modified estimators for some noisy dynamical...

American Journal of Computational and Applied Mathematics 2013, 3(4): 238-249 DOI: 10.5923/j.ajcam.20130304.07

Modified Estimators for Some Noisy Dynamical Systems

Salah H Abid*, Shahad S Al-Azawie

Al-Mustansirya University – Education College-Mathematics Dept

Abstract Lele in (1994)[8], Used the estimat ing function method to analyze and estimate the exponential and the logistic noisy dynamical systems. also, He used simulat ion to study the small-sample properties. Hassan in (2012)[7], derived an estimator o f the Tent noisy dynamical system based on estimating function method. In th is paper we will modify Lele̕s estimator of noisy logistic and noisy exponential dynamical systems and Hasan̕s estimator of noisy tent dynamical system, to be more efficient based on a simulat ion experiment. The results shows that the modified estimators performs better than unmodified estimators based on the mean square error criterion.

Keywords Lele's estimator, Hassan's estimstor, Noisy dynamical system

1. Introduction The dynamical system is a mathemat ic formula describes

changes of an initial state space for the time. Which could be discrete or continuous. In the noisy dynamical system we can define the "noisy chaotic" by using stochastic Lyapunov exponent (SLE).

Godambe (1984)[4], Studied utilizing the theory of estimating equations[3], this paper develops concepts of parameter defin ing function and effective parameter. The paper provides theory and techniques for choosing from a given set of robust parameters one that is most effective, or one that can most efficiently be estimated.

Berliner (1991)[2], Studied statistical prediction for dynamical system with error.considered parametric models that lead to chaos on a subset of the parameter space. He considered a likelihood based approach for estimation of the parameters of the underlying determin istic system in the presence of measurement error. He also considered Bayesian prediction for these so called "unpredictable processes" , also he suggested some novel characteristics of chaotic data analysis.

Hassan (2012)[7], Compare among some of the methods used to estimate these models by using simulat ion. and propose an asymmetric dynamical system which is based on a mixture of two maps (asymmetric logistic and asymmetric tent).

In this research we are interested of noisy Dynamical systems according to logistic, exponential, and tent maps, so we will introduce then as follows.

* Corresponding author: [email protected] (Salah H Abid) Published online at http://journal.sapub.org/ajcam Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved

2. Estimating Function Method Estimating function[5], have proved to be a promising

alternative to the maximum likelihood estimat ion. they usually lead to simple numerical calculat ions and also posses a robustuess property[4], In that they need only the specification of the first few, usually two, moments instead of the complete specification of the underlying distribution.

In the fo llowing we briefy describe the estimat ing function approach and then apply it to the three maps logistic, exponential, and tent.

Let θ be a one dimensional parametre taking values in Θ. Let 𝒳𝒳 denote the sample space. An estimating function for θ is defined[3] as any function ƒ: 𝒳𝒳 x Θ → ℛ such that EΘ [ƒ(y,θ)] =0.

An estimator 𝜃𝜃^ is obtained by solving the empirical version ƒ(y,θ)=0 under suilab le regularity conditions. These estimators are consist and asymptotically normal.

1) Logistic map with additive Gaussion noise. The dynamical system of the logistic map is .[10]

𝐺𝐺a(𝒳𝒳) = 𝒳𝒳𝑡𝑡+1 = a 𝒳𝒳𝑡𝑡 (1-𝒳𝒳𝑡𝑡 ) (1)

Where all t in T= {0,1,2,…}, 𝒳𝒳𝑡𝑡 ∈[0,1], and a∈[0,4]. The above dynamical system has the following

properties.[10] a) If 0 < a < 1 then the system has one fixed point q=0 is a

stable. b) if 1 < a < 3 then the system has two fixed point q1=0 is

unstable and q2= 1-(1/a) is stable and appears 1-cycle. c) if 3 < a < 3.449 then the system has one fixed point q =

1-(1/a) unstable and appers 2-cycles. d) if a=3.57 then the system becomes chaotic. The noisy dynamical system of logistic map.[8]

American Journal of Computational and Applied Mathematics 2013, 3(4): 238-249 239

𝐺𝐺𝑎𝑎 (Yt,ɳ𝑡𝑡 ) =Yt+1 = 𝒳𝒳𝑡𝑡+1 + ɳ t (2)

Where Ga: ℛ x → ℛ, a ∈[0,4], and {ɳt} (iid) random varible we assume that ɳt ∼ Gauss (0,𝜎𝜎2).

