1 Investigation

1.1 Data

matchup<-read.csv("matchups.csv")
matchup$atimdifmin<-abs(matchup$timdifmin)
matchup$logchlorI<-log(matchup$chlorI)
matchup$logchlorS<-log(matchup$chlorS)
matchup$dlogchlor<-matchup$logchlorI-matchup$logchlorS
matchup$Error<-matchup$dlogchlor^2
datatable(matchup, options = list(pageLength = 25, 
                                  dom = 'tip', 
                                  scrollY = 500,
                                  scroller = TRUE, 
                                  scrollX = TRUE,
                                  fixedColumns = TRUE), width = "100%", height = "100%")

1.2 Column Names

names(matchup)

 [1] "timeI"      "lonI"       "latI"       "satid"      "lonS"      
 [6] "latS"       "chlorI"     "chlorS"     "distkm"     "timdifmin" 
[11] "sdlnchlor"  "nchlor"     "senzr"      "solzr"      "windspeed" 
[16] "tlg869"     "taua869"    "atimdifmin" "logchlorI"  "logchlorS" 
[21] "dlogchlor"  "Error"

1.3 Location Plots

#pal <- colorFactor(wheel("tomato", num = length(matchup$chla_hplcM)), domain = unique(matchup$chla_hplcM))
pal <- colorNumeric(
  palette = "Reds",
  domain = matchup$chlorI)
pal1 <- colorNumeric(
  palette = "Blues",
  domain = matchup$chlorS)

m <- leaflet(matchup) %>%
     setView(lng = -70, lat = 50, zoom = 2.3)%>%
     addProviderTiles(providers$Esri.NatGeoWorldMap)  %>%  # Add default OpenStreetMap map tiles
     addCircleMarkers(lng=~lonI, lat=~latI,
                   #icon = icons, 
                   popup=~chlorI,
                   label =~chlorI,
                   radius=7,
                   color = ~pal(chlorI),
                   stroke = FALSE, fillOpacity = 1) %>%  # Add default OpenStreetMap map tiles
     addCircleMarkers(lng=~lonS, lat=~latS,
                   #icon = icons, 
                   popup=~chlorS,
                   label =~chlorS,
                   radius=7,
                   color = ~pal1(chlorS),
                   stroke = FALSE, fillOpacity = 1)
m  # Print the map

#mapshot(m, file = "Rplot.png")

locationSat<- data.frame(Longitude=matchup[,2], Latitude=matchup[,3], Chlorophyll=matchup[,7])
locationSat$Location<-"In Situ"
locationM<-data.frame(Longitude=matchup[,5], Latitude=matchup[,6], Chlorophyll=matchup[,8])
locationM$Location<-"Satellite"
location<-rbind.data.frame(locationSat,locationM)
location$logChlorophyll<-log(location$Chlorophyll)

pal <- colorFactor(c("navy","red"), domain = c("Satellite", "In Situ"))

m <- leaflet(location) %>%
     setView(lng = -70, lat = 50, zoom = 2.3)%>%
     addProviderTiles(providers$Esri.NatGeoWorldMap)  %>%  # Add default OpenStreetMap map tiles
     addCircleMarkers(lng=~Longitude, lat=~Latitude,
                   #icon = icons, 
                   popup=~Location,
                   label =~Chlorophyll,
                   radius=6,
                   color = ~pal(Location),
                   stroke = FALSE, fillOpacity = 1) 
m  # Print the map

#mapshot(m, file = "Rplot.png")

1.4 Histograms

matchup$rchi<-rchisq(nrow(matchup), 1, ncp = 0)

p1 <- ggplot(data=location, aes(x=logChlorophyll, fill=Location)) + 
              geom_density(alpha=.3, col="black") + 
              labs(title="Histogram of Log Chlorophyll")

p2 <- ggplot(data=matchup, aes(x=Error)) + 
  geom_histogram(aes(y =..density..), position="identity",
                 breaks=seq(min(matchup$Error), max(matchup$Error)+1, length.out = 20), 
                 alpha=.5, color="light blue", fill="red") + 
               geom_density(alpha=.3, fill="brown", col="black") + 
  labs(title="Histogram of Error")


p3 <- ggplot(data = matchup, aes(x = logchlorS, y = logchlorI)) + 
    geom_point(color='red')+
    xlim(-4,4) +
    ylim(-4,4) +
    geom_smooth(method = "loess", se = TRUE) + 
    geom_abline(intercept = 0, slope = 1, color="black", 
                 linetype="dashed", size=1)+
  labs(title="Log Chlorophyll Measurements")+
  xlab("Satellite")+ylab("In Situ")

p4 <- ggplot(data=matchup, aes(x=rchi)) + 
  geom_histogram(aes(y =..density..), position="identity",
                 breaks=seq(min(matchup$rchi), max(matchup$rchi)+1, length.out = 20), 
                 alpha=.5, color="light blue", fill="red") + 
               geom_density(alpha=.3, fill="brown", col="black") + 
  labs(title="Histogram of Chi Squared df=1")+
  xlab("Simulated")

# grid.arrange(p1, p2,
#              p3, p4,# Second row with 2 plots in 2 different columns
#              nrow = 2)   
# 
# 
# grid.arrange(# First row with one plot spaning over 2 columns
#              arrangeGrob(p1, p2,ncol = 2), 
#              p3,# Second row with 2 plots in 2 different columns
#              nrow = 2)   
# 
# grid.arrange(p1, p2, p3, nrow = 1)

grid.arrange(p1, p2,
             p3, p4,# Second row with 2 plots in 2 different columns
             nrow = 2)

# subplot(ggplotly(p1), ggplotly(p2), ggplotly(p3), ggplotly(p4), margin = 0.05, nrows = 2)

1.4.1 Testing Log-Normality

shapiro.test(location[location$Location=="In Situ",]$logChlorophyll)


    Shapiro-Wilk normality test

data:  location[location$Location == "In Situ", ]$logChlorophyll
W = 0.96495, p-value = 0.00001699

shapiro.test(location[location$Location=="Satellite",]$logChlorophyll)


    Shapiro-Wilk normality test

data:  location[location$Location == "Satellite", ]$logChlorophyll
W = 0.93484, p-value = 0.00000001187

Failed.

1.4.2 Comparing with Gamma and Generalised Gamma

matchup$rgamma<-rGG(nrow(matchup), mu=exp(-0.8), sigma= 1.49, nu= 1)
matchup$rgGamma<-rGG(nrow(matchup), mu=exp(-1.119), sigma= 1.66, nu= 0.68)


p1 <- ggplot(data=matchup, aes(x=Error)) + 
  geom_histogram(aes(y =..density..), position="identity",
                 breaks=seq(min(matchup$Error), max(matchup$Error)+1, length.out = 20), 
                 alpha=.5, color="light blue", fill="red") + 
               geom_density(alpha=.3, fill="brown", col="black") + 
  labs(title="Histogram of Error")

p2 <- ggplot(data=matchup, aes(x=rgamma)) + 
  geom_histogram(aes(y =..density..), position="identity",
                 breaks=seq(min(matchup$rgamma), max(matchup$rgamma)+1, length.out = 20), 
                 alpha=.5, color="light blue", fill="red") + 
               geom_density(alpha=.3, fill="brown", col="black") + 
  labs(title="Histogram of Gamma")+
  xlab("Simulated")

p3 <- ggplot(data=matchup, aes(x=rgGamma)) + 
  geom_histogram(aes(y =..density..), position="identity",
                 breaks=seq(min(matchup$rgGamma), max(matchup$rgGamma)+1, length.out = 20), 
                 alpha=.5, color="light blue", fill="red") + 
               geom_density(alpha=.3, fill="brown", col="black") + 
  labs(title="Histogram of GGamma")+
  xlab("Simulated")

grid.arrange(p1, p2,
             p3, p4,# Second row with 2 plots in 2 different columns
             nrow = 2)

