## 1. Introduction

## 2. Methods

^{100}random permutations with the same total edge length, and plotted this on the theoretical distribution of macrostates for a landscape with that dimensionality, number of patch types and proportion of each patch type. Finally, I fit linear models to predict the mean and standard deviation of the distributions of macrostates (total edge length) of a landscape as a function of the dimensionality of the landscape, and the Fragstats metrics patch richness (number of cover types) and SHDI (the Shannon diversity of the landscape; [14]).

## 3. Results

#### 3.1. Comparing Fully Permuted and Row–Column Permuted Distributions of Total Edge Length

#### 3.2. The Vector-Randomized Distribution of Total Edge Length Is Normally Distributed

#### 3.3. Relationship between lnW, pln(p) and lnp

^{100}permutations of the 256 cell lattice, using the parametric normal probability density function, and confirmed that the slope is constant at 1, and only the intercept changes (Figure 5). Therefore, the linear relationship between lnp and lnW gives us a means to compute the relative Boltzmann entropy for each test landscape. Specifically, the ratios of the lnp values for the different test landscapes provide a measure of the relative difference in the Boltzmann configurational entropies.

#### 3.4. Configurational Entropy of the 12 Test Landscapes

#### 3.5. Shape of the Entropy–Total Edge Length Curve and Position of the Test Landscapes Along It

^{100}permutations of the 256 cell landscape lattice. This follows from the observation in [11] that a perfect checkerboard has only one microstate (if an odd number of rows or columns) or two microstates (if an even number of rows and columns) and so has the lowest possible entropy.

#### 3.6. Boundary Limits of the Normal Approximation for Permuted Total Edge Length

#### 3.7. Normal Distributions of the Permuted Total Edge Length of Real Landscapes

#### 3.8. Comparison of Entropy Curves of Real Landscapes

#### 3.9. Predicting the Parametric Normal Distribution of Sample Landscapes

#### 3.10. Predicting the Value of Configurational Entropy Based on Landscape Extent and Pattern

## 4. Discussion

^{100}permutations of the lattice) of configurational entropy, as well as the index of the proportion of maximum possible entropy described above.

