# Potential of Multiway PLS (N-PLS) Regression Method to Analyse Time-Series of Multispectral Images: A Case Study in Agriculture

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## Abstract

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^{2}obtained on the prediction set (0.661), and the root mean square of error (RMSE), which was 10.7%. Limitations of the approach when dealing with time-series of large-scale images which represent a source of challenges are discussed; however, the N–PLS regression seems to be a suitable choice for analysing complex multispectral imagery data with different spectral domains and with a clear temporal evolution, such as an extreme weather event.

## 1. Introduction

**X**of independent variables and a response array

**y**[20]. In this paper, PLS regression is extended to multiway data (N-PLS), with the main emphasis on three-way data, by including time as a dimension. It is important to note that N-PLS imposes a trilinear structure on the data [21] and when introducing time into the analysis, this implies that the emphasis is exclusively on the development of a correlational structure over time [22], i.e., time will determine both the variability between individuals and the variability between variables (wavelengths). The N-PLS methodology allows a joint evaluation of time and variables, in order to determine the interaction between them, to know if and how the variables modify their “behaviour” at different occasions. To the authors’ knowledge, such an approach has not been proposed in the analysis of remotely sensed satellite imagery.

## 2. Materials and Methods

#### 2.1. Type of Problem the N-Way Partial Least Squares Aims to Address in Remote Sensing

_{n}different dates, with each image being constituted of M

_{m}spectral bands or wavelengths (variables). Given the data dimensionality, multiway arrays have been proven to be a natural and efficient representation of the data, in particular, tensor subspace learning methods have been shown to outperform their corresponding vector subspace methods, especially for small sample size problems [24]. Therefore, the N-PLS algorithm should constitute a relevant approach to retrieve the information contained in the spectral bands of satellite imagery to highlight the target phenomenon taking into account its effect on the temporal evolution of the imagery.

#### 2.2. N-Way Partial Least Squares

**X**) and relates it to a dependent variable (

**y**) using a relatively low number of parameters, which makes the prediction more robust [26]. This situation leads to an analysis that is able to extract the maximum information possible from samples measured at different times based on cubic structure.

**X**= ||X

_{i,j,k}||. The first of the three dimensions of cube

**X**corresponds to the objects (I), the second to time (J) and the third to spectral bands (K). Thus, the data were organized in a three-way array of independent variables

**X**(I × J × K) and a response vector y of size (I × 1) that is defined by the objects dimensions (Figure 2).

**X**(remote sensing data) to the response vector

**y**(ground truth data). As Abdi [25] specified, PLS regression performs a simultaneous decomposition of

**X**and

**y**by means of a set of latent variables that explain as much as possible of the covariance between

**X**and

**y**. As Bergant et al. [27] detailed, if

**X**(I × JK) is a properly unfolded two-way form of a three-way array

**X**(I × J × K), the 3-PLS1 method can be written in mathematical form (Equation 1), where

**X**and

**y**are cantered along the first dimension, I.

**W**, allowing for the scores in

**S**to be expressed directly in terms of the

**X**-columns, is the essential part of the method [28]. The regression coefficients

**b**(Equation (1)) can be estimated afterwards by a least square approach [27]. The vector

**r**in Equation (1) presents the part of

**y**not explained by the model. For the estimation of

**W**, the algorithm for three-way

**X**and a single response

**y**proposed by Jong [28] was considered. If the initial values of ${\mathit{y}}_{\mathit{a}}$ (a is the latent variable counter) are set to the original values $\mathit{y}$ (a = 0 and ${\mathit{y}}_{\mathbf{0}}$ = $\mathit{y}$), the algorithm can be summarized as follows [27]:

- Compute the reshaped covariance matrix
**Ž**= ${y}_{a}^{T}$**X**(1 × JK). - Define the first singular weight vectors ${\mathit{w}}_{a}^{J}$ and ${\mathit{w}}_{a}^{K}$ from $\mathit{Z}$: [${\mathit{w}}_{a}^{J}$, ${\mathit{w}}_{a}^{K}$] = svd($\mathit{Z}$,1). From hence, store them as additional columns in separate weight arrays
**W**^{J}= [${\mathit{w}}_{1}^{J}$…${\mathit{w}}_{a}^{J}$ ] and**W**^{K}[${\mathit{w}}_{1}^{K}$…${\mathit{w}}_{a}^{K}$]. - Calculate
**S**=**XW**. - Calculate the regression coefficients regressing $\mathit{y}$ on
**S**as**b**= (**S**^{T}**S**)^{−1}**S**^{T}$\mathit{y}$. - Calculate the residuals $\mathrm{r}$ = $\mathit{y}$ −
**Sb**. - Increase a to a + 1 and continue from step 1 to the appropriate description of $\mathit{y}$. The inclusion of an additional latent variable (a + 1) in the model is terminated when the joint analysis of RMSEC (Root Mean Square Error of Calibration) and RMSECV (Root Mean Square Error of Cross-Validation) [30] indicates overfitting due to sampling variability.

