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Technical Brief

Computational Analysis of the U.S. Forest Fires OPEN ACCESS

[+] Author and Article Information
António M. Lopes

UISPA-LAETA/INEGI,
Faculty of Engineering,
University of Porto,
Rua Dr. Roberto Frias,
Porto 4200-465, Portugal
e-mail: aml@fe.up.pt

J. A. Tenreiro Machado

Department of Electrical Engineering,
Institute of Engineering,
Polytechnic of Porto,
R. Dr. António Bernardino de Almeida, 431,
Porto 4249-015, Portugal
e-mail: jtm@isep.ipp.pt

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 29, 2016; final manuscript received December 28, 2016; published online January 24, 2017. Assoc. Editor: Bogdan I. Epureanu.

J. Comput. Nonlinear Dynam 12(4), 044503 (Jan 24, 2017) (5 pages) Paper No: CND-16-1311; doi: 10.1115/1.4035672 History: Received June 29, 2016; Revised December 28, 2016

This paper analyses forest fires (FF) in the U.S. during 1984–2013, based on data collected by the monitoring trends in burn severity (MTBS) project. The study adopts the tools of dynamical systems to tackle information about space, time, and size. Computational visualization methods are used for reducing the information dimensionality and to unveil the relationships embedded in the data.

FIGURES IN THIS ARTICLE
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Forest fires (FF) affect many regions of the globe. The FF consume vast areas of vegetation, compromising ecosystems and the CO2 cycle [1]. In the last years, the effects of FF became dramatic, motivating the intensive study of these phenomena [2]. The FF burnt areas are influenced by natural and human factors, such as the orography, type of vegetation and weather, and the efficiency of the detection and suppression schemes. Understanding FF dynamics in space and time, and the resulting patterns, will help not only in the identification of future hazards but also in deciding strategies for FF prevention, detection, and suppression [3]. The FF dynamics involve size, space, and time, and their statistical distributions unveil long-range memory, as well as self-similarity. This behavior leads to power-law (PL) characteristics [4], often explained by the self-organized criticality (SOC) evolution [5]. Nevertheless, some authors suggest that PL distributions may be too simple to describe completely FF [6].

In this paper, we study the FF dynamics using tools adopted in the field of complex systems. A public domain FF catalog containing data about the events occurred in the U.S. during the period 1984–2013 is tackled. Instead of modeling individual occurrences or the dynamics of the fire propagation, we study the collective behavior of the events in space and time. We adopt clustering techniques and fractional entropy to combine information about space, time, and amplitude, generating “maps” that reflect the FF patterns. We then use computer visualization tools to identify the emerging patterns and to unveil relationships between them.

Bearing these ideas in mind, the paper is organized as follows. Section 2 describes briefly the dataset. Section 3 introduces the main mathematical tools for processing the FF data. Section 4 analyzes the visualization schemes and discusses the results. Finally, Sec. 5 outlines the main conclusions.

Data characterizing FF in the U.S. is collected by the monitoring trends in burn severity (MTBS) project [7] and is available at the website.2 The MTBS covers FF greater than 1000 acres in the west and 500 acres in the east since 1984. The data analyzed herein were retrieved in October, 2015 and, by this date; data for year 2014 were still incomplete. The MTBS catalog includes more than 19,000 events, where each record corresponds to one FF, with information about the date, time (with 1 day resolution), coordinates (latitude and longitude), and size (burnt area in acres). We decided to discard events with size smaller than 1000 acres, because they are available just for the east region of the U.S., and to avoid “quantification errors” due to the uncertain registration of small size FF.

