LM_GAM_Synth_Ealgal_prediction

AIM : How to best predict Ebed algal (Brown, Red and Green) using Ebed PAR int, Ebed and Kd at different lambda (412, 443, 488, 531, 555, 645, 667)? We investigate the question using a linear multiple regression and GAM approach.

Document Note : The maps, plots and app within this document are interactive so make sure you give them a play like zooming in and out in the maps but also on the plots. Clicking on the legend allows to only select and display the time series needed.

Table of contents

1. Synthetic data

2. Predicting E algal Brown

   • Linear Regression
   • Multiple linear regression
   • GAM

3. Predicting E algal Red

   • Linear Regression
   • Multiple linear regression
   • GAM

4. Predicting E algal Green

   • Linear Regression
   • Multiple linear regression
   • GAM

5. Conclusion

Synthetic data

vars <- read_csv('Synth_Ealgal_Elam_Kdlam_EbedInt.csv')

p <- vars %>% 
  pivot_longer(-n,names_to = 'Vars',values_to = 'Values') %>%
  ggplot(data=.) +
  geom_histogram(aes(x=Values)) +
  facet_wrap(~Vars, scales = 'free') +
  theme_light()
p

Figure 1 - Distribution of variables.

Observations:

• Why some high Ebed values? Units? To look for in synthetic+Ebedspectral_28_for_francois.pro?

Predicting E algal Brown

Linear regression

vars_brown <- vars %>% 
  pivot_longer(-EbedAction_brown,names_to = 'Vars',values_to = 'Values')
  
p <- ggplot(data=vars_brown,aes(x=Values,y=EbedAction_brown)) +
  geom_point() +
  facet_wrap(~Vars, scales = 'free') +
  geom_smooth(method = 'lm') +
  theme_light()
p

Figure 2 - Linear model to explain E algal Brown.

kable(do(group_by(vars_brown,Vars), glance(lm(EbedAction_brown ~ Values, data = .)))
 ,caption = 'Table 1 - Linear model.')

Table 1 - Linear model.

Vars	r.squared	adj.r.squared	sigma	statistic	p.value	df	logLik	AIC	BIC	deviance	df.residual	nobs
Ebed_412	0.8624664	0.8623286	26.497017	6.258407e+03	0.0000000	1	-4694.970	9395.939	9410.663	700687.747	998	1000
Ebed_443	0.9624955	0.9624579	13.836756	2.561213e+04	0.0000000	1	-4045.266	8096.532	8111.255	191072.916	998	1000
Ebed_488	0.9623554	0.9623177	13.862578	2.551309e+04	0.0000000	1	-4047.131	8100.261	8114.984	191786.737	998	1000
Ebed_531	0.8982067	0.8981047	22.795662	8.806178e+03	0.0000000	1	-4544.508	9095.016	9109.739	518602.902	998	1000
Ebed_555	0.8784377	0.8783159	24.911047	7.211783e+03	0.0000000	1	-4633.249	9272.498	9287.221	619319.127	998	1000
Ebed_645	0.9135667	0.9134801	21.005504	1.054847e+04	0.0000000	1	-4462.722	8931.444	8946.167	440348.724	998	1000
Ebed_667	0.8525104	0.8523626	27.439314	5.768578e+03	0.0000000	1	-4729.914	9465.829	9480.552	751410.139	998	1000
Ebed_par	0.9959511	0.9959470	4.546341	2.454877e+05	0.0000000	1	-2932.260	5870.520	5885.244	20627.875	998	1000
EbedAction_green	0.9987831	0.9987819	2.492381	8.191397e+05	0.0000000	1	-2331.176	4668.352	4683.075	6199.539	998	1000
EbedAction_red	0.9964931	0.9964896	4.231127	2.835814e+05	0.0000000	1	-2860.406	5726.812	5741.535	17866.632	998	1000
Kd_412	0.0215241	0.0205437	70.675288	2.195363e+01	0.0000032	1	-5676.034	11358.067	11372.790	4985006.370	998	1000
Kd_443	0.0249092	0.0239321	70.552932	2.549440e+01	0.0000005	1	-5674.301	11354.602	11369.325	4967760.787	998	1000
Kd_488	0.0264839	0.0255085	70.495938	2.715002e+01	0.0000002	1	-5673.493	11352.985	11367.708	4959737.859	998	1000
Kd_531	0.0247064	0.0237291	70.560268	2.528160e+01	0.0000006	1	-5674.405	11354.809	11369.533	4968793.871	998	1000
Kd_555	0.0232567	0.0222780	70.612688	2.376286e+01	0.0000013	1	-5675.147	11356.295	11371.018	4976179.445	998	1000
Kd_645	0.0201623	0.0191805	70.724456	2.053598e+01	0.0000066	1	-5676.729	11359.458	11374.181	4991944.761	998	1000
Kd_667	0.0194159	0.0184333	70.751387	1.976072e+01	0.0000098	1	-5677.110	11360.219	11374.943	4995747.290	998	1000
n	0.0007533	-0.0002479	71.421487	7.523792e-01	0.3859325	1	-5686.536	11379.073	11393.796	5090826.771	998	1000

