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appendix.qmd
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# Appendix {.unnumbered}
This appendix features a list of abbreviations and symbols used throughout this thesis (@tbl-abbreviations) as well as the numerical values of the performance results of the models tested (@tbl-ResultsNarea, [-@tbl-ResultsNmass], and [-@tbl-ResultsHSI]).
| **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]{.smallcaps} | Leaf-level radiative transfer model. |
| [Less]{.smallcaps} | Canopy-level radiative transfer model. |
: List of abbreviations. The abbreviations used are informed by the closest literature. {#tbl-abbreviations}
|**Method** |**R^2^** |**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]{.smallcaps} |0.684 ± 0.004 |0.176 ± 0.0012 |m |
: 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 (R^2^) 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. {#tbl-ResultsNarea tbl-colwidths="[80,30,30,10]"}
|**Method** |**R^2^** |**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 |
: 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 (R^2^) 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. {#tbl-ResultsNmass tbl-colwidths="[80,30,30,10]"}
|**Method** |**R^2^** |**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]{.smallcaps} |0.116 ± 0.011 |0.237 ± 0.0015 |kl |
|[Less]{.smallcaps} + interpolation |0.235 ± 0.009 |0.22 ± 0.0013 |g |
: 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 (R^2^) 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. {#tbl-ResultsHSI tbl-colwidths="[80,30,30,10]"}