For studying possible correlations between FF and meteorological variables, we consider the temperature and precipitation time series. The data about temperature are available at the National Aeronautics and Space Administration (NASA) website.3 Each record consists of the average temperatures per month, expressed in celsius degree. Some occasional gaps of 1 month in the data (represented on the original time series by the value 999.9) are substituted by a linear interpolation between the two adjacent values. The values are interpolated linearly in order to have daily temperatures. Precipitation time series are from the U.S. Department of Agriculture, Agricultural Research Service, available online at the website.4 The available data cover the period from 1950 to 2010 with 1 day time resolution. Precipitation values are expressed in millimeters. These variables do not cover all possible factors involved in FF but allow a preliminary analysis for validating the postulate of possible patterns of their space–time dynamics.

Nonhierarchical Clustering.

Clustering groups objects that are similar in some sense. The K-means is a nonhierarchical clustering algorithm used in machine learning and data mining [8]. The method starts with a collection of N objects XN = {x1, x2,…, xN} so that xp, 1 ≤ p ≤ N, represents a point in d-dim space (xpd), and K clusters, specified in advance by the user. The K-means groups the objects into K ≤ N clusters, CK = {c1, c2,…, cK}, minimizing a function, J, that measures the sum of distances between the points and the centers of their clusters, given by the set MK = {μ1, μ2,…, μK} Display Formula

(1)J=p=1Ni=1Krpixpμi2

where rpi {0, 1} is a parameter that reflects if object xp belongs to cluster i, 1 ≤ i ≤ K [8]. The result consists of partitioning the space d into K cells [9]. Given the centers of the clusters, MK, a Voronoi tessellation subdivides d into K cells, such that for each μi, the corresponding cells are all points closer to μi than to any other center.

Fractional Entropy.

The concept of entropy is very important not only in the area of information processing but also in statistical mechanics [10]. Generalizations of the classic Boltzmann–Gibbs–Shannon formulation were proposed for explaining the statistical nature of complex systems that follow PL distributions [11]. These generalizations usually do not obey the fourth Khinchin axiom [12].

Recently, entropy was interpreted in the perspective of fractional calculus [13,14]. The generalized fractional-order entropy Sα,α, is given by [15] Display Formula

(2)Sα=i{piαΓ(α+1)[lnpi+ψ(1)ψ(1α)]}pi

where Dα{·} is the derivative of order α, and Γ(·) and ψ(·) represent the gamma and digamma functions, respectively [15]. For α = 0, expression (2) yields the classical Boltzmann–Gibbs–Shannon entropy.

Multidimensional Scaling.

Multidimensional scaling (MDS) is a computational scheme for visualizing information by exploring similarities between data [1618]. For a total of N items in space d, the MDS algorithm requires the definition of a similarity index Cij, between elements (i, j) = {1, 2,…, N} and the calculation of a matrix C = [Cij], N × N dimensional, of item-to-item similarities. MDS assigns a “point” to each item in m (m ≤ d). By rearranging the item coordinates in m, MDS arrives to an optimal configuration for approximating the observed similarities in d. The method uses a minimization iterative algorithm that evaluates successive configurations for optimizing a goodness-of-fit. Low-dimensional spaces (e.g., m = 2, or m = 3) allow the resulting “points” to be visualized in a “map.” Since we depart from relative (i.e., item-to-item) measurements, we can rotate or translate the final MDS map, given that the distances between points remain identical. The quality of the MDS approach can be evaluated by means of the so-called Shepard and stress plots.

We consider fire events in the Continental United States (CONUS) in the period 1984–2013. As mentioned in Sec. 2, all occurrences with less than 1000 acres in burnt area are discarded for catalog uniformity. The data may be interpreted as a sequence of samples in space and time, measuring the FF burnt area. As such, the data series describe a four-dim problem, since we have two dimensions for space (latitude and longitude), one for time, and one for FF amplitude. The comparison of four-dim data poses computational and algorithmic challenges, not only due to the large number of records but also due to the sparsity of their four-dim vector coordinates. The FF evolve in space–time and, therefore, the coordinates represent only the final part of the event.