Bifurcation diagram of the system (1) and (2) are in figures (1-a) and (1-b).

(a)

(b)

Figure (1). The bifurcation diagram of the logistic system, x-axis represent the values of a and y-axis represent the values of Хt+1. (a) in deterministic case, (b) in noisy case with additive Gaussian(0, 0.0025) noise

2) Exponential map with multiplicative log-Gaussian noise.

The dynamical system of the exponential map[9] Ea(𝒳𝒳𝑡𝑡 ) = 𝒳𝒳𝑡𝑡+1 = 𝒳𝒳𝑡𝑡 exp (a(1-𝒳𝒳𝑡𝑡 )) (3)

Where all t in T={0,1,2,…}, 𝒳𝒳𝑡𝑡 ∈[0,∞), and a ∈ (0,∞) The above dynamical system has the following properties.

[9] a) If 0 < a < 2 Then the system has two fixed point

q1=0 is unstable and q2=1 is stable. b) If 2 < a < 2.44 Then the system has two fixed point

q1=0 and q2=1 bath are unstable. c) If a=2.6924 Then the system chaotic. The noisy dynamical system of the exponential map.[8]

Ea( Yt ,𝘡𝘡t) = Yt+1 = 𝒳𝒳𝑡𝑡+1 𝘡𝘡t (4)

Where Ea:[0,∞) x →[0,∞), a ∈ (0,∞), and {𝘡𝘡t} iid random variables we assume that 𝘡𝘡t ∼ log Gauss (−𝜎𝜎

2

2, 𝜎𝜎2).


(a)

(b)

Figure (2). The bifurcation diagram of the Exponential system, x-axis represent the values of a and y-axis represent the values of Хt+1. (a) in deterministic case, (b) in noisy case with additive log Gaussian(-0.05, 0.1) noise

3) Tent map with additive Gassus noise The dynamical system of the tent map.[6]

Ta(𝒳𝒳𝑡𝑡 ) = 𝒳𝒳𝑡𝑡+1 = �2𝑎𝑎 𝒳𝒳𝑡𝑡 𝑖𝑖𝑖𝑖 0 < 𝑎𝑎 < 1

2

2𝑎𝑎 (1 − 𝒳𝒳𝑡𝑡 ) 𝑖𝑖𝑖𝑖 12

< 𝑎𝑎 < 1� (5)

Where all t in T={0,1,2,...}, 𝒳𝒳𝑡𝑡 ∈[0,1] , and a ∈[0,1] The above dynamical system has the following

properties.[6] a) if 0 < a < 1/2 then the system has one fixed point q =0 is

stuble. b) if a=1/2, if 0 ≤ 𝒳𝒳 ≤ 1/2 then all points represent fixed

points and if 1/2 ≤ 𝒳𝒳 ≤ 1 then all points represent eventually fixed points.

240 Salah H Abid et al.: Modified Estimators for Some Noisy Dynamical Systems

c) If 1/2 ≤ a ≤ 1 then the system has two fixed point q1=0 and q2= 2𝑎𝑎

1 +2𝑎𝑎 both are unstuble and the system becomes

chaotic. The noisy dynamical system of the tent map

Ta(Yt,ɳt) = Yt+1 = �2𝑎𝑎𝒳𝒳𝑡𝑡 + ɳ𝑡𝑡 𝑖𝑖𝑖𝑖 𝒳𝒳𝑡𝑡 < 1

2

2𝑎𝑎(1 − 𝒳𝒳𝑡𝑡) + ɳ𝑡𝑡 𝑖𝑖𝑖𝑖 𝒳𝒳𝑡𝑡 > 12

� (6)

Where Ta: ℛ x → ℛ , a∈[0,1], and {ɳ t} iid random variable , we assume that ɳt∼ Gauss (0,𝜎𝜎2) as in the logistic system.


(a)

(b)

Figure (3). The bifurcation diagram of the Tent system, x-axis represent the values of a and y-axis represent the values of Хt+1. (a) in deterministic case, (b) in noisy case with additive Gaussian(0, 0.0025) noise

3. Measurement Error Since the calculat ion of the parameter values is triv ial, the

practical calculation of anoise introduced series

{Yt,t=0,1,2,…} , can be done using the mesnring device by two kinds of mesurment erro rs.

