# Quantifying Leaf Chlorophyll Concentration of Sorghum from Hyperspectral Data Using Derivative Calculus and Machine Learning

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Site and Plant Material

#### 2.2. Data Collection

#### 2.2.1. Leaf Chlorophyll Concentration Measurements

^{2}using a calibration coefficient [66]. Only sunlit representative leaves from each plot were selected for measurements. The LCC measurements were taken at noon on two days, (9 November 2016 and 11 November 2016) while the sorghum plants were at the grain development growth stage.

#### 2.2.2. Hyperspectral Reflectance Measurements

#### 2.3. Fractional Derivative Calculation

#### 2.4. Calculation of Vegetation Indices

#### 2.5. Feature Selection Methods

#### 2.5.1. Pearson’s Correlation Coefficient (PCC)

#### 2.5.2. Variable Importance in the Projection (VIP)

#### 2.5.3. Mean Decrease Impurity (MDI)

#### 2.6. Machine Learning Algorithms

#### 2.7. Modeling Pipeline and Evaluation

## 3. Results

#### 3.1. Descriptive Statistics of Collected Samples

^{2}with a mean value of 50.26 µg/cm

^{2}. The sample distribution had a standard deviation of 7.54 with a coefficient of variation (CV) of 15%. Figure 3a also shows normally distributed LCC sample values. The descriptive statistics for spectral features are visually represented in Figure 3b. The mean spectral curve (350–2500 nm) of corresponding LCC samples shows a typical reflectance pattern of healthy vegetation: moderately strong reflectance at green region (approximately 500–650 nm), very strong reflectance at NIR region (approximately 750–1000 nm), and two water absorption regions at around 1500 nm and 2000 nm. This reveals that the sample leaves selected for this study were healthy and representative for the analysis.

#### 3.2. Spectral Features After Fractional Derivative Analysis

#### 3.3. Feature Importance Scores

_{4}, Datt

_{1}, MTCI, NDVI, REP, RI

_{db}, SR

_{750/710}, VOG

_{2}, and VOG

_{3}were found as showing higher scores. With increasing derivative orders, the scores for different features became noisier (Figure 7a,b). In terms of MDI (Figure 7c), very few features were highlighted in each derivative order, for example, only Vog

_{2}and Vog

_{3}were found highly important in original spectra, with order 0.2 and order 0.4, respectively. After order 1.4, the number of important features increased abruptly.

#### 3.4. Model Results of LCC Estimation

^{2}, RMSE, and RMSE%) were only calculated for the combination of feature selection method and number of features that yielded the lowest cross validation MSE score from the training set. These metrics were calculated with the validation dataset and all derivative orders of two different datasets: reflectance-based and VI-based spectra. The validation metrics of LCC estimation are demonstrated in Table 3. In addition, the model R

^{2}and RMSE are illustrated in Figure 8 with respect to different derivative order.

^{2}ranging from 0.578 to 0.734 and RMSE% ranging from 8.125 to 10.227). The predictive performance of all models showed improvement with increasing derivative order up to a particular point. For instance, PLSR (R

^{2}of 0.701 and RMSE% of 8.603) showed the highest result at order 0.2, RFR (R

^{2}of 0.683 and RMSE% of 8.865) and SVR (R

^{2}of 0.734 and RMSE% of 8.125) yielded peaks at order 1.0, and ELR (R

^{2}of 0.704 and RMSE% of 8.567) performed the best at order 0.4. After the respective orders, each model started to decline in their performance (Figure 8a,c). Overall, the SVR showed consistently good performance until the derivative order reached 1.8 (R

^{2}ranging from 0.457 to 0.734 and RMSE% ranging from 8.125 to 11.605). Table 3 also shows the best combination of feature selection method and number of features for each model and derivative order. The best performing model within the reflectance-based analysis (i.e., SVR with order 1.0) used 75 features selected by MDI. Overall, the MDI was found as the optimal feature selection method for most of the well performed models.

