Appendix

This appendix features a list of abbreviations and symbols used throughout this thesis (Table 1) as well as the numerical values of the performance results of the models tested (Table 2, 3, and 4).

Table 1: List of abbreviations. The abbreviations used are informed by the closest literature.
Abbreviation Meaning
\(\lambda\) Wavelength
\(\boldsymbol \rho\) Reflectance vector
\(\rho_\lambda\) Spectral reflectance at wavelength \(\lambda\).
\(\boldsymbol \tau\) Transmittance vector
\(\tau_\lambda\) Spectral transmittance at wavelength \(\lambda\).
\(\text{N}_{\%}\) Mass-based nitrogen, i.e., (foliar) nitrogen concentration.
\(\text{N}_{\text {area}}\) Area-based nitrogen.
LMA Leaf mass per area.
LAI Leaf area index.
\(\mathbf X\) Design matrix, matrix of input variables to a model.
\(\mathbf x_{(i)}\) The \(i\)th row of the design matrix \(\mathbf X\), i.e., one single observation.
\(\mathbf y\) Response vector, vector of output variables from a model.
\(\mathbf {\hat y}\) Estimated values of \(\mathbf y\).
\({\bar y}\) Arithmetic mean of all values in \(\mathbf y\).
\(\mathcal D\) The entire data set of \(\mathbf X\) and \(\mathbf y\).
\(n\) Number of observations in \(\mathcal D\).
\(d\) Number of predictor variables, i.e., number of columns in \(\mathbf X\).
\(p\) Number of parameters in a model.
\(\boldsymbol \Sigma\) Covariance matrix of \(\mathbf X\).
\(\boldsymbol \beta\) Coefficient vector (vector of parameters of a linear model).
\(\boldsymbol \theta\) Parameter vector (vector of parameters of any model).
\(\mu_\mathbf{x}\) Arithmetic mean of \(\mathbf{x}\).
\(\sigma_\mathbf{x}\) Standard deviation of \(\mathbf{x}\).
\(J\) Cost function.
Prospect Leaf-level radiative transfer model.
Less Canopy-level radiative transfer model.
Table 2: Performances of different modelling methods trying to predict nitrogen per area-based on spectroradiometric data on a leaf level. The root mean squared error (RMSE) and coefficient of determination (R2) were computed based on a 20-fold 50 times repeated cross validation, where the mean and the standard deviation are reported (\(\mu ± \sigma\)). The compact letter display (CLD) is based on a pairwise Wilcoxon rank sum tests.
Method R2 RMSE [g mm-2] CLD
NDVI -0.016 ± 0.024 0.316 ± 0.0037 a
NDRE 0.123 ± 0.007 0.293 ± 0.0011 b
EVI 0.115 ± 0.006 0.295 ± 0.001 c
GCI -0.055 ± 0.015 0.322 ± 0.0023 d
GNDVI -0.048 ± 0.013 0.321 ± 0.002 d
MCARI 0.01 ± 0.014 0.312 ± 0.0021 e
RECI 0.128 ± 0.004 0.292 ± 8e-04 f
TCARI -0.016 ± 0.005 0.316 ± 7e-04 a
WDRVI 0.011 ± 0.016 0.312 ± 0.0025 e
R434 0.255 ± 0.008 0.27 ± 0.0014 g
Combined vegetation indices 0.464 ± 0.019 0.229 ± 0.004 h
Recursive feature selection 0.544 ± 0.01 0.211 ± 0.0024 i
Square feature selection 0.61 ± 0.007 0.195 ± 0.0019 j
Lasso 0.542 ± 0.04 0.212 ± 0.0089 i
Ridge 0.496 ± 0.007 0.222 ± 0.0015 k
Elastic-net 0.537 ± 0.049 0.213 ± 0.0106 i
Principal components 0.614 ± 0.021 0.195 ± 0.0052 jl
Partial least squares 0.626 ± 0.02 0.192 ± 0.0052 l
Random forest 0.492 ± 0.01 0.223 ± 0.0021 k
Extreme gradient boosting 0.548 ± 0.03 0.21 ± 0.007 i
Cubist 0.276 ± 0.046 0.266 ± 0.0083 g
Prospect-Pro 0.684 ± 0.004 0.176 ± 0.0012 m
Table 3: Performances of different modelling methods trying to predict foliar nitrogen concentration based on spectroradiometric data on a leaf level. The root mean squared error (RMSE) and coefficient of determination (R2) were computed based on a 20-fold 50 times repeated cross validation, where the mean and the standard deviation are reported (\(\mu ± \sigma\)). The compact letter display (CLD) is based on a pairwise Wilcoxon rank sum tests.