(1) Additive error. Here Yt = 𝒳𝒳𝑡𝑡 + 𝜎𝜎ɳ𝑡𝑡 , 𝜎𝜎 > 0, where ɳ𝑡𝑡 ’𝑠𝑠 are independent identically distributed random variables with mean 0 and variance 1. [3] assumed these to be Gaussian.

(2) Mult iplicative error. Here Yt = 𝒳𝒳𝑡𝑡+1 𝘡𝘡t; where 𝘡𝘡t‘s are independent identically random variables with mean 1, variance 𝜎𝜎2 , and range on[0,∞).

By apply ing the estimating function method on the logistic and exponential system Lele derived the Following[8],

i) For a noisy logistic dynamical system, he wrote, ƒ(y,a)= ∑ (𝑦𝑦𝑡𝑡 +1−𝑛𝑛−1𝑡𝑡=0 a(𝑦𝑦𝑡𝑡 − 𝑦𝑦𝑡𝑡2 + 𝜎𝜎2)) and then got the EF estimator

of as

â= ∑ 𝑦𝑦𝑡𝑡+1𝑛𝑛−1𝑡𝑡=0

∑ (𝑦𝑦𝑡𝑡−𝑦𝑦𝑡𝑡2+𝜎𝜎2 )𝑛𝑛−1𝑡𝑡=0

(7)

ii) For a noisy dynamical exponential system, he wrote. ƒ(y,a)=∑ {[(𝑙𝑙𝑙𝑙𝑙𝑙𝑌𝑌𝑡𝑡+1 − 𝑙𝑙𝑙𝑙𝑙𝑙𝑌𝑌𝑡𝑡𝑛𝑛−1

𝑡𝑡=0 )2 −2𝜎𝜎2 ]−𝑎𝑎2 (1−2𝑌𝑌𝑡𝑡 + 𝑌𝑌𝑡𝑡1 +𝑒𝑒𝜎𝜎2 )}

and then got the EF estimator of as

â = abs[∑ (log 𝑦𝑦𝑡𝑡+1−𝑙𝑙𝑙𝑙𝑙𝑙𝑦𝑦𝑡𝑡𝑛𝑛−1𝑡𝑡=0 )2−𝜎𝜎2

∑ (1−2𝑦𝑦𝑡𝑡𝑛𝑛−1𝑡𝑡=0 + 𝑦𝑦𝑡𝑡

1+𝑒𝑒𝜎𝜎2)]1/2 (8)

(iii) By using the same argument Hassan in 2012 derived the estimator of the parameter of noisy tent dynamical system to be[7],

â= 𝑛𝑛2∑ 𝑦𝑦𝑡𝑡+1𝑛𝑛−1𝑡𝑡=0

𝑛𝑛 ∑ 𝑦𝑦𝑡𝑡−∑ 𝑦𝑦𝑡𝑡2𝑛𝑛−1𝑡𝑡=0

𝑛𝑛−1𝑡𝑡=0

(9)

4. The Empirical Study A simulat ion experiment was conducted to modify the

estimating function estimator of parameter of logistic, exponential and tent random dynamical system. We wrote the programs for this goal by asing Matlab R2008a, with run size equal to 500.

After execute the Simulat ion experiment, we got a tremendous amount of results. These results are summarized in tables, but put these tables in this research was inappropriate because of their big magnitude . As Possible alternative,we summed up these results in figures.

Figures (4.a),(4.b),(5.a),(5.b),(6.a) and (6.b) for logistic map, figures (7.a),(7.b),(8.a),(8.b),(9.a) and (9.b) for exponential map and figures (10.a),(10.b),(11.a),(11.b),(12.a) and (12.b) for tent map.

If any person wish, it is possible to email us to provide the results tables of the simulat ion experiment.

The figures show us the clear differences between the real values of a and their estimates (depending of the lele's estimators for the cases of logistic and exponential maps and hassan's estimator for tent map) , especially in the cases of tent and exponential maps . So, as a goal of our work, we took upon ourselves the issue of modification of lele's and hassan's estimators


Following the results we reached 1) Figures (4), (5) and (6) represent the estimations of the

parameter o f the noisy logistic dynamical system for the following combinations of 𝜎𝜎2 , a, and L, where 𝜎𝜎2 is the variance of the noisy variant, a is the bifurcation parameter of the logistic system and L is the length of the noisy orbit.