#subplot(ggplotly(p1), ggplotly(p2), ggplotly(p3), ggplotly(p4), margin = 0.05, nrows = 2
        #, shareX = TRUE, shareY = TRUE
#        )

2 Modelling on Data

2.1 GLM Model

m1<-glm(Error ~ distkm +  atimdifmin + sdlnchlor + nchlor + senzr + solzr + windspeed + taua869,
        family=Gamma(link="log"), data = matchup)
summary(m1)


Call:
glm(formula = Error ~ distkm + atimdifmin + sdlnchlor + nchlor + 
    senzr + solzr + windspeed + taua869, family = Gamma(link = "log"), 
    data = matchup)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-4.2000  -1.6247  -0.7419   0.3381   2.6932  

Coefficients:
              Estimate Std. Error t value  Pr(>|t|)    
(Intercept) -1.5331065  0.6742112  -2.274   0.02392 *  
distkm      -1.1194265  0.4138218  -2.705   0.00735 ** 
atimdifmin   0.0021509  0.0005061   4.250 0.0000314 ***
sdlnchlor    2.9764107  0.8041374   3.701   0.00027 ***
nchlor      -0.0473464  0.0497581  -0.952   0.34236    
senzr        0.6653076  0.3262554   2.039   0.04260 *  
solzr        0.4933609  0.4448334   1.109   0.26858    
windspeed    0.0149680  0.0426444   0.351   0.72592    
taua869     -0.6293843  1.6230513  -0.388   0.69855    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for Gamma family taken to be 1.742496)

    Null deviance: 673.33  on 232  degrees of freedom
Residual deviance: 585.39  on 224  degrees of freedom
AIC: -63.835

Number of Fisher Scoring iterations: 12

#xtable(summary(m1), caption="Coefficient", digits=4)

2.1.1 GLM Model Checking

autoplot(m1, colour="red", which = 1:6, ncol = 2, label.size = 3, title="Model checking plots")

shapiro.test(m1$residuals)


    Shapiro-Wilk normality test

data:  m1$residuals
W = 0.75227, p-value < 2.2e-16

Residuals are not centred around zero. The Shapiro–Wilk test tests the null hypothesis that a sample came from a normally distributed population. Here we reject the null.

# Evaluate Collinearity
vif(m1)

    distkm atimdifmin  sdlnchlor     nchlor      senzr      solzr 
  1.146489   1.111858   1.023226   1.024226   1.224432   1.156866 
 windspeed    taua869 
  1.105677   1.104678

sqrt(vif(m1)) > 2 # problem?

    distkm atimdifmin  sdlnchlor     nchlor      senzr      solzr 
     FALSE      FALSE      FALSE      FALSE      FALSE      FALSE 
 windspeed    taua869 
     FALSE      FALSE

No problem with Collinearity.

# Evaluate Nonlinearity
# component + residual plot
crPlots(m1)

Partial-residual plots, for linear and generalized linear models. Evaluate Nonlinearity.

durbinWatsonTest(m1)

 lag Autocorrelation D-W Statistic p-value
   1       0.5024233     0.9938355       0
 Alternative hypothesis: rho != 0

Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation at lag 1 in the residuals (prediction errors) from a regression analysis.

2.2 GAM Model

m2 <- gam(Error ~ s(distkm) +  s(atimdifmin) + s(sdlnchlor) + nchlor + s(senzr) + s(solzr) + s(windspeed) + s(taua869), family=Gamma(link="log"), data = matchup)
summary(m2)


Call: gam(formula = Error ~ s(distkm) + s(atimdifmin) + s(sdlnchlor) + 
    nchlor + s(senzr) + s(solzr) + s(windspeed) + s(taua869), 
    family = Gamma(link = "log"), data = matchup)
Deviance Residuals:
    Min      1Q  Median      3Q     Max 
-4.3812 -1.4933 -0.5005  0.4067  2.6204 

(Dispersion Parameter for Gamma family taken to be 1.4866)

    Null Deviance: 673.3335 on 232 degrees of freedom
Residual Deviance: 509.6107 on 203.0003 degrees of freedom
AIC: -63.9112 

Number of Local Scoring Iterations: 30 

Anova for Parametric Effects
               Df  Sum Sq Mean Sq F value    Pr(>F)    
s(distkm)       1   8.206  8.2056  5.5196 0.0197652 *  
s(atimdifmin)   1  17.612 17.6116 11.8466 0.0007013 ***
s(sdlnchlor)    1  13.957 13.9571  9.3884 0.0024801 ** 
nchlor          1   1.126  1.1256  0.7571 0.3852611    
s(senzr)        1   1.849  1.8488  1.2436 0.2660933    
s(solzr)        1   0.206  0.2059  0.1385 0.7101867    
s(windspeed)    1   0.007  0.0073  0.0049 0.9440965    
s(taua869)      1   5.890  5.8897  3.9618 0.0478862 *  
Residuals     203 301.787  1.4866                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Anova for Nonparametric Effects
              Npar Df Npar F   Pr(F)   
(Intercept)                            
s(distkm)           3 2.5677 0.05556 . 
s(atimdifmin)       3 0.5929 0.62034   
s(sdlnchlor)        3 1.0060 0.39114   
nchlor                                 
s(senzr)            3 2.9483 0.03388 * 
s(solzr)            3 1.8143 0.14573   
s(windspeed)        3 3.4808 0.01688 * 
s(taua869)          3 5.2361 0.00168 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#suumerym1<-summary(m2)
#gamtabs(m2, caption='Summary of m1')
#xtable(suumerym1$anova, caption="ANOVA table for GAM", digits=4)

2.2.1 GAM Model Checking

autoplot(m2, colour="red", which = 1:6, ncol = 2, label.size = 3, title="Model checking plots")

shapiro.test(m2$residuals)


    Shapiro-Wilk normality test

data:  m2$residuals
W = 0.8104, p-value = 3.853e-16

vif(m2)

    s(distkm) s(atimdifmin)  s(sdlnchlor)        nchlor      s(senzr) 
     1.146489      1.111858      1.023226      1.024226      1.224432 
     s(solzr)  s(windspeed)    s(taua869) 
     1.156866      1.105677      1.104678

sqrt(vif(m2)) > 2

    s(distkm) s(atimdifmin)  s(sdlnchlor)        nchlor      s(senzr) 
        FALSE         FALSE         FALSE         FALSE         FALSE 
     s(solzr)  s(windspeed)    s(taua869) 
        FALSE         FALSE         FALSE

crPlots(m2)

durbinWatsonTest(m2)

 lag Autocorrelation D-W Statistic p-value
   1       0.4300903      1.139699       0
 Alternative hypothesis: rho != 0

Similarly, residuals are not centred around zero. The Shapiro–Wilk test rejects the null.

