# Gap-Filling of NDVI Satellite Data Using Tucker Decomposition: Exploiting Spatio-Temporal Patterns

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Region and Data Set

#### 2.1. Study Region

#### 2.2. Data Sets and Pre-Processing

^{2}) of the Iberian Peninsula. Thus, for SPAIN2, the actual ground truth values were not known.

## 3. Methods

#### 3.1. Tucker Decomposition

**Figure 6.**A visual representation of the Tucker decomposition (

**a**). The third order tensor $\mathcal{X}$ is decomposed into a core tensor $\mathcal{G}$ and the loading matrices

**A**,

**B**, and

**C**. The residuals are represented with $\mathcal{E}$, a tensor that is the same size as $\mathcal{X}$. The Tucker decomposition allows for different ranks along the tensor modes, which are denoted as P, Q, and R. The imputation process of EM Tucker is described in (

**b**).

**A**,

**B,**and

**C**and the core tensor $\mathcal{G}$ in an iterative fashion, as shown in Equations (5)–(8).

**B**and

**C**in order to form “factor landscapes”, which are then scaled by scores in

**A**to reconstruct the original data as accurately as possible. Because of this, the spatial loading vectors in

**B**and

**C**can interact or “cross-talk”. The interaction pattern of these loading vectors is encoded in the core tensor $\mathcal{G}$. An illustrative overview of the EM Tucker decomposition as well as a flowchart describing the imputation process is provided in Figure 6.

#### 3.2. EM Tucker for Imputation of Missing Values

**A**,

**B**, and

**C**are typically initialized using higher order singular value decomposition (HOSVD) [30]. However, HOSVD cannot be used directly in the presence of missing data, as the loss cannot be determined; therefore, prior imputation or marginalization is required. In this study, the imputation approach is chosen, which entails replacing missing data with sensible values before the Tucker decomposition can be applied. One example of a comprehensive imputation approach includes the expectation maximization (EM) algorithm [56], which assumes normal distribution, equal variance, and independence of the residuals.

**1**denotes a tensor containing ones, ${\mathcal{M}}^{\left(s\right)}$ denotes the so-called interim model reconstruction at iteration s, and $\mathcal{Y}$ is a tensor containing elements ${y}_{ijk}$ such that

#### 3.3. Model Selection

#### 3.4. Metrics

#### 3.5. Reference Methods

## 4. Results

## 5. Discussion

**A**together with the abnormal residuals in $\mathcal{E}$ could be used to identify outlying behavior (see Figure 6a). However, it was not within the scope of this study to evaluate the methods towards their ability to detect anomalies or outliers. We refer the reader to other studies that highlight the potential of tensor decompositions for anomaly detection in satellite data [64].

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AIC | Akaike information criterion |

ALS | Alternating least squares |

AVHRR | Advanced very-high-resolution radiometer |

BIC | Bayesian information criterion |

EM | Expectation maximization |

HOSVD | Higher order singular value decomposition |

KNN | K-nearest neighbors |

MAR | Missing at random |

MCAR | Missing completely at random |

MODIS | Moderate resolution imaging spectroradiometer |

NDVI | Normalized difference vegetation index |

PCA | Principal component analysis |

RMSE | Root mean square error |

RRMSE | Relative root mean square error |

SI | Single imputation |

SSIM | Structural similarity index |

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**Figure 1.**Seasonal NDVI differences on the Iberian Peninsula in 2008: 23 January (

**a**), 2 April (

**b**), 18 June (

**c**), 24 August (

**d**), and 15 November (

**e**). The smaller black rectangle near the middle of the peninsula shows study region 1, and the larger black rectangle in the south shows study region 2. The last subfigure (

**f**) depicts a mask, which was used to separate ocean from land, with white representing land, and black representing ocean.

**Figure 2.**Enlarged study regions as indicated by the black rectangles in Figure 1 referring to 23 January. (

**a**) shows study region 1, and (

**b**) shows study region 2. Note that the latter is three times larger than the former. Study region 2 contains natural missing data (white), unlike study region 1.

**Figure 3.**A single time frame from study region 1 corresponding to 23 January is shown in (

**a**). This particular day was selected because it contained no missing data. The chosen frame was replicated 30 times (

**b**) to form a 30 × 30 × 30 data set (SIM1). Subsequently, missing data were added artificially.

**Figure 4.**Selected time frames in the SPAIN1 subsection from the Iberian data set (study region 1), which contained no missing data. The important thing to notice here is that the frames change over time.

**Figure 5.**Three different levels of missing data for one frame in SIM1 are displayed as examples. Upper row indicates data missing completely at random (MCAR). Lower row indicates data missing at random (MAR).