Following the values of 𝜎𝜎2 , a and L used in simulation experiment, to get Lele’s estimator of noisy logistic dynamical system, 𝜎𝜎2= 0.01, 0.16 , 0.25, a = 0.25 , 0.5 , 1, 1.5, 1.75, 2 , 2.25 , 2.5 , 3 , 3.75, 3.95 , 4 and L=20, 35, 50, 65, 85, 100, 160, 220, 280, 340, 400, 500, 550, 600, 650, 700, 750, 800, 850, 1000.

According to the results one can easly modify Lele̕s estimator of a in the noisy Logistic dynamical system to be ã𝐿𝐿= 0.948759 â – 0.098929 𝜎𝜎2 + 0.00000606 L.

2) Figures (7), (8) and (9) represent the estimations of the parameter of the noisy exponential dynamical system for the following combinations of 𝜎𝜎2 , a and L, where 𝜎𝜎2 is the variance of the noisy variant, a is the bifurcation parameter of the exponential system and L is the length of the noisy orbit.

Following the values of 𝜎𝜎2 , a and L used in simulation experiment to get Lele’s estimator of noisy exponential dynamical system, 𝜎𝜎2= 0.09, 0.21 , 0.36, a = 0.5 , 1 , 1.25, 1.5, 2 , 2.5 ,3 , 3.75, 3.95 , 4 , 4.5 , 5 and L =20 , 35 , 50 , 65 ,

85 , 100 , 160 , 220 , 280, 340, 400 , 500, 550 , 600, 650 , 700 , 750 , 800 , 850 , 1000.

According to the above results one can easly modify Lele̕s estimator of a in the noisy Exponential dynamical system to be ãE = 1.092332 â + 0.0083 𝜎𝜎2 + 0.00025 L

3) Figures (10), (11) and (12) represent the estimations of the parameter of the noisy tent dynamical system for the following combinations of 𝜎𝜎2 , a, and L, where 𝜎𝜎2 is the variance of the noisy variant, a is the bifurcation parameter of the tent system and L is the length of the noisy orbit.

Following the values of 𝜎𝜎2 , a, and L used in simulation experiment to get Hassan’s estimator o f noisy tent dynamical system, 𝜎𝜎2= 0.01 , 0.16 , 0.25, a = 0.25 , 0.5, 1, 1.5, 1.75, 2 , 2.25 , 2.5 , 3 , 3.75, 3.95 , 4 and L=20, 35, 50, 65, 85, 100, 160, 220, 280, 340, 400, 500, 550, 600, 650, 700, 750, 800, 850, 1000.

According to the above results one can easly modify Hassan’s estimator of a in the noisy Tent dynamical system to be ãT = 0.9432 â + 0.27 𝜎𝜎2 + 0.000043 L

4) An additional Experiments was conducted to compare between â and ã for noisy logistic, noisy exponential and noisy tent dynamical systems in different situations.

Tables (1), (2), (3), (4), (5) and (6) contain the mean square error results. It is clear that the performance of ã is better than the performance of â.

(a)

0

1

2

3

4

5 L

50 100 280 500 650

800 20

65 160

340 550

700 850

35

85

220

400

600

750

1000

50

100

280

500

650

800

20 65

160 340

550 700

850 35

85 220 400 600 750

1000 50 100 280 500

650 800

20 65

160 340

550 700

850

35

85

220

400

600

750

1000

50

100

280

500

650

800

20 65

160 340

550 700

850 35

85 220 400 600 750

a â


(b) Figure 4. represent the estimations of the parameter of the noisy Logistic dynamical system (𝝈𝝈𝟐𝟐 = 𝟎𝟎.𝟎𝟎𝟎𝟎)

(a)

(b)

Figure 5. represent the estimations of the parameter of the noisy Logistic dynamical system (𝝈𝝈𝟐𝟐 = 𝟎𝟎.𝟎𝟎𝟏𝟏)