In the sequel, we formulate a strategy to reduce dimensionality while preserving the most important information. In the first step, the two-dim space coordinates for the complete set of data are discretized using a limited number K of Voronoi cells. The number K is to be adjusted numerically and leads, implicitly, to the spacial coordinate reduction toward a one-dim small set. In a second step, events within the same Voronoi cell are “joined” and processed in the time domain. To compensate for the corresponding space discretization errors and to manage the FF size variable, histograms (evolving in time) are built and characterized by means of entropy. Finally, entropies are compared by means of a measure. Therefore, the data processing is projected from four-dim down to a two-dim space.

The CONUS territory is divided into the K cells Vi,i=1,,K, and the corresponding series xi(t) are obtained, where i is associated with space and t represents time, with 1 day resolution. We then calculate the vectors si(k)=[Si,kα], where k denotes time sampled with a period of 1 year and Si,kα denotes the fractional entropy of the histogram of amplitudes (burnt area) for events in cell Vi and year k. We proceed by constructing a matrix C using the Canberra distance, Cij, between entropy vectors si(k) and sj(k) [19] Display Formula

(3)Cij=k|Si,kαSj,kα||Si,kα|+|Sj,kα|,i,j=1,,K
The matrix M feeds the MDS algorithm generating a map that we can interpret in terms of FF patterns.

For defining the CONUS cells Vi, we adopt a Voronoi tessellation. The cell events and centers are calculated by a K-means clustering algorithm, based on the FF geographical coordinates. The value of K is chosen as a compromise between spatial resolution and statistical significance of the data captured by each Voronoi cell. A value of K too low produces poor spatial resolution, while a K too high leads to a small number of events per cell. We adopt the silhouette criterion to compare different solutions [20]. Therefore, given the coordinates of the events during the period 1984–2013, we calculate the silhouette average values, σ, versus K, for K ∈ [7, 30]. The better tessellation corresponding to the higher value of σ is obtained for {σ, K} = {0.66, 20}. Figure 1 depicts the Voronoi tessellation for K = 20 and the geographical location of the events (dots).

Correlation Between FF and Weather.

FF are related with several atmospheric variables reflecting the periodical variation of the seasons. Therefore, it seems natural to explore possible relationships between them, while seeking for the emergence of hidden patterns. We do not seek here to correlate FF with all possible variables, since human and social issues are clearly important but difficult to quantify. Instead, we want to demonstrate some relationships with variables that have undoubtedly time and space regularities justifying, therefore, the idea of possible hidden patterns within FF data.

In this line of thought, for each cell Vi, we calculate the lagged cross-correlation between the FF and the temperature data, or the FF and the precipitation data (to be denoted by rT and rP, respectively). The FF series is kept fixed, while the second time series is shifted by multiples of 1 day. Figure 2 depicts the results for the Voronoi cell V1. We observe that rT is smooth, while rP is considerably noisy. This is due to the different nature of the time series. Temperature evolution in time is continuous, while FF and precipitation have discontinuities corresponding to days with no events that lead to data series characterized by burst impulses in some periods of time. Both for rT and rP, we verify an annual periodicity and that maximum correlations occur for the time delays τT = −25 and τP = 154 days, respectively. This means that FF are positively correlated with temperature and inversely correlated with precipitation. By other words, this simply means that the FF activity is sensitive to warmer and drier periods of the year. More important are, however, τT and τP quantifying the time delay for the propagation of effects that somehow describe the characteristics of each Voronoi cell.

Figure 3 depicts the values of τT (smooth line) and τP (noisy line), superimposed over the U.S. map. The large (small) delays in temperature (precipitation) reflect simply that the variable has a slow (fast) influence in time upon the FF evolution. We observe three clusters that are clearly related with their geographical location. The occidental cluster comprises cells {V1,V2,V3,V4,V5,V6,V7,V9,V10}, where FF activity is maximum at warmer and drier months, that is, in the summer season. The oriental cluster includes {V12,V14,V15,V16,V17,V18,V19,V20}, exhibiting higher values of τT, meaning that important FF take place also in colder periods of the year. The midcluster is the transition between the occidental and the oriental clusters and contains cells {V8,V11,V13}.

