# Geographically Weighted Negative Binomial Regression Model Predicts Wildfire Occurrence in the Great Xing’an Mountains Better Than Negative Binomial Model

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. The Logical Framework of the Study

#### 2.3. Data Collection

#### 2.3.1. Response Variable

#### 2.3.2. Predictor Variables

^{2}), road density (km/km

^{2}), settlements proportion (%), GDP (Gross Domestic Product) (yuan/km

^{2}), and population density (person/km

^{2}). Table 1 lists the descriptive statistics of the response and predictor variables and Figure 4 shows the spatial distributions of the predictor variables across the Great Xing’an Mountains.

_{soil}is the NDVI (value range is 0.2442-0.8225) value of the bare soil or no vegetation cover (value range is 0.1539–0.2449), and NDVI

_{veg}represents the NDVI value of the pixel completely covered by the vegetation (value range is 0.8199–0.8923). We employed the “Zonal Statistics as Table” tool in ArcGIS 10.2 to extract the average vegetation cover in fire season of each unit grid.

^{2}) and railway density (km/km

^{2}) and the proportion of residential areas (%) of each grid. Data on population density and GDP were extracted from the Resource and Environment Data Cloud Platform [6], which has spatial distribution grid data for 2000, 2005, and 2010 with 1 km resolution. To supplement the data for the missing years, we computed the annual population and GDP growth rate according to the national statistical yearbook from 2000–2016. We then created the population and GDP raster data for 2000–2016 through the raster calculator tool in ArcGIS 10.2, while “Zonal Statistics as Table” in ArcGIS software was employed to compute the average population density and average per capita GDP for each grid.

#### 2.4. Model Description

_{i}) = μ and var(Y

_{i}) = μ + ҡμ

^{2}, respectively, and ҡ (ҡ ≥ 0) is the dispersion parameter.

_{0}and β

_{k}are parameters of GWNBR model at position i; (xi, yi) is the geographic coordinate of position i; ҡ

_{i}is the discrete parameter of position i.

_{ij}is the weight value of an observation at location j for estimating the coefficient at location i, d

_{ij}is the distance between regression point i and neighbor j, and hi is the kernel bandwidth size. The global NB and GWNBR models for the wildfire count data of the Great Xing’an Mountain over 16 years were fitted using SAS software 9.4 [35].

#### 2.5. Model Evaluation and Comparison

_{ij}is the corresponding element in the spatial weight matrix W. The Moran’s I ranges from −1 to 1. If the value of Moran’s I index is greater than 0, it indicates that the residual value of the study area is spatially positively correlated; less than 0 indicates that the residual value has a spatially negative correlation, and equal to 0 indicates lack of spatial autocorrelation of the residual.

