Proposing Robust IRWs Technique to Estimate Segmented Regression Model for the Bed load Transport of Tigris River with Change Point of Water Discharge Amount at Baghdad City

Abstract

Segmented regression consists of several sections separated by different points of membership, showing the heterogeneity arising from the process of separating the segments within the research sample. This research is concerned with estimating the location of the change point between segments and estimating model parameters, and proposing a robust estimation method and compare it with some other methods that used in the segmented regression. One of the traditional methods (Muggeo method) has been used to find the maximum likelihood estimator in an iterative approach for the model and the change point as well. Moreover, robust estimation method (IRWm method) has used which depends on the use of the robust M-estimator technique in segmentation idea and using the Tukey weight function. Our contribution to this research lies in the suggestion to use the S-estimator technique and using the Tukey weight function, to obtain a robust method against cases of violation of the normal distribution condition for random errors or the effect of outliers, and this method will be called IRWs. The aforementioned methods have been applied to a real data set related to the bed-load of Tigris River/ Baghdad city as a response variable and the amount of water discharge as an explanatory variable. The results of the comparison showed the superiority of the proposed method.

Full text

Journal of Economics and Administrative Sciences
Vol.26 (NO. 121) 2020, pp. 415-427
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Available online at http://jeasiq.uobaghdad.edu.iq
IRWs







)dromar72@coadec.uobaghdad.edu.iq(
)mohammad.stat.145@gmail.com(
Published:11/3/2020 Accepted:3/5/2020 Received :August / 2020 
 NC 4.0)-(CC BY NonCommercial 4.0 International-Attribution 
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 
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

         Muggeo    
     IRWm
    M-estimator       Tukey
      S-estimator   Tukey
IRWs

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         M
S
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

Journal of Economics and Administrative Sciences
Vol.26 (NO. 121) 2020, pp. 415-427
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-1Introduction
     Segmented linear regression model   


join point
(change point)(break points)
   [10]broken line model     ([16] joinpoint regression model)[17]piecewise linear model






      
     ([11][12]Quandt,1958,1960)
([9]Hudson,1966), ([6]Feder,1975), ([7]Gallant and Fuller,1973)   
 
      [10]Muggeo,2003)  
(Linear reparametrization technique)

    


([17]F. Zhang and Q. Li,2017)
   robust rank-based estimator bent linear regression     
       Muggeo,2003)     rank dispersion
([4]O.A.Ali, M.A.Abbas,2019)
     (M-Estimator)    
HuberTukeyHampel(Muggeo)
              
 (Tukey)   [1]Sukru A., Birdal S.,2020
[9] Hudson 1966 [10] Muggeo2003  (MML) 
  (two-segmented linear regression model)
 


               
(S-estimator)(Muggeo)(S-estimator)   MM-scales  
(M-estimator)


Journal of Economics and Administrative Sciences
Vol.26 (NO. 121) 2020, pp. 415-427
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2
2-1
      
            
[8:pp.569]



( 


 



(1)
  


0

 
2-2Estimation methods1-2-2(Muggeo)
    (maximum likelihood)      
(ML)

([10]Vito M. R. Muggeo,2003)

 
[10:pp3057]
       
 
 
(3)
 

reparameterization


x


(Muggeo)


 (Taylor expansion)    
   
 
[17:pp.6]

Journal of Economics and Administrative Sciences
Vol.26 (NO. 121) 2020, pp. 415-427
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

(5)(6)

 (5)




 



    (ML)

[10:pp.3059-3061]
 

  
(10)  

(9)  (ML)
 


(

(9)3
 24
4s

4
ML
sth
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2-2-2MRobust IRWm-estimator
  (IRWm-estimator)        ([4]O.A.Ali, M.A.Abbas,2019)(M-estimator)
(Muggeo)
 (loss function
            [4:pp.393-394]



[4:pp.394-395]
   
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
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


     

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0.01
  

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



       M
(9)(Muggeo) a 
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

 
(9)Muggeo


Journal of Economics and Administrative Sciences
Vol.26 (NO. 121) 2020, pp. 415-427

b  c  






M
[5:pp56]



 









 



(14)





 
(15) (WLS)

[5:pp.57]


n n
d ac[17:pp.7] 









 (11)(Muggeo)



 (Muggeo)
[10:pp.3061]     
sth

(IRWm-estimator)
p


(Tukey) ([4]O.A.Ali, M.A.Abbas,2019)
 (Tukey bisquare)[14:pp.6]









 

c = 4.685

Journal of Economics and Administrative Sciences
Vol.26 (NO. 121) 2020, pp. 415-427

3-2-2    (S-estimator)   

  S       MM-scales 
[13]Rousseeuw and Yohai,1984        M
         weaknesses   [15:pp.354]
         (Muggeo)
M(2-2-2)


S
(IRWs-estimator)

   






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    
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      S
(9)(Muggeo) a
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 
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(9)Muggeo
 b
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 c

  

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
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   
S[15:pp. 354]


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 
 
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 
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(Tukey)
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 

  
(19)(2-2-2)
s(WLS)

[15:pp. 355]