2.2.2 Partial Plots

plot(m2, se = TRUE, col="red", lwd=2, pages=1 ,residuals=TRUE)

2.3 GAMLSS Model with Gamma

m3 <- gamlss(Error ~ cs(distkm) +  cs(atimdifmin) + cs(sdlnchlor) + nchlor + cs(senzr) + cs(solzr) + cs(windspeed) + cs(taua869), 
           family=GA(sigma.link="log"), 
           control=gamlss.control(c.crit = 0.0001, n.cyc = 50), 
           data = matchup,
           trace=FALSE,
           )

GAMLSS-RS iteration 1: Global Deviance = -133.8051 
GAMLSS-RS iteration 2: Global Deviance = -133.4522 
GAMLSS-RS iteration 3: Global Deviance = -133.4274 
GAMLSS-RS iteration 4: Global Deviance = -133.4229 
GAMLSS-RS iteration 5: Global Deviance = -133.4219 
GAMLSS-RS iteration 6: Global Deviance = -133.4215 
GAMLSS-RS iteration 7: Global Deviance = -133.4213 
GAMLSS-RS iteration 8: Global Deviance = -133.4212 
GAMLSS-RS iteration 9: Global Deviance = -133.4207 
GAMLSS-RS iteration 10: Global Deviance = -133.4212 
GAMLSS-RS iteration 11: Global Deviance = -133.4208 
GAMLSS-RS iteration 12: Global Deviance = -133.4212 
GAMLSS-RS iteration 13: Global Deviance = -133.4209 
GAMLSS-RS iteration 14: Global Deviance = -133.4211 
GAMLSS-RS iteration 15: Global Deviance = -133.4209 
GAMLSS-RS iteration 16: Global Deviance = -133.4211 
GAMLSS-RS iteration 17: Global Deviance = -133.4209 
GAMLSS-RS iteration 18: Global Deviance = -133.421 
GAMLSS-RS iteration 19: Global Deviance = -133.421

summary(m3)

******************************************************************
Family:  c("GA", "Gamma") 

Call:  gamlss(formula = Error ~ cs(distkm) + cs(atimdifmin) +  
    cs(sdlnchlor) + nchlor + cs(senzr) + cs(solzr) +  
    cs(windspeed) + cs(taua869), family = GA(sigma.link = "log"),  
    data = matchup, control = gamlss.control(c.crit = 0.0001,  
        n.cyc = 50), trace = FALSE) 

Fitting method: RS() 

------------------------------------------------------------------
Mu link function:  log
Mu Coefficients:
                 Estimate Std. Error t value Pr(>|t|)   
(Intercept)    -0.7781941  0.6862996  -1.134  0.25818   
cs(distkm)     -0.8906453  0.4249891  -2.096  0.03736 * 
cs(atimdifmin)  0.0013807  0.0004908   2.813  0.00539 **
cs(sdlnchlor)   2.4870284  0.8189289   3.037  0.00271 **
nchlor         -0.0439038  0.0514856  -0.853  0.39481   
cs(senzr)       0.3251230  0.3385427   0.960  0.33802   
cs(solzr)      -0.2287731  0.4491539  -0.509  0.61107   
cs(windspeed)   0.0169463  0.0444586   0.381  0.70348   
cs(taua869)    -2.5299200  1.6602942  -1.524  0.12913   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

------------------------------------------------------------------
Sigma link function:  log
Sigma Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.28168    0.03874   7.271 7.68e-12 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas: 
 i) Std. Error for smoothers are for the linear effect only. 
ii) Std. Error for the linear terms maybe are not accurate. 
------------------------------------------------------------------
No. of observations in the fit:  233 
Degrees of Freedom for the fit:  31.00109
      Residual Deg. of Freedom:  201.9989 
                      at cycle:  19 
 
Global Deviance:     -133.421 
            AIC:     -71.41878 
            SBC:     35.56717 
******************************************************************

2.3.1 GAMLSS with Gamma Model Checking

plot(m3)

******************************************************************
          Summary of the Quantile Residuals
                           mean   =  0.01608588 
                       variance   =  1.024702 
               coef. of skewness  =  -0.3359576 
               coef. of kurtosis  =  2.744791 
Filliben correlation coefficient  =  0.9949545 
******************************************************************

shapiro.test(m3$residuals)


    Shapiro-Wilk normality test

data:  m3$residuals
W = 0.98908, p-value = 0.07487

Rsq(m3)

[1] 0.2975244

AIC(m3)

[1] -71.41878

vif(m3)

    cs(distkm) cs(atimdifmin)  cs(sdlnchlor)         nchlor      cs(senzr) 
      1.200207       1.180365       1.044411       1.131815       1.316032 
     cs(solzr)  cs(windspeed)    cs(taua869) 
      1.159958       1.171829       1.172611

sqrt(vif(m3)) > 2

    cs(distkm) cs(atimdifmin)  cs(sdlnchlor)         nchlor      cs(senzr) 
         FALSE          FALSE          FALSE          FALSE          FALSE 
     cs(solzr)  cs(windspeed)    cs(taua869) 
         FALSE          FALSE          FALSE

durbinWatsonTest(m3$residuals)

[1] 1.341672

wp(m3, ylim.all=.6); title("Residuals for Gamma Modelling with GAMLSS")

To check the adequacy of the fitted GAM we use a worm plot, which is a de-trended QQ-plot of the residuals expect the dots (which look like a worm) to be close to the middle horizontal line and 95% of them to lie between the upper and lower dotted curves, which act as 95% pointwise confidence intervals, with no systematic departure.

term.plot(m3, pages=1, ask=FALSE, partial.resid = TRUE, col.term = "black", 
         col.se = "orange", col.shaded = "gray", col.res = "red", 
         col.rug = "gray", lwd.term = 2, lty.se = 2, lwd.se = 1)

2.4 GAMLSS Model with Generalised Gamma

m3g<-gamlss(Error ~ cs(distkm) +  cs(atimdifmin) + cs(sdlnchlor) + nchlor + cs(senzr) + cs(solzr) + cs(windspeed) + cs(taua869), 
            family=GG(mu.link ="log" , sigma.link="log", nu.link="identity"), 
            control=gamlss.control(c.crit = 0.0001, n.cyc = 50), 
            data = matchup)

GAMLSS-RS iteration 1: Global Deviance = -134.9773 
GAMLSS-RS iteration 2: Global Deviance = -136.036 
GAMLSS-RS iteration 3: Global Deviance = -137.0451 
GAMLSS-RS iteration 4: Global Deviance = -137.8288 
GAMLSS-RS iteration 5: Global Deviance = -138.4587 
GAMLSS-RS iteration 6: Global Deviance = -138.9699 
GAMLSS-RS iteration 7: Global Deviance = -139.392 
GAMLSS-RS iteration 8: Global Deviance = -139.7429 
GAMLSS-RS iteration 9: Global Deviance = -140.0376 
GAMLSS-RS iteration 10: Global Deviance = -140.2761 
GAMLSS-RS iteration 11: Global Deviance = -140.4867 
GAMLSS-RS iteration 12: Global Deviance = -140.6686 
GAMLSS-RS iteration 13: Global Deviance = -140.8256 
GAMLSS-RS iteration 14: Global Deviance = -140.9615 
GAMLSS-RS iteration 15: Global Deviance = -141.0796 
GAMLSS-RS iteration 16: Global Deviance = -141.1824 
GAMLSS-RS iteration 17: Global Deviance = -141.2722 
GAMLSS-RS iteration 18: Global Deviance = -141.3507 
GAMLSS-RS iteration 19: Global Deviance = -141.4196 
GAMLSS-RS iteration 20: Global Deviance = -141.48 
GAMLSS-RS iteration 21: Global Deviance = -141.533 
GAMLSS-RS iteration 22: Global Deviance = -141.5796 
GAMLSS-RS iteration 23: Global Deviance = -141.6205 
GAMLSS-RS iteration 24: Global Deviance = -141.6563 
GAMLSS-RS iteration 25: Global Deviance = -141.6877 
GAMLSS-RS iteration 26: Global Deviance = -141.7151 
GAMLSS-RS iteration 27: Global Deviance = -141.7389 
GAMLSS-RS iteration 28: Global Deviance = -141.7595 
GAMLSS-RS iteration 29: Global Deviance = -141.7773 
GAMLSS-RS iteration 30: Global Deviance = -141.7925 
GAMLSS-RS iteration 31: Global Deviance = -141.8053 
GAMLSS-RS iteration 32: Global Deviance = -141.816 
GAMLSS-RS iteration 33: Global Deviance = -141.8247 
GAMLSS-RS iteration 34: Global Deviance = -141.8316 
GAMLSS-RS iteration 35: Global Deviance = -141.8369 
GAMLSS-RS iteration 36: Global Deviance = -141.8407 
GAMLSS-RS iteration 37: Global Deviance = -141.8431 
GAMLSS-RS iteration 38: Global Deviance = -141.8441 
GAMLSS-RS iteration 39: Global Deviance = -141.844 
GAMLSS-RS iteration 40: Global Deviance = -141.8427 
GAMLSS-RS iteration 41: Global Deviance = -141.8403 
GAMLSS-RS iteration 42: Global Deviance = -141.8369 
GAMLSS-RS iteration 43: Global Deviance = -141.8325 
GAMLSS-RS iteration 44: Global Deviance = -141.8271 
GAMLSS-RS iteration 45: Global Deviance = -141.8209 
GAMLSS-RS iteration 46: Global Deviance = -141.8138 
GAMLSS-RS iteration 47: Global Deviance = -141.8059 
GAMLSS-RS iteration 48: Global Deviance = -141.7971 
GAMLSS-RS iteration 49: Global Deviance = -141.7875 
GAMLSS-RS iteration 50: Global Deviance = -141.7771