Observations:

• Very good linear relationships with Ebed_par
• Good-ish linear relationships with Ebed(\(\lambda\)). Some more expo fit-like.
• Also with EbedAction Green and Red, not used in the next. We are also interested in explaining deviation in their relationships, but another story.
• Negative relationship but way weaker relationship with Kds, we are not including them in the regression this time.

Multiple linear regression

vars_mlm_b <- lm(EbedAction_brown ~ Ebed_par + Ebed_412 + Ebed_443 + Ebed_488 + Ebed_531 + Ebed_555 + Ebed_645 + Ebed_667,data=vars)
summary(vars_mlm_b)

## 
## Call:
## lm(formula = EbedAction_brown ~ Ebed_par + Ebed_412 + Ebed_443 + 
##     Ebed_488 + Ebed_531 + Ebed_555 + Ebed_645 + Ebed_667, data = vars)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -22.4222  -0.0133  -0.0029   0.0338  17.9193 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.005097   0.045407   0.112  0.91064    
## Ebed_par      0.267175   0.025381  10.527  < 2e-16 ***
## Ebed_412    108.038560   5.211571  20.731  < 2e-16 ***
## Ebed_443     12.384190   6.253666   1.980  0.04795 *  
## Ebed_488     58.360596   7.646015   7.633 5.39e-14 ***
## Ebed_531     10.872783   5.804605   1.873  0.06134 .  
## Ebed_555     -9.660195   9.230408  -1.047  0.29556    
## Ebed_645    -42.784415  15.141026  -2.826  0.00481 ** 
## Ebed_667     28.058157   5.921033   4.739 2.46e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.085 on 991 degrees of freedom
## Multiple R-squared:  0.9998, Adjusted R-squared:  0.9998 
## F-statistic: 5.412e+05 on 8 and 991 DF,  p-value: < 2.2e-16

Observations:

• Most significant predictor is Ebed_par again
• Interesting significance of Ebed_412, Ebed_488, Ebed_645 and Ebed_667 (Blue and Red light, cf aaction spectra) BUT non linear relationship with 500<\(\lambda\)<600, green light.
• \(R^2 = 0.9998\) almost like just with taking Ebed_par alone…
• Green Ebed, non linear relationship so better use other regression method like GAM?

layout(matrix(c(1,2,3,4),2,2)) # optional 4 graphs/page
plot(vars_mlm_b)

Figure 3 - Assumptions check of linear model to explain E algal Brown.

Observations:

• Q-Q plot suggests more extreme values than “pure” normal distribution.
• To include uniform depth distribution so Ebed will be normally distributed?

GAM

p <- vars %>% 
  dplyr::select(Ebed_412,Ebed_443,Ebed_488,Ebed_531,Ebed_555,Ebed_645,Ebed_667,Ebed_par,EbedAction_brown) %>% 
  pivot_longer(-EbedAction_brown) %>% 
  ggplot(data=.,aes(x=value,y=EbedAction_brown)) +
     geom_point() +
     facet_wrap(~name, scales = 'free') +
     geom_smooth(method = 'gam') +
     theme_light()
p

Figure 4 - GAM model to explain E algal Brown. No interactions.

vars_brown <- vars %>% 
  dplyr::select(Ebed_412,Ebed_443,Ebed_488,Ebed_531,Ebed_555,Ebed_645,Ebed_667,Ebed_par,EbedAction_brown) %>% 
  pivot_longer(-EbedAction_brown)

kable(do(group_by(vars_brown,name), glance(gam(EbedAction_brown ~ value, data = .)))
 ,caption = 'Table 2 - GAM no interactions.')