#### Next Steps

## 5. Conclusions

## Supplementary Materials

## Conflicts of Interest

## References

- Vranken, I.; Baudry, J.; Aubinet, M.; Visser, M.; Bogaert, J. A review on the use of entropy in landscape ecology: Heterogeneity, unpredictability, scale dependence and their links with thermodynamics. Landsc. Ecol.
**2015**, 30, 51–65. [Google Scholar] [CrossRef][Green Version] - Cushman, S.A. Thermodynamics in landscape ecology: The importance of integrating measurement and modeling of landscape entropy. Landsc. Ecol.
**2015**, 30, 7–10. [Google Scholar] [CrossRef] - Turner, M.G. Landscape Ecology: The Effect of Pattern on Process. Annu. Rev. Ecol. Syst.
**1989**, 20, 171–197. [Google Scholar] [CrossRef] - Forman, R.T.T.; Godron, M. Heterogeneity and typology. In Landscape Ecology; Forman, R.T.T., Godron, M., Eds.; Wiley: New York, NY, USA, 1986; pp. 463–493. [Google Scholar]
- O’Neill, R.V.; De Angelis, D.L.; Waide, J.B.; Allen, T.F.H. A Hierarchical Concept of Ecosystems; Princeton University Press: Princeton, NJ, USA, 1986. [Google Scholar]
- Naveh, Z. Biocybernetic and thermodynamic perspectives of landscape functions and land use patterns. Landsc. Ecol.
**1987**, 1, 75–83. [Google Scholar] [CrossRef] - O’neill, R.V.; Johnson, A.R.; King, A.W. A hierarchical framework for the analysis of scale. Landsc. Ecol.
**1989**, 3, 193–205. [Google Scholar] [CrossRef] - Wu, J.G.; Loucks, O.L. From blanace of nature to hierarchical patch dynamics: A paradigm shift in ecology. Q. Rev. Biol.
**1995**, 70, 439–466. [Google Scholar] [CrossRef] - Zhang, H.; Wu, J.G. A statistical thermodynamic model of the organizational order of vegetation. Ecol. Model.
**2002**, 153, 69–80. [Google Scholar] [CrossRef] - Zurlini, G.; Petrosillo, I.; Jones, B.; Zaccarelli, N. Highlighting order and disorder in social-ecological landscapes to foster adaptive capacity and sustainability. Landsc. Ecol.
**2013**, 28, 1161–1173. [Google Scholar] [CrossRef] - Cushman, S.A. Calculating the configurational entropy of a landscape mosaic. Landsc. Ecol.
**2016**, 31, 481–489. [Google Scholar] [CrossRef] - Gardner, R.H. RULE: A program for the generation of random maps and the analysis of spatial patterns. In Landscape Ecological Analysis: Issues and Applications; Klopatek, J.M., Gardner, R.H., Eds.; Springer: New York, NY, USA, 1999; pp. 280–303. [Google Scholar]
- Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - McGarigal, K.; Cushman, S.A.; Ene, E. FRAGSTATS v4: Spatial Pattern Analysis Program for Categorical and Continuous Maps; University of Massachusetts: Amherst, MA, USA, 2012. [Google Scholar]
- Neel, M.C.; Mcgarigal, K.; Cushman, S. Behavior of classlevel landscape metrics across gradients of class aggregation and area. Landsc. Ecol.
**2004**, 19, 435–445. [Google Scholar] [CrossRef] - LANDFIRE. Existing Vegetation Type Layer, LANDFIRE 1.1.0, U.S. Department of the Interior, Geological Survey. 2008. Available online: http://landfire.cr.usgs.gov/viewer/ (accessed on 28 October 2010).
- Cushman, S.A.; McGarigal, K.; Neel, M.C. Parsimony in landscape metrics: Strength, universality and consistency. Landsc. Ecol.
**2008**, 8, 691–703. [Google Scholar] [CrossRef] - Wiens, J. Spatial scaling in ecology. Funct. Ecol.
**1989**, 3, 385–397. [Google Scholar] [CrossRef] - Levin, S.A. The problem of pattern and scale in ecology: The Robert H. MacArthur Award Lecture. Ecology
**1992**, 73, 1943–1967. [Google Scholar] [CrossRef] - McGarigal, K.; Wan, H.I.; Zeller, K.A.; Timm, B.C.; Cushman, S.A. Multi-scale habitat selection modeling: A review and outlook. Landsc. Ecol.
**2016**, 31, 1161–1175. [Google Scholar] [CrossRef] - Gao, P.; Zhang, H.; Li, Z. A hierarchy-based solution to calculate the configurational entropy of landscape gradients. Landsc. Ecol.
**2017**, 32, 1–14. [Google Scholar] [CrossRef] - McGarigal, K.; Cushman, S.A. The gradient concept of landscape structure. In Issues and Perspectives in Landscape Ecology; Wiens, J., Moss, M., Eds.; Cambridge University Press: Cambridge, UK, 2005; pp. 112–119. [Google Scholar]
- Cushman, S.A.; Gutzweiler, K.; Evans, S.; Mcgarigal, K. The Gradient Paradigm: A Conceptual and Analytical Framework for Landscape Ecology. In Spatial Complexity Informatics & Wildlife Conservation; Springer: Berlin, Germany, 2010; pp. 83–108. [Google Scholar]
- Swanson, F.J.; Kratz, T.K.; Caine, N.; Woodmansee, R.G. Landform effects on ecosystem patterns and processes. BioScience
**1988**, 38, 92–98. [Google Scholar] [CrossRef] - Hansen, A.J.; Neilson, R.P.; Dale, V.H.; Flather, C.H.; Iverson, L.R.; Currie, D.J.; Shafer, S.; Cook, R.; Bartlein, P.J. Global change in forests: Responses of species, communities and biomes. BioScience
**2001**, 51, 765–779. [Google Scholar] [CrossRef] - Gleason, H.A. The individualistic concept of the plant association. Am. Midl. Nat.
**1939**, 21, 92–110. [Google Scholar] [CrossRef] - Curtis, J.T.; McIntosh, R.P. An upland forest continuum I the prairie-forest border region of Wisconsin. Ecology
**1951**, 32, 476–496. [Google Scholar] [CrossRef]

**Figure 1.**The 12 test landscapes. (

**a**) random; (

**b**) H1; (

**c**) H2; (

**d**) H3; (

**e**) H4; (

**f**) H5; (

**g**) H6; (

**h**) H7; (

**i**) H8; (

**j**) H9; (

**k**) H10; (

**l**) checker board.

**Figure 2.**Location and pattern of the 25 sample landscapes in the San Francisco Peaks region of Northern Arizona. The extent of the 20 × 20 sample landscape is shown in semitransparent background overlaid on a hillshade of topography. The individual sample landscapes are shown with random colormap corresponding to each class in the cover type-seral stage classification.

**Figure 3.**Match between the vector-permuted distribution of total edge length and the predictions of the normal probability function.

**Figure 4.**Relationship between configurational Boltzmann entropy (lnW) calculated from the vector-permuted distribution of all 12 test landscapes.

**Figure 5.**Verification that the linear relationship with slope 1 is consistent for a large number (10 × 10

^{100}) of computed configurations. W** is the number of microstates with a given total edge length expected from 10 × 10

^{100}permutations of the lattice, using the normal probability density function.