In conclusion, this simple analysis revealed correlations between FF and the local temperature and precipitation. However, other natural and human factors that influence FF do not have available numerical records or are difficult to quantify, demanding more powerful tools to unveil the intricate dynamics of FF.

MDS Visualization of FF.

The fractional entropy vectors si(k) for the MDS were obtained by adopting the parameter value, α = 0.7. In fact, close to this value, Sα has a maximum and we verify that this corresponds to a good discrimination both in time (i.e., between years) and space (i.e., between cells).

In Fig. 4, we show the data series x1(t) and the corresponding entropy s1(k) for cell V1. The annual periodicity of the FF is well observed in the graph of x1(t), while the entropy s1(k) unveils a cyclical behavior with a larger period. Similar patterns are observed for all cells and are further discussed in Sec. 4.3.

Figure 5 depicts the three-dim map of items produced by the MDS. Each point represents a Voronoi cell, Vi, embedding information about space, time, and size of the FF.

Figure 6 represents the Shepard, for the three-dim plot, and stress plots that assess the MDS results. The Shepard diagram shows a good distribution of points around the 45 deg line, which means a good fit of the distances to the dissimilarities. The stress plot reveals that a three-dim space leads to a stress value of almost one half of the two-dim map.

The MDS map of Fig. 5 reveals limitations when searching for some clusters. Therefore, to overcome this problem, we need to include an additional scheme for clarifying possible patterns.

Clustering and Visualization of FF.

Here, we propose a method to improve the visualization and the identification of patterns in the MDS portraits. The approach is based on the superposition of a graph over the MDS map. This graph has nodes for the K points in the MDS that represent items. The standard MDS analysis is based on the objects (interpreted as clusters) emerging in the final map. We can rely either in the direct visualization of the plot or in the implementation of some additional algorithm to extract the clusters. In the present work, we consider a novel approach for defining more complex objects that generalize the usual simple clusters of points.

We start by connecting the points that are closer in the MDS plot. This scheme produces several sets, P, of interconnected points (nodes) that represent the building blocks of prototype complex objects. The resulting sets, P, of interconnected points are compared by means of the distances between their nodes, and connection is established only between those two nodes i and j that are closer (i.e., between Pi and its neighbor Pj). This scheme creates a second level of interconnection composed by more complex building blocks. The scheme is repeated iteratively until there is a continuous route interconnecting all points in the MDS and no further processing is required. With this new object, we can interpret the MDS plot not only on the clusters but also on the structure of the interconnection between them that produce an m-dimensional object.

Figure 7 depicts the three-dim MDS map and the superimposed object. We observe the emergence of two clusters: A={V1,V2,V3,V4,V5,V7,V8,V11,V16,V18} and B={V6,V9,V10,V13,V14,V15,V17,V19,V20}. Cells V7 and V17 form the “centers” of A and B, respectively, while cell V12 connects the two clusters.

With this method, we verify the emergence of an object within the MDS points that reflects all input variables (space, time, and size) and leads to a more assertive interpretation. The method is not affected by the borders of the region under study, integrates different types of variables, and allows a numerical tuning by means of the number of Voronoi cells K and the order α of the fractional entropy. The strategy seems promising for handling other series of complex data and to reveal possible hidden patterns.

FF Entropy Forecasting.

The entropy time series si(k),i=1,,20 are studied for unveiling existing temporal patterns, as well as for forecasting future behavior. Here, we adopt nonlinear least-squares [21] to fit an empirical model ŝi(k) into the data. This model follows a Fourier-like series and is given by Display Formula

(4)ŝi(k)=a0,i+n=1Nbn,isin(nωih+dn,i),i=1,,20

where a0,i, bn,I, and ωi are parameters to be determined. We should note that we allow ωi to vary, that is, to have a different value for each distinct entropy time series si(k). The number of terms, N = 10, was chosen as a compromise between model complexity and goodness of fit, measured by the coefficient of determination R2. For the 20 Voronoi cells, we obtained R2 ∈ [0.803, 0.966].