## 3. Results

#### 3.1. Comparison of Significant Explanatory Variables between Two Models

#### 3.2. Comparison of Model Fitting between GWNBR and Global NB

#### 3.3. Spatial Autocorrelation of Residuals

#### 3.4. Spatial Distribution of Fire Occurrence and Residual

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Mckenzie, D.; Shankar, U.; Keane, R.E.; Stavros, E.N.; Heilman, W.E.; Fox, D.G.; Riebau, A.C. Smoke consequences of new wildfire regimes driven by climate change. Earth’s Future
**2014**, 2, 35–59. [Google Scholar] [CrossRef] [Green Version] - Guo, F.; Innes, L.J.; Wang, G.; Ma, X.; Sun, L.; Hu, H.; Su, Z. Historic distribution and driving factors of human-caused fires in the Chinese boreal forest between 1972 and 2005. J. Plant Ecol.
**2015**, 8, 480–490. [Google Scholar] [CrossRef] - Wu, Z.; He, H.; Yang, J.; Liu, Z.; Liang, Y. Relative effects of climatic and local factors on fire occurrence in boreal forest landscapes of northeastern China. Sci. Total Environ.
**2014**, 493, 472–480. [Google Scholar] [CrossRef] [PubMed] - Guo, F.; Selvalakshmi, S.; Lin, F.; Wang, G.; Wang, W.; Su, Z.; Liu, A. Geospatial information on geographical and human factors improved anthropogenic fire occurrence modeling in the Chinese boreal forest. Can. J. For. Res.
**2016**, 46, 582–594. [Google Scholar] [CrossRef] [Green Version] - Guo, F.; Wang, G.; Innes, J.L.; Ma, Z.; Liu, A.; Lin, Y. Comparison of six generalized linear models for occurrence of lightning-induced fires in northern Daxing’an Mountains. China J. For. Res.
**2016**, 27, 379–388. [Google Scholar] [CrossRef] - Guo, F.; Su, Z.; Wang, G.; Sun, L.; Tigabu, M.; Yang, X.; Hu, H. Understanding fire drivers and relative impacts in different Chinese forest ecosystems. Sci. Total Environ.
**2017**, 605, 411–425. [Google Scholar] [CrossRef] [PubMed] - Martinez, J.; Vega-Garcia, C.; Chuvieco, E. Human-caused wildfire riskrating for prevention planning in Spain. J. Environ. Manag.
**2009**, 90, 1241–1252. [Google Scholar] [CrossRef] [PubMed] - Niklasson, M.; Granstrom, A. Numbers and sizes of fires: Long-term spatially explicit fire history in a Swedish boreal landscape. Ecology
**2000**, 81, 1484–1499. [Google Scholar] [CrossRef] - Oliveira, S.; Oehler, F.; San-Miguel-Ayanz, J.; Camia, A.; Pereira, J.M.C. Modeling spatial patterns of fire occurrence in Mediterranean Europe using Multiple Regression and Random Forest. For. Ecol. Manag.
**2012**, 275, 117–129. [Google Scholar] [CrossRef] - Pradhan, B.; Suliman, M.D.H.B.; Awang, M.A.B. Forest fire susceptibility and risk mapping using remote sensing and geographical information systems (GIS). Disaster Prev. Manag.
**2007**, 16, 344–352. [Google Scholar] [CrossRef] - Wallenius, T.H.; Kuuluvainen, T.; Vanha-Majamaa, I. Fire history in relation to site type and vegetation in Vienansalo wilderness in eastern Fennoscandia, Russia. Can. J. For. Res.
**2004**, 34, 1400–1409. [Google Scholar] [CrossRef] - Pereira, M.G.; Trigo, R.M.; da Camara, C.C.; Pereira, J.M.C.; Leite, S.M. Synoptic patterns associated with large summer forest fires in Portugal. Agric. For. Meteorol.
**2005**, 129, 11–25. [Google Scholar] [CrossRef] - Syphard, A.D.; Radeloff, V.C.; Keuler, N.S.; Taylor, R.S.; Hawbaker, T.J.; Stewart, S.I.; Clayton, M.K. Predicting spatial patterns of fire on a southern California landscape. Int. J. Wildland Fire
**2008**, 17, 602–613. [Google Scholar] [CrossRef] - Hu, H. Forest Fire Ecology and Management, 2nd ed.; China Forestry Publishing House: Beijing, China, 2005; p. 66. [Google Scholar]
- Mandallaz, D.; Ye, R. Prediction of forest fires with Poisson model. Can. J. For. Res.