^{2}ranging from 0.618 to 0.744 and RMSE% ranging from 7.971 to 9.734). The best performing model was found with ELR at original spectra (R

^{2}of 0.744 and RMSE% of 7.971), which was even higher than the best model found with reflectance-based analysis (i.e., SVR at order 1.0 resulting in R

^{2}of 0.734 and RMSE% of 8.125). The ELR with original spectra used 15 features as input which were selected by PCC. Overall, most of the well-performing models at lower derivative orders showed PCC as an optimal feature selection method. However, according to Figure 8b, the model performance decreased with increasing derivative orders within the VI-based analysis. Therefore, the LCC estimation worked better with derivative spectra at 1.0 order when direct reflectance from wavelengths was used, whereas the original spectra showed good performance when the model inputs were VIs.

## 4. Discussion

#### 4.1. Performance Analysis of Derivative Spectra and VIs in LCC Estimation

^{2}of 0.729 and RMSE% of 8.201) used fewer features (n = 25) compared to the highest performing model that used more features (n = 75), yet the results were only slightly less than the best model. Therefore, we find it inconclusive to state that either fractional derivative or integer-order derivative is better in estimating LCC from sorghum.

^{2}) without any derivative analysis (i.e., original spectrum, Figure 10a) and derivative spectrum from order 0.2 to 2.0 with a smaller spectral window (i.e., the NIR region of 700–1000 nm). The selected features with the best models found at each derivative order are also highlighted. Figure 10 is a close-up version of Figure 4 that highlights how the increasing derivative order amplify certain information in the spectral curve and how important features are then selected by different feature selection methods. According to Figure 10a, the original spectra show an increasing slope until around 760 nm and start to flatten out until 1000 nm. With increasing derivative order, the flatten curve starts to show abrupt peaks on it and the important features start to appear in a distributed manner. For example, with order 0.6 (Figure 10d), important features are seen all over the spectrum instead of clustering at the lower end of the spectrum as in the case of original spectra (Figure 10a). This is the reason that the correlation coefficient between LCC and derivative spectra significantly increased with increasing derivative orders (Figure 5).

#### 4.2. Impact of Feature Selection Methods in Modeling Pipeline

#### 4.3. Performance of Machine Learning Models in LCC Estimation

## 5. Conclusions

- In terms of reflectance-based analysis, increasing derivative order can show improved model performance until a certain order; however, it is inconclusive to state that a particular derivative order is optimal for estimating LCC of sorghum. Further assessment with data from multiple study sites and growth stages is required to make such an inference.
- VI-based modeling with original spectra outperformed reflectance-based modeling with derivative-augmented spectra.
- Sensitive feature selection is a crucial step in any machine learning pipeline. MDI score was found effective in selecting sensitive features from a large feature space (reflectance-based analysis), whereas PCC worked better with a smaller feature space (VI-based analysis).
- When single wavelengths were used in the analysis from different FD orders, SVR outperformed all other models. However, PLSR and ELR required fewer model parameters and computational time, which can be advantageous in model training. Alternatively, ELR with VIs from original spectra yielded slightly better results compared to all other models. Therefore, ELR worked better when hand-crafted features (VIs) were used.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Location of test site and data collection. (

**a**) Experimental field; (

**b**) the field scanner operating in the field; (

**c**) in-situ data collection of leaf chlorophyll concentration (LCC) using Dualex 4 Scientific (yellow box) and spectral data using PSR-3500 spectroradiometer (blue box); (

**d**) location of study area in Pinal County, AZ; (

**e**) a top view image of the field collected from ArcGIS Online.

**Figure 3.**Distribution of collected LCC samples collected with Dualex 4 Scientific (

**a**) and mean hyperspectral spectra with 1 and 2 standard deviations collected using Spectral Evolution PSR-3500 (

**b**). (

**a**) The left axis represents the frequency of LCC samples, whereas the right axis represents the probability density; the target variable for this study (i.e., LCC) has a normal distribution. (

**b**) The mean spectral curve of sorghum leaf samples exhibits a health vegetation reflectance curve.

**Figure 4.**Varying spectral features of the minimum (red line), median (green line), and maximum (blue line) LCC samples after different fractional-order derivative treatment, i.e., original spectra in (