Table 2 - GAM no interactions.

name	df	logLik	AIC	BIC	deviance	df.residual	nobs
Ebed_412	2	-4694.970	9395.939	9410.663	700687.75	998	1000
Ebed_443	2	-4045.266	8096.532	8111.255	191072.92	998	1000
Ebed_488	2	-4047.131	8100.261	8114.984	191786.74	998	1000
Ebed_531	2	-4544.508	9095.016	9109.739	518602.90	998	1000
Ebed_555	2	-4633.249	9272.498	9287.221	619319.13	998	1000
Ebed_645	2	-4462.722	8931.444	8946.167	440348.72	998	1000
Ebed_667	2	-4729.914	9465.829	9480.552	751410.14	998	1000
Ebed_par	2	-2932.260	5870.520	5885.244	20627.87	998	1000

Observations:

• Best predictor based on lowest AIC: Ebed_par, then Ebed_443, then Ebed_488, then Ebed_645, then Ebed_531, then Ebed_555, then Ebed_412, then Ebed_667.
• What about interactions?

vars_gam_b <- gam(EbedAction_brown ~ s(Ebed_par, bs='cr') + s(Ebed_412, bs='cr') + s(Ebed_443, bs='cr') + s(Ebed_488, bs='cr') + s(Ebed_531, bs='cr') + s(Ebed_555, bs='cr') + s(Ebed_645, bs='cr') + s(Ebed_667, bs='cr'), data=vars)

plot_gam(vars_gam_b, ncol = 3) + ylab("Pred. Ebed Algal Brown") + theme_light()#all predictors

Figure 4 - GAM model to explain E algal Brown.

Observations:

• Interesting negative relationships between Ealgal Brown with 555 and 645 (match with action spectra of Brown)
• Parabolic relationships with 443 and 531. Not sure how to interpret these, especially the 443.
• Positive relationships with 412, 488, 667 and PAR, makes sense and consistent with action spectra.

summary(vars_gam_b)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## EbedAction_brown ~ s(Ebed_par, bs = "cr") + s(Ebed_412, bs = "cr") + 
##     s(Ebed_443, bs = "cr") + s(Ebed_488, bs = "cr") + s(Ebed_531, 
##     bs = "cr") + s(Ebed_555, bs = "cr") + s(Ebed_645, bs = "cr") + 
##     s(Ebed_667, bs = "cr")
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 19.82191    0.02686   737.9   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df       F  p-value    
## s(Ebed_par) 3.294  3.724  27.206  < 2e-16 ***
## s(Ebed_412) 6.998  7.098 108.838  < 2e-16 ***
## s(Ebed_443) 5.731  5.993  26.288  < 2e-16 ***
## s(Ebed_488) 2.845  3.172  43.267  < 2e-16 ***
## s(Ebed_531) 4.032  4.503   7.530 3.67e-06 ***
## s(Ebed_555) 1.386  1.638   9.025 0.000588 ***
## s(Ebed_645) 3.123  3.295  11.545  < 2e-16 ***
## s(Ebed_667) 4.879  5.020  37.885  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =      1   Deviance explained =  100%
## GCV = 0.74645  Scale est. = 0.7216    n = 1000

Observations:

• Deviance explained = 100%
• All predictors significant

plot_gam_3d(model = vars_gam_b, main_var = Ebed_412, second_var = Ebed_555, palette='bilbao', direction = 1)

Figure 4bis - Prediction of Ebed_brown using a GAM using multiple predictors (Ebed_par, Ebed_412, Ebed_443, Ebed_488, Ebed_531 ,Ebed_555, Ebed_645, Ebed_667). Showing Ebed_412 and Ebed_555.

Observations:

• High Ebed_555 but low Ebed 412 lead to low Ebed Brown!!

plot_gam_3d(model = vars_gam_b, main_var = Ebed_412, second_var = Ebed_667, palette='bilbao', direction = 1)

Figure 4bis - Prediction of Ebed_brown using a GAM using multiple predictors (Ebed_par, Ebed_412, Ebed_443, Ebed_488, Ebed_531 ,Ebed_555, Ebed_645, Ebed_667). Showing Ebed_412 and Ebed_667.