Journal of Economics and Administrative Sciences
Vol.26 (NO. 121) 2020, pp. 415-427

n n
 d
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ac[17:pp.7] 




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
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 (11)(Muggeo)



            (Muggeo)
[10:pp.3061]     
sth

(IRWs-estimator)
2-3The standard error for the change point
             Wald    Wald    
 
[17:pp.8]

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
 
var(.)cov(. , ,)


 converges(Muggeo) 


[10:pp.3061]



3The Application 
3-1Real Data Description 
 (Sediment transport)  


      
[2:pp.7-8]
8

([3]Ammar A. Ali, Nadhir A. Al-Ansari, Qusay Al-Suhail and
Sven Knutsson,2017)   16 
(1)
[3:pp.67] CS6-1
8(1)(2)

Journal of Economics and Administrative Sciences
Vol.26 (NO. 121) 2020, pp. 415-427

:(1)
[3:p p.60]
(1)CS6-1 
(van Rijn 1984)
1

1000
900
800
700
600
500
400
300
X
4.011
3.122
2.529
2.157
2.003
1.866
1.954
2.251
Y
[3]
1
 XDischarge    Y bed load  

Journal of Economics and Administrative Sciences
Vol.26 (NO. 121) 2020, pp. 415-427

(2)    CS6-1   
(van Rijn 1984)
3-2Results Discussion

          
(2)
(3)(S.E.(  ))
   (MSE)       
(2)

(2)
CS6-1 
Parameters of the
segmented linear
regression model
Methods
Muggeo
IRWm-
estimator
IRWs-estimator
Tukey Bisquare
Tukey Bisquare

737.0557
737.4091
733.4267


2.1157
2.1049
1.8560


-0.00014
-0.00012
0.00025


-3.4483
-3.4475
-3.3990


0.00741
0.00741
0.00741
S.E.(  )
34.151
33.203
0.124
MSE
0.02743
0.02522
7.6698e-8


Journal of Economics and Administrative Sciences
Vol.26 (NO. 121) 2020, pp. 415-427


Muggeo

IRWm-estimator(Tukey Bisquare)

IRWs-estimator(Tukey Bisquare) 
(3)CS6-1  

 (2)    (MSE
    S.E.(  )  (IRWs-estimator)   (Tukey bisquare)(IRWm-estimator)(Tukey bisquare)
4Conclusions and Recommendations
4-1Conclusions

  (IRWs-estimator)(Tukey)
    ML (Muggeo)  (IRWm-estimator) 
(MSE)S.E.(  )      (IRWs-estimator)   (Tukey) 
              
         
  (IRWs-estimator) (Tukey)

.

Journal of Economics and Administrative Sciences
Vol.26 (NO. 121) 2020, pp. 415-427

4-2Recommendations


(IRWs-estimator)
 3-4Future Studies
     
              




              

Acknowledgments
We introduce our thanks to the editors and the reviewers of (JEAS) journal for
their efforts in enriching the research with their valuable comments. Thanks also
to Dr. Ammar Adel Ali for providing us with data on the hydrology of the Tigris
River at Baghdad city. References
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Journal of Economics and Administrative Sciences
Vol.26 (NO. 121) 2020, pp. 415-427

Proposing Robust IRWs Technique to Estimate Segmented Regression Model
for the Bed load Transport of Tigris River with Change Point of Water
Discharge Amount at Baghdad City
Mohammed Ahmed Abbas
Omar Abdulmohsin Ali
Researcher. Sunni Endowment
Diwan, Department of Religious
Education and Islamic Studies,
Baghdad, Iraq
Assist. Prof. Department of
Statistics, College of Management
and Economics, Baghdad
University, Baghdad, Iraq
(mohammad.stat.145@gmail.com)
(dromar72@coadec.uobaghdad.edu.iq)
Published:11/3/2020 Accepted:3/5/2020 Received :August / 2020
NonCommercial 4.0 -Creative Commons AttributionThis work is licensed under a NC 4.0)-(CC BY International 
Abstract
Segmented regression consists of several sections separated by different
points of membership, showing the heterogeneity arising from the process of
separating the segments within the research sample. This research is concerned
with estimating the location of the change point between segments and estimating
model parameters, and proposing a robust estimation method and compare it
with some other methods that used in the segmented regression. One of the
traditional methods (Muggeo method) has been used to find the maximum
likelihood estimator in an iterative approach for the model and the change point
as well. Moreover, robust estimation method (IRW method) has used which
depends on the use of the robust M-estimator technique in segmentation idea and
using the Tukey weight function. The research’s contribution lies in the
suggestion to use the S-estimator technique and using the Tukey weight function,
to obtain a robust method against cases of violation of the normal distribution
condition for random errors or the effect of outliers, and this method will be
called IRWs. The mentioned methods have been applied to a real data set related
to the bed-load of Tigris River/ Baghdad city as a response variable and the
amount of water discharge as an explanatory variable. The results of the
comparison showed the superiority of the proposed method.
Keywords: segmented linear regression, change point, reparameterization,
M-estimator, S-estimator, bed-load transport, hydrology of water bodies.