**1997**, 27, 1685–1694. [Google Scholar] [CrossRef] - Griffith, D.A.; Haining, R. Beyond mule kicks: the Poisson distribution in geographical analysis. Geogr. Anal.
**2006**, 38, 123–139. [Google Scholar] [CrossRef] - Podur, J.J.; Martell, D.L.; Stanford, D. A compound Poisson model for the annual area burned by forest fires in the province of Ontario. Environmetrics
**2009**, 21, 457–469. [Google Scholar] [CrossRef] - Cameron, A.C.; Trivedi, P.K. Regression Analysis of Count Data; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Anselin, L.; Griffith, D.A. Do spatial effects really matter in regression analysis? Pap. Reg. Sci.
**1988**, 65, 11–34. [Google Scholar] [CrossRef] - Foody, G.M. Geographical weighting as a further refinement to regression modelling: an example focused on the NDVI-rainfall relationship. Remote Sens. Environ.
**2003**, 88, 283–293. [Google Scholar] [CrossRef] - Wang, Q.; Ni, J.; Tenhunen, J. Application of a geographically-weighted regression analysis to estimate net primary production of Chinese forest ecosystems. Glob. Ecol. Biogeogr.
**2005**, 14, 379–393. [Google Scholar] [CrossRef] - Koutsias, N.; Martínez, J.; Chuvieco, E.; AlligÖwer, B. Modeling wildland fire occurrence in southern Europe by a geographically weighted regression approach. In Proceedings of the 5th International Workshop on Remote Sensing and GIS Applications to Forest Fire Management: Fire Effects Assessment, Zaragoza, Spain, 16–18 June 2005; pp. 57–60. [Google Scholar]
- Padilla, M.; Vega-Garcia, C. On the comparative importance of fire danger rating indices and their integration with spatial and temporal variables for predicting daily human-caused fire occurrences in Spain. Int. J. Wildland Fire
**2011**, 20, 46–58. [Google Scholar] [CrossRef] - Chuvieco, E.; Aguado, I.; Yebra, M.; Nieto, H.; Salas, P.J.; Martín, M.P. Development of a framework for fire risk assessment using remote sensing and geographic information system technologies. Ecol. Model.
**2010**, 221, 46–58. [Google Scholar] [CrossRef] [Green Version] - Fotheringham, A.S.; Brunsdon, C.; Charlton, M.E. Geographically Weighted Regression: The Analysis of Spatially Varying Relationships; John Wiley and Sons: New York, NY, USA, 2002. [Google Scholar]
- Rodrigues, M.; De La Riva, J.; Fotheringham, S. Modeling the spatial variation of the explanatory factors of human-caused wildfires in Spain using geographically weighted logistic regression. Appl. Geogr.
**2014**, 48, 52–63. [Google Scholar] [CrossRef] - Justice, C.O.; Giglio, L.; Korontzi, S.; Owens, J.; Morisette, J.T.; Roy, D.; Descloitres, J.; Alleaume, S.; Petitcolin, F.; Kaufman, Y. The MODIS fire products. Remote Sens. Environ.
**2002**, 83, 244–262. [Google Scholar] [CrossRef] - Faivre, N.; Jin, Y.; Goulden, M.L.; Randerson, J.T. Controls on the spatial pattern of wildfire ignitions in Southern California. Int. J. Wildland Fire
**2014**, 23, 799. [Google Scholar] [CrossRef] - Zhang, H.; Qi, P.; Guo, G. Improvement of fire danger modelling with geographically weighted logistic model. Int. J. Wildland Fire
**2014**, 23, 1130–1146. [Google Scholar] [CrossRef] - Purevdorj, T.; Tateishi, R.; Ishiyama, T.; Honda, Y. Relationships between percent vegetation cover and vegetation indices. Int. J. Remote Sens.