**a**), order 0.2 in (

**b**), and so on until order 2.0 in (

**k**) with 0.2 order as increment. Each plot also demarcates the regions of visible (VIS), near-infrared (NIR), and short wave infrared (SWIR) bands with grey dashed lines. The wavelengths are shown from 450 to 1800 nm since typical vegetation spectra show noise at around 400 nm and 2500 nm. With increasing derivative order, the range of derivative reflectance factor starts increasing which can be observed in (

**b**–

**k**).

**Figure 5.**The correlation coefficient between LCC and original spectral data (

**a**) and fractional derivative augmented spectra (

**b**–

**k**). The dashed lines in each plot represent the limit of statistical significance at 99% confidence. The data points located beyond these limits are significantly correlated with LCC. With increasing derivative order, several wavelengths showed increased and statistically significant correlation coefficients (

**c**–

**h**). However, from order 1.6 (

**i**), the pattern of correlation becomes noisy and insignificant.

**Figure 6.**Feature importance scores for wavelengths of different derivative orders calculated from three feature selection methods: (

**a**) Pearson’s correlation coefficient (PCC), (

**b**) partial least squares regression based variable importance in the projection (VIP), and (

**c**) random forest regression based mean decrease impurity (MDI). The 0.0 order in the x-axis represents the original spectra without any derivative treatment. The feature importance score was scaled from 0–1 for each method and derivative order.

**Figure 7.**Feature importance score for vegetation indices of different derivative orders calculated from three feature selection methods: (

**a**) Pearson’s correlation coefficient (PCC), (

**b**) partial least squares regression based variable importance in the projection (VIP), and (

**c**) random forest regression based mean decrease impurity (MDI). The 0.0 order in the x-axis represents the original spectra without any derivative treatment. The y-axis represents different VIs analyzed in this study; however, VIs are not represented in any logical order.

**Figure 8.**Model R

^{2}and RMSE for reflectance-based analysis (left side, i.e.,

**a**,

**c**), and VI-based analysis (right side, i.e.,

**b**,

**d**). The 0.0 order in the x-axis represents the original spectra without any derivative treatment. For reflectance-based modeling, model performance increases; however, the performance starts decreasing after certain derivative order. For VI-based modeling, the model performance was found better with original spectra (order 0.0). With increasing derivative order, model performance starts declining.

**Figure 9.**Boxplot for measured (M) and predicted LCC from different derivative orders and original spectra (order 0.0) using all models (i.e., the x-axis). The figures on the left side (

**a**–

**d**) contain boxplots from reflectance-based analysis, whereas the figures on the right side (

**e**–

**h**) show the boxplots generated from VI-based analysis.

**Figure 10.**Original spectral curve of a leaf sample (

**a**), and corresponding fractional derivative (FD)-transformed spectra (

**b**–

**k**). The spectra only show the NIR region (700–1000 nm) since it is considered the most important region for feature selection. The red circles show the position of features which were selected as input for the best performing model of corresponding order. The n represents the number of features found within the NIR range.

**Figure 11.**Scatterplots showing the relationship between different VIs and LCC values. The entire sample (n = 349) is shown in these plots. The VI values are scaled from 0 to 1 for visual enhancement in the figure. The scale transformation did not change the pattern of relationship between LCC and VIs.

VI | Equation | Reference |
---|---|---|

ARI_{1} | $1/{R}_{550}-1/{R}_{700}$ | [77] |

ARI_{2} | ${R}_{800}\left(1/{R}_{550}-1/{R}_{700}\right)$ | [77] |

Cart_{1} | ${R}_{695}/{R}_{420}$ | [78] |

Cart_{2} | ${R}_{695}/{R}_{760}$ | [78] |

Cart_{3} | ${R}_{605}/{R}_{760}$ | [78] |

Cart_{4} | ${R}_{710}/{R}_{760}$ | [78] |

Cart_{5} | ${R}_{695}/{R}_{670}$ | [78] |

CCI | $({R}_{777}-{R}_{747})/{R}_{673}$ | [79] |

Datt_{1} | $({R}_{850}-{R}_{710})/({R}_{850}-{R}_{680})$ | [80] |

Datt_{2} | ${R}_{850}/{R}_{710}$ | [80] |

Datt_{3} | ${R}_{754}/{R}_{704}$ | [80] |

EVI | $2.5\left(({R}_{800}-{R}_{670}\right)/({R}_{800}-6{R}_{670}-7.5{R}_{475}+1))$ | [81,82] |

