ARMAOutlierIdentification Class

Detects and determines outliers and simultaneously estimates the model parameters in a time series whose underlying outlier free series follows a general seasonal or nonseasonal ARMA model. This class also allows computation of forecasts.

Inheritance Hierarchy

SystemObject
Imsl.StatARMAOutlierIdentification

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class ARMAOutlierIdentification

<SerializableAttribute>
Public Class ARMAOutlierIdentification

[SerializableAttribute]
public ref class ARMAOutlierIdentification

[<SerializableAttribute>]
type ARMAOutlierIdentification =  class end

The ARMAOutlierIdentification type exposes the following members.

Constructors

	Name	Description
	ARMAOutlierIdentification	Constructor for ARMAOutlierIdentification.

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Methods

	Name	Description
	Compute	Detects and determines outliers and simultaneously estimates the model parameters for the given time series.
	ComputeForecasts	Computes forecasts, associated probability limits and $\psi$ weights for an outlier contaminated time series whose underlying outlier free series obeys a general seasonal or non-seasonal ARMA model.
	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.)
	GetAR	Returns the final autoregressive parameter estimates.
	GetDeviations	Returns the deviations used for calculating the forecast confidence limits.
	GetForecast	Returns forecasts for the original outlier contaminated series.
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetMA	Returns the final moving average parameter estimates.
	GetOmegaWeights	Returns the $\omega$ weights for the detected outliers.
	GetOutlierFreeForecast	Returns forecasts for the outlier free series.
	GetOutlierFreeSeries	Returns the outlier free series.
	GetOutlierStatistics	Returns the outlier statistics.
	GetPsiWeights	Returns the $\psi$ weights of the infinite order moving average form of the model.
	GetResidual	Returns the residuals.
	GetTauStatistics	Returns the t value for each detected outlier.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Fields

	Name	Description
	ADDITIVE	Indicates detection of an additive outlier.
	INNOVATIONAL	Indicates detection of an innovational outlier.
	LEVEL_SHIFT	Indicates detection of a level shift outlier.
	TEMPORARY_CHANGE	Indicates detection of a temporary change outlier.
	UNABLE_TO_IDENTIFY	Indicates detection of an outlier that cannnot be categorized.

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Properties

	Name	Description
	AccuracyTolerance	The tolerance value controlling the accuracy of the parameter estimates.
	AIC	Returns Akaike's information criterion (AIC).
	AICC	Returns Akaike's Corrected Information Criterion (AICC).
	BIC	Returns the Bayesian Information Criterion (BIC).
	Confidence	The confidence level for calculating confidence limit deviations via method GetDeviations.
	Constant	Returns the constant parameter estimate.
	CriticalValue	The critical value used as a threshold during outlier detection.
	Delta	The dampening effect parameter.
	NumberOfOutliers	Returns the number of outliers detected.
	RelativeError	The stopping criterion for use in the nonlinear equation solver.
	ResidualStandardError	Returns the residual standard error of the outlier free series.

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Remarks

Consider a univariate time series $\{Y_t\}$ that can be described by the following multiplicative seasonal ARIMA model of order $(p,0,q) \times (0,d,0)_s$ :

$Y_t - \mu = \frac{\theta(B)}{\Delta_s^d \phi(B)} a_t,\, t = 1,\ldots,n$

Here, $\Delta_s^d=(1-B^s)^d,\, \theta(B)=1-\theta_1B-\ldots-\theta_qB^q,\, \phi(B)=1-\phi_1B-\ldots-\phi_pB^p$ . B is the lag operator, $B^kY_t=Y_{t-k}$ , $\{a_t\}$ is a white noise process, and $\mu$ denotes the mean of the series $\{Y_t\}$ .

Outlier detection and parameter estimation

In general, $\{Y_t\}$ is not directly observable due to the influence of outliers. Chen and Liu (1993) distinguish between four types of outliers: innovational outliers (IO), additive outliers (AO), temporary changes (TC) and level shifts (LS). If an outlier occurs as the last observation of the series, then Chen and Liu's algorithm is unable to determine the outlier's classification. In class ARMAOutlierIdentification, such an outlier is called a UI (unable to identify) and is treated as an innovational outlier.

In order to take the effects of multiple outliers occurring at time points $t_1,\ldots,t_m$ into account, Chen and Liu consider the following model:

$Y_t^\ast - \mu = \sum_{j=1}^m\omega_jL_j(B)I_t(t_j)+\frac{\theta(B)}{\Delta_s^d\phi(B)}a_t$

Here, $\{Y_t^\ast\}$ is the observed outlier contaminated series, and $\omega_j$ and

denote the magnitude and dynamic pattern of outlier j, respectively.

is an indicator function that determines the temporal course of the outlier effect, $I_{t_j}(t_j)=1, I_t(t_j)=0$ otherwise. Note that

operates on

via $B^kI_t=I_{t-k}, \, k=0,1,\ldots$ .

The last formula shows that the outlier free series $\{Y_t\}$ can be obtained from the original series $\{Y_t^\ast\}$ by removing all occurring outlier effects:

$Y_t = Y_t^\ast-\sum_{j=1}^m\omega_jL_j(B)I_t(t_j)$

The different types of outliers are characterized by different values for

$L_j(B)=\frac{\theta(B)}{\Delta_s^d\phi(B)}$ for an innovational outlier,
for an additive outlier,
$L_j(B)=(1-B)^{-1}$ for a level shift outlier and
$L_j(B)=(1-\delta B)^{-1},\, 0 \lt \delta \lt 1,$ for a temporary change outlier.