0

1

2

3

4

5

L 40

0

35 550

65 650

100

750

220

850

340

20 500

50 600

85 700

160

800

280

1000

40

0

a â

0

1

2

3

4

5

6 L

50 100 280 500 650 800

20 65

160 340

550 700

850 35

85

220

400

600

750

1000

50

100

280

500

650 800

20 65

160 340

550 700

850 35

85 220 400 600 750 1000

50 100 280 500 650 800

20 65

160 340

550 700

850 35

85

220

400

600

750

1000

50

100

280

500

650 800

20 65

160 340

550 700

850 35

85 220 400 600 750

a â

0 1 2 3 4 5 6 7

L 34

0 10

00

340

1000

34

0 10

00

340

1000

34

0 10

00

340

1000

34

0 10

00

340

1000

34

0 10

00

340

1000

34

0 10

00

340

a â


(a)

(b)

Figure 6. represent the estimations of the parameter of the noisy Logistic dynamical system (𝝈𝝈𝟐𝟐 = 𝟎𝟎.𝟐𝟐𝟐𝟐)

Table (1). MSE of â and ã for the noisy logistic map. x0 =0.1 and 𝜎𝜎2 =0.01

4 3.95 3.75 3.5 3.25 2.5 1.5 1 0.5 a L 0.0232

0.012844

0.0064

0.004654

0.0057

0.004247

0.0042

0.002255

0.0024

0.002159

7.4934e-004

0.000648

1.6066e-004

0.00015

0.0018

0.001487

2.3210

1.301327

MSE â

MSEã 20

0.0055

0.004302

0.0025

0.001919

0.0017

0.001644

0.0012

0.001174

7.6522e-004

0.000505

2.7885e-004

0.000275

4.3429e-005

3.33E-05

0.0015

0.001278

0.0915

0.089236

MSE â

MSEã 50

0.0023

0.001774

0.0013

0.001129

8.8806e-004

0.000514

6.3375e-004

0.000423

4.5336e-004

0.00033

1.3350e-004

0.000111

2.1796e-005

1.9E-05

9.2401e-004

0.000787

0.1444

0.120662

MSE â

MSEã 100

4.1657e-004

0.000305

2.6942e-004

0.000166

1.5598e-004

0.00013

1.3079e-004

0.000114

8.4720e-005

6.63E-05

3.1406e-005

2.38E-05

4.2058e-006

3.72E-06

0.0586

0.030925

31.9170

22.75633

MSE â

MSEã 500

1.8837e-004

0.000183

1.4182e-004

0.000118

8.8197e-005

5.69E-05

5.8172e-005

5.79E-05

4.0029e-005

3.14E-05

1.4131e-005

9.63E-06

2.0052e-006

1.06E-06

0.0292

0.021681

0.0564

0.031825

MSE â

MSEã 1000

0

1

2

3

4

5

6 L

50 100 280 500 650

800 20

65 160

340 550

700 850

35

85

220

400

600

750

1000

50

100

280

500

650

800

20 65

160 340

550 700

850 35

85 220 400 600 750

1000 50 100 280 500

650 800

20 65

160 340

550 700

850

35

85

220

400

600

750

1000

50

100

280

500

650

800

20 65

160 340

550 700

850 35

85 220 400 600 750

a â

0 1 2 3 4 5 6

L 40

0

35 550

65 650

100

750

220

850

340

20 500

50 600

85 700

160

800

280

1000

40

0

a â


Table (2). MSE of â and ã for the noisy logistic map . x0 =0.1 and 𝜎𝜎2 =0.25 4 3.95 3.75 3.5 3.25 2.5 1.5 1 0.5 a L

2.9515e+003

1799.506

6.4825e+003

5612.99

88.5177

63.06825

5.7150

4.037179

21.7557

14.39221

5.5728

5.424468

0.4041

0.363264

8.8175

8.334044

89.9661

62.0753

MSEâ

MSEã 20

20.6545

11.74544

703.1697

517.6904

5.6402

4.277203

8.1866

5.167619

0.1523

0.147272

0.0345

0.0296

1.2902

1.085389

0.9808

0.864939

34.2197

20.87824

MSEâ

MSEã 50

115.0710

58.05488

11.0949

6.238037

0.1411

0.127803

0.1822

0.143012

0.0411

0.025583

0.0091

0.006045

0.0032

0.001883

4.4193

3.555483

10.8455

8.045718

MSEâ

MSEã 100

0.0248

0.017794

0.0166

0.014269

0.0100

0.007776

0.0074

0.005965

0.0049

0.004735

0.0017

0.001296

5.4341e-004

0.000465

7.4235

4.334012

317.8962

170.4927

MSEâ

MSEã 500

0.0099

0.009153

0.0073

0.005744

0.0045

0.004098

0.0032

0.00191

0.0021

0.001525

8.3087e-004

0.000794

2.5331e-004

0.000232

19.2254

12.90214

40.7226

22.44794

MSEâ

MSEã 1000

(a)

