# On Integrating Size and Shape Distributions into a Spatio-Temporal Information Entropy Framework

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## Abstract

**:**

## 1. Introduction

**(i)**defining patches rules,

**(ii)**extracting the multiway information crossing spatio-temporal patch characteristics and C, and,

**(iii)**quantifying and mapping the spatio-temporal information from entropy decomposition and related methods. This framework, termed the patch size and shape entropy (PsishENT) framework, is based on the Shannon entropy and existing spatio-temporal approaches of the Shannon entropy itself [6,8,15] on the rendered information in (ii) (Section 3). As part of (iii) a multiway correspondence analysis can be used [21,22] (Section 5) which is related to the concept of mutual information reminded in Section 2. This multiway analysis provides a decomposition for which, each part has an interpretation similar to a product of the spatial, temporal and categorical distributions, therefore providing after a transformation a simple entropy decomposition (see Section 2 and Section 5). These three major steps of the framework are detailed within their potential sub-steps in the next few sections before summarising the approach in Section 7.

## 2. Using Shannon’s Multivariate Decomposition Entropy

#### With Spatial and Temporal Supports

## 3. Taking into Account Spatio-Temporal Relative Proximities

#### With a Symmetric or Non-Symmetric Spatio-Temporal Approach

## 4. Constructing the Spatial and Temporal Patches Characteristics

## 5. Using Multiway Correspondence Analysis

^{th}rank-one tensor, which contributes at ${\sigma}_{r}^{2}/(1+{\chi}^{2}/N)\phantom{\rule{0.166667em}{0ex}}\%$ of the whole decomposition or ${\sigma}_{r}^{2}/({\chi}^{2}/N)\phantom{\rule{0.166667em}{0ex}}\%$ to the departure from complete independence used in 2-way correspondence analysis [28] and multiway correspondence analysis (FCAk) [21]. Therefore, $CT{R}_{r}$ quantifies the role of each combination $stc$ within the rank-one tensor and is expressing its spatio-temporal structuring in interaction with C. Ratios such as, $H{R}_{CT{R}_{r}}^{\left(S\right)}=H\left({p}_{S}{v}_{{S}_{r}}^{2}\right)/H\left(CT{R}_{r}\right)$ or $H{R}_{CT{R}_{r}}^{\ast \left(S\right)}=(H\left({p}_{S}{v}_{{T}_{r}}^{2}\right)+H\left({p}_{S}{v}_{{C}_{r}}^{2}\right))/H\left(CT{R}_{r}\right)$ highlight the entropic contribution to the relative importance from S in the information structuring extracted from the rank-one tensor. Linked the $CT{R}_{r}$ is the rank-one tensor itself for which a non-negative approximation would allow a similar entropy decomposition.

## 6. Cartographic Representations of the Quantified Information

## 7. The PsishENT Operational Framework

**(i)**, after possible transformations of the initial data (not shown here) the definitions of spatial and temporal patches are made, based on rules (i.e., topology, fuzziness etc.), which generate categorical variables $Si$ and/or $Sh$, $Ti$ and/or $Th$ which may result in classes of sizes or shapes after aggregation rules (Section 4). In

**(ii)**, choosing the variables involved (dimensions of the multiway table) and the statistic to compute cell values in the multiway table, includes various choices, i.e., a positive value for each multiway indices, e.g., $C=c$, $Si=1$, $Ti=3$ (Section 3).

**(iii)**, a series of analyses based on entropy decomposition theorem (Section 2) and other methods (Section 5) that embed distribution decompositions that are related to for example criteria of independence, homogeneity, uniformity, can be performed to produce results in forms of summary table (e.g., break down of entropy), maps and curves (e.g., time series of a statistic based on an entropy), see Section 6 or from the equations listed in previous sections.