summary(m3g)

******************************************************************
Family:  c("GG", "generalised Gamma Lopatatsidis-Green") 

Call:  gamlss(formula = Error ~ cs(distkm) + cs(atimdifmin) +  
    cs(sdlnchlor) + nchlor + cs(senzr) + cs(solzr) +  
    cs(windspeed) + cs(taua869), family = GG(mu.link = "log",  
    sigma.link = "log", nu.link = "identity"), data = matchup,  
    control = gamlss.control(c.crit = 0.0001, n.cyc = 50)) 

Fitting method: RS() 

------------------------------------------------------------------
Mu link function:  log
Mu Coefficients:
                 Estimate Std. Error t value      Pr(>|t|)    
(Intercept)    -0.2457093  0.5642770  -0.435        0.6637    
cs(distkm)     -0.5865982  0.2634197  -2.227        0.0271 *  
cs(atimdifmin)  0.0019294  0.0003154   6.118 0.00000000488 ***
cs(sdlnchlor)   2.5109015  0.5222761   4.808 0.00000298807 ***
nchlor         -0.0706765  0.0333637  -2.118        0.0354 *  
cs(senzr)       0.2748830  0.2371412   1.159        0.2478    
cs(solzr)       0.1397952  0.3020100   0.463        0.6439    
cs(windspeed)  -0.0285686  0.0331152  -0.863        0.3893    
cs(taua869)    -1.6342807  1.0142710  -1.611        0.1087    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

------------------------------------------------------------------
Sigma link function:  log
Sigma Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.2318     0.3646  -0.636    0.526

------------------------------------------------------------------
Nu link function:  identity 
Nu Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)    3.720      2.959   1.257     0.21

------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas: 
 i) Std. Error for smoothers are for the linear effect only. 
ii) Std. Error for the linear terms maybe are not accurate. 
------------------------------------------------------------------
No. of observations in the fit:  233 
Degrees of Freedom for the fit:  32.00109
      Residual Deg. of Freedom:  200.9989 
                      at cycle:  50 
 
Global Deviance:     -141.7771 
            AIC:     -77.77493 
            SBC:     32.66206 
******************************************************************

2.4.1 GAMLSS with Generalised Gamma Model Checking

plot(m3g)

******************************************************************
          Summary of the Quantile Residuals
                           mean   =  0.0092253 
                       variance   =  0.9875956 
               coef. of skewness  =  0.0893399 
               coef. of kurtosis  =  2.561804 
Filliben correlation coefficient  =  0.9965565 
******************************************************************

shapiro.test(m3g$residuals)


    Shapiro-Wilk normality test

data:  m3g$residuals
W = 0.99169, p-value = 0.2096

Rsq(m3g)

[1] 0.3089104

AIC(m3g)

[1] -77.77493

GAIC(m3,m3g)

          df       AIC
m3g 32.00109 -77.77493
m3  31.00109 -71.41878

vif(m3g)

    cs(distkm) cs(atimdifmin)  cs(sdlnchlor)         nchlor      cs(senzr) 
      1.318779       1.212528       1.163455       1.337637       1.864737 
     cs(solzr)  cs(windspeed)    cs(taua869) 
      1.503492       1.758193       1.300490

sqrt(vif(m3g)) > 2

    cs(distkm) cs(atimdifmin)  cs(sdlnchlor)         nchlor      cs(senzr) 
         FALSE          FALSE          FALSE          FALSE          FALSE 
     cs(solzr)  cs(windspeed)    cs(taua869) 
         FALSE          FALSE          FALSE

durbinWatsonTest(m3g$residuals)

[1] 1.29997

wp(m3g, ylim.all=.6); title("Generelised Gamma Model Checking")

term.plot(m3g, pages=1, ask=FALSE, partial.resid = TRUE, col.term = "black", 
         col.se = "orange", col.shaded = "gray", col.res = "red", 
         col.rug = "gray", lwd.term = 2, lty.se = 2, lwd.se = 1)

2.5 GAMLSS Model with Generalised Gamma add Regression on Scale and Shape

m3gs<-gamlss(Error ~ cs(distkm) +  cs(atimdifmin) + cs(sdlnchlor) + nchlor + cs(senzr) + cs(solzr) + cs(windspeed) + cs(taua869),
            sigma.fo=~cs(distkm) +  cs(atimdifmin),
            nu.fo=~cs(distkm) +  cs(atimdifmin) + cs(sdlnchlor) + nchlor + cs(senzr) + cs(solzr) + cs(windspeed) + cs(taua869),
            family=GG(mu.link ="log"), 
            control=gamlss.control(c.crit = 0.001, n.cyc = 40), 
            data = matchup)

GAMLSS-RS iteration 1: Global Deviance = -160.1331 
GAMLSS-RS iteration 2: Global Deviance = -164.1527 
GAMLSS-RS iteration 3: Global Deviance = -166.4022 
GAMLSS-RS iteration 4: Global Deviance = -168.0836 
GAMLSS-RS iteration 5: Global Deviance = -169.3969 
GAMLSS-RS iteration 6: Global Deviance = -170.4602 
GAMLSS-RS iteration 7: Global Deviance = -171.3324 
GAMLSS-RS iteration 8: Global Deviance = -172.0727 
GAMLSS-RS iteration 9: Global Deviance = -172.7086 
GAMLSS-RS iteration 10: Global Deviance = -173.2656 
GAMLSS-RS iteration 11: Global Deviance = -173.7622 
GAMLSS-RS iteration 12: Global Deviance = -174.2097 
GAMLSS-RS iteration 13: Global Deviance = -174.6196 
GAMLSS-RS iteration 14: Global Deviance = -175.0028 
GAMLSS-RS iteration 15: Global Deviance = -175.3634 
GAMLSS-RS iteration 16: Global Deviance = -175.7069 
GAMLSS-RS iteration 17: Global Deviance = -176.0393 
GAMLSS-RS iteration 18: Global Deviance = -176.3598 
GAMLSS-RS iteration 19: Global Deviance = -176.6698 
GAMLSS-RS iteration 20: Global Deviance = -176.9655 
GAMLSS-RS iteration 21: Global Deviance = -177.2515 
GAMLSS-RS iteration 22: Global Deviance = -177.5276 
GAMLSS-RS iteration 23: Global Deviance = -177.7883 
GAMLSS-RS iteration 24: Global Deviance = -178.0332 
GAMLSS-RS iteration 25: Global Deviance = -178.2645 
GAMLSS-RS iteration 26: Global Deviance = -178.4807 
GAMLSS-RS iteration 27: Global Deviance = -178.6805 
GAMLSS-RS iteration 28: Global Deviance = -178.867 
GAMLSS-RS iteration 29: Global Deviance = -179.038 
GAMLSS-RS iteration 30: Global Deviance = -179.1948 
GAMLSS-RS iteration 31: Global Deviance = -179.3396 
GAMLSS-RS iteration 32: Global Deviance = -179.4724 
GAMLSS-RS iteration 33: Global Deviance = -179.5929 
GAMLSS-RS iteration 34: Global Deviance = -179.7048 
GAMLSS-RS iteration 35: Global Deviance = -179.8046 
GAMLSS-RS iteration 36: Global Deviance = -179.8962 
GAMLSS-RS iteration 37: Global Deviance = -179.9627 
GAMLSS-RS iteration 38: Global Deviance = -180.0556 
GAMLSS-RS iteration 39: Global Deviance = -180.1126 
GAMLSS-RS iteration 40: Global Deviance = -180.1939

summary(m3gs)