Observations:

• High Ebed_412 and high/low Ebed_667 lead to high Ebed Brown!!
• 667 less important of a predictor than 412?

Predicting E algal red

Linear regression

vars_red <- vars %>% 
  pivot_longer(-EbedAction_red,names_to = 'Vars',values_to = 'Values')
  
p <- ggplot(data=vars_red,aes(x=Values,y=EbedAction_red)) +
  geom_point() +
  facet_wrap(~Vars, scales = 'free') +
  geom_smooth(method = 'lm') +
  theme_light()
p

Figure 5 - Linear model to explain E algal Red

kable(do(group_by(vars_red,Vars), glance(lm(EbedAction_red ~ Values, data = .)))
 ,caption = 'Table 2 - Linear model.')

Table 2 - Linear model.

Vars	r.squared	adj.r.squared	sigma	statistic	p.value	df	logLik	AIC	BIC	deviance	df.residual	nobs
Ebed_412	0.8310128	0.8308435	31.017469	4.907772e+03	0.0000000	1	-4852.488	9710.976	9725.699	960159.21	998	1000
Ebed_443	0.9456264	0.9455719	17.594363	1.735649e+04	0.0000000	1	-4285.516	8577.032	8591.755	308942.49	998	1000
Ebed_488	0.9725025	0.9724749	12.511980	3.529621e+04	0.0000000	1	-3944.624	7895.248	7909.972	156236.55	998	1000
Ebed_531	0.9227212	0.9226437	20.975373	1.191627e+04	0.0000000	1	-4461.287	8928.573	8943.296	439086.33	998	1000
Ebed_555	0.9050944	0.9049993	23.244771	9.517716e+03	0.0000000	1	-4564.018	9134.035	9148.759	539238.74	998	1000
Ebed_645	0.8912972	0.8911882	24.877107	8.182993e+03	0.0000000	1	-4631.886	9269.771	9284.494	617632.71	998	1000
Ebed_667	0.8221632	0.8219850	31.819272	4.613887e+03	0.0000000	1	-4878.010	9762.019	9776.743	1010441.11	998	1000
Ebed_par	0.9973445	0.9973418	3.888256	3.748219e+05	0.0000000	1	-2775.898	5557.796	5572.520	15088.29	998	1000
EbedAction_brown	0.9964931	0.9964896	4.468307	2.835814e+05	0.0000000	1	-2914.947	5835.894	5850.617	19925.83	998	1000
EbedAction_green	0.9928302	0.9928231	6.388980	1.381979e+05	0.0000000	1	-3272.512	6551.024	6565.748	40737.43	998	1000
Kd_412	0.0244693	0.0234918	74.524642	2.503291e+01	0.0000007	1	-5729.067	11464.135	11478.858	5542814.35	998	1000
Kd_443	0.0283046	0.0273309	74.378002	2.907079e+01	0.0000001	1	-5727.098	11460.196	11474.919	5521023.05	998	1000
Kd_488	0.0300933	0.0291215	74.309511	3.096498e+01	0.0000000	1	-5726.176	11458.353	11473.076	5510859.55	998	1000
Kd_531	0.0282283	0.0272546	74.380922	2.899016e+01	0.0000001	1	-5727.137	11460.274	11474.997	5521456.47	998	1000
Kd_555	0.0266544	0.0256791	74.441132	2.732950e+01	0.0000002	1	-5727.946	11461.892	11476.616	5530399.24	998	1000
Kd_645	0.0230979	0.0221191	74.577005	2.359678e+01	0.0000014	1	-5729.770	11465.540	11480.263	5550606.27	998	1000
Kd_667	0.0222846	0.0213049	74.608044	2.274693e+01	0.0000021	1	-5730.186	11466.372	11481.095	5555227.55	998	1000
n	0.0007774	-0.0002238	75.424172	7.764425e-01	0.3784435	1	-5741.065	11488.131	11502.854	5677428.15	998	1000

Observations:

• Very good linear relationships with Ebed_par
• Also with EbedAction Green and Brown, not used in the next. We are also interested in explaining deviation in their relationships, but another story.
• Less Good-ish linear relationships with Ebed(\(\lambda\)). Some more expo fit-like.
• Negative relationship but way weaker relationship with Kds

Multiple linear regression

vars_mlm_r <- lm(EbedAction_red ~ Ebed_par + Ebed_412 + Ebed_443 + Ebed_488 + Ebed_531 + Ebed_555 + Ebed_645 + Ebed_667,data=vars)
summary(vars_mlm_r)