**Figure 6.**Location of the 12 test landscapes along the parabolic curve of entropy as a function of total edge length in the landscape.

**Figure 7.**Plot of the Shapiro–Wilk W statistic for the distribution of total edge length for a sample of 100,000 permutations of 25 test landscapes of varying dimensionality (20 × 20, 30 × 30, 40 × 40, 50 × 50) and minority class proportion (2%, 4%, 6% 8%, 10%). The two black contours show critical values of the test statistic at alpha 0.05 for sample sizes of 1000 (lower) and 1500 (upper).

**Figure 8.**Normal distribution of the permuted total edge length for a sample landscape. (

**a**) Sample landscape; (

**b**) Overlay of the normal probability function predicted frequency distribution over the observed frequency distribution of total edge length resulting from 100,000 spatial randomizations of this landscape.

**Figure 9.**The parabolic entropy curve for a sample landscape. The actual entropy of the sample landscape is indicated with the arrow on the far left side of the curve, indicating that this landscape has very low entropy in comparison to the distribution of microstates across all possible configurational macrostates. W** is the number of microstates (configurations of the lattice) with a given macrostate (total edge length) expected in 10 × 10

^{100}permutations of the lattice.

**Figure 10.**Ordering of the five replicate 100 × 100 dimensionality landscapes from low to high entropy at each of the five levels of dimensionality. SHDI is the Shannon Diversity index value for the landscape as calculated by FRAGSTATS. W** is the number of microstates (configurations of the lattice) with a given macrostate (total edge length) expected in 10 × 10

^{100}permutations of the lattice.

**Figure 11.**Scatter plots of the relationships between landscape dimensionality, patch richness and Shannon diversity and the mean (

**a**–

**c**) and standard devation (

**d**–

**f**) of the normal distribution of permuted total edge lengths across the 25 sample landscapes in the San Francisco Peaks region.

**Table 1.**Investigation of the boundary limits of landscape extent and proportionality for the normal approximation. Five levels of landscape dimensionality and five levels of landscape proportion of a minority class in a two-class landscape were investigated.

Rows and Columns | Number Pixels | 2% Minority | 4% Minority | 6% Minority | 8% Minority | 10% Minority |
---|---|---|---|---|---|---|

20 | 400 | 8 | 16 | 24 | 32 | 40 |

30 | 900 | 18 | 36 | 54 | 72 | 90 |

40 | 1600 | 32 | 64 | 96 | 128 | 160 |

50 | 2500 | 50 | 100 | 150 | 200 | 250 |

60 | 3600 | 72 | 144 | 216 | 288 | 360 |

**Table 2.**Calculating the relative Boltzmann entropy (lnp) for the 12 test landscapes. H1—test landscape with H parameter = 0.1; H2—test landscape with H parameter = 0.2; H3—test landscape with H parameter = 0.3; H4—test landscape with H parameter = 0.4; H5—test landscape with H parameter = 0.5; H6—test landscape with H parameter = 0.6; H7—test landscape with H parameter = 0.7; H8—test landscape with H parameter = 0.8; H9—test landscape with H parameter = 0.9; H10—test landscape with H parameter = 1.0; R—simple random test landscape; C—checker-board test landscape.

lnp | |
---|---|

H10 | −167.1559064 |

H9 | −152.5994528 |

H8 | −147.8977559 |

H7 | −137.2196798 |

H6 | −122.675764 |

H5 | −111.6424486 |

H4 | −96.09550409 |

H3 | −77.238565 |

H2 | −66.53959246 |

H1 | −47.06930617 |

R | −4.251511739 |

C | −250.09384 |

**Table 3.**Comparing relative configurational Boltzmann entropy (lnp) among the 12 test landscapes. Values are column test landscape entropy as a proportion of row test landscape entropy.

H10 | H9 | H8 | H7 | H6 | H5 | H4 | H3 | H2 | H1 | R | C | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