**1998**, 19, 3519–3535. [Google Scholar] [CrossRef] - Gutman, G.; Ignatov, A. The derivation of the green vegetation fraction from noaa/avhrr data for use in numerical weather prediction models. Int. J. Remote Sens.
**1998**, 9, 1533–1543. [Google Scholar] [CrossRef] - McCulloch, C.E.; Searle, S.R. Generalized, Linear and Mixed Models; John Wiley & Sons: New York, NY, USA, 2001; pp. 1–184. [Google Scholar]
- Myers, R.H.; Montgomery, D.C.; Vining, G.G. Generalized Linear Models; John Wiley & Sons: New York, NY, USA, 2002; pp. 100–194. [Google Scholar]
- Da Silva, A.R.; Rodrigues, T.C.V. Geographically weighted negative binomial regression-incorporating overdispersion. Stat. Comput.
**2014**, 5, 769–783. [Google Scholar] [CrossRef] - SAS Institute, Inc. STAT 9.4 Users’ Manual; SAS Institute, Inc.: Cary, NC, USA, 2013. [Google Scholar]
- Burnham, K.P.; Anderson, D.R. Multimodel inference: Understanding AIC and BIC in model selection. Sociol. Method. Res.
**2004**, 33, 261–304. [Google Scholar] [CrossRef] - Bailey, T.C.; Gatrell, A.C. Interactive Spatial Data Analysis; Longman Scientific and Technical: Essex, UK, 1995. [Google Scholar]
- Chen, V.Y.J.; Deng, W.S.; Yang, T.C.; Matthews, S.A. Geographically weighted quantile regression (GWQR): An application to mortality data. Geogr. Anal.
**2012**, 44, 134–150. [Google Scholar] [CrossRef] - Wu, W.; Zhang, L. Comparison of spatial and non-spatial logistic regression models for modeling the occurrence of cloud cover in northeastern Puerto Rico. Appl. Geogr.
**2013**, 37, 52–62. [Google Scholar] [CrossRef] - Calkin, D.E.; Gebert, K.M.; Jones, J.G.; Neilson, R.P. Forest Service large fire area burned and suppression expenditure trends, 1970–2002. J. For.
**2005**, 103, 179–183. [Google Scholar] [CrossRef] - Gabban, A.; San-Miguel-Ayanz, J.; Viegas, D.X. A comparative analysis of the use of NOAA-AVHRR NDVI and FWI data for forest fire risk assessment. Int. J. Remote Sens.
**2008**, 29, 5677–5687. [Google Scholar] [CrossRef] - Lampin-Maillet, C.; Jappiot, M.; Long, M.; Bouillon, C.; Morge, D.; Ferrier, J.P. Mapping wildland-urban interfaces at large scales integrating housing density and vegetation aggregation for fire prevention in the South of France. J. Environ. Manag.
**2010**, 91, 732–741. [Google Scholar] [CrossRef] [Green Version] - Liu, Z.; Yang, J.; Chang, Y.; Weisberg, P.J.; He, H.S. Spatial patterns and drivers of fire occurrence and its future trend under climate change in a boreal forest of Northeast China. Glob. Chang. Biol.
**2012**, 18, 2041–2056. [Google Scholar] [CrossRef] - Hu, T.; Zhou, G. Drivers of lightning-and human-caused fire regimes in the Great Xing’an Mountains. For. Ecol. Manag.
**2014**, 329, 49–58. [Google Scholar] [CrossRef] - Guo, F.T.; Wang, G.Y.; Su, Z.W.; Liang, H.L.; Wang, W.H.; Lin, F.F.; Liu, A.Q. What drives forest fire in Fujian, China? Evidence from logistic regression and Random Forests. Int. J. Wildland Fire
**2016**, 25, 505–519. [Google Scholar] [CrossRef] - Jetz, W.; Rahbek, C.; Lichstein, J.W. Local and global approaches to spatial data analysis in ecology. Glob. Ecol. Biogeogr.
**2005**, 14, 97–98. [Google Scholar] [CrossRef] [Green Version] - Guo, L.; Ma, Z.; Zhang, L. Comparison of bandwidth selection in application of geographically weighted regression: a case study. Can. J. For. Res.
**2008**, 38, 2526–2534. [Google Scholar] [CrossRef]