#### 2.3. Model Construction

#### 2.3.1. Structuration of Time-Series Data

**X**, had the same number of steps (N days of measurements) to prevent temporal data gaps due to clouds and inconsistent numbers of available satellite images. The interpolation at a date t was performed band-by-band, by a convolution of the chronology measured with a Gaussian filter [31] centred on t and with full width at half maximum (P).

**X**and vector

**y**using the calibration dataset (see Section 2.3.2). For this analysis, values of P and N ranged from 10 to 50 and 5 to 30, respectively.

#### 2.3.2. Calibration and Validation of the Model

**y**, as follows:

- The vector y was sorted in ascending order.
- After sorting, every fourth individual was placed in the validation set, the others retained in the calibration set.

**X**cube and the

**y**from calibration set. As proposed by Bergant et al. [27], the final calibration fit can be written as in Equation (2).

**ŷ = Sb = XWb = XB**

_{NPLS}+ b_{0.}**ŷ**is the vector of estimated responses and

**b**the intercept of the linear regression model. The matrix of regression coefficients

_{0}**B**(Equation (2)) can be used on new data for the estimation of unknown response values. The prediction performance of the empirical model was quantified by the standard coefficient of determination (R

_{NPLS}= Wb^{2}), the bias and the standard error parameters [34,35].

#### 2.4. Case-Study

#### 2.4.1. Study Area

^{2}, from the Spanish border to the delta of the Rhône.

#### 2.4.2. Remote Sensing Data

#### Data Acquisition and Preprocessing

#### Spectral Bands

#### 2.4.3. Ground-Truth Data

#### 2.4.4. Modelling

#### Model Construction

**X**(I × J × K).

**X**was incomplete and there was a need for interpolation to obtain a continuous data cube. J, defined as the number of dates selected to represent the time-series, can be determined by optimising the parameters N and P. For this model, J was determined by using ranges of N and P described in Section 2.3.1. Once J was determined, interpolation was performed (again following the method in Section 2.3.1) to have a consistent time step.

- a cube
**X**(107, J, 12) where the first dimension corresponds to the vineyard blocks (I), the second dimension corresponds to time (J), which is optimised during modelling, and the third dimension of the three-way array**X**corresponds to spectral bands (K) averaged for each field, - a vector
**y**(107), corresponding to the estimated percentage yield loss by the winegrowers from the 107 blocks.

#### Model Validation

#### Model Evaluation

^{2}, the bias and the standard error parameters. The regression b-coefficients resulting from the application of the methodology were plotted (i) against time as a function of wavelengths and (ii) against spectral bands as a function of time identify the time and the wavelengths that best highlighted this phenomenon. To further illustrate this, a 3D view of the b-coefficients vs. time vs. spectral bands was also generated. Figure 7 summarizes the implementation and the workflow scheme of the N-PLS adapted to the case study.

## 3. Results

#### 3.1. Optimisation of Model Parameters over the Study Site

**X**cube and the

**y**losses from the calibration set when varying the Gaussian filter width (P) and interval between dates (N). This procedure led to an observation of the lowest error of cross-validation (RMSECV) at P = 30 and N = 15. N = 15 equates to J = 7, i.e., 7 dates over the study period, and resulted in a cube

**X**of dimensions (107, 7, 12). Figure 8 highlights three situations in which the cross-validation error (RMSECV) increased when applied to these data, as follows: (i) when the Gaussian filter is too weak (P > 25); (ii) when the time step is short (N < 10 days); (iii) when the time step is long (N > 20 days).

#### 3.2. Quality of the N-PLS Model

^{2}, the bias and the standard error of prediction of yield losses in the calibration (Figure 10a) and validation (Figure 10b) analyses. The N-PLS model showed a performance accuracy (R

^{2}) of 0.56 in the calibration set and 0.66 in the validation set, with a standard error of cross-validation in the calibration set of 12.4% and a standard error of prediction of losses in validation set of 10.7%.

**B**corresponding to the 5 latent variables selected for the model. The N-PLS model estimated yield loss for each block by making the inner product of b-coefficients with a spectral-temporal profile (7 × 12) contained in the cube

_{NPLS}**X**(107,7,12). Thus, the b-coefficients had the same dimension as the spectral-temporal profile and could be analysed to identify how specific parts or periods of the spectral-temporal profile were more or less related to yield losses.