******************************************************************
Family:  c("GG", "generalised Gamma Lopatatsidis-Green") 

Call:  gamlss(formula = Error ~ cs(distkm) + cs(atimdifmin) +  
    cs(sdlnchlor) + nchlor + cs(senzr) + cs(solzr) +  
    cs(windspeed) + cs(taua869), sigma.formula = ~cs(distkm) +  
    cs(atimdifmin), nu.formula = ~cs(distkm) + cs(atimdifmin) +  
    cs(sdlnchlor) + nchlor + cs(senzr) + cs(solzr) +  
    cs(windspeed) + cs(taua869), family = GG(mu.link = "log"),  
    data = matchup, control = gamlss.control(c.crit = 0.001,  
        n.cyc = 40)) 

Fitting method: RS() 

------------------------------------------------------------------
Mu link function:  log
Mu Coefficients:
                 Estimate Std. Error t value    Pr(>|t|)    
(Intercept)     0.3882071  0.6029718   0.644     0.52059    
cs(distkm)     -1.2348256  0.4490625  -2.750     0.00663 ** 
cs(atimdifmin)  0.0010555  0.0004631   2.279     0.02394 *  
cs(sdlnchlor)   2.9871105  0.5631621   5.304 0.000000362 ***
nchlor         -0.0958215  0.0371596  -2.579     0.01080 *  
cs(senzr)       0.4634794  0.3035628   1.527     0.12874    
cs(solzr)      -0.2792880  0.4058419  -0.688     0.49232    
cs(windspeed)  -0.0834379  0.0377890  -2.208     0.02863 *  
cs(taua869)    -0.9264336  1.5121624  -0.613     0.54095    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

------------------------------------------------------------------
Sigma link function:  log
Sigma Coefficients:
                 Estimate Std. Error t value Pr(>|t|)  
(Intercept)    -0.3831722  0.2031942  -1.886   0.0611 .
cs(distkm)      0.3107327  0.2578122   1.205   0.2298  
cs(atimdifmin)  0.0007217  0.0002852   2.530   0.0123 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

------------------------------------------------------------------
Nu link function:  identity 
Nu Coefficients:
                Estimate Std. Error t value Pr(>|t|)   
(Intercept)     5.803869   1.805927   3.214  0.00158 **
cs(distkm)     -1.359094   1.145804  -1.186  0.23728   
cs(atimdifmin) -0.002892   0.001858  -1.557  0.12142   
cs(sdlnchlor)  -0.659691   1.579212  -0.418  0.67669   
nchlor         -0.205343   0.124253  -1.653  0.10032   
cs(senzr)       0.415159   0.677779   0.613  0.54104   
cs(solzr)      -0.477691   0.953435  -0.501  0.61703   
cs(windspeed)  -0.191006   0.089612  -2.131  0.03454 * 
cs(taua869)     1.492608   3.125778   0.478  0.63363   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas: 
 i) Std. Error for smoothers are for the linear effect only. 
ii) Std. Error for the linear terms maybe are not accurate. 
------------------------------------------------------------------
No. of observations in the fit:  233 
Degrees of Freedom for the fit:  68.99744
      Residual Deg. of Freedom:  164.0026 
                      at cycle:  40 
 
Global Deviance:     -180.1939 
            AIC:     -42.19901 
            SBC:     195.9138 
******************************************************************

2.5.1 GAMLSS with Generalised Gamma Regression on SS Model Checking

plot(m3gs)

******************************************************************
          Summary of the Quantile Residuals
                           mean   =  -0.02206921 
                       variance   =  0.9542674 
               coef. of skewness  =  0.05649195 
               coef. of kurtosis  =  2.390731 
Filliben correlation coefficient  =  0.9963909 
******************************************************************

shapiro.test(m3gs$residuals)


    Shapiro-Wilk normality test

data:  m3gs$residuals
W = 0.99152, p-value = 0.1961

Rsq(m3gs)

[1] 0.4139585

GAIC(m3,m3g,m3gs)

           df       AIC
m3g  32.00109 -77.77493
m3   31.00109 -71.41878
m3gs 68.99744 -42.19901

vif(m3gs)

    cs(distkm) cs(atimdifmin)  cs(sdlnchlor)         nchlor      cs(senzr) 
      2.160861       2.999938       1.118570       1.394376       2.121984 
     cs(solzr)  cs(windspeed)    cs(taua869) 
      1.985239       1.636725       1.771172

sqrt(vif(m3gs)) > 2

    cs(distkm) cs(atimdifmin)  cs(sdlnchlor)         nchlor      cs(senzr) 
         FALSE          FALSE          FALSE          FALSE          FALSE 
     cs(solzr)  cs(windspeed)    cs(taua869) 
         FALSE          FALSE          FALSE

durbinWatsonTest(m3gs$residuals)

[1] 1.520991

wp(m3gs, ylim.all=.6); title("GG(mu, sigma)")

2.5.2 Partials for mu

term.plot(m3gs, pages=1, ask=FALSE, partial.resid = TRUE, col.term = "black", 
         col.se = "orange", col.shaded = "gray", col.res = "red", 
         col.rug = "gray", lwd.term = 2, lty.se = 2, lwd.se = 1)

2.5.3 Partials for sigma

term.plot(m3gs, pages=1, what="sigma", ask=FALSE, partial.resid = TRUE, col.term = "black", 
         col.se = "orange", col.shaded = "gray", col.res = "red", 
         col.rug = "gray", lwd.term = 2, lty.se = 2, lwd.se = 1)

2.5.4 Partials for nu

term.plot(m3gs, pages=1, what="nu", partial.resid = TRUE, col.term = "black", 
         col.se = "orange", col.shaded = "gray", col.res = "red", 
         col.rug = "gray", lwd.term = 2, lty.se = 2, lwd.se = 1)

plotSimpleGamlss(Error, distkm, model=m3gs, data=matchup,
#x.val=seq(6,16,2), val=5, N=1000, ylim=c(0,25),
cols=heat_hcl(100))

drop1(m3gs)