Expression (4) may be interpreted as the output of a nonlinear system that is excited by different natural and human-generated inputs. The term a0,i models a bias (or operating point), while the harmonics describe the system dynamic behavior. The higher harmonics are associated with system nonlinearities. Moreover, parameter ωi varies in the interval ωi ∈ [0.978, 1.044] rad/s, with average value ω=1.004 rad/s, corresponding to a period T = 6.2 years. We verify that this value of T is close to half the solar cycle and, therefore, this natural phenomenon seems to represent the main cause of FF variability in time besides the standard 1 year evolution.

Figure 8 depicts the original, si(k) (filled circles), and the estimated values, ŝi(k) (continuous line), of the entropy for all cells in clusters A and B. The plots were extended toward year 2018, allowing for entropy forecasting (nonfilled circles).

Figure 9 details the results for cell V1, where ŝ1(k)=[8.1569,4.3422,8.1767,5.9344,4.9631] for years 2104– 2018.

This paper analyzed FF in the perspective of the collective behavior of the events in space and time. Data from a public domain FF catalog, containing information of events for the U.S., during the period 1984–2013, were tackled. Clustering techniques and fractional entropy were adopted to combine the available information about space, time, and burnt area. Furthermore, computer visualization tools were used to identify patterns and to unveil relationships between the data. The mathematical and computational tools allow novel perspectives of the study that may be used to better understand the FF dynamics.

The authors thank the Monitoring Trends in Burn Severity (MTBS) project, the National Aeronautics and Space Administration (NASA), and the U.S. Department of Agriculture, Agricultural Research Service, for the data.

Flannigan, M ., 2015, “ Carbon Cycle: Fire Evolution Split by Continent,” Nat. Geosci., 8(3), pp. 167–168. [CrossRef]
Keeley, J. E. , 2009, “ Fire Intensity, Fire Severity and Burn Severity: A Brief Review and Suggested Usage,” Int. J. Wildland Fire, 18(1), pp. 116–126. [CrossRef]
San-Miguel-Ayanz, J. , Schulte, E. , Schmuck, G. , and Camia, A. , 2013, “ The European Forest Fire Information System in the Context of Environmental Policies of the European Union,” For. Policy Econ., 29, pp. 19–25. [CrossRef]
Machado, J. , and Lopes, A. , 2014, “ Complex Dynamics of Forest Fires,” Math. Probl. Eng., 2014, p. 575872. [CrossRef]
Bak, P. , Chen, K. , and Tang, C. , 1990, “ A Forest-Fire Model and Some Thoughts on Turbulence,” Phys. Lett. A, 147(5), pp. 297–300. [CrossRef]
Reed, W. J. , and McKelvey, K. S. , 2002, “ Power-Law Behaviour and Parametric Models for the Size-Distribution of Forest Fires,” Ecol. Modell., 150(3), pp. 239–254. [CrossRef]
Eidenshink, J. , Schwind, B. , Brewer, K. , Zhu, Z.-L. , Quayle, B. , and Howard, S. , 2007, “ Project for Monitoring Trends in Burn Severity,” Fire Ecol., 3(1), pp. 3–21. [CrossRef]
Jain, A. K. , 2010, “ Data Clustering: 50 Years Beyond K-Means,” Pattern Recognit. Lett., 31(8), pp. 651–666. [CrossRef]
Okabe, A. , Boots, B. , Sugihara, K. , and Chiu, S. N. , 1999, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Wiley, Hoboken, NJ.
Lopes, A. M. , and Machado, J. T. , 2015, “ Entropy Analysis of Industrial Accident Data Series,” ASME J. Comput. Nonlinear Dyn., 11(3), p. 031006. [CrossRef]
Asgarani, S. , and Mirza, B. , 2015, “ Two-Parameter Entropies, Sk,r, and Their Dualities,” Phys. A, 417, pp. 185–192. [CrossRef]
Khinchin, A. I. , 1957, Mathematical Foundations of Information Theory, Dover, Mineola, NY.
Kenneth, M. , and Ross, B. , 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
Machado, J. , Lopes, A. , Duarte, F. , Ortigueira, M. , and Rato, R. , 2014, “ Rhapsody in Fractional,” Fract. Calculus Appl. Anal., 17(4), pp. 1188–1214.
Machado, J. T. , 2014, “ Fractional Order Generalized Information,” Entropy, 16(4), pp. 2350–2361. [CrossRef]
Borg, I. , and Groenen, P. J. , 2005, Modern Multidimensional Scaling: Theory and Applications, Springer, Berlin.
Lopes, A. M. , and Machado, J. T. , 2014, “ Analysis of Temperature Time-Series: Embedding Dynamics Into the MDS Method,” Commun. Nonlinear Sci. Numer. Simul., 19(4), pp. 851–871. [CrossRef]
Lopes, A. M. , Machado, J. T. , Pinto, C. M. , and Galhano, A. M. , 2013, “ Fractional Dynamics and MDS Visualization of Earthquake Phenomena,” Comput. Math. Appl., 66(5), pp. 647–658. [CrossRef]
Cha, S.-H. , 2007, “ Comprehensive Survey on Distance/Similarity Measures Between Probability Density Functions,” Int. J. Mathe. Models Meth. App. Sci., 4(1), pp. 300–307.
Rousseeuw, P. J. , 1987, “ Silhouettes: A Graphical Aid to the Interpretation and Validation of Cluster Analysis,” J. Comput. Appl. Math., 20, pp. 53–65. [CrossRef]
Draper, N. R. , Smith, H. , and Pownell, E. , 2014, Applied Regression Analysis, Wiley, New York.
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References

