A study of the stability of an amplitude-dependent exponential autoregressive model with application

In this paper we studied and found the stability conditions of the (AEXPAR) model by using a dynamical approach and by adopting the linear approximation technique. ةعسلا ىلع دمتعم يسأ يتاذ رادحنا جذومن ةیرارقتسا ةسارد قیبطتلا عم صلختسملا مادختساب هتیرا رقتسا طورش داجیاوٕ (AEXPAR) ةیرا رقتسا ةسا رد ثحبلا اذه يف مت جذومن يكیمانید بولسأ ) يكرح ( ةیطخلا بیرقتلا ةقیرط مادختسابو . 1-Introduction: Linear models, in time-series analysis as well as in many other areas, have been widely investigated and applied in handling real-life data. In the last four decades, many authors have questioned the omnipresence of linear approaches, and thus time-series analysis has moved towards the nonlinear area due to the complexity of most measured time series characterizing. Much attention has been given, in this respect, to bilinear models, threshold models, random coefficient autoregressive models. Haggan and T. Ozaki studied in (1981) Modeling nonlinear random vibrations using an amplitude-dependent autoregressive time series model.[3] Ozaki in(1982)proposed a dynamic approach in order to find the necessary and sufficient condition for stability of exponential Prof. \ College of Computers Sciences and Math.\ University of Mosul * Asst. Prof. \ College of Education\ University of Mosul ** College of Computers Sciences and Math.\ University of Mosul *** Received:1/10 /2011 ____________________Accepted: 21 /12 / 2011 A study of the stability of an amplitude-dependent... ] 53 [ autoregressive models. This approach is a local linearization technique used to find the approximated linear autoregressive model near the nonzero singular point of a nonlinear model.By using a variational equation near the nonzero singular point of an exponential autoregressive model. Ozaki finds the necessary and sufficient condition for existence and stability of a nonzero singular point of (EXPAR) model, furthermore he finds the stability condition of a limit cycle if it exists[1] T.Ozaki,J.C.Jimenez,R.Biscay and P.Valdes studied in (1999) the Nonlinear time series models and Neural dynamical systems. Especially continuous time deterministic and stochastic dynamical system models and nonlinear autoregression model. (EXPAR) considered from the view point of prediction [7] Kentaro Ishizuka, Hiroko Kato and Tomohiro Nakatani studied in(2005) a speech signal analysis approach that uses the exponential autoregressive (EXPAR) model. In real speech signal, the amplitude and frequency are fluctuating randomly.These fluctuations are non-Gaussian and have nonlinear dynamics. [4] Jelloul Allal and Saı ̈d El Melhaoui studied in(2005) the problem of detecting the eventual existence of an exponential component in an AR(1) model, that is, the problem of testing ordinary AR(1) dependence against the alternative of an exponential autoregression [EXPAR(1)] model,was considered. A local asymptotic normality property was established for EXPAR(1) models in the vicinity of AR(1) ones. Two problems arose in this context, which were quite typical in the study of nonlinear time-series models. The first was a problem of parameter identification in the EXPAR(1) model. A special parameterization was developed so as to overcome this technical problem. The second problem was related to the fact that the underlying innovation density had to be treated as a nuisance. The problem at hand, indeed, appeared to be nonadaptive. These problems were solved using semi-parametrically efficient pseudoGaussian methods (which did not require Gaussian observations).[2] Mika Meitz and Pentti Saikkonen studied in(2006) the stability of nonlinear autoregressive models with conditionally heteroskedastic errors. They consider a nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model.[5] Definition 1 :A singular point of ( ) p t t t t x x x f x = ,......, , 2 1 is defined as a point, which every trajectory of ( ) p t t t t x x x f x = ,......, , 2 1 beginning sufficiently near it approaches either for ¥ ® t or for -¥ ® t . If it approaches it for ¥ ® t we call it stable singular point and if it approaches it for -¥ ® t we call it unstable singular point. Obviously a singular point z satisfies ( ) z z z z z ,........, , , f = .[6] The Fourth Scientific Conference of the College of Computer Science & Mathematics ] 54 [ Definition 2: A limit cycle of ( ) p t t t t x x x f x = ,......, , 2 1 is defined as an isolated and closed trajectory q t t t x x x + + + ,......, , 2 1 where q is a positive integer. Closed means that if initial values ( ) p x x ,....., 1 belong to the limit cycle then ( ) ( ) p q k p q k q k x x x x x ,......., ,......, , 1 2 1 = + + + for any integer k. Isolated means that every trajectory beginning sufficiently near the limit cycle approaches either for ¥ ® t or for -¥ ® t . If it approaches it for ¥ ® t we call it stable limit cycle and if it approaches it for -¥ ® t we call it unstable limit cycle.[6] Theorem 1: Let { } t x be expressed by the exponential model ( ) t t x t x e x t e p f + + = 1 1 1 2 1 A limit cycle of period q q t t t t x x x x + + + ,......, , , 2 1 of the model is orbitally stable if