GAMLSS-RS iteration 1: Global Deviance = -150.2221 
GAMLSS-RS iteration 2: Global Deviance = -154.422 
GAMLSS-RS iteration 3: Global Deviance = -156.9242 
GAMLSS-RS iteration 4: Global Deviance = -159.2718 
GAMLSS-RS iteration 5: Global Deviance = -160.6686 
GAMLSS-RS iteration 6: Global Deviance = -162.1146 
GAMLSS-RS iteration 7: Global Deviance = -163.3887 
GAMLSS-RS iteration 8: Global Deviance = -164.4779 
GAMLSS-RS iteration 9: Global Deviance = -165.3866 
GAMLSS-RS iteration 10: Global Deviance = -166.124 
GAMLSS-RS iteration 11: Global Deviance = -166.7414 
GAMLSS-RS iteration 12: Global Deviance = -167.2311 
GAMLSS-RS iteration 13: Global Deviance = -167.6311 
GAMLSS-RS iteration 14: Global Deviance = -167.9637 
GAMLSS-RS iteration 15: Global Deviance = -168.2017 
GAMLSS-RS iteration 16: Global Deviance = -168.4769 
GAMLSS-RS iteration 17: Global Deviance = -168.6862 
GAMLSS-RS iteration 18: Global Deviance = -168.8664 
GAMLSS-RS iteration 19: Global Deviance = -169.0257 
GAMLSS-RS iteration 20: Global Deviance = -169.1696 
GAMLSS-RS iteration 21: Global Deviance = -169.2898 
GAMLSS-RS iteration 22: Global Deviance = -169.4021 
GAMLSS-RS iteration 23: Global Deviance = -169.5053 
GAMLSS-RS iteration 24: Global Deviance = -169.601 
GAMLSS-RS iteration 25: Global Deviance = -169.6889 
GAMLSS-RS iteration 26: Global Deviance = -169.7718 
GAMLSS-RS iteration 27: Global Deviance = -169.8485 
GAMLSS-RS iteration 28: Global Deviance = -169.9216 
GAMLSS-RS iteration 29: Global Deviance = -169.9903 
GAMLSS-RS iteration 30: Global Deviance = -170.0541 
GAMLSS-RS iteration 31: Global Deviance = -170.1142 
GAMLSS-RS iteration 32: Global Deviance = -170.172 
GAMLSS-RS iteration 33: Global Deviance = -170.2259 
GAMLSS-RS iteration 34: Global Deviance = -170.2945 
GAMLSS-RS iteration 35: Global Deviance = -170.3299 
GAMLSS-RS iteration 36: Global Deviance = -170.3743 
GAMLSS-RS iteration 37: Global Deviance = -170.4178 
GAMLSS-RS iteration 38: Global Deviance = -170.4592 
GAMLSS-RS iteration 39: Global Deviance = -170.4985 
GAMLSS-RS iteration 40: Global Deviance = -170.5364 
GAMLSS-RS iteration 1: Global Deviance = -153.9124 
GAMLSS-RS iteration 2: Global Deviance = -157.9864 
GAMLSS-RS iteration 3: Global Deviance = -160.523 
GAMLSS-RS iteration 4: Global Deviance = -162.7597 
GAMLSS-RS iteration 5: Global Deviance = -163.9974 
GAMLSS-RS iteration 6: Global Deviance = -165.2687 
GAMLSS-RS iteration 7: Global Deviance = -166.3586 
GAMLSS-RS iteration 8: Global Deviance = -167.2372 
GAMLSS-RS iteration 9: Global Deviance = -168.0203 
GAMLSS-RS iteration 10: Global Deviance = -168.7076 
GAMLSS-RS iteration 11: Global Deviance = -169.32 
GAMLSS-RS iteration 12: Global Deviance = -169.8649 
GAMLSS-RS iteration 13: Global Deviance = -170.3537 
GAMLSS-RS iteration 14: Global Deviance = -170.7882 
GAMLSS-RS iteration 15: Global Deviance = -171.1753 
GAMLSS-RS iteration 16: Global Deviance = -171.5182 
GAMLSS-RS iteration 17: Global Deviance = -171.8257 
GAMLSS-RS iteration 18: Global Deviance = -172.1019 
GAMLSS-RS iteration 19: Global Deviance = -172.3472 
GAMLSS-RS iteration 20: Global Deviance = -172.5694 
GAMLSS-RS iteration 21: Global Deviance = -172.7665 
GAMLSS-RS iteration 22: Global Deviance = -172.9428 
GAMLSS-RS iteration 23: Global Deviance = -173.1012 
GAMLSS-RS iteration 24: Global Deviance = -173.2205 
GAMLSS-RS iteration 25: Global Deviance = -173.3697 
GAMLSS-RS iteration 26: Global Deviance = -173.4681 
GAMLSS-RS iteration 27: Global Deviance = -173.5949 
GAMLSS-RS iteration 28: Global Deviance = -173.689 
GAMLSS-RS iteration 29: Global Deviance = -173.7762 
GAMLSS-RS iteration 30: Global Deviance = -173.8526 
GAMLSS-RS iteration 31: Global Deviance = -173.9099 
GAMLSS-RS iteration 32: Global Deviance = -173.9708 
GAMLSS-RS iteration 33: Global Deviance = -174.0249 
GAMLSS-RS iteration 34: Global Deviance = -174.076 
GAMLSS-RS iteration 35: Global Deviance = -174.1232 
GAMLSS-RS iteration 36: Global Deviance = -174.1685 
GAMLSS-RS iteration 37: Global Deviance = -174.2101 
GAMLSS-RS iteration 38: Global Deviance = -174.2484 
GAMLSS-RS iteration 39: Global Deviance = -174.3068 
GAMLSS-RS iteration 40: Global Deviance = -174.319 
GAMLSS-RS iteration 1: Global Deviance = -153.6997 
GAMLSS-RS iteration 2: Global Deviance = -158.2326 
GAMLSS-RS iteration 3: Global Deviance = -160.0709 
GAMLSS-RS iteration 4: Global Deviance = -161.3278 
GAMLSS-RS iteration 5: Global Deviance = -162.1725 
GAMLSS-RS iteration 6: Global Deviance = -162.908 
GAMLSS-RS iteration 7: Global Deviance = -163.5326 
GAMLSS-RS iteration 8: Global Deviance = -164.0613 
GAMLSS-RS iteration 9: Global Deviance = -164.5193 
GAMLSS-RS iteration 10: Global Deviance = -164.9242 
GAMLSS-RS iteration 11: Global Deviance = -165.2867 
GAMLSS-RS iteration 12: Global Deviance = -165.6164 
GAMLSS-RS iteration 13: Global Deviance = -165.9191 
GAMLSS-RS iteration 14: Global Deviance = -166.198 
GAMLSS-RS iteration 15: Global Deviance = -166.4614 
GAMLSS-RS iteration 16: Global Deviance = -166.7129 
GAMLSS-RS iteration 17: Global Deviance = -166.9455 
GAMLSS-RS iteration 18: Global Deviance = -167.1632 
GAMLSS-RS iteration 19: Global Deviance = -167.3716 
GAMLSS-RS iteration 20: Global Deviance = -167.5659 
GAMLSS-RS iteration 21: Global Deviance = -167.7528 
GAMLSS-RS iteration 22: Global Deviance = -167.932 
GAMLSS-RS iteration 23: Global Deviance = -168.106 
GAMLSS-RS iteration 24: Global Deviance = -168.2757 
GAMLSS-RS iteration 25: Global Deviance = -168.4405 
GAMLSS-RS iteration 26: Global Deviance = -168.6008 
GAMLSS-RS iteration 27: Global Deviance = -168.7536 
GAMLSS-RS iteration 28: Global Deviance = -168.9083 
GAMLSS-RS iteration 29: Global Deviance = -169.0564 
GAMLSS-RS iteration 30: Global Deviance = -169.2003 
GAMLSS-RS iteration 31: Global Deviance = -169.3394 
GAMLSS-RS iteration 32: Global Deviance = -169.4728 