## 8. Illustrative Example of Land Cover Forecasts

## 9. Discussion & Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

2D | 2-dimensional Euclidean representation of the geographical space |

2D+1 | 2D footprint with a positive height |

CTR | Relative Contribution (in correspondence analysis) |

IPCC | International Panel on CLimate Change |

FCAk | Factorial Correspondence Analysis of a k-ways table |

PTAk | Principal Tensor Analysis of k-way array |

RCP | Representative Concentration Pathway |

LSM | Land Surface Model |

PsishENT | Patch size and shape Entropy |

NNTF | Non-Negative Tensor Factorisation |

## Appendix A. Plant Functional Types

pfts | Full Name |
---|---|

pft1 | bare ground |

pft2* | tropical broadleaf evergreen |

pft3* | tropical broadleaf raingreen |

pft4 | temperate needleleaf evergreen |

pft5 | temperate broadleaf evergreen |

pft6 | temperate broadleaf summergreen |

pft7 | boreal needleleaf evergreen |

pft8 | boreal broadleaf summergreen |

pft9 | boreal needleleaf summergreen |

pft10 | C3 grass |

pft11* | C4 grass |

pft12 | NonVascular moss and lichen |

pft13 | boreal broadleaf shrubs |

pft14 | C3 arctic grass |

pft15* | C3 agriculture |

pft16* | C4 agriculture |

## Appendix B. PsishENT Analysis Using Distribution of Patches

${\mathit{H}}^{\mathit{u}}(.)$ | Year 2020 | Year 2050 | Year 2100 |
---|---|---|---|

patches distribution | |||

$Si$ | 0.9609153 | 0.9768040 | 0.9409071 |

$C\mid Si$ | 0.7314626 | 0.6296681 | 0.7008305 |

C | 0.8994154 | 0.8859183 | 0.9554671 |

$Si\mid C$ | 0.7539514 | 0.6610335 | 0.6271249 |

$Si,C$ | 0.8342513 | 0.7851758 | 0.8083785 |

grid-cells distribution | |||

$Si$ | 0.7030593 | 0.7933917 | 0.7640653 |

$C\mid Si$ | 0.6548033 | 0.5613683 | 0.6314215 |

C | 0.8520292 | 0.8745148 | 0.9297879 |

$Si\mid C$ | 0.4600228 | 0.4075092 | 0.3963961 |

$Si,C$ | 0.6764207 | 0.6653087 | 0.6908424 |

**Table A3.**Margins of the multiway table $Si\times Ti\times C$ using the patches distribution and Signed CTRs (rounded %) for the rank-one tensor of the FC3 representing $44.3\%$ and $15.4\%$ of the variability (see entropy decomposition in Table A5).

Margins & Tensor $44.3\%$ | Tensor $15.4\%$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$Si$ | $Ti$ | C | $Si$ | $Ti$ | C | ||||||

1 | 16 | 1 | 4 | pft1 | 20 | 1 | 16 | 1 | 23 | pft1 | 73 |

2 | 9 | 2 | 4 | pft4 | 4 | 2 | 9 | 2 | 18 | pft4 | −4 |

$>2$ | 15 | $>2$ | 3 | pft5 | 2 | $>2$ | 15 | $>2$ | 11 | pft5 | −2 |

$>7$ | 19 | $>4$ | 7 | pft6 | 7 | $>7$ | 19 | $>4$ | 9 | pft6 | −6 |

$>25$ | 15 | $>7$ | 21 | pft7 | 5 | $>25$ | 15 | $>7$ | 7 | pft7 | −1 |

$>50$ | 25 | $>20$ | 11 | pft8 | 7 | $>50$ | 25 | $>20$ | −2 | pft8 | −4 |

$>100$ | 11 | $>30$ | 26 | pft9 | 3 | $>100$ | 11 | $>30$ | −21 | pft9 | −5 |

$>60$ | 24 | pft10 | 16 | $>60$ | −9 | pft10 | −1 | ||||

pft12 | 9 | pft12 | −2 | ||||||||

pft13 | 15 | pft13 | 0 | ||||||||

pft14 | 12 | pft14 | −2 |

**Table A4.**Margins of the multiway table $Si\times Ti\times C$ using the patches distribution and Signed CTRs (rounded %) for the rank-one tensor of the FC3 representing $8.7\%$ and $4.7\%$ of the variability (see entropy decomposition in Table A5).

Margins & Tensor $8.7\%$ | Tensor $4.7\%$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$Si$ | $Ti$ | C | $Si$ | $Ti$ | C | ||||||