Class ARMAOutlierIdentification is an implementation of Chen and Liu's algorithm. It determines the coefficients in $\phi(B)$ and $\theta(B)$ and the outlier effects in the model for the observed series jointly in three stages. The magnitude of the outlier effects is determined by least squares estimates. Outlier detection itself is realized by examination of the maximum value of the standardized statistics of the outlier effects. For a detailed description, see Chen and Liu's original paper (1993).

Intermediate and final estimates for the coefficients in $\phi(B)$ and $\theta(B)$ are computed by the Compute methods from classes ARMA and ARMAMaxLikelihood. If the roots of $\phi(B)$ or $\theta(B)$ lie on or within the unit circle, then the algorithm stops with an appropriate exception. In this case, different values for p and q should be tried.

Forecasting

From the relation between original and outlier free series,

$Y_t^\ast = Y_t+\sum_{j=1}^m\omega_jL_j(B)I_t(t_j)$

it follows that the Box-Jenkins forecast at origin t for lead time l, $\hat{Y}_t^\ast(l)$ , can be computed as

$\hat{Y}_t^\ast(l) = \hat{Y}_t(l)+\sum_{j=1}^m \omega_jL_j(B)I_{t+l}(t_j) ,\; l=1,\ldots,{\rm{nForecast}}$

Therefore, computation of the forecasts for $\{Y^\ast_t\}$ is done in two steps:

Computation of the forecasts for the outlier free series $\{Y_t\}$ .
Computation of the forecasts for the original series $\{Y^\ast_t\}$ by adding the multiple outlier effects to the forecasts for $\{Y_t\}$ .

Step 1: Computation of the forecasts for the outlier free series $\{Y_t\}$

Since

$\varphi(B)(Y_t-\mu) = \theta(B)a_t$

where

$\varphi(B):=\Delta_s^d\phi(B)=1-\varphi_1B-\ldots-\varphi_{p+sd}B^{p+sd}$

the Box-Jenkins forecast at origin t for lead time l, $\hat{Y}_t(l)$ , can be computed recursively as

$\hat{Y}_t(l)=(1-\sum_{j=1}^{p+sd}\varphi_j)\mu+\sum_{j=1}^{p+sd}\varphi_j\hat{Y}_t(l-j)-\sum_{j=1}^q\theta_ja_{t+l-j}$

Here,

$\hat{Y}_t(l-j)= \left\{ \begin{array}{ccc} Y_{t+l-j} & {\mbox{for}} & l-j \le 0 \\ \hat{Y}_t(l-j) & {\mbox{for}} & l-j \gt 0 \end{array} \right.$

and

$a_k = \left\{ \begin{array}{ccc} 0 & {\mbox{for}} & k \le \max\{1, p+sd\} \\ Y_k-\hat{Y}_{k-1}(1) & {\mbox{for}} & k = \max\{1, p+sd\}+1,\ldots,n \end{array} \right.$

Step 2: Computation of the forecasts for the original series $\{{Y_t}^*\}$ by adding the multiple outlier effects to the forecasts for $\{Y_t\}$

The formulas for for the different types of outliers are as follows:

Innovational outlier (IO)

$L_j(B) = \frac{\theta(B)}{\Delta_s^d\phi(B)}:=\psi(B)=\sum_{k=0}^\infty\psi_kB^k,\, \psi_0=1$

Additive outliers (AO)

Level shifts (LS)

$L_j(B) = \frac{1}{1-B}=\sum_{k=0}^{\infty}B^k$

Temporary changes (TC)

$L_j(B) = \frac{1}{1-\delta B}=\sum_{k=0}^\infty\delta^kB^k$

Assuming the outlier occurs at time point

, the outlier impact is therefore:

Innovational outliers (IO)

$\omega_jL_j(B)I_t(t_j) = \left\{ \begin{array}{ccc} 0 & {\mbox{for}} & t \lt t_j \\ \omega_j \psi_k & {\mbox{for}} & t=t_j+k, \, k \ge 0 \end{array} \right.$

Additive outliers (AO)

$\omega_jL_j(B)I_t(t_j) = \left\{ \begin{array}{ccc} 0 & {\mbox{for}} & t \ne t_j \\ \omega_j & {\mbox{for}} & t=t_j \end{array} \right.$

Level shifts (LS)

$\omega_jL_j(B)I_t(t_j) = \left\{ \begin{array}{ccc} 0 & {\mbox{for}} & t \lt t_j \\ \omega_j & {\mbox{for}} & t=t_j+k, \, k \ge 0 \end{array} \right.$

Temporary changes (TC)

$\omega_jL_j(B)I_t(t_j) = \left\{ \begin{array}{ccc} 0 & {\mbox{for}} & t \lt t_j \\ \omega_j \delta^k & {\mbox{for}} & t=t_j+k, \, k \ge 0 \end{array} \right.$

From these formulas, the forecasts $\hat{Y}_t^\ast(l)$ can be computed easily. The $100(1-\alpha)$ percent probability limits for $Y_{t+l}^\ast$ and $Y_{t+l}$ are given by

$\hat{Y}_t^\ast(l)\, {\rm{(or }}\,\,\, \hat{Y}_t(l) {\rm{, resp.)}} \pm u_{\alpha/2}(1+\sum_{j=1}^{l-1}\psi_j^2)^{1/2}s_a$

where $u_{\alpha/2}$ is the $100(1-\alpha/2)$ percentile of the standard normal distribution,

is an estimate of the variance $\sigma_a^2$ of the random shocks, and the $\psi$ weights $\{\psi_j\}$ are the coefficients in

$\psi(B) := \sum_{k=0}^\infty \psi_k B^k := \frac{\theta(B)}{\Delta_s^d\phi(B)}, \, \psi_0=1 \,.$

For a detailed explanation of these concepts, see chapter 5:"Forecasting" in Box, Jenkins and Reinsel (1994).

Reference

Imsl.Stat Namespace

Other Resources

Example 1

Example 2

Example 3