GNDVI_{1} | $({R}_{750}-{R}_{550})/({R}_{750}+{R}_{550})$ | [83] |

GNDVI_{2} | $({R}_{800}-{R}_{550})/({R}_{800}+{R}_{550})$ | [83] |

MCARI_{1} | $(({R}_{700}-{R}_{670})-0.2({R}_{700}-{R}_{550}))\left({R}_{700}/{R}_{670}\right)$ | [84] |

MCARI_{2} | $1.2(2.5({R}_{800}-{R}_{670})-1.3({R}_{800}-{R}_{550}))$ | [85] |

mNDVI | $({R}_{750}-{R}_{705})/({R}_{750}+{R}_{705}-2{R}_{445})$ | [80,86] |

mSR | $({R}_{750}-{R}_{445})/({R}_{705}-{R}_{445})$ | [80,86] |

MTCI | $({R}_{754}-{R}_{709})/({R}_{709}-{R}_{681})$ | [87] |

MTVI_{1} | $1.2(1.2({R}_{800}-{R}_{550})-2.5({R}_{670}-{R}_{550}))$ | [85] |

NDCI | $({R}_{762}-{R}_{527})/({R}_{762}+{R}_{527})$ | [88] |

NDVI | $({R}_{750}-{R}_{705})/({R}_{750}+{R}_{705})$ | [89] |

PRI | $({R}_{531}-{R}_{570})/({R}_{531}+{R}_{570})$ | [90] |

PSRI | $({R}_{678}-{R}_{500})/{R}_{750}$ | [91] |

REP | $700+40\left(({R}_{670}-{R}_{780}\right)/2-{R}_{700}))/({R}_{740}-{R}_{700})$ | [92] |

RI_{db} | ${R}_{735}/{R}_{720}$ | [93] |

SIPI | $({R}_{800}-{R}_{445})/({R}_{800}+{R}_{680})$ | [94] |

SPVI_{1} | $0.4\times 3.7({R}_{800}-{R}_{670})-1.2\left|{R}_{530}-{R}_{670}\right|$ | [95,96] |

SPVI_{2} | $0.4\times 3.7({R}_{800}-{R}_{670})-1.2\left|{R}_{550}-{R}_{670}\right|$ | [95] |

SR_{440/690} | ${R}_{440}/{R}_{690}$ | [97] |

SR_{700/670} | ${R}_{700}/{R}_{670}$ | [98] |

SR_{750/550} | ${R}_{750}/{R}_{550}$ | [98] |

SR_{750/700} | ${R}_{750}/{R}_{700}$ | [99] |

SR_{750/710} | ${R}_{750}/{R}_{710}$ | [100] |

SR_{752/690} | ${R}_{752}/{R}_{690}$ | [100] |

SR_{800/680} | ${R}_{800}/{R}_{680}$ | [86] |

SRPI | ${R}_{430}/{R}_{680}$ | [101] |

TCARI | $3\left(({R}_{700}-{R}_{670}\right)-0.2({R}_{700}-{R}_{550})({R}_{700}/{R}_{670}))$ | [18] |

TCARI_{2} | $3\left(({R}_{750}-{R}_{705}\right)-0.2({R}_{750}-{R}_{550})({R}_{750}/{R}_{705}))$ | [20] |

TVI | $0.5(120({R}_{750}-{R}_{550})-200({R}_{670}-{R}_{550}))$ | [102] |

VOG_{1} | ${R}_{740}/{R}_{720}$ | [103] |

VOG_{2} | $({R}_{734}-{R}_{747})/({R}_{715}+{R}_{726})$ | [103] |

VOG_{3} | $({R}_{734}-{R}_{747})/({R}_{715}+{R}_{720})$ | [103] |

Sample Size | Maximum | Minimum | Mean | SD | CV (%) | |
---|---|---|---|---|---|---|

LCC (µg/cm^{2}) | 349 | 70.30 | 30.80 | 50.26 | 7.54 | 15.00 |

**Table 3.**Validation results of partial least squares regression (PLSR), random forest regression (RFR), support vector regression (SVR), and extreme learning regression (ELR) for LCC with different derivative orders.