1-Introduction:
Linear models, in time-series analysis as well as in many other areas, have been widely investigated and applied in handling real-life data.In the last four decades, many authors have questioned the omnipresence of linear approaches, and thus time-series analysis has moved towards the nonlinear area due to the complexity of most measured time series characterizing.Much attention has been given, in this respect, to bilinear models, threshold models, random coefficient autoregressive models.
Haggan and T. Ozaki studied in (1981) Modeling nonlinear random vibrations using an amplitude-dependent autoregressive time series model.[3] Ozaki in(1982)proposed a dynamic approach in order to find the necessary and sufficient condition for stability of exponential autoregressive models.This approach is a local linearization technique used to find the approximated linear autoregressive model near the nonzero singular point of a nonlinear model.By using a variational equation near the nonzero singular point of an exponential autoregressive model.Ozaki finds the necessary and sufficient condition for existence and stability of a nonzero singular point of (EXPAR) model, furthermore he finds the stability condition of a limit cycle if it exists [1] T.Ozaki,J.C.Jimenez,R.Biscay and P.Valdes studied in (1999) the Nonlinear time series models and Neural dynamical systems.Especially continuous time deterministic and stochastic dynamical system models and nonlinear autoregression model.(EXPAR) considered from the view point of prediction [7] Kentaro Ishizuka, Hiroko Kato and Tomohiro Nakatani studied in(2005) a speech signal analysis approach that uses the exponential autoregressive (EXPAR) model.In real speech signal, the amplitude and frequency are fluctuating randomly.These fluctuations are non-Gaussian and have nonlinear dynamics.[4] Jelloul Allal and Saı¨d El Melhaoui studied in(2005) the problem of detecting the eventual existence of an exponential component in an AR(1) model, that is, the problem of testing ordinary AR(1) dependence against the alternative of an exponential autoregression [EXPAR(1)] model,was considered.A local asymptotic normality property was established for EXPAR (1) models in the vicinity of AR(1) ones.Two problems arose in this context, which were quite typical in the study of nonlinear time-series models.The first was a problem of parameter identification in the EXPAR(1) model.A special parameterization was developed so as to overcome this technical problem.The second problem was related to the fact that the underlying innovation density had to be treated as a nuisance.The problem at hand, indeed, appeared to be nonadaptive.These problems were solved using semi-parametrically efficient pseudo-Gaussian methods (which did not require Gaussian observations).[2] Mika Meitz and Pentti Saikkonen studied in(2006) the stability of nonlinear autoregressive models with conditionally heteroskedastic errors.They consider a nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model.[5] Definition 1 :A singular point of ( ) ,......., ,......, , f o r a n y i n t e g e r k .Isolated means that every trajectory beginning sufficiently near the limit cycle approaches either for . If it approaches it for ¥ ® t we call it stable limit cycle and if it approaches it for -¥ ® t we call it unstable limit cycle.[6] Theorem 1: A limit cycle of period q For the prove see [6]

Amplitude-dependent exponential autoregressive[AEXPAR] models:
This model is a class of nonlinear autoregressive models.These models were independently introduced by Jones(1976) and [Ozaki and Oda (1978)].The basic form of an AEXPAR model of order (k) is Where The proposed model: Let the model in equation (1) be given, for k=1 and by replacing j b by ( ) Then the proposed model will be as follows:

Statistical properties for the proposed model The singular point
To find the singular point of the model ( 2) we use the definition (1)of the singular point and suppose that the white noise is neglected then we have The stability condition for the singular point: We will find the stability condition for the non-zero singular point as follows : put ,s=t,t-1 in equation ( 4) then we have: Taylor expansion of as follows: Where A and B are constant defined as follows: ( ) 6) is a first order linear autoregressive model which is stable if all roots of the characteristic equation

Lies inside the unit cycle
The stability condition of the limit cycle: Let the limit cycle of period q of the propposed model(2) has the form respectively we find the following difference equation Where ( ) It is not easy to solve difference equation with periodic coefficients.What is required to know whether t z of (7) converges to zero or not , and this can be checked by seeing whether t q t z z + less than one or not.
From the relation (7) we get: Then by using theorem(1) the model ( 2) is stable iff x T

Examples:
We shall give two examples to explain the statistical properties for the proposed model

Example(1)
Let the time series { } t x is given by the following model: We can see from the above figures that it doesn't depend on the initial value, this implies that the proposed model has a limit cycle.

Example(2)
Let the time series { } t x is given by the following model:

Conclusions:
In this paper we studied and found the stability conditions of the (AEXPAR) model by using a dynamical approach and by adopting the linear approximation technique.We find that the proposed model has a stable non-zero singular point if it belongs to the interval [0,1](i.e. 1 0 < < z ) a n d o t h e r w i s e i t w i l l b e unstable, so that the proposed model is stable and it has a limit cycle if all the roots of the characteristic equation lie inside the unit cycle i.e.