**Figure 4.**Spatial distributions of vegetation cover (

**a**), elevation (

**b**), slope (

**c**), average precipitation (

**d**), average temperature (

**e**), average relative humidity (

**f**), settlement proportion (

**g**), road density (

**h**), railway density (

**i**), GDP (

**j**), and population density (

**k**).

**Figure 5.**Spatial distributions of the significant estimate coefficients of (

**a**) vegetation cover, (

**b**) elevation, (

**c**) slope, (

**d**) average precipitation, (

**e**) average temperature, (

**f**) average relative humidity, (

**g**) settlement proportion, (

**h**) road density, (

**i**) railway density, (

**j**) GDP of the GWNBR model.

**Figure 6.**Spatial distributions of the model predictions based on global NB (

**a**) and GWNBR (

**b**) models; residuals from global NB (

**c**) and GWNBR (

**d**) models; and the difference between the absolute residuals of GWNBR and global NB (

**e**).

Variables | Mean | Median | Minimum Value | Maximum Value | Standard Deviation |
---|---|---|---|---|---|

Fire Occurrence (point) | 2.54 | 0 | 0 | 47 | 5.6 |

Elevation (m) | 551.8 | 519.9 | 159.4 | 1209.1 | 193.2 |

Slope (degree) | 4.6 | 4.2 | 0.76 | 13.5 | 1.9 |

Railway Density (km/km^{2}) | 0.01 | 0 | 0 | 0.49 | 0.05 |

Road Density (km/km^{2}) | 0.12 | 0 | 0 | 0.89 | 0.16 |

Settlement Proportion (%) | 0.001 | 0 | 0 | 0.558 | 0.016 |

Average Rel. Humidity (%) | 85.9 | 85.7 | 74.9 | 89.2 | 1.2 |

Average Temperature (°C) | 2.1 | 2.0 | −0.1 | 4.3 | 0.7 |

Average Precipitation (mm/day) | 2.1 | 2.1 | −4.6 | 2.6 | 0.2 |

Vegetation Cover (%) | 0.6 | 0.6 | 0.4 | 0.7 | 0.04 |

GDP (Gross Domestic Product) (yuan/km^{2}) | 6.9 | 1.9 | 0.03 | 853.7 | 34.2 |

Population Density (person/km^{2}) | 6.1 | 0 | 0 | 2688.4 | 69.1 |

**Table 2.**Coefficient estimates of the full variables models from global negative Binomial (NB) and geographically weighted negative Binomial regression (GWNBR) model.

Statistics | β_{Intercept} | β_{Elevation} | β_{Slope} | β_{Railway density} | β_{Road density} | β_{Settlement proportion} | β_{Average relative humidity} | β_{Average temperature} | β_{Average precipitation} | β_{Vegetation cover} | β_{GDP} | β_{Population density} |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Global negative binominal (NB) | ||||||||||||

Estimate | −120.6 | −0.0002 | 0.27 | 0.92 | −1.07 | −14.63 | 1.67 | 1.91 | −3.16 | −38.15 | −0.005 | 0.0001 |

Standard deviation | 9.4 | 0.0003 | 0.03 | 0.72 | 0.20 | 4.47 | 0.12 | 0.14 | 0.54 | 1.10 | 0.001 | 0.0008 |

Estimate −1Std | −129.9 | −0.0005 | 0.24 | 0.20 | −1.28 | −19.10 | 1.56 | 1.76 | −3.70 | −39.25 | −0.006 | −0.0007 |

Estimate + 1Std | −111.1 | 0.0001 | 0.29 | 1.64 | −0.87 | −10.17 | 1.79 | 2.05 | −2.62 | −37.05 | −0.004 | 0.0009 |

Geographically weighted negative binominal regression (GWNBR) | ||||||||||||

Minimum value | −275.3 | −0.0029 | −0.03 | −0.79 | −1.55 | −22.01 | 1.48 | 1.89 | −14.22 | −41.49 | −0.007 | −0.0012 |

Lower quartiles | −201.5 | −0.001 | 0.09 | 0.08 | −1.24 | −18.47 | 1.66 | 2.09 | −8.18 | −39.42 | −0.005 | −0.0004 |

Mean | −161.1 | 0.0003 | 0.22 | 0.82 | −1.10 | −15.50 | 2.18 | 2.48 | −5.58 | −36.17 | −0.004 | −0.0001 |

Media | −142.1 | 0.001 | 0.21 | 0.92 | −1.10 | −15.97 | 1.92 | 2.15 | −3.67 | −36.86 | −0.004 | −0.0001 |

Upper quartiles | −118.5 | 0.0015 | 0.35 | 1.51 | −0.94 | −12.74 | 2.68 | 2.87 | −2.97 | −33.53 | −0.004 | 0.0002 |

Maximum value | −106.3 | 0.0027 | 0.45 | 2.32 | −0.64 | −7.55 | 3.62 | 3.88 | −2.65 | −25.77 | −0.003 | 0.0005 |