## 4. Discussion

^{2}) of 0.66 and a RMSE of approximately 10%. The theory for the use of remote sensing instruments to monitor electromagnetic radiation reflectance changes in crops is well demonstrated in the scientific literature [38]. In situations where crops interact with any given aspect of their environment, such as seasonal climatic variations or meteorological extreme events, the interactions between plants and light reflectance translates into changes in plant signal patterns that can be interpreted using satellite data [39]. Besides the model of yield loss estimation, the N-PLS analysis showed the interest of adopting a systemic analysis which accounts simultaneously for the spectral and temporal characteristics of the considered data. The approach allowed the identification of the spectral bands that responded most strongly to the phenomenon of interest while keeping the information of the period of influence.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**(

**a**) Representation of three-way array (

**X**) and (

**b**) response vector

**y**to compute a tri-linear PLS1 procedure.

**Figure 3.**Location of the study area in Southern France (

**a**) and the location of the vineyards that contained the 107 blocks of interest for the study (

**b**).

**Figure 4.**Mean monthly temperature (

**a**) and maximum monthly temperatures records (

**b**) from 2009 to 2019 across the Languedoc-Roussillon region, France, highlighting a peak in the maximum monthly temperature corresponding to the extreme weather event that occurred in June 2019, while the mean monthly temperature for this month was not extreme. The vertical black dashed line highlights the month of the heat wave. Source: Historique Météo-France.

**Figure 5.**(

**a**) Maximum temperature record map 28 June 2019 France. (

**b**) Maximum temperature record map 28 June 2019 Languedoc-Roussillon. Temperatures ≥ 40 °C are shown in red, temperatures < 40 °C and ≥30 °C are shown in orange and temperatures < 30 °C are shown in yellow. Source: Météo-France.

**Figure 8.**Evolution of RMSECV as a function of varying intervals between dates (N) and Gaussian filter width (P) obtained for 2 blocks repeated 5 times with a N-PLS performed with a

**X**cube and the

**y**yield losses. Parameter optimisation was achieved at, P = 30 and N = 15 as indicated by the white dash circle.

**Figure 9.**Evolution of the RMSEC and the RMSECV for a cross-validation of 2 blocks repeated 10 times of a N-PLS between the

**X**cube and the

**y**losses. The black frame indicates the optimal number of latent variables (5 LV).

**Figure 10.**Results of the N-PLS prediction of losses on individuals in the calibration set (

**a**), with 80 vineyard blocks and in the validation test (

**b**), with 27 vineyard blocks. The standard error of cross-validation in the calibration set was 12.4 and the standard error of prediction of yield losses in validation set was 10.7.

**Figure 11.**N-PLS b-coefficients corresponding to 5 latent variables and the following two different dimensions: (

**a**) plotted according to the spectral dimension and (

**b**) plotted according to the temporal dimension. Black dot-dash line highlights the most relevant date of the heat wave.

**Figure 12.**Combined representation of the temporal and spectral profiles as a 3D view of N-PLS b-coefficients corresponding to 5 latent variables. Black dot-dash line to highlight the most relevant date of the heat wave.

Sentinel-2 Band | Central Wavelength (nm) | Bandwidth (nm) | Spatial Resolution (m) |
---|---|---|---|

Band 1–Aerosol | 442.7 | 21 | 60 |

Band 2–Blue | 492.4 | 66 | 10 |

Band 3–Green | 559.8 | 36 | 10 |

Band 4–Red | 664.6 | 31 | 10 |

Band 5–Vegetation Red Edge | 704.1 | 15 | 20 |

Band 6–Vegetation Red Edge | 740.5 | 15 | 20 |

Band 7–Vegetation Red Edge | 782.8 | 20 | 20 |

Band 8–NIR | 832.8 | 106 | 10 |

Band 8A–Vegetation Red Edge | 864.1 | 21 | 20 |

Band 9–VNIR | 945.1 | 20 | 60 |

Band 11–SWIR | 1613.1 | 91 | 20 |

Band 12–SWIR | 2202.4 | 175 | 20 |

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**MDPI and ACS Style**

Lopez-Fornieles, E.; Brunel, G.; Rancon, F.; Gaci, B.; Metz, M.; Devaux, N.; Taylor, J.; Tisseyre, B.; Roger, J.-M.
Potential of Multiway PLS (N-PLS) Regression Method to Analyse Time-Series of Multispectral Images: A Case Study in Agriculture. *Remote Sens.* **2022**, *14*, 216.
https://doi.org/10.3390/rs14010216

**AMA Style**

Lopez-Fornieles E, Brunel G, Rancon F, Gaci B, Metz M, Devaux N, Taylor J, Tisseyre B, Roger J-M.
Potential of Multiway PLS (N-PLS) Regression Method to Analyse Time-Series of Multispectral Images: A Case Study in Agriculture. *Remote Sensing*. 2022; 14(1):216.
https://doi.org/10.3390/rs14010216

**Chicago/Turabian Style**

Lopez-Fornieles, Eva, Guilhem Brunel, Florian Rancon, Belal Gaci, Maxime Metz, Nicolas Devaux, James Taylor, Bruno Tisseyre, and Jean-Michel Roger.
2022. "Potential of Multiway PLS (N-PLS) Regression Method to Analyse Time-Series of Multispectral Images: A Case Study in Agriculture" *Remote Sensing* 14, no. 1: 216.
https://doi.org/10.3390/rs14010216