1 | 6 | 1 | 4 | pft1 | 4 | 1 | 0 | 1 | 1 | pft1 | 4 |

2 | 5 | 2 | 4 | pft4 | −13 | 2 | 0 | 2 | 3 | pft4 | 15 |

$>2$ | 4 | $>2$ | 3 | pft5 | 4 | $>2$ | −3 | $>2$ | 1 | pft5 | 0 |

$>7$ | 12 | $>4$ | 7 | pft6 | −7 | $>7$ | 1 | $>4$ | 3 | pft6 | 9 |

$>25$ | −3 | $>7$ | 21 | pft7 | 2 | $>25$ | 50 | $>7$ | 8 | pft7 | 0 |

$>50$ | −2 | $>20$ | 11 | pft8 | 0 | $>50$ | −2 | $>20$ | −3 | pft8 | −26 |

$>100$ | −69 | $>30$ | 26 | pft9 | −9 | $>100$ | −44 | $>30$ | −62 | pft9 | 10 |

$>60$ | 24 | pft10 | 3 | $>60$ | 18 | pft10 | 1 | ||||

pft12 | 10 | pft12 | 0 | ||||||||

pft13 | −39 | pft13 | −35 | ||||||||

pft14 | 9 | pft14 | 0 |

**Table A5.**CTR-tensor entropy decomposition for the FCA3 of the multiway table $Si\times Ti\times C$: the four best rank-one tensors, Equation (24), representing altogether $73\%$ of variability.

${\mathit{H}}^{\mathit{u}}(.)$ | Tensor$44.3\%$ | Tensor$15.4\%$ | Tensor$8.7\%$ | Tensor$4.66\%$ |

$Si$ | 0.985 | 0.985 | 0.571 | 0.483 |

$Ti$ | 0.870 | 0.919 | 0.870 | 0.598 |

C | 0.920 | 0.471 | 0.806 | 0.683 |

$CTR-tensor$ | 0.923 | 0.771 | 0.755 | 0.594 |

repeat of Table 4 | ||||

${\mathit{H}}^{\mathit{u}}(.)$ | Tensor$40.9\%$ | Tensor$16.70\%$ | Tensor$9.54\%$ | Tensor$3.55\%$ |

$Si$ | 0.786 | 0.661 | 0.786 | 0.929 |

$Ti$ | 0.596 | 0.596 | 0.908 | 0.830 |

C | 0.894 | 0.830 | 0.452 | 0.356 |

$CTR-tensor$ | 0.765 | 0.703 | 0.701 | 0.683 |

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**Figure 4.**Spatial spread of dominant pfts in each grid cell for years 2020, 2050 and 2100 (list of pfts given in Figure 2 and in Appendix A).

**Figure 5.**Frequencies of the 7 classes of spatial patches over 846 inland grid cells for all pfts where a 1 patch is a grid-cell with fraction >15% (wider solid lines are smoother fit of the time series) in thinner lines.

**Figure 6.**Map of the ratios to conditional entropy ${H}^{u}(C\mid Si)$ of Table 1 from occurring local patch sizes (ranges: 2020 2%–77%, 2050 2%–80%, 2100 0%–92%).

**Figure 7.**Map of the ratios to conditional entropy ${H}^{u}(Si\mid C)$ of Table 1 from occurring local patches of C (ranges: 2020 1%–87%, 2050 0%–89%, 2100 1%–67%).

**Figure 8.**C-signed CTR-tensor spatial scores rebuilt for the dominant pft in June for years 2020, 2050 and 2100.

**Table 1.**Decomposition of the normalised Shannon entropy, Equation (8), for the spatial patch sizes classes $Si$ and the pft categories variable C (11 categories out of 14, see Appendix A) at 2020, 2050 and 2100. ($\frac{log\left(\right|S\left|\right)}{log\left(\right|S\left|\right)+log\left(\right|C\left|\right)}=0.4479736$ and $\frac{log\left(\right|C\left|\right)}{log\left(\right|S\left|\right)+log\left(\right|C\left|\right)}=0.5520264$)

${\mathit{H}}^{\mathit{u}}(.)$ | Year 2020 | Year 2050 | Year 2100 |
---|---|---|---|

$Si$ | 0.7030593 | 0.7933917 | 0.7640653 |

$C\mid Si$ | 0.6548033 | 0.5613683 | 0.6314215 |

C | 0.8520292 | 0.8745148 | 0.9297879 |

$Si\mid C$ | 0.4600228 | 0.4075092 | 0.3963961 |

$Si,C$ | 0.6764207 | 0.6653087 | 0.6908424 |

**Table 2.**Margins of the multiway table $Si\times Ti\times C$ and Signed CTRs (rounded %) for the rank-one tensor of the FC3 representing $40.9\%$ and $16.7\%$ of the variability (see entropy decomposition in Table 4).