Parameters | Estimate | Standard Error | p-Value |
---|---|---|---|

Intercept | −120.43 | 9.53 | <0.0001 |

Slope | 0.25 | 0.02 | <0.0001 |

Road density | −0.96 | 0.18 | <0.0001 |

Settlement proportion | −14.03 | 4.22 | 0.0009 |

Average relative humidity | 1.67 | 0.12 | <0.0001 |

Average temperature | 1.92 | 0.14 | <0.0001 |

Average precipitation | −3.09 | 0.56 | <0.0001 |

Vegetation cover | −38.10 | 1.09 | <0.0001 |

GDP | −0.01 | 0.001 | <0.0001 |

Dispersion | 2.22 | 0.09 |

**Table 4.**Statistics of model fitting and residuals for the global NB and geographically weighted regression (GWR) models with all variables and significant variables.

All variables | Significant Variables | |||
---|---|---|---|---|

Statistics | Global NB | GWNBR | Global NB | GWNBR |

AIC (Akaike information criterion) | 29442.17 | 22582.42 | 30138.93 | 23576.01 |

BIC (Bayesian Information Criterion) | 29959.05 | 22609.66 | 30172.17 | 22760.81 |

Prediction Accuracy | 62.99% | 68.23% | 62.4% | 68.6% |

MSE (Mean Square Error) | 348.21 | 82.65 | 363.04 | 100.58 |

Mean of absolute residuals | 5.94 | 2.77 | 5.9598 | 2.8911 |

Std of Residuals | 18.56 | 9.04 | 18.9521 | 9.9755 |

Moran’s Index | 0.042 | 0.02 | 0.041 | 0.0204 |

Z-score | 118.2 | 56.3 | 114.5 | 57.3 |

P-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

**Table 5.**The interquartile range (IQR) of the GWNBR coefficient estimates with the standard error (SE) of the global NB estimates, indicating spatial non-stationary in the relationships between response variable and its accompanying predictor variables.

Coefficient | IQR(GWR) | SE (Global NB) | 2SE (Glzobal NB) | Status |
---|---|---|---|---|

Intercept | 83.07 | 9.41 | 18.83 | Non-stationary |

Elevation | 0.002 | 0.0003 | 0.001 | Non-stationary |

Slope | 0.26 | 0.03 | 0.06 | Non-stationary |

Railway density | 1.43 | 0.72 | 1.44 | Stationary |

Road density | 0.29 | 0.20 | 0.41 | Stationary |

Settlement proportion | 5.74 | 4.47 | 8.94 | Stationary |

Average relative humidity | 1.02 | 0.12 | 0.23 | Non-stationary |

Average temperature | 0.78 | 0.14 | 0.29 | Non-stationary |

Average precipitation | 5.20 | 0.54 | 1.08 | Non-stationary |

Vegetation cover | 5.89 | 1.10 | 2.20 | Non-stationary |

GDP | 0.001 | 0.001 | 0.002 | Stationary |

Population density | 0.001 | 0.001 | 0.002 | Stationary |

Dispersion | 0.61 | 0.09 | 0.1724 | - |

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## Share and Cite

**MDPI and ACS Style**

Su, Z.; Hu, H.; Tigabu, M.; Wang, G.; Zeng, A.; Guo, F.
Geographically Weighted Negative Binomial Regression Model Predicts Wildfire Occurrence in the Great Xing’an Mountains Better Than Negative Binomial Model. *Forests* **2019**, *10*, 377.
https://doi.org/10.3390/f10050377

**AMA Style**

Su Z, Hu H, Tigabu M, Wang G, Zeng A, Guo F.
Geographically Weighted Negative Binomial Regression Model Predicts Wildfire Occurrence in the Great Xing’an Mountains Better Than Negative Binomial Model. *Forests*. 2019; 10(5):377.
https://doi.org/10.3390/f10050377

**Chicago/Turabian Style**

Su, Zhangwen, Haiqing Hu, Mulualem Tigabu, Guangyu Wang, Aicong Zeng, and Futao Guo.
2019. "Geographically Weighted Negative Binomial Regression Model Predicts Wildfire Occurrence in the Great Xing’an Mountains Better Than Negative Binomial Model" *Forests* 10, no. 5: 377.
https://doi.org/10.3390/f10050377