(b)

Figure 7. represent the estimations of the parameter of the noisy exponential dynamical system (𝝈𝝈𝟐𝟐 = 𝟎𝟎.𝟎𝟎𝟎𝟎)

0

1

2

3

4

5 L

50 100 280 500 650 800

20 65

160 340

550 700

850 35

85

220

400

600

750

1000

50

100

280

500

650

800 20

65 160

340 550

700 850

35 85 220 400 600 750

1000 50 100 280 500 650

800 20

65 160

340 550

700 850

35

85

220

400

600

750

1000

50

100

280

500

650

800 20

65 160

340 550

700 850

35 85 220 400 600 750

a â

0

2

4

6

L 40

0

35 550

65 650

100

750

220

850

340

20 500

50 600

85 700

160

800

280

1000

40

0

a â


(a)

(b)

Figure 8. represent the estimations of the parameter of the noisy exponential dynamical system (𝝈𝝈𝟐𝟐 = 𝟎𝟎.𝟐𝟐𝟎𝟎)

Table (3). MSE of â and ã for the noisy exponential map, x0=1.3 and 𝜎𝜎2 =0.09

4.5 3.95 3.75 3.5 3.25 3 2.5 2 1.5 1 a L 54.2231

28.9951

52.2230

28.91195

23.2288

21.0158

0.0730

0.054396

0.5261

0.282436

0.0585

0.052616

0.0124

0.010725

0.2615

0.244354

0.2937

0.242576

0.1119

0.06274

MSEâ

MSEã

20

26.3294

21.5545

22.9436

19.17189

9.4107

7.36115

0.0397

0.030468

0.0768

0.074285

0.0456

0.0446

0.0031

0.002047

0.2928

0.288578

0.2646

0.202666

0.0930

0.07923

MSEâ

MSEã

50

52.3632

27.2726

48.8188

25.76285

15.3062

10.91308

0.0505

0.038958

0.1197

0.103916

0.0247

0.014287

0.0013

0.000867

0.3286

0.238882

0.2616

0.218027

0.0877

0.076333

MSEâ

MSEã

100

168.3539

95.3636

165.2245

87.18276

9.1237

6.774211

0.1610

0.090849

0.0719

0.05264

0.0100

0.006157

4.7988e-004

0.000399

0.3966

0.345253

0.2569

0.200902

0.0813

0.061525

MSEâ

MSEã

500

139.5894

133.6568

137.2736

133.6198

7.3700

6.625224

0.2511

0.237332

0.0478

0.032981

0.0098

0.009502

4.0305e-004

0.000334

0.4140

0.267062

0.2573

0.256027

0.0818

0.064265

MSEâ

MSEã

1000

0

1

2

3

4

5 L

50 100 280 500 650 800

20 65

160 340

550 700

850 35

85

220

400

600

750

1000

50

100

280

500

650

800 20

65 160

340 550

700 850

35 85 220 400 600 750

1000 50 100 280 500 650

800 20

65 160

340 550

700 850

35

85

220

400

600

750

1000

50

100

280

500

650

800 20

65 160

340 550

700 850

35 85 220 400 600 750

a â

0

2

4

6 L

400

35 550

65 650

100

750

220

850

340

20 500

50 600

85 700

160

800

280

1000

40

0

a â


Table (4). MSE of â and ã for the noisy exponential map , x0=1.3 and 𝜎𝜎2 =0.36

4.5 3.95 3.75 3.5 3.25 3 2.5 2 1.5 1 a L 1050.3672

879.3231

887.8290

679.3851

878.3548

646.7934

663.6303

558.2831

4.0658e+003

3585.509

0.2419

0.147589

71.2222

43.42361

0.4760

0.412153

15.6349

11.13976

8.6025

6.076962

MSEâ

MSEã

20

140.0025

118.3200

136.8388

107.4073

67.0978

41.76513

0.4994

0.331766

0.6400

0.376634

0.0406

0.032664

0.0458

0.033977

0.1676

0.095308

0.0136

0.010013

0.1570

0.11906

MSEâ

MSEã 50

339.3998

266.6006

332.6524

