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back to ["River width computation"] [[BR]]

__Other model__ :[[BR]]
- Elvira model : model 4 with river >2nd order[[BR]]
Log10(Wet width + 1) = 0.22734+ 0.20045 (log10 catchment area) + 0.25939 (log 10 Shreve index)[[BR]]

- Thornton model : W = 4.98 CA^0.47 [[BR]]
4.
__Models tested__:[[BR]]
'''Linear regression'''
 * Model 1 : width ~ a*catchment_area + b
 * Model 2 : log(width) ~ a*log(up_area)+ b
 * Model 3 : log(width) ~ a*log(up_area) + uga + b 
 * Model 4 : log(width) ~ a*log(up_area) + b*log(shreve index) + c
 * Model 5 : log(width) ~ a*log(up_area) + b*log(shreve index) + uga + c

'''Non linear regression''' with library(nlme)
 * Model 1: width~alpha*up_area^beta
 * Model 2 : width~gamma*shreve^lambda  (à tester : width~gamma*strahler^lambda

   
 *  Data extraction :
   * year > 1987  (RHP)
   * Period : april-october
   * complete sampling method
   * largeur lame eau >0
   * CV largeur lame eau <= 5%
   * shreve >1 ?  (pris en compte)

NB : 
f=a(x^b )[[BR]]
On pose A=log(a), F=log(f), X=log(x)[[BR]]
F=log(f)=log(a)+b log(x)=A+bX [[BR]]
La représentation graphique donne une droite, ce qui permet de déterminer A et b, puis de calculer a grâce à la relation A=log(a). La méthode dite "de régression linéaire par les moindres carrés" est utilisée.[[BR]] 

__Model Validation for linear regression__
 * Residuals versus fitted values to verify homogeneity
 * QQ-plot or histogram of the residuals for normality
 * Residuals versus each explanatory variable to check independence

["River width graphics"]

__Model results__
1618 stations 
||df||AIC||
||lm1||  3|| 9706.075||
||lm2||  3|| 2595.026||
||lm3|| 12|| 2472.912||
||lm4||  4|| 2375.609||
||lm5|| 13|| 2354.614||

--> model lm2 width=1.88*area^0.24
                              Estimate Std. Error t value Pr(>|t|) [[BR]]   
(Intercept)                    0.63593    0.03924   16.20   <2e-16 ***[[BR]]
log(ers0_station2$st_up_area)  0.24624    0.00892   27.61   <2e-16 ***[[BR]]

--> model lm4 retenu avec 41% de la variation expliquée[[BR]]
                              Estimate Std. Error t value Pr(>|t|) [[BR]]   
(Intercept)                    0.90824    0.04071  22.312  < 2e-16 ***[[BR]]
log(ers0_station2$st_up_area)  0.09331    0.01297   7.195 9.53e-13 ***[[BR]]
log(ers0_station2$shreve)      0.20489    0.01331  15.390  < 2e-16 ***[[BR]]



 * '''nls1<-nls(st_cs_largeurlameeau~gamma*st_up_area^lambda, data=ers0_station2,start=list(gamma=5,lambda=0.5))'''
summary(nls1)
       Estimate Std. Error t value Pr(>|t|) [[BR]]   
gamma   2.25662    0.12533   18.00   <2e-16 ***[[BR]]
lambda  0.23981    0.01031   23.26   <2e-16 ***[[BR]]
[[Image(source:data/Docs/trac/RiverWidth/nls1_width_up_area.jpeg)]] [[BR]]


 * '''nls2<-nls(st_cs_largeurlameeau~gamma*shreve^lambda, data=ers0_station2,start=list(gamma=3.33,lambda=0.28))
summary(nls2)'''[[BR]]
       Estimate Std. Error t value Pr(>|t|) [[BR]]   
gamma  4.108061   0.121048   33.94   <2e-16 ***[[BR]]
lambda 0.227287   0.008639   26.31   <2e-16 ***[[BR]]
[[Image(source:data/Docs/trac/RiverWidth/nls2_width_shreve.jpeg)]] [[BR]]


 * '''nls3<-nls(st_cs_largeurlameeau~alpha*st_up_area^beta +gamma*shreve^lambda, data=subset(ers0_station2,ers0_station2$st_up_area<=8000),ers0_station2,start=list(alpha=1.37,beta=0.32,gamma=3.33,lambda=0.28))'''[[BR]]
summary(nls3)[[BR]]
Parameters:[[BR]]
       Estimate Std. Error t value Pr(>|t|)   [[BR]] 
alpha   0.44572    0.32403   1.376    0.169    [[BR]]
beta    0.36923    0.09191   4.017 6.16e-05 ***[[BR]]
gamma   2.60166    0.62737   4.147 3.55e-05 ***[[BR]
lambda  0.23017    0.03480   6.614 5.07e-11 ***[[BR]]
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 [[BR]]

Residual standard error: 4.077 on 1611 degrees of freedom[[BR]]

Number of iterations to convergence: 22 [[BR]]
Achieved convergence tolerance: 8.651e-06 [[BR]]

--> st_cs_largeurlameeau ~ 0.44 * st_up_area^0.36+ 2.6 * shreve^0.23 [[BR]]


 * '''loess'''
loess.smooth(ers0_station2$st_up_area[ers0_station2$st_up_area<800],ers0_station2$st_cs_largeurlameeau[ers0_station2$st_up_area<800])[[BR]]
[[Image(source:data/Docs/trac/RiverWidth/loess_width_up_area.jpeg)]] [[BR]]

["Some comments"]