Method R2 RMSE [%] CLD
NDVI 0.184 ± 0.005 0.22 ± 7e-04 a
NDRE 0.177 ± 0.005 0.221 ± 7e-04 bc
EVI 0.177 ± 0.006 0.221 ± 7e-04 bc
GCI 0.089 ± 0.006 0.233 ± 8e-04 d
GNDVI 0.093 ± 0.007 0.232 ± 8e-04 d
MCARI 0.151 ± 0.005 0.225 ± 7e-04 e
RECI 0.174 ± 0.007 0.222 ± 9e-04 b
TCARI 0.148 ± 0.006 0.225 ± 8e-04 e
WDRVI 0.177 ± 0.005 0.221 ± 6e-04 bc
R434 0.065 ± 0.007 0.236 ± 9e-04 f
Combined vegetation indices 0.14 ± 0.02 0.226 ± 0.0026 e
Recursive feature selection 0.235 ± 0.006 0.213 ± 9e-04 g
Square feature selection 0.22 ± 0.015 0.215 ± 0.0021 h
Lasso 0.407 ± 0.012 0.188 ± 0.002 ij
Ridge 0.182 ± 0.009 0.221 ± 0.0012 ac
Elastic-net 0.407 ± 0.01 0.188 ± 0.0016 i
Principal components 0.417 ± 0.017 0.186 ± 0.0028 j
Partial least squares 0.493 ± 0.022 0.174 ± 0.0037 k
Random forest 0.27 ± 0.02 0.208 ± 0.0028 l
Extreme gradient boosting 0.288 ± 0.05 0.206 ± 0.0072 l
Cubist 0.384 ± 0.034 0.191 ± 0.0053 i
Prospect-Pro 0.453 ± 0.005 0.18 ± 8e-04 m
Table 4: Performances of different modelling methods trying to predict foliar nitrogen concentration based on hyperspectral images. The root mean squared error (RMSE) and coefficient of determination (R2) were computed based on a 20-fold 50 times repeated cross validation, where the mean and the standard deviation are reported (\(\mu ± \sigma\)). The compact letter display (CLD) is based on a pairwise Wilcoxon rank sum tests.
Method R2 RMSE [%] CLD
NDVI -0.026 ± 0.027 0.257 ± 0.0034 a
NDRE -0.003 ± 0.026 0.254 ± 0.0033 bc
EVI -0.082 ± 0.011 0.264 ± 0.0013 d
GCI -0.01 ± 0.026 0.255 ± 0.0032 abc
GNDVI -0.012 ± 0.027 0.255 ± 0.0034 abc
MCARI -0.001 ± 0.019 0.254 ± 0.0024 b
RECI -0.007 ± 0.023 0.254 ± 0.0029 abc
TCARI 0.068 ± 0.006 0.245 ± 8e-04 e
WDRVI -0.021 ± 0.023 0.256 ± 0.0029 ac
R434 0.198 ± 0.007 0.227 ± 9e-04 f
Combined vegetation indices 0.237 ± 0.011 0.221 ± 0.0015 g
Recursive feature selection 0.142 ± 0.022 0.235 ± 0.003 h
Square feature selection 0.256 ± 0.014 0.218 ± 0.0021 i
Lasso 0.249 ± 0.012 0.22 ± 0.0018 i
Ridge 0.206 ± 0.008 0.226 ± 0.0012 j
Elastic-net 0.25 ± 0.012 0.219 ± 0.0017 i
Principal components 0.236 ± 0.013 0.221 ± 0.0019 g
Partial least squares 0.228 ± 0.013 0.223 ± 0.0019 g
Random forest 0.102 ± 0.032 0.24 ± 0.0042 k
Extreme gradient boosting 0.127 ± 0.018 0.237 ± 0.0024 hl
Cubist 0.175 ± 0.013 0.23 ± 0.0018 m
Less 0.116 ± 0.011 0.237 ± 0.0015 kl
Less + interpolation 0.235 ± 0.009 0.22 ± 0.0013 g