H10 | 1.00 | 0.91 | 0.88 | 0.82 | 0.73 | 0.67 | 0.57 | 0.46 | 0.40 | 0.28 | 0.03 | 1.50 |

H9 | 1.10 | 1.00 | 0.97 | 0.90 | 0.80 | 0.73 | 0.63 | 0.51 | 0.44 | 0.31 | 0.03 | 1.64 |

H8 | 1.13 | 1.03 | 1.00 | 0.93 | 0.83 | 0.75 | 0.65 | 0.52 | 0.45 | 0.32 | 0.03 | 1.69 |

H7 | 1.22 | 1.11 | 1.08 | 1.00 | 0.89 | 0.81 | 0.70 | 0.56 | 0.48 | 0.34 | 0.03 | 1.82 |

H6 | 1.36 | 1.24 | 1.21 | 1.12 | 1.00 | 0.91 | 0.78 | 0.63 | 0.54 | 0.38 | 0.03 | 2.04 |

H5 | 1.50 | 1.37 | 1.32 | 1.23 | 1.10 | 1.00 | 0.86 | 0.69 | 0.60 | 0.42 | 0.04 | 2.24 |

H4 | 1.74 | 1.59 | 1.54 | 1.43 | 1.28 | 1.16 | 1.00 | 0.80 | 0.69 | 0.49 | 0.04 | 2.60 |

H3 | 2.16 | 1.98 | 1.91 | 1.78 | 1.59 | 1.45 | 1.24 | 1.00 | 0.86 | 0.61 | 0.06 | 3.24 |

H2 | 2.51 | 2.29 | 2.22 | 2.06 | 1.84 | 1.68 | 1.44 | 1.16 | 1.00 | 0.71 | 0.06 | 3.76 |

H1 | 3.55 | 3.24 | 3.14 | 2.92 | 2.61 | 2.37 | 2.04 | 1.64 | 1.41 | 1.00 | 0.09 | 5.31 |

R | 39.32 | 35.89 | 34.79 | 32.28 | 28.85 | 26.26 | 22.60 | 18.17 | 15.65 | 11.07 | 1.00 | 58.82 |

C | 0.67 | 0.61 | 0.59 | 0.55 | 0.49 | 0.45 | 0.38 | 0.31 | 0.27 | 0.19 | 0.02 | 1.00 |

**Table 4.**Linear model predicting the mean total edge length in the permuted distribution across the 25 sample landscapes as a function of landscape dimensionality (number of rows and columns), patch richness (number of patch types, PR) and Shannon diversity (SHDI). lm (formula = mean ~ dimension + SHDI + PR, data = data).

Estimate | Std. Error | t | Pr ( > |t|) | |
---|---|---|---|---|

(Intercept) | −3.37 × 10^{3} | 4.67 × 10^{2} | −7.205 | 4.22 × 10^{−7} |

dimension | 1.66 | 4.63 × 10^{−2} | 35.775 | <2.00 × 10^{−16} |

SHDI | 2.50 × 10^{3} | 3.59 × 10^{−2} | 6.945 | 7.35 × 10^{−7} |

PR | −7.05 × 10 | 1.90 × 10 | −3.712 | 0.00129 |

Adjusted R-squared: 0.9914 | F-statistic: 921 | p-value: < 2.2 × 10^{−16} |

**Table 5.**Linear model predicting the standard deviation of total edge length in the permuted distribution across the 25 sample landscapes as a function of landscape dimensionality (number of rows and columns), patch richness (number of patch types, PR) and Shannon diversity (SHDI). lm (formula = std ~ dim

^{0.5}+ ln(PR) + SHDI, data = data).

Estimate | Std. Error | t | Pr ( > |t|) | |
---|---|---|---|---|

(Intercept) | 4.93602 | 5.17294 | 0.954 | 0.351 |

dim^{0.5} | 0.48459 | 0.03947 | 12.278 | 4.77 × 10^{−11} |

ln(PR) | −0.44567 | 2.98704 | −0.149 | 0.883 |

SHDI | −2.56384 | 2.33961 | −1.096 | 0.286 |

Adjusted R-squared: 0.944 | F-statistic: 120 | p-value: 2.628 × 10^{−13} |

**Table 6.**Model-averaged coefficients from a Generalized Linear Model (GLM) model predicting configurational entropy (lnW**) as a function of landscape dimension (20 × 20, 40 × 40, 60 × 60, 80 × 80, 100 × 100) and several landscape metrics. AI—Aggregation Index, SHEI—Shannon Landscape Evenness Index, ED—Edge Density, PR—Patch Richness. W** is the number of microstates (configurations of the lattice) with a given macrostate (total edge length) expected in 10 × 10

^{100}permutations of the lattice.

Estimate | Std. Error | z | Pr ( > |z|) | AIC Variable Importance | |
---|---|---|---|---|---|

(Intercept) | 2.08 × 10^{4} | 1.99 × 10^{4} | 1.03 | 0.30307 | |

dimension | −1.90 | 2.05 × 10^{−1} | 8.758 | 2.00 × 10^{−16} | 1.0 |

ED | 1.89 | 6.56 × 10^{−1} | 2.711 | 0.00672 | 0.49 |

SHEI | −3.05 × 10^{4} | 1.08 × 10^{4} | 2.668 | 0.00764 | 0.89 |

AI | −3.46 × 10^{2} | 1.24 × 10^{2} | 2.633 | 0.00846 | 0.40 |

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).