258.6818

30.9401

24.93934

0.2454

0.237142

0.2942

0.22435

0.0199

0.017041

0.0405

0.023645

0.0452

0.024241

0.0189

0.009535

3.3008e+007

18558537

MSEâ

MSEã 100

267.6769

177.7820

219.5193

172.7233

11.2903

10.28271

0.2728

0.162838

0.1321

0.095957

0.0038

0.003634

0.0326

0.029897

0.0365

0.024495

0.0060

0.003307

0.0253

0.018153

MSEâ

MSEã 500

167.6952

97.9209

160.0870

85.38522

8.7292

5.361828

0.3610

0.249382

0.0950

0.087962

0.0019

0.00161

0.0329

0.026572

0.0317

0.02507

0.0054

0.005349

0.0250

0.01393

MSEâ

MSEã

1000

(a)

(b) Figure 9. represent the estimations of the parameter of the noisy exponential dynamical system (𝝈𝝈𝟐𝟐 = 𝟎𝟎.𝟑𝟑𝟏𝟏)

0

1

2

3

4

5 L

50 100 280 500 650 800

20 65

160 340

550 700

850 35

85

220

400

600

750

1000

50

100

280

500

650 800

20 65

160 340

550 700

850 35

85 220 400 600 750 1000

50 100 280 500 650 800

20 65

160 340

550 700

850 35

85

220

400

600

750

1000

50

100

280

500

650 800

20 65

160 340

550 700

850 35

85 220 400 600 750

a â

0

2

4

6

L 40

0

35 550

65 650

100

750

220

850

340

20 500

50 600

85 700

160

800

280

1000

40

0

a â


(a)

(b)

Figure 10. represent the estimations of the parameter of the noisy tent dynamical system (𝝈𝝈𝟐𝟐 = 𝟎𝟎.𝟎𝟎𝟎𝟎)

Table (5). MSE of â and ã for the noisy tent map , x0=0.1 and 𝜎𝜎2 =0.01

1 0.875 0.75 0.625 0.5 0.375 0.25 0.125 a L 0.0674

0.049017

0.0387

0.028838

0.0212

0.011381

0.0116

0.010433

0.0169

0.014616

0.0554

0.051767

0.3713

0.306668

0.8464

0.474555

MSEâ

MSEã

20

0.0563

0.054457

0.0318

0.031102

0.0131

0.008652

0.0054

0.005322

0.0167

0.012791

0.2946

0.25098

0.2166

0.21124

0.6831

0.378181

MSEâ

MSEã

50

0.1329

0.088654

0.0299

0.021736

0.0101

0.008418

0.0034

0.002959

0.0166

0.014131

0.0759

0.063423

0.1028

0.080411

0.4540

0.348431

MSEâ

MSEã

100

0.1809

0.141468

0.0277

0.020962

0.0084

0.007427

0.0020

0.001055

0.0166

0.011836

0.1331

0.102679

1.0362

0.899562

0.0239

0.013824

MSEâ

MSEã

500

0.1854

0.12635

0.0284

0.014986

0.0081

0.006014

0.0018

0.001016

0.0166

0.012153

0.0145

0.008928

0.0090

0.007478

0.0223

0.019413

MSEâ

MSEã

1000

0

0.2

0.4

0.6

0.8

1 L

35 65 100 220 340 500

600 700

800 100

35 65

100 220

340

500

600

700

800

1000

35

65

100

220

340

500 600

700 800

100 35

65 100

220 340 500 600 700 800

1000 35 65 100 220 340

500 600

700 800

100 35

65 100

220

340

500

600

700

800

1000

35

65

100

220

340

500 600

700 800

100 35

65 100

220 340 500 600 700 800

a â

0

0.5

1

1.5 L 85 340

650

100

85 34

0 65

0 10

00

85 340

650

100

85 34

0 65

0 10

00

85 340

650

100

85 34

0 65

0 10

00

85 340

650

100

85 34

0 65

0

a â


(a)

(b)

Figure 11. represent the estimations of the parameter of the noisy tent dynamical system (𝝈𝝈𝟐𝟐 = 𝟎𝟎.𝟎𝟎𝟏𝟏)