Margins & Tensor $40.9\%$ | Tensor $16.7\%$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{S}\mathit{i}$ | $\mathit{T}\mathit{i}$ | C | $\mathit{S}\mathit{i}$ | $\mathit{T}\mathit{i}$ | C | ||||||

1 | 2 | 1 | 1 | pft1 | 10 | 1 | 3 | 1 | 1 | pft1 | 5 |

2 | 2 | 2 | 1 | pft4 | 6 | 2 | 3 | 2 | 1 | pft4 | 0 |

$>2$ | 5 | $>2$ | 1 | pft5 | 2 | $>2$ | 7 | $>2$ | 1 | pft5 | 10 |

$>7$ | 14 | $>4$ | 2 | pft6 | 4 | $>7$ | 35 | $>4$ | 2 | pft6 | 0 |

$>25$ | 12 | $>7$ | 8 | pft7 | 4 | $>25$ | 2 | $>7$ | 8 | pft7 | 2 |

$>50$ | 23 | $>20$ | 7 | pft8 | 4 | $>50$ | 2 | $>20$ | 7 | pft8 | 6 |

$>100$ | 42 | $>30$ | 21 | pft9 | 23 | $>100$ | −48 | $>30$ | 21 | pft9 | −21 |

$>60$ | 59 | pft10 | 13 | $>60$ | 59 | pft10 | 7 | ||||

pft12 | 5 | pft12 | 21 | ||||||||

pft13 | 22 | pft13 | −22 | ||||||||

pft14 | 7 | pft14 | 6 |

**Table 3.**Signed CTRs (rounded %) for the rank-one tensor of the FC3 of the table $Si\times Ti\times C$ representing $9.54\%$ and $3.55\%$ of the variability (see entropy decomposition in Table 4).

Tensor $9.54\%$ | Tensor $3.55\%$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{S}\mathit{i}$ | $\mathit{T}\mathit{i}$ | C | $\mathit{S}\mathit{i}$ | $\mathit{T}\mathit{i}$ | C | ||||||

1 | 2 | 1 | 18 | pft1 | 60 | 1 | −16 | 1 | 27 | pft1 | −81 |

2 | 2 | 2 | 15 | pft4 | 0 | 2 | −9 | 2 | 29 | pft4 | 1 |

$>2$ | 5 | $>2$ | 11 | pft5 | −1 | $>2$ | −26 | $>2$ | 14 | pft5 | 5 |

$>7$ | 14 | $>4$ | 9 | pft6 | 1 | $>7$ | −15 | $>4$ | 13 | pft6 | 3 |

$>25$ | 12 | $>7$ | 18 | pft7 | 3 | $>25$ | −5 | $>7$ | 9 | pft7 | 1 |

$>50$ | 23 | $>20$ | 6 | pft8 | 2 | $>50$ | 7 | $>20$ | 0 | pft8 | 1 |

$>100$ | 42 | $>30$ | 1 | pft9 | −29 | $>100$ | 22 | $>30$ | −2 | pft9 | 0 |

$>60$ | −24 | pft10 | 0 | $>60$ | −8 | pft10 | 3 | ||||

pft12 | 0 | pft12 | 3 | ||||||||

pft13 | −4 | pft13 | 0 | ||||||||

pft14 | 0 | pft14 | 2 |

**Table 4.**CTR-tensor entropy decomposition for the FCA3 of the multiway table $Si\times Ti\times C$: the four best rank-one tensors, Equation (24), representing altogether $70.69\%$ of variability

${\mathit{H}}^{\mathit{u}}(.)$ | Tensor $40.9\%$ | Tensor $16.70\%$ | Tensor $9.54\%$ | Tensor $3.55\%$ |
---|---|---|---|---|

$Si$ | 0.786 | 0.661 | 0.786 | 0.929 |

$Ti$ | 0.596 | 0.596 | 0.908 | 0.830 |

C | 0.894 | 0.830 | 0.452 | 0.356 |

$CTR-tensor$ | 0.765 | 0.703 | 0.701 | 0.683 |

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## Share and Cite

**MDPI and ACS Style**

Leibovici, D.G.; Claramunt, C.
On Integrating Size and Shape Distributions into a Spatio-Temporal Information Entropy Framework. *Entropy* **2019**, *21*, 1112.
https://doi.org/10.3390/e21111112

**AMA Style**

Leibovici DG, Claramunt C.
On Integrating Size and Shape Distributions into a Spatio-Temporal Information Entropy Framework. *Entropy*. 2019; 21(11):1112.
https://doi.org/10.3390/e21111112

**Chicago/Turabian Style**

Leibovici, Didier G., and Christophe Claramunt.
2019. "On Integrating Size and Shape Distributions into a Spatio-Temporal Information Entropy Framework" *Entropy* 21, no. 11: 1112.
https://doi.org/10.3390/e21111112