## 
## Call:
## lm(formula = EbedAction_red ~ Ebed_par + Ebed_412 + Ebed_443 + 
##     Ebed_488 + Ebed_531 + Ebed_555 + Ebed_645 + Ebed_667, data = vars)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -20.9027  -0.1642   0.0228   0.0582  24.1409 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.03109    0.09108  -0.341 0.732956    
## Ebed_par      0.17325    0.05091   3.403 0.000693 ***
## Ebed_412     21.61340   10.45372   2.068 0.038943 *  
## Ebed_443    118.60969   12.54403   9.455  < 2e-16 ***
## Ebed_488     41.75247   15.33689   2.722 0.006596 ** 
## Ebed_531     24.49114   11.64327   2.103 0.035677 *  
## Ebed_555     50.49340   18.51498   2.727 0.006501 ** 
## Ebed_645    -54.08429   30.37089  -1.781 0.075252 .  
## Ebed_667     87.28131   11.87681   7.349 4.17e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.176 on 991 degrees of freedom
## Multiple R-squared:  0.9992, Adjusted R-squared:  0.9992 
## F-statistic: 1.499e+05 on 8 and 991 DF,  p-value: < 2.2e-16

Observations:

• Most significant predictor is Ebed_par again
• Interesting significance of Ebed_555 this time compared to brown
• \(R^2 = 0.9992\) almost like just with taking Ebed_par alone…
• Worth keeping Kds in regression? Non linear relationship so better use other regression method like GAM?

layout(matrix(c(1,2,3,4),2,2)) # optional 4 graphs/page
plot(vars_mlm_r)

Figure 6 - Assumptions check of linear model to explain E algal Red

Observations:

• Q-Q plot suggests more extreme values than “pure” normal distribution.

GAM

vars_gam_r <- gam(EbedAction_red ~ s(Ebed_par, bs='cr') + s(Ebed_412, bs='cr') + s(Ebed_443, bs='cr') + s(Ebed_488, bs='cr') + s(Ebed_531, bs='cr') + s(Ebed_555, bs='cr') + s(Ebed_645, bs='cr') + s(Ebed_667, bs='cr'), data=vars)

plot_gam(vars_gam_r, ncol = 2) + ylab("Pred. Ebed Algal Red") + theme_light()#all predictors

Figure 7 - GAM model to explain E algal Red

summary(vars_gam_r)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## EbedAction_red ~ s(Ebed_par, bs = "cr") + s(Ebed_412, bs = "cr") + 
##     s(Ebed_443, bs = "cr") + s(Ebed_488, bs = "cr") + s(Ebed_531, 
##     bs = "cr") + s(Ebed_555, bs = "cr") + s(Ebed_645, bs = "cr") + 
##     s(Ebed_667, bs = "cr")
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 22.63920    0.05657   400.2   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##                edf Ref.df      F p-value    
## s(Ebed_par) 1.6920 2.0213 32.630 < 2e-16 ***
## s(Ebed_412) 7.1283 7.2622 33.913 < 2e-16 ***
## s(Ebed_443) 5.1762 5.6051 11.706 < 2e-16 ***
## s(Ebed_488) 2.7561 3.0627 13.127 < 2e-16 ***
## s(Ebed_531) 3.4140 3.8800  4.446 0.00208 ** 
## s(Ebed_555) 0.9864 0.9875  5.507 0.01991 *  
## s(Ebed_645) 2.7264 2.8857 14.646 < 2e-16 ***
## s(Ebed_667) 4.9484 5.0670 17.616 < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Rank: 72/73
## R-sq.(adj) =  0.999   Deviance explained = 99.9%
## GCV = 3.2987  Scale est. = 3.2003    n = 1000

Observations:

• Deviance explained = 99.9%
• Again, Ebed_Par best explain and very linear
• Ebed_555 less significant than Ebed_blue, how to interpret that?

plot_gam_3d(model = vars_gam_r, main_var = Ebed_555, second_var = Ebed_443, palette='bilbao', direction = 1)

Figure 7bis - Prediction of Ebed_red using a GAM using multiple predictors (Ebed_par, Ebed_412, Ebed_443, Ebed_488, Ebed_531, Ebed_555, Ebed_645, Ebed_667). Showing Ebed_443 and Ebed_555.