Table (6). MSE of â and ã for the noisy tent map , x0=0.1 and 𝜎𝜎2 =0.25

1 0.875 0.75 0.625 0.5 0.375 0.25 0.125 a L 0.0662

0.05951

0.0403

0.03809

0.0223

0.015387

0.0127

0.012313

0.2522

0.209046

93.5043

60.31743

2.0778

2.067524

0.2354

0.184938

MSEâ

MSEã

20

0.0549

0.048415

0.0321

0.019585

0.0140

0.008536

0.0066

0.005715

0.6113

0.435547

4.7030

3.322284

0.3560

0.235507

0.4324

0.420891

MSEâ

MSEã

50

0.1313

0.097405

0.0302

0.017174

0.0104

0.007657

0.0042

0.003185

0.0179

0.011299

0.1762

0.170383

0.0330

0.028313

0.2348

0.197527

MSEâ

MSEã

100

0.1813

0.091468

0.0272

0.015293

0.0087

0.00788

0.0022

0.010395

0.0167

0.010395

0.0499

0.03315

0.0639

0.037605

0.0124

0.009976

MSEâ

MSEã

500

0.1853

0.159277

0.0285

0.022163

0.0079

0.006368

0.0019

0.001836

0.0166

0.012659

0.0309

0.02646

0.0081

0.004729

0.0244

0.013086

MSEâ

MSEã

1000

0

0.2

0.4

0.6

0.8

1 L

35 65 100 220 340 500

600 700

800 1000 35

65 100

220

340

500

600

700

800

1000

35

65

100

220

340

500 600

700 800

1000 35

65 100

220 340 500 600 700 800

1000 35 65 100 220 340

500 600

700 800

1000 35

65 100

220

340

500

600

700

800

1000

35

65

100

220

340

500 600

700 800

1000 35

65 100

220 340 500 600 700 800

a â

0

0.5

1

1.5 L

160

600

20 220

650

35 280

700

50 340

750

65 400

800

85 500

850

100

550

1000

16

0 60

0

a â


(a)

(b) Figure 12. represent the estimations of the parameter of the noisy tent dynamical system (𝝈𝝈𝟐𝟐 = 𝟎𝟎.𝟐𝟐𝟐𝟐)

REFERENCES [1] Alligood , K, Sauer , T. , and Yorke , J , (1997) "Chaos: An

Introduction to Dynamical systems ", Springer , New York.

[2] Berliner , L. (1991) "Likelihood and Bayesian Prediction of Chaotic systems" , Journal of the American Statistical Association , Vol. 86, No 416, pp 938-952.

[3] Godambe , V. P (1960), " An Optimum Property of Regular Maximum Likelihood Estimation" , Annals of Mathematical Statistics , 31 , 1208-1211.

[4] Godambe , V. Thompson , M. (1984), " Robust Estimation Through Equations" , Biometrika Trust , Vol. 71, No. 1, pp115 – 125.

[5] Godambe, V. P. (1985), "The Foundations of Finite sample Estimation in Stochastic Processes", Biometrika, 72, 419-428.

[6] Gulick, D. (1991), "Encounter with chaos ", Mc Graw-Hill, New York.

[7] Hassan, M ,H. (2012), "Some Aspects of Noisy Chaotic systems". Al-mustansiriya University.

[8] Lele, S. (1994), "Estimating Function in Chaotic Systems", Journal of the American Statistical Association , Vol. 89, No. 426 pp 512-516.

[9] May , R. (1976), "Simple Mathematical Models With Very Complicated Dynamics", Nature, Vol. 261, pp 459-467.

[10] Sprott, C. J. (2003), "Chaos and Time – Series Analysis". University of Wisconsin – Madison.

0

0.2

0.4

0.6

0.8

1 L

35 65 100 220 340 500

600 700

800 1000 35

65 100

220

340

500

600

700

800

1000

35

65

100

220

340

500 600

700 800

1000 35

65 100

220 340 500 600 700 800

1000 35 65 100 220 340

500 600

700 800

1000 35

65 100

220

340

500

600

700

800

1000

35

65

100

220

340

500 600

700 800

1000 35

65 100

220 340 500 600 700 800

a â

0

0.5

1

1.5 L

160

600

20 220

650

35 280

700

50 340

750

65 400

800

85 500

850

100

550

1000

16

0 60

0

a â

modified estimators for some noisy dynamical...

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