**Figure 7.**Tucker model reconstructions for a single time frame shown for different ranks in the spatial modes (P, Q). The tensor that was decomposed here consisted of 30 identical 30 × 30 frames (SIM1). The reconstruction becomes more accurate as the ranks in both spatial modes are increased.

**Figure 8.**RRMSE for different models using the homogeneous tensor, SIM1, with artificially added MCAR (

**a**) and MAR (

**b**) elements. RRMSE values were calculated at positions indicated by the light grey vertical lines.

**Figure 9.**RRMSE for different models using the noisy tensor, SIM2, with artificially added MCAR (

**a**) and MAR (

**b**) elements.

**Figure 10.**RRMSE for different models applied to SPAIN1. The tensor contained artificially added MCAR (

**a**) and MAR (

**b**) elements.

**Figure 11.**Average correlation between the ground truth and the gap-filled data for SPAIN1 based on each of the methods at different levels of artificially added missing data, both for MCAR (

**a**) and MAR (

**b**). The correlation for the mean imputation method was always zero for all levels of missing data (the black line).

**Figure 12.**Structural similarity indices (SSIM) for SPAIN1 for all gap-filling methods at different levels of artificially added missing data, both for MCAR (

**a**) and MAR (

**b**).

**Figure 13.**Selected time frames from SPAIN2 (study region 2) are shown across rows (Sierra Nevada mountains appear in light blue (NDVI = 0) in winter in the low-center cell). The first column shows the SPAIN2 frames prior to imputation, where missing values are represented in white. The different imputation methods are shown across the remaining columns. Imputing the missing values, the SPAIN2 tensor was divided into nine equally large 30 × 30 × 66 sub-tensors, and the imputation was conducted on each sub-tensor individually. The blue grid, drawn on top of the each of the frames, shows this division.

**Table 1.**An overview of all data sets used in this study. The resulting data sets will be referred to by the corresponding aliases.

Alias | Description | Dimension |
---|---|---|

SIM1 | Constructed by repeating a single time frame from study region 1 with no missing data. Missing data were added artificially. Used for model evaluation. | 30 × 30 × 30 |

SIM2 | Constructed by adding noise to SIM1. Missing data were added artificially. Used for model evaluation. | 30 × 30 × 30 |

SPAIN1 | All time frames from study region 1 with no missing data. Missing data were added artificially. Used for model evaluation. | 30 × 30 × 54 |

SPAIN2 | Study region 2 with natural missing data. No ground truth data available. Used to demonstrate the performance of the models visually. | 90 × 90 × 66 |

Alias | Description | Software |
---|---|---|

Single mean imputation | Tensor mean imputed for missing values | No external code used |

Single imputation Tucker (SI Tucker) | Tensor mean was imputed for missing values prior to decomposition | “tucker” function, N-Way Toolbox, Matlab [54] |

Hybrid method | Running-window temporal imputation. Remaining missing data then imputed with KNN | “knnimpute” function, Bioinformatics toolbox, Matlab [63] |

EM PCA | Column mean was imputed prior to iterative PCA decomposition | “imputeEM” function, mvdlab package, R [55] |

EM Tucker | A combination of row and column mean was imputed prior to iterative decomposition | “tucker” function, N-Way Toolbox, Matlab [54] |

Method | Total Computation Time [s] |
---|---|

Simple mean imputation | 0.03 |

Single imputation Tucker | 5.11 |

Hybrid method | 1.26 |

EM PCA | 6.06 |

EM Tucker | 363.94 |

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**MDPI and ACS Style**

Þórðarson, A.F.; Baum, A.; García, M.; Vicente-Serrano, S.M.; Stockmarr, A.
Gap-Filling of NDVI Satellite Data Using Tucker Decomposition: Exploiting Spatio-Temporal Patterns. *Remote Sens.* **2021**, *13*, 4007.
https://doi.org/10.3390/rs13194007

**AMA Style**

Þórðarson AF, Baum A, García M, Vicente-Serrano SM, Stockmarr A.
Gap-Filling of NDVI Satellite Data Using Tucker Decomposition: Exploiting Spatio-Temporal Patterns. *Remote Sensing*. 2021; 13(19):4007.
https://doi.org/10.3390/rs13194007

**Chicago/Turabian Style**

Þórðarson, Andri Freyr, Andreas Baum, Mónica García, Sergio M. Vicente-Serrano, and Anders Stockmarr.
2021. "Gap-Filling of NDVI Satellite Data Using Tucker Decomposition: Exploiting Spatio-Temporal Patterns" *Remote Sensing* 13, no. 19: 4007.
https://doi.org/10.3390/rs13194007