Ord. | Metrics | Reflectance-based | VI-based | ||||||
---|---|---|---|---|---|---|---|---|---|

PLSR | RFR | SVR | ELR | PLSR | RFR | SVR | ELR | ||

0.0 | R^{2} | 0.671 | 0.443 | 0.676 | 0.558 | 0.673 | 0.618 | 0.717 | 0.744 |

RMSE | 4.493 | 5.842 | 4.459 | 5.207 | 4.477 | 4.841 | 4.169 | 3.964 | |

RMSE% | 9.035 | 11.747 | 8.966 | 10.471 | 9.002 | 9.734 | 8.382 | 7.971 | |

Features | VIP-75 | MDI-50 | VIP-75 | VIP-50 | VIP-30 | MDI-10 | PCC-25 | PCC-15 | |

0.2 | R^{2} | 0.701 | 0.509 | 0.706 | 0.548 | 0.714 | 0.625 | 0.708 | 0.698 |

RMSE | 4.279 | 5.486 | 4.249 | 5.265 | 4.187 | 4.794 | 4.231 | 4.306 | |

RMSE% | 8.603 | 11.032 | 8.543 | 10.588 | 8.418 | 9.639 | 8.509 | 8.658 | |

Features | VIP-75 | MDI-75 | VIP-175 | VIP-50 | PCC-10 | VIP-30 | PCC-15 | PCC-10 | |

0.4 | R^{2} | 0.653 | 0.654 | 0.720 | 0.704 | 0.674 | 0.696 | 0.651 | 0.579 |

RMSE | 4.616 | 4.605 | 4.142 | 4.261 | 4.468 | 4.320 | 4.623 | 5.081 | |

RMSE% | 9.281 | 9.259 | 8.330 | 8.567 | 8.984 | 8.686 | 9.295 | 10.217 | |

Features | VIP-25 | MDI-125 | MDI-100 | VIP-25 | PCC-20 | MDI-10 | PCC-15 | PCC-15 | |

0.6 | R^{2} | 0.653 | 0.661 | 0.680 | 0.608 | 0.672 | 0.675 | 0.678 | 0.650 |

RMSE | 4.614 | 4.560 | 4.427 | 4.901 | 4.482 | 4.464 | 4.445 | 4.624 | |

RMSE% | 9.278 | 9.169 | 8.902 | 9.855 | 9.012 | 8.975 | 8.938 | 9.296 | |

Features | VIP-50 | MDI-50 | MDI-175 | MDI-25 | PCC-15 | MDI-20 | VIP-15 | VIP-15 | |

0.8 | R^{2} | 0.621 | 0.648 | 0.729 | 0.589 | 0.670 | 0.672 | 0.660 | 0.640 |

RMSE | 4.820 | 4.649 | 4.078 | 5.018 | 4.499 | 4.483 | 4.566 | 4.697 | |

RMSE% | 9.692 | 9.347 | 8.201 | 10.090 | 9.047 | 9.014 | 9.182 | 9.445 | |

Features | VIP-200 | MDI-50 | MDI-25 | MDI-25 | PCC-15 | MDI-5 | PCC-10 | VIP-10 | |

1.0 | R^{2} | 0.632 | 0.683 | 0.734 | 0.578 | 0.655 | 0.616 | 0.555 | 0.644 |

RMSE | 4.747 | 4.409 | 4.041 | 5.086 | 4.596 | 4.850 | 5.226 | 4.673 | |

RMSE% | 9.546 | 8.865 | 8.125 | 10.227 | 9.241 | 9.753 | 10.508 | 9.397 | |

Features | VIP-200 | MDI-75 | MDI-75 | PCC-25 | PCC-10 | MDI-20 | PCC-20 | VIP-10 | |

