *AR - Auto-Regressive Modelling:* 

      The argument 'x~ar(p)' calls the underlying functions  'ar.mle'
     or code{ar.ols} depending on the  'method''s choice.  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.  
      '[stats:ar]' 

     *MA - Moving-Average Modelling:* 

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

     *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. 
      '[stats:arima]' 

     *ARFIMA/FRACDIFF Modelling:* 

            The argument 'x~arfima()' 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 following 
     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. 

_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 R/Rmetrics and
     SPlus/Finmetrics, this approach will be quite natural.

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

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

     For the first selection 'x~ar()' the function 'armaFit()'  uses
     the AR modelling algorithm as implemented in R's 'stats'  package.

     For the second 'x~ma()', third 'x~arma()', and fourth  selection
     'x~arima()' the function 'armaFit()' uses the  ARMA modelling
     algorithm also as implemented in R's 'stats'  package.

     For the last selection 'x~arfima()' the function 'armaFit()' uses
     the fractional ARIMA modelling algorithm from R's contributed 
     'fracdiff' package. 

     Note, that the 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 still use the functions from R's '"stats"'
     package: 'arima.sim' that simulates from an ARIMA time series 
     model, 'ar, arima, arima0' that fit an AR, ARIMA model to an 
     univariate time series, 'predict' that forecasts from a fitted 
     model, and 'tsdiag' that plots time-series diagnostics.  No
     function from these packages is masked, modified or overwritten.

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

     For 'arfima' models the following methods are not yet implemented:
     'plot', 'fitted', 'residuals',  'predict', and 'predictPlot'.

_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:

     ## armaSim -
        # Simulation:
        x = armaSim(model = list(ar = c(0.5, -0.5), ma = 0.1), n = 1000)
        
     ## armaFit -
        # Estimate the Parameters:
        fit = armaFit(~ arma(2, 1), data = x)
        print(fit)
           
     ## summary -
        # Diagnostic Analysis:
        par(mfrow = c(2, 2), cex = 0.7)
        summary(fit, which =  "all")
        
     ## plot - 
        # Interactive Plots:
        # par(mfrow = c(1, 1))
        # plot(fit)
        
     ## predict -
        # Forecast 5 Steps Ahead:
        par(mfrow = c(1, 1))
        predict(fit, 5)
        
     ## armaFit -
        # Alternative Calls:
        TS = as.timeSeries(data(msft.dat))
        armaFit(formula = diff(log(Close)) ~ ar(5), data = TS)
        armaFit(Close ~ ar(5), data = returnSeries(TS, digits = 12))
        TS.RET = returnSeries(TS, digits = 12)
        armaFit(Close ~ ar(5), TS.RET)
        armaFit(Close ~ ar(5), as.data.frame(TS.RET))
        armaFit(~ ar(5), as.vector(TS.RET[, "Close"]))
        armaFit(~ ar(5), as.ts(TS.RET)[, "Close"])
        attach(TS.RET)
        armaFit(Close ~ ar(5))
        detach(TS.RET)

