IDL Analyst Reference Guide: Random Number Generation

IMSL_RANDOM_ARMA

The IMSL_RANDOM_ARMA function generates a time series from a specific IMSL_ARMA model.

Syntax

Result = IMSL_RANDOM_ARMA(n, nparams [, /ACCEPT_REJECT] [, AR_LAGS=array] [, CONST=value] [, /DOUBLE] [, INPUT_NOISE=array] [, MA_LAGS=array] [, OUTPUT_NOISE=variable] [, /VAR_NOISE] [, W_INIT=array])

Result = IMSL_RANDOM_ARMA(n, nparams, ar [, /ACCEPT_REJECT] [, AR_LAGS=array] [, CONST=value] [, /DOUBLE] [, INPUT_NOISE=array] [, MA_LAGS=array] [, OUTPUT_NOISE=variable] [, /VAR_NOISE] [, W_INIT=array])

Result = IMSL_RANDOM_ARMA(n, nparams, ma [, /ACCEPT_REJECT] [, AR_LAGS=array] [, CONST=value] [, /DOUBLE] [, INPUT_NOISE=array] [, MA_LAGS=array] [, OUTPUT_NOISE=variable] [, /VAR_NOISE] [, W_INIT=array])

Result = IMSL_RANDOM_ARMA(n, nparams, ar, ma [, /ACCEPT_REJECT] [, AR_LAGS=array] [, CONST=value] [, /DOUBLE] [, INPUT_NOISE=array] [, MA_LAGS=array] [, OUTPUT_NOISE=variable] [, /VAR_NOISE] [, W_INIT=array])

Return Value

One-dimensional array of length n containing the generated time series.

Arguments

n

Number of observations to be generated. Parameter n must be greater than or equal to one.

nparams

One-dimensional array containing the parameters p and q consecutively. nparams(0) = p, where p is the number of autoregressive parameters. Parameter p must be greater than or equal to zero. nparams(1) = q, where q is the number of moving average parameters. Parameter q must be greater than or equal to zero.

ar

One-dimensional array of length p containing the autoregressive parameters.

ma

One-dimensional array of length q containing the moving average parameters.

Keywords

ACCEPT_REJECT

If present and nonzero, the random noises will be generated from a normal distribution using an acceptance/rejection method. If keyword Accept_Reject is not used, the random noises will be generated using an inverse normal CDF method. This argument will be ignored if keyword Input_Noise is used.

AR_LAGS

One-dimensional array of length p containing the order of the nonzero autoregressive parameters. Default: Ar_Lags = [1, 2, ..., p]

CONST

Overall constant. See the Discussion section. Default: Const = 0

DOUBLE

If present and nonzero, double precision is used.

INPUT_NOISE

One-dimensional array of length n + max (Ar_Lags(i)) containing the random noises. Keywords Input_Noise and Var_Noise can not be used together. Keywords Input_Noise and Output_Noise cannot be used together.

MA_LAGS

One-dimensional array of length q containing the order of the nonzero moving average parameters. Default: Ma_Lags = [1, 2, ..., q]

OUTPUT_NOISE

Named variable into which a one-dimensional array of length n + max (Ma_Lags(i)) containing the random noises is stored.

VAR_NOISE

If present (and Input_Noise is not used), noise a_t is generated from a normal distribution with mean 0 and variance Var_Noise. Var_Noise and Input_Noise cannot be used together. Default: Var_Noise = 1.0

W_INIT

One-dimensional array of length max (Ar_Lags(i)) containing the initial values of the time series. Default: W_Init(*) = Const/(1 – ar(0) – ar(1) – ... – ar(p - 1))

Discussion

The IMSL_RANDOM_ARMA function simulates an IMSL_ARMA(p, q) process, {Wt}, for t = 1, 2, ..., n. The model is:

Let m be the mean of the time series {W_t}. The overall constant q0 (Const) is:

Time series whose innovations have a nonnormal distribution may be simulated by providing the appropriate innovations in Input_Noise and start values in W_Init.

The time series is generated according to the following model:

X(i) = Const + ar(0) * X(i – Ar_Lags(0)) + ... + ar(p – 1) * X(i – Ar_Lags(p – 1)) +

A(I) – ma(0) * A(i – Ma_Lags(0)) – ... – ma(q – 1) * A(i – Ma_Lags(q – 1))

where the constant is related to the mean of the series:

as follows:

and where:

X(t) = W(t), t = 0, 1, ..., n - 1

and:

W(t) = W_Init(t + p), t = –p, –p + 1, ..., -2,-1

and A is either Input_Noise (if Input_Noise is used) or Output_Noise (otherwise).

Examples

Example 1

In this example, IMSL_RANDOM_ARMA is used to generate a time series of length five, using an IMSL_ARMA model with three autoregressive parameters and two moving average parameters. The start values are 0.1000, 0.0500, and 0.0375.

IMSL_RANDOMOPT, SET = 123457  
n  =  5  
nparams  =  [3, 2]  
ar  =  [0.5, 0.25, 0.125]  
ma  =  [-0.5, -0.25]  
r  =  IMSL_RANDOM_ARMA(n, nparams, ar, ma)  
PM, r, FORMAT = '(5F10.3)',$  
   TITLE = '                   IMSL_ARMA random deviates'  
  
                    IMSL_ARMA random deviates  
   0.637     0.317    -0.366    -2.122    -1.407  

IMSL_RANDOMOPT, SET = 123457  
n  =  5  
nparams  =  [3, 2]  
ar  =  [0.5, 0.25, 0.125]  
ma  =  [-0.5, -0.25]  
wi  =  [0.1, 0.05, 0.0375]  
theta0  =  1  
avar  =  0.1  
r  =  IMSL_RANDOM_ARMA(n, nparams, ar, ma, /ACCEPT_REJECT, $  
   W_INIT = wi, CONST = theta0, VAR_NOISE = avar)  
PM, r, FORMAT = '(5F10.3)', $  
   TITLE = '                 IMSL_ARMA random deviates:'  
  
                    IMSL_ARMA random deviates:  
   1.467     1.788     2.459     3.330     3.941