1.2 | R^{2} | 0.528 | 0.673 | 0.708 | 0.573 | 0.526 | 0.514 | 0.543 | 0.494 |

RMSE | 5.380 | 4.480 | 4.235 | 5.119 | 5.393 | 5.461 | 5.296 | 5.572 | |

RMSE% | 10.818 | 9.009 | 8.515 | 10.294 | 10.844 | 10.981 | 10.649 | 11.203 | |

Features | VIP-175 | VIP-75 | VIP-150 | VIP-50 | MDI-5 | MDI-15 | MDI-5 | MDI-5 | |

1.4 | R^{2} | 0.536 | 0.602 | 0.662 | 0.492 | 0.056 | 0.286 | 0.282 | 0.249 |

RMSE | 5.332 | 4.937 | 4.550 | 5.579 | 7.607 | 6.614 | 6.633 | 6.786 | |

RMSE% | 10.721 | 9.927 | 9.149 | 11.219 | 15.295 | 13.299 | 13.337 | 13.645 | |

Features | VIP-200 | MDI-25 | PCC-150 | MDI-25 | VIP-15 | MDI-15 | MDI-5 | PCC-5 | |

1.6 | R^{2} | 0.446 | 0.588 | 0.573 | 0.420 | −0.020 | 0.066 | −0.023 | 0.075 |

RMSE | 5.830 | 5.028 | 5.119 | 5.962 | 7.906 | 7.567 | 7.919 | 7.530 | |

RMSE% | 11.724 | 10.110 | 10.294 | 11.988 | 15.898 | 15.215 | 15.924 | 15.141 | |

Features | VIP-175 | PCC-25 | VIP-150 | PCC-50 | MDI-10 | MDI-10 | MDI-5 | MDI-5 | |

1.8 | R^{2} | 0.281 | 0.339 | 0.457 | 0.109 | −0.065 | −0.028 | −0.087 | −0.296 |

RMSE | 6.637 | 6.368 | 5.771 | 7.393 | 8.082 | 7.940 | 8.164 | 8.915 | |

RMSE% | 13.347 | 12.805 | 11.605 | 14.865 | 16.251 | 15.966 | 16.417 | 17.926 | |

Features | PCC-200 | MDI-25 | VIP-150 | MDI-25 | PCC-5 | MDI-25 | PCC-10 | VIP-10 | |

2.0 | R^{2} | 0.128 | 0.035 | 0.116 | 0.166 | −0.280 | −0.239 | −0.089 | −0.040 |

RMSE | 7.311 | 7.691 | 7.361 | 7.151 | 8.860 | 8.715 | 8.173 | 7.986 | |

RMSE% | 14.701 | 15.465 | 14.802 | 14.380 | 17.816 | 17.525 | 16.434 | 16.058 | |

Features | VIP-150 | MDI-75 | VIP-100 | VIP-50 | MDI-5 | MDI-10 | MDI-5 | VIP-30 |

^{2}: coefficient of determination; RMSE: root mean squared error; RMSE%: relative RMSE.

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## Share and Cite

**MDPI and ACS Style**

Bhadra, S.; Sagan, V.; Maimaitijiang, M.; Maimaitiyiming, M.; Newcomb, M.; Shakoor, N.; Mockler, T.C.
Quantifying Leaf Chlorophyll Concentration of Sorghum from Hyperspectral Data Using Derivative Calculus and Machine Learning. *Remote Sens.* **2020**, *12*, 2082.
https://doi.org/10.3390/rs12132082

**AMA Style**

Bhadra S, Sagan V, Maimaitijiang M, Maimaitiyiming M, Newcomb M, Shakoor N, Mockler TC.
Quantifying Leaf Chlorophyll Concentration of Sorghum from Hyperspectral Data Using Derivative Calculus and Machine Learning. *Remote Sensing*. 2020; 12(13):2082.
https://doi.org/10.3390/rs12132082

**Chicago/Turabian Style**

Bhadra, Sourav, Vasit Sagan, Maitiniyazi Maimaitijiang, Matthew Maimaitiyiming, Maria Newcomb, Nadia Shakoor, and Todd C. Mockler.
2020. "Quantifying Leaf Chlorophyll Concentration of Sorghum from Hyperspectral Data Using Derivative Calculus and Machine Learning" *Remote Sensing* 12, no. 13: 2082.
https://doi.org/10.3390/rs12132082