Predicting E algal Green

Linear regression

vars_green <- vars %>% 
  pivot_longer(-EbedAction_green,names_to = 'Vars',values_to = 'Values')
  
p <- ggplot(data=vars_green,aes(x=Values,y=EbedAction_green)) +
  geom_point() +
  facet_wrap(~Vars, scales = 'free') +
  geom_smooth(method = 'lm') +
  theme_light()
p

Figure 8 - Linear model to explain E algal Green

kable(do(group_by(vars_green,Vars), glance(lm(EbedAction_green ~ Values, data = .)))
 ,caption = 'Table 3 - Linear model.')

Table 3 - Linear model.

Vars	r.squared	adj.r.squared	sigma	statistic	p.value	df	logLik	AIC	BIC	deviance	df.residual	nobs
Ebed_412	0.8783311	0.8782092	24.120173	7.204587e+03	0.0000000	1	-4600.986	9207.972	9222.695	580619.186	998	1000
Ebed_443	0.9703817	0.9703521	11.900643	3.269742e+04	0.0000000	1	-3894.530	7795.060	7809.783	141342.056	998	1000
Ebed_488	0.9526864	0.9526390	15.041239	2.009530e+04	0.0000000	1	-4128.733	8263.466	8278.190	225786.387	998	1000
Ebed_531	0.8794141	0.8792932	24.012583	7.278256e+03	0.0000000	1	-4596.516	9199.031	9213.754	575450.945	998	1000
Ebed_555	0.8587624	0.8586209	25.987573	6.068108e+03	0.0000000	1	-4675.556	9357.112	9371.835	674003.220	998	1000
Ebed_645	0.9285967	0.9285252	18.477773	1.297895e+04	0.0000000	1	-4334.506	8675.012	8689.735	340745.233	998	1000
Ebed_667	0.8723838	0.8722559	24.702648	6.822322e+03	0.0000000	1	-4624.848	9255.696	9270.419	609000.389	998	1000
Ebed_par	0.9929335	0.9929264	5.812911	1.402312e+05	0.0000000	1	-3178.019	6362.038	6376.761	33722.354	998	1000
EbedAction_brown	0.9987831	0.9987819	2.412195	8.191397e+05	0.0000000	1	-2298.475	4602.949	4617.673	5807.048	998	1000
EbedAction_red	0.9928302	0.9928231	5.855213	1.381979e+05	0.0000000	1	-3185.270	6376.540	6391.263	34214.949	998	1000
Kd_412	0.0188827	0.0178996	68.493765	1.920762e+01	0.0000130	1	-5644.680	11295.361	11310.084	4682013.123	998	1000
Kd_443	0.0220131	0.0210332	68.384406	2.246362e+01	0.0000025	1	-5643.082	11292.165	11306.888	4667074.195	998	1000
Kd_488	0.0235881	0.0226097	68.329321	2.410963e+01	0.0000011	1	-5642.277	11290.553	11305.276	4659558.301	998	1000
Kd_531	0.0220133	0.0210334	68.384401	2.246378e+01	0.0000025	1	-5643.082	11292.165	11306.888	4667073.436	998	1000
Kd_555	0.0207235	0.0197422	68.429481	2.111970e+01	0.0000049	1	-5643.741	11293.483	11308.206	4673228.660	998	1000
Kd_645	0.0180742	0.0170903	68.521981	1.837006e+01	0.0000200	1	-5645.092	11296.184	11310.908	4685871.414	998	1000
Kd_667	0.0173608	0.0163762	68.546869	1.763216e+01	0.0000292	1	-5645.455	11296.911	11311.634	4689275.905	998	1000
n	0.0007404	-0.0002608	69.124139	7.394850e-01	0.3900324	1	-5653.842	11313.683	11328.406	4768590.291	998	1000

Observations:

• Very good linear relationships with Ebed_par
• Also with EbedAction Brown and Red, not used in the next. We are also interested in explaining deviation in their relationships, but another story.
• Negative relationship but way weaker relationship with Kds

Multiple linear regression

vars_mlm_g <- lm(EbedAction_green ~ Ebed_par + Ebed_412 + Ebed_443 + Ebed_488 + Ebed_531 + Ebed_555 + Ebed_645 + Ebed_667,data=vars)
summary(vars_mlm_g)

