ARMAMaxLikelihood Class |
Namespace: Imsl.Stat
The ARMAMaxLikelihood type exposes the following members.
Name | Description | |
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ARMAMaxLikelihood |
Constructor for ARMAMaxLikelihood.
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Name | Description | |
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Compute |
Computes the exact maximum likelihood estimates for the autoregressive
and moving average parameters of an ARMA time series.
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Equals | Determines whether the specified object is equal to the current object. (Inherited from Object.) | |
Finalize | Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.) | |
Forecast |
Returns forecasts for lead times
at origins
z.Length-BackwardOrigin-1+j where
.
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GetAR |
Returns the final autoregressive parameter estimates.
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GetDeviations |
Returns the deviations for each forecast used for calculating the
forecast confidence limits.
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GetForecast |
Returns forecasts
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GetGradients |
Returns the gradients for the final parameter estimates.
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GetHashCode | Serves as a hash function for a particular type. (Inherited from Object.) | |
GetMA |
Returns the final moving average parameter estimates.
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GetPsiWeights |
Returns the psi weights used for calculating forecasts from the infinite
order moving average form of the ARMA model.
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GetResiduals |
The current values of the vector of residuals.
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GetTimeSeries |
Returns the time series used to construct ARMAMaxLikelihood.
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GetType | Gets the Type of the current instance. (Inherited from Object.) | |
IsInvertible |
Tests whether the coefficients in ma are invertible.
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IsStationary |
Tests whether the coefficients in ar are stationary.
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MemberwiseClone | Creates a shallow copy of the current Object. (Inherited from Object.) | |
SetAR |
Sets the initial values for the autoregressive terms to the p
values in ar.
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SetMA |
Sets the initial values for the moving average terms to the q
values in ma.
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ToString | Returns a string that represents the current object. (Inherited from Object.) |
Name | Description | |
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BackwardOrigin |
The maximum backward origin.
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Confidence |
The confidence level for calculating confidence
limit deviations returned from GetDeviations.
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Constant |
The constant parameter in the ARMA series.
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GradientTolerance |
The gradient tolerance for the convergence algorithm.
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InnovationVariance |
The estimated innovation variance of this series.
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Likelihood |
The final estimate for , where
is equal to the likelihood function evaluated using the
final parameter estimates.
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MaxIterations |
The maximum number of iterations.
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Mean |
The mean used for centering the series.
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P |
The number of autoregressive terms in the ARMA model
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Q |
The number of moving average terms in the ARMA model
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Tolerance |
The tolerance for the convergence algorithm.
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ARMAMaxLikelihood computes estimates of parameters for a nonseasonal ARMA model given a sample of observations, , for where z.Length. The class is derived from the maximum likelihood estimation algorithm described by Akaike, Kitagawa, Arahata and Tada (1979), and the XSARMA routine published in the TIMSAC-78 Library.
The stationary time series with mean can be represented by the nonseasonal autoregressive moving average (ARMA) model by the following relationship:
where B is the backward shift operator defined by and The ARMAMaxLikelihood class estimates the AR coefficients and the MA coefficients using maximum likelihood estimation.ARMAMaxLikelihood checks the initial estimates for both the autoregressive and moving average coefficients to ensure that they represent a stationary and invertible series respectively.
If
are the initial estimates for a stationary series then all (complex) roots of the following polynomial will fall outside the unit circle:If
are initial estimates for an invertible series then all (complex) roots of the polynomial will fall outside the unit circle.By default, the order of the lags for the autoregressive terms is and for the moving average terms. However, this cannot be overridden.
Initial values for the AR and MA coefficients can be supplied via the SetAR and SetMA methods. Otherwise, initial estimates are computed internally by the method of moments. The class computes the roots of the associated polynomials. If the AR estimates represent a nonstationary series, ARMAMaxLikelihood issues a warning message and replaces the intial AR estimates with initial values that are stationary. If the MA estimates represent a noninvertible series, a terminal error is issued and new initial values must be sought.
ARMAMaxLikelihood also validates the final estimates of the AR coefficients to ensure that they too represent a stationary series. This is done to guard against the possibility that the internal log-likelihood optimizer converged to a nonstationary solution. If nonstationary estimates are encountered, an exception will be thrown.
For model selection, the ARMA model with the minimum value for AIC might be preferred, , where L is the value of the maximum likelihood function evaluated at the parameter estimates.
ARMAMaxLikelihood can also handle white noise processes, i.e. processes.