GAMLSS-RS iteration 33: Global Deviance = -169.602 
GAMLSS-RS iteration 34: Global Deviance = -169.7242 
GAMLSS-RS iteration 35: Global Deviance = -169.8409 
GAMLSS-RS iteration 36: Global Deviance = -169.951 
GAMLSS-RS iteration 37: Global Deviance = -170.0518 
GAMLSS-RS iteration 38: Global Deviance = -170.1447 
GAMLSS-RS iteration 39: Global Deviance = -170.236 
GAMLSS-RS iteration 40: Global Deviance = -170.3203 
GAMLSS-RS iteration 1: Global Deviance = -159.3744 
GAMLSS-RS iteration 2: Global Deviance = -162.6632 
GAMLSS-RS iteration 3: Global Deviance = -164.489 
GAMLSS-RS iteration 4: Global Deviance = -165.803 
GAMLSS-RS iteration 5: Global Deviance = -166.8782 
GAMLSS-RS iteration 6: Global Deviance = -167.7962 
GAMLSS-RS iteration 7: Global Deviance = -168.5918 
GAMLSS-RS iteration 8: Global Deviance = -169.2915 
GAMLSS-RS iteration 9: Global Deviance = -169.9182 
GAMLSS-RS iteration 10: Global Deviance = -170.4845 
GAMLSS-RS iteration 11: Global Deviance = -170.9961 
GAMLSS-RS iteration 12: Global Deviance = -171.4704 
GAMLSS-RS iteration 13: Global Deviance = -171.9063 
GAMLSS-RS iteration 14: Global Deviance = -172.3152 
GAMLSS-RS iteration 15: Global Deviance = -172.7047 
GAMLSS-RS iteration 16: Global Deviance = -173.0761 
GAMLSS-RS iteration 17: Global Deviance = -173.4316 
GAMLSS-RS iteration 18: Global Deviance = -173.7734 
GAMLSS-RS iteration 19: Global Deviance = -174.1 
GAMLSS-RS iteration 20: Global Deviance = -174.4176 
GAMLSS-RS iteration 21: Global Deviance = -174.7143 
GAMLSS-RS iteration 22: Global Deviance = -174.987 
GAMLSS-RS iteration 23: Global Deviance = -175.245 
GAMLSS-RS iteration 24: Global Deviance = -175.4825 
GAMLSS-RS iteration 25: Global Deviance = -175.7016 
GAMLSS-RS iteration 26: Global Deviance = -175.898 
GAMLSS-RS iteration 27: Global Deviance = -176.0759 
GAMLSS-RS iteration 28: Global Deviance = -176.2371 
GAMLSS-RS iteration 29: Global Deviance = -176.3824 
GAMLSS-RS iteration 30: Global Deviance = -176.5133 
GAMLSS-RS iteration 31: Global Deviance = -176.6304 
GAMLSS-RS iteration 32: Global Deviance = -176.7343 
GAMLSS-RS iteration 33: Global Deviance = -176.833 
GAMLSS-RS iteration 34: Global Deviance = -176.9184 
GAMLSS-RS iteration 35: Global Deviance = -176.9955 
GAMLSS-RS iteration 36: Global Deviance = -177.0684 
GAMLSS-RS iteration 37: Global Deviance = -177.1349 
GAMLSS-RS iteration 38: Global Deviance = -177.1967 
GAMLSS-RS iteration 39: Global Deviance = -177.2555 
GAMLSS-RS iteration 40: Global Deviance = -177.3098 
GAMLSS-RS iteration 1: Global Deviance = -156.14 
GAMLSS-RS iteration 2: Global Deviance = -159.8965 
GAMLSS-RS iteration 3: Global Deviance = -161.6843 
GAMLSS-RS iteration 4: Global Deviance = -162.8207 
GAMLSS-RS iteration 5: Global Deviance = -163.6423 
GAMLSS-RS iteration 6: Global Deviance = -164.2758 
GAMLSS-RS iteration 7: Global Deviance = -164.7856 
GAMLSS-RS iteration 8: Global Deviance = -165.212 
GAMLSS-RS iteration 9: Global Deviance = -165.5761 
GAMLSS-RS iteration 10: Global Deviance = -165.8959 
GAMLSS-RS iteration 11: Global Deviance = -166.1763 
GAMLSS-RS iteration 12: Global Deviance = -166.4442 
GAMLSS-RS iteration 13: Global Deviance = -166.6811 
GAMLSS-RS iteration 14: Global Deviance = -166.8964 
GAMLSS-RS iteration 15: Global Deviance = -167.0957 
GAMLSS-RS iteration 16: Global Deviance = -167.2806 
GAMLSS-RS iteration 17: Global Deviance = -167.4525 
GAMLSS-RS iteration 18: Global Deviance = -167.6212 
GAMLSS-RS iteration 19: Global Deviance = -167.7594 
GAMLSS-RS iteration 20: Global Deviance = -167.9042 
GAMLSS-RS iteration 21: Global Deviance = -168.0346 
GAMLSS-RS iteration 22: Global Deviance = -168.1568 
GAMLSS-RS iteration 23: Global Deviance = -168.2726 
GAMLSS-RS iteration 24: Global Deviance = -168.3784 
GAMLSS-RS iteration 25: Global Deviance = -168.4754 
GAMLSS-RS iteration 26: Global Deviance = -168.5665 
GAMLSS-RS iteration 27: Global Deviance = -168.6524 
GAMLSS-RS iteration 28: Global Deviance = -168.7333 
GAMLSS-RS iteration 29: Global Deviance = -168.8111 
GAMLSS-RS iteration 30: Global Deviance = -168.883 
GAMLSS-RS iteration 31: Global Deviance = -168.9506 
GAMLSS-RS iteration 32: Global Deviance = -169.014 
GAMLSS-RS iteration 33: Global Deviance = -169.0754 
GAMLSS-RS iteration 34: Global Deviance = -169.1326 
GAMLSS-RS iteration 35: Global Deviance = -169.1878 
GAMLSS-RS iteration 36: Global Deviance = -169.2407 
GAMLSS-RS iteration 37: Global Deviance = -169.2873 
GAMLSS-RS iteration 38: Global Deviance = -169.3322 
GAMLSS-RS iteration 39: Global Deviance = -169.3755 
GAMLSS-RS iteration 40: Global Deviance = -169.4147 
GAMLSS-RS iteration 1: Global Deviance = -155.1702 
GAMLSS-RS iteration 2: Global Deviance = -157.9764 
GAMLSS-RS iteration 3: Global Deviance = -159.4814 
GAMLSS-RS iteration 4: Global Deviance = -160.727 
GAMLSS-RS iteration 5: Global Deviance = -161.852 
GAMLSS-RS iteration 6: Global Deviance = -162.8924 
GAMLSS-RS iteration 7: Global Deviance = -163.8693 
GAMLSS-RS iteration 8: Global Deviance = -164.7932 
GAMLSS-RS iteration 9: Global Deviance = -165.6772 
GAMLSS-RS iteration 10: Global Deviance = -166.4884 
GAMLSS-RS iteration 11: Global Deviance = -167.2422 
GAMLSS-RS iteration 12: Global Deviance = -167.9494 
GAMLSS-RS iteration 13: Global Deviance = -168.6069 
GAMLSS-RS iteration 14: Global Deviance = -169.2169 
GAMLSS-RS iteration 15: Global Deviance = -169.7838 
GAMLSS-RS iteration 16: Global Deviance = -170.3097 
GAMLSS-RS iteration 17: Global Deviance = -170.8008 
GAMLSS-RS iteration 18: Global Deviance = -171.2603 
GAMLSS-RS iteration 19: Global Deviance = -171.689 
GAMLSS-RS iteration 20: Global Deviance = -172.0896 
GAMLSS-RS iteration 21: Global Deviance = -172.4633 
GAMLSS-RS iteration 22: Global Deviance = -172.8147 
GAMLSS-RS iteration 23: Global Deviance = -173.1427 
GAMLSS-RS iteration 24: Global Deviance = -173.4514 