## 
## Call:
## lm(formula = EbedAction_green ~ Ebed_par + Ebed_412 + Ebed_443 + 
##     Ebed_488 + Ebed_531 + Ebed_555 + Ebed_645 + Ebed_667, data = vars)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -22.4114  -0.0277  -0.0119   0.0782  23.5325 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.01738    0.05931   0.293  0.76960    
## Ebed_par      0.32439    0.03315   9.786  < 2e-16 ***
## Ebed_412     61.73225    6.80678   9.069  < 2e-16 ***
## Ebed_443     21.71382    8.16785   2.658  0.00798 ** 
## Ebed_488     49.00755    9.98638   4.907 1.08e-06 ***
## Ebed_531     -8.27486    7.58133  -1.091  0.27533    
## Ebed_555    -32.72179   12.05574  -2.714  0.00676 ** 
## Ebed_645    -81.03303   19.77553  -4.098 4.51e-05 ***
## Ebed_667     79.55153    7.73340  10.287  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.417 on 991 degrees of freedom
## Multiple R-squared:  0.9996, Adjusted R-squared:  0.9996 
## F-statistic: 2.971e+05 on 8 and 991 DF,  p-value: < 2.2e-16

Observations:

• Most significant predictor is Ebed_par again, but also Blue light an red peak absorption (Ebed_667)
• Interesting negative slope for 531<\(\lambda\)<645
• \(R^2 = 0.9996\) almost like just with taking Ebed_par alone…
• Worth keeping Kds in regression? Non linear relationship so better use other regression method like GAM?

layout(matrix(c(1,2,3,4),2,2)) # optional 4 graphs/page
plot(vars_mlm_g)

Figure 9 - Assumptions check of linear model to explain E algal Green

Observations:

• Q-Q plot suggests more extreme values than “pure” normal distribution.

GAM

vars_gam_g <- gam(EbedAction_green ~ s(Ebed_par, bs='cr') + s(Ebed_412, bs='cr') + s(Ebed_443, bs='cr') + s(Ebed_488, bs='cr') + s(Ebed_531, bs='cr') + s(Ebed_555, bs='cr') + s(Ebed_645, bs='cr') + s(Ebed_667, bs='cr'), data=vars)

plot_gam(vars_gam_g, ncol = 2) + ylab("Pred. Ebed Algal Green") + theme_light()#all predictors

Figure 10 - GAM model to explain E algal Green

summary(vars_gam_g)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## EbedAction_green ~ s(Ebed_par, bs = "cr") + s(Ebed_412, bs = "cr") + 
##     s(Ebed_443, bs = "cr") + s(Ebed_488, bs = "cr") + s(Ebed_531, 
##     bs = "cr") + s(Ebed_555, bs = "cr") + s(Ebed_645, bs = "cr") + 
##     s(Ebed_667, bs = "cr")
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 18.05158    0.04083   442.1   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df      F  p-value    
## s(Ebed_par) 2.387  2.710 10.118 2.03e-05 ***
## s(Ebed_412) 7.037  7.140 29.033  < 2e-16 ***
## s(Ebed_443) 5.695  5.970 28.148  < 2e-16 ***
## s(Ebed_488) 2.402  2.697 29.490  < 2e-16 ***
## s(Ebed_531) 4.124  4.591  8.122 9.30e-07 ***
## s(Ebed_555) 2.082  2.577  0.932   0.4213    
## s(Ebed_645) 3.120  3.293  2.354   0.0647 .  
## s(Ebed_667) 4.963  5.081  1.672   0.1371    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =      1   Deviance explained =  100%
## GCV = 1.7237  Scale est. = 1.6672    n = 1000

Observations:

• Deviance explained = 100%
• Again, Ebed_Par best explain and \(\lambda\)<531
• Not significant Ebed green light
• Less linear effects of Ebed than for Brown and Red??

plot_gam_3d(model = vars_gam_g, main_var = Ebed_531, second_var = Ebed_443, palette='bilbao', direction = -1)

Figure 11 - Prediction of Ebed_green using a GAM using multiple predictors (Ebed_par, Ebed_412, Ebed_443, Ebed_488, Ebed_531, Ebed_555, Ebed_645, Ebed_667). Showing Ebed_443 and Ebed_531.

Conclusion - Model comparison

What can be changed?