class: center, middle, inverse, title-slide # From empirical to theoretical descriptors
of landscapes complexity ### Jakub Nowosad
https://nowosad.github.io
### the 10th IALE World Congress, July 1-5, 2019, Milan, Italy --- class: center, middle <!-- Flash presentation (5 min.). --> # Landscapes complexity <!-- what's that? --> <!-- what it is for? --> ## There is a relationship between an area's pattern composition and configuration and ecosystem characteristics, such as vegetation diversity, animal distributions, and water quality within this area. #### (*Hunsaker i Levine, 1995; Fahrig i Nuttle, 2005; Klingbeil i Willig, 2009; Holzschuh et al., 2010; Fahrig et al., 2011; Carrara et al., 2015; Arroyo-Rodŕıguez et al. 2016; Duflot et al., 2017*, many others...). <!-- our idea that patterns should be described from theoretical perspective and with minimal number of metrics. More metrics are needed for specific applications, but for general description of patterns two are sufficient. --> --- # Empirical descriptors - Nowosad J., Stepinski T. (2018) Global inventory of landscape patterns and dimensions of landscape spatial configuration. Ecological Indicators, 89:159-167; DOI: 10.1016/j.ecolind.2018.02.007 <img src="figs/world_lc.png" width="90%" style="display: block; margin: auto;" /> ## ESA CCI Land cover - 300 m --- # Empirical descriptors - **How to describe landscapes?** <table> <thead> <tr> <th style="text-align:left;"> Type </th> <th style="text-align:left;"> Landscape-level metrics </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> Shape </td> <td style="text-align:left;"> PAFRAG – Parameter-area fractal dimension; CONTIG AM – Contiguity index area-weighted average; CONTIG RA – Contiguity index range </td> </tr> <tr> <td style="text-align:left;"> Aggregation </td> <td style="text-align:left;"> AI – Aggregation index; CONTAG – Contagion; IJI – Interspersion Juxtaposition index; PLATJ – Percentage of like adjacencies, PD – Patch density; DIVISION – Landscape division index; LPI – largest patch index </td> </tr> <tr> <td style="text-align:left;"> Connectivity </td> <td style="text-align:left;"> COHESION – Patch cohesion index </td> </tr> <tr> <td style="text-align:left;"> Diversity </td> <td style="text-align:left;"> SHDI – Shannon's diversity index; SIDI – Simpson's diversity index; MSIDI – Modified Simpson's's diversity index; SHEI – Shannon's evenness index; SIEI – Simpson's evenness index; MSIEI – Modified Simpson's evenness index </td> </tr> </tbody> </table> --- # Empirical descriptors .pull-left[ <img src="figs/pap2_02.png" width="80%" style="display: block; margin: auto;" /> **Major loadings of the first two rotated principal components.** Positive loadings - green lines; negative loadings - red lines; magnitude is indicated by color intensity and thickness of the line ] -- .pull-right[ <img src="figs/pap2_03.png" width="80%" style="display: block; margin: auto;" /> Examples of dependence of landscape spatial configuration on two parameters, **complexity C** and **aggregation A** ] <!-- landscape metrics --> <!-- problems with correlations, etc --> --- # Empirical descriptors **Issues with this approach:** .lc[ - Highly correlated landscape metrics are used - Each new dataset requires recalculation of both, landscape metrics and principal components analysis (PCA) - PCA results interpretation is not straightforward <img src="figs/pap2_04_legend.png" width="88%" style="display: block; margin: auto;" /> ] .rc[ <img src="figs/pap2_04_map.png" width="96%" style="display: block; margin: auto;" /> ] --- # Theoretical descriptors - Different patterns generate different **co-occurrence matrices** <!