GAMLSS-RS iteration 25: Global Deviance = -173.7445 
GAMLSS-RS iteration 26: Global Deviance = -174.015 
GAMLSS-RS iteration 27: Global Deviance = -174.272 
GAMLSS-RS iteration 28: Global Deviance = -174.5137 
GAMLSS-RS iteration 29: Global Deviance = -174.7424 
GAMLSS-RS iteration 30: Global Deviance = -174.9561 
GAMLSS-RS iteration 31: Global Deviance = -175.1585 
GAMLSS-RS iteration 32: Global Deviance = -175.3523 
GAMLSS-RS iteration 33: Global Deviance = -175.5311 
GAMLSS-RS iteration 34: Global Deviance = -175.7024 
GAMLSS-RS iteration 35: Global Deviance = -175.8643 
GAMLSS-RS iteration 36: Global Deviance = -176.0168 
GAMLSS-RS iteration 37: Global Deviance = -176.1615 
GAMLSS-RS iteration 38: Global Deviance = -176.2991 
GAMLSS-RS iteration 39: Global Deviance = -176.4076 
GAMLSS-RS iteration 40: Global Deviance = -176.5568 
GAMLSS-RS iteration 1: Global Deviance = -155.1453 
GAMLSS-RS iteration 2: Global Deviance = -158.0946 
GAMLSS-RS iteration 3: Global Deviance = -159.9088 
GAMLSS-RS iteration 4: Global Deviance = -161.2037 
GAMLSS-RS iteration 5: Global Deviance = -162.104 
GAMLSS-RS iteration 6: Global Deviance = -162.7422 
GAMLSS-RS iteration 7: Global Deviance = -163.2251 
GAMLSS-RS iteration 8: Global Deviance = -163.5072 
GAMLSS-RS iteration 9: Global Deviance = -163.8026 
GAMLSS-RS iteration 10: Global Deviance = -164.0046 
GAMLSS-RS iteration 11: Global Deviance = -164.1596 
GAMLSS-RS iteration 12: Global Deviance = -164.2874 
GAMLSS-RS iteration 13: Global Deviance = -164.3955 
GAMLSS-RS iteration 14: Global Deviance = -164.4801 
GAMLSS-RS iteration 15: Global Deviance = -164.5519 
GAMLSS-RS iteration 16: Global Deviance = -164.6141 
GAMLSS-RS iteration 17: Global Deviance = -164.6615 
GAMLSS-RS iteration 18: Global Deviance = -164.7026 
GAMLSS-RS iteration 19: Global Deviance = -164.7375 
GAMLSS-RS iteration 20: Global Deviance = -164.769 
GAMLSS-RS iteration 21: Global Deviance = -164.7959 
GAMLSS-RS iteration 22: Global Deviance = -164.8223 
GAMLSS-RS iteration 23: Global Deviance = -164.8455 
GAMLSS-RS iteration 24: Global Deviance = -164.8686 
GAMLSS-RS iteration 25: Global Deviance = -164.8899 
GAMLSS-RS iteration 26: Global Deviance = -164.9155 
GAMLSS-RS iteration 27: Global Deviance = -164.9441 
GAMLSS-RS iteration 28: Global Deviance = -164.9832 
GAMLSS-RS iteration 29: Global Deviance = -165.0264 
GAMLSS-RS iteration 30: Global Deviance = -165.0786 
GAMLSS-RS iteration 31: Global Deviance = -165.14 
GAMLSS-RS iteration 32: Global Deviance = -165.2086 
GAMLSS-RS iteration 33: Global Deviance = -165.2888 
GAMLSS-RS iteration 34: Global Deviance = -165.3818 
GAMLSS-RS iteration 35: Global Deviance = -165.485 
GAMLSS-RS iteration 36: Global Deviance = -165.6046 
GAMLSS-RS iteration 37: Global Deviance = -165.7404 
GAMLSS-RS iteration 38: Global Deviance = -165.8844 
GAMLSS-RS iteration 39: Global Deviance = -166.0313 
GAMLSS-RS iteration 40: Global Deviance = -166.1777 
GAMLSS-RS iteration 1: Global Deviance = -149.3315 
GAMLSS-RS iteration 2: Global Deviance = -155.0145 
GAMLSS-RS iteration 3: Global Deviance = -158.0749 
GAMLSS-RS iteration 4: Global Deviance = -160.424 
GAMLSS-RS iteration 5: Global Deviance = -162.4715 
GAMLSS-RS iteration 6: Global Deviance = -163.7648 
GAMLSS-RS iteration 7: Global Deviance = -164.9552 
GAMLSS-RS iteration 8: Global Deviance = -165.9587 
GAMLSS-RS iteration 9: Global Deviance = -166.8115 
GAMLSS-RS iteration 10: Global Deviance = -167.5782 
GAMLSS-RS iteration 11: Global Deviance = -168.174 
GAMLSS-RS iteration 12: Global Deviance = -168.71 
GAMLSS-RS iteration 13: Global Deviance = -169.186 
GAMLSS-RS iteration 14: Global Deviance = -169.5975 
GAMLSS-RS iteration 15: Global Deviance = -169.961 
GAMLSS-RS iteration 16: Global Deviance = -170.3048 
GAMLSS-RS iteration 17: Global Deviance = -170.605 
GAMLSS-RS iteration 18: Global Deviance = -170.807 
GAMLSS-RS iteration 19: Global Deviance = -171.0254 
GAMLSS-RS iteration 20: Global Deviance = -171.2238 
GAMLSS-RS iteration 21: Global Deviance = -171.4015 
GAMLSS-RS iteration 22: Global Deviance = -171.5627 
GAMLSS-RS iteration 23: Global Deviance = -171.7053 
GAMLSS-RS iteration 24: Global Deviance = -171.8372 
GAMLSS-RS iteration 25: Global Deviance = -171.9573 
GAMLSS-RS iteration 26: Global Deviance = -172.0628 
GAMLSS-RS iteration 27: Global Deviance = -172.1503 
GAMLSS-RS iteration 28: Global Deviance = -172.2361 
GAMLSS-RS iteration 29: Global Deviance = -172.3175 
GAMLSS-RS iteration 30: Global Deviance = -172.3917 
GAMLSS-RS iteration 31: Global Deviance = -172.4624 
GAMLSS-RS iteration 32: Global Deviance = -172.5284 
GAMLSS-RS iteration 33: Global Deviance = -172.5999 
GAMLSS-RS iteration 34: Global Deviance = -172.6466 
GAMLSS-RS iteration 35: Global Deviance = -172.7039 
GAMLSS-RS iteration 36: Global Deviance = -172.7547 
GAMLSS-RS iteration 37: Global Deviance = -172.8014 
GAMLSS-RS iteration 38: Global Deviance = -172.8461 
GAMLSS-RS iteration 39: Global Deviance = -172.8883 
GAMLSS-RS iteration 40: Global Deviance = -172.9253

Single term deletions for
mu

Model:
Error ~ cs(distkm) + cs(atimdifmin) + cs(sdlnchlor) + nchlor + 
    cs(senzr) + cs(solzr) + cs(windspeed) + cs(taua869)
                   Df     AIC     LRT  Pr(Chi)   
<none>                -42.199                    
cs(distkm)     3.9963 -40.534  9.6575 0.046492 * 
cs(atimdifmin) 3.9969 -44.318  5.8749 0.208376   
cs(sdlnchlor)  3.9946 -40.315  9.8735 0.042455 * 
nchlor         0.9991 -41.313  2.8841 0.089348 . 
cs(senzr)      3.9952 -39.410 10.7791 0.029062 * 
cs(solzr)      3.9997 -46.561  3.6371 0.457285   
cs(windspeed)  3.9968 -36.176 14.0162 0.007224 **
cs(taua869)    4.0009 -42.932  7.2686 0.122422   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Uncertainty Characterisation in Ocean Colour Estimation

Kayvan Nejabati Zenouz

Nov 20, 2019