-- - **The co-occurrence matrix representation** is not only compact --> - This representation is compact and **allows to calculate several metrics based on the information theory applied to bivariate random variable (x,y)**, where x is a category of a focus cell and y is category of a cell adjacent to it .pull-left[ <img src="index_files/figure-html/unnamed-chunk-8-1.png" width="83%" style="display: block; margin: auto;" /> ] .pull-right[ Co-occurrence matrices ``` ## [[1]] ## 1 2 ## 1 1142 106 ## 2 106 2126 ## ## [[2]] ## 1 2 ## 1 546 304 ## 2 304 2326 ## ## [[3]] ## 1 2 3 4 ## 1 712 47 144 62 ## 2 47 684 183 108 ## 3 144 183 370 35 ## 4 62 108 35 556 ``` ] --- # Theoretical descriptors - **Marginal entropy [H(x)]** - Diversity (thematic complexity, composition) of spatial categories <img src="index_files/figure-html/unnamed-chunk-10-1.png" style="display: block; margin: auto;" /> --- # Theoretical descriptors - **Conditional entropy [H(y|x)]** - Configurational complexity (geometric intricacy) of a spatial pattern - The high value of **conditional entropy** shows that cells of one category are adjacent to cells of many different categories <img src="index_files/figure-html/unnamed-chunk-11-1.png" style="display: block; margin: auto;" /> --- # Theoretical descriptors - **Joint entropy [H(x,y)]** - An overall spatio-thematic complexity metric <!-- It represents the uncertainty in determining a category of the focus cell and the category of the adjacent cell. --> <!-- In other words, it measures diversity of values in a co-occurrence matrix -- the smaller the diversity, the larger the value of **joint entropy**. --> <img src="index_files/figure-html/unnamed-chunk-12-1.png" style="display: block; margin: auto;" /> --- # Theoretical descriptors - **Mutual information [I(y,x)]** - Quantifies the information that one random variable (x) provides about another random variable (y) <!-- It tells how much easier is to predict a category of an adjacent cell if the category of the focus cell is known. --> - It disambiguates landscape pattern types characterized by the same value of overall complexity <img src="index_files/figure-html/unnamed-chunk-13-1.png" style="display: block; margin: auto;" /> --- # Theoretical descriptors - **Relative mutual information [U]** - Due to the spatial autocorrelation, the value of mutual information tends to grow with a diversity of the landscape (**marginal entropy**) <!-- To adjust this tendency, it is possible to calculate **relative mutual information** by dividing the **mutual information** by the **marginal entropy**. --> - **U** is calculated by dividing the **mutual information** by the **marginal entropy** (0-1 range) <img src="index_files/figure-html/unnamed-chunk-14-1.png" style="display: block; margin: auto;" /> --- # Theoretical descriptors - Only two of the five IT metrics are relatively independent (weakly correlated) - **2D parametrization of landscape configurations based on two weakly correlated IT metrics groups similar patterns into distinct regions of the parameters space** - This provides the basis for classification of landscapes into landscape pattern configuration types (LPCTs) <img src="figs/pap5_02c.png" width="100%" style="display: block; margin: auto;" /> --- class: left, middle, clear .pull-left[ ## Summary: - **Information theory provides a consistent framework for the analysis of landscape patterns** - **Article:** Nowosad J., Stepinski T. (2019) Information Theory as a consistent framework for quantification and classification of landscape patterns, Landscape Ecology - **Blog post:** ["Information theory provides a consistent framework for the analysis of spatial patterns"](https://nowosad.github.io/post/ent-bp1/)<!-- explains basic concepts behind information theory-based metrics allowing for numerical description of spatial patterns.--> - **Software:** H(x), H(y|x), H(x,y), and I(y,x) are implemented as the `lsm_l_ent()`, `lsm_l_condent()`, `lsm_l_joinent()`, and `lsm_l_mutinf` functions in the R package **landscapemetrics** (Hesselbarth et al., 2019). ] .pull-right[ ## Me: Twitter:
jakub_nowosad Email: nowosad.jakub@gmail.com ## Resources: https://nowosad.github.io http://sil.uc.edu ] <br> <br> .footnote[ **Slides:** http://bit.ly/iale19 ]