Flannigan, M ., 2015, “ Carbon Cycle: Fire Evolution Split by Continent,” Nat. Geosci., 8(3), pp. 167–168. [CrossRef]
Keeley, J. E. , 2009, “ Fire Intensity, Fire Severity and Burn Severity: A Brief Review and Suggested Usage,” Int. J. Wildland Fire, 18(1), pp. 116–126. [CrossRef]
San-Miguel-Ayanz, J. , Schulte, E. , Schmuck, G. , and Camia, A. , 2013, “ The European Forest Fire Information System in the Context of Environmental Policies of the European Union,” For. Policy Econ., 29, pp. 19–25. [CrossRef]
Machado, J. , and Lopes, A. , 2014, “ Complex Dynamics of Forest Fires,” Math. Probl. Eng., 2014, p. 575872. [CrossRef]
Bak, P. , Chen, K. , and Tang, C. , 1990, “ A Forest-Fire Model and Some Thoughts on Turbulence,” Phys. Lett. A, 147(5), pp. 297–300. [CrossRef]
Reed, W. J. , and McKelvey, K. S. , 2002, “ Power-Law Behaviour and Parametric Models for the Size-Distribution of Forest Fires,” Ecol. Modell., 150(3), pp. 239–254. [CrossRef]
Eidenshink, J. , Schwind, B. , Brewer, K. , Zhu, Z.-L. , Quayle, B. , and Howard, S. , 2007, “ Project for Monitoring Trends in Burn Severity,” Fire Ecol., 3(1), pp. 3–21. [CrossRef]
Jain, A. K. , 2010, “ Data Clustering: 50 Years Beyond K-Means,” Pattern Recognit. Lett., 31(8), pp. 651–666. [CrossRef]
Okabe, A. , Boots, B. , Sugihara, K. , and Chiu, S. N. , 1999, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Wiley, Hoboken, NJ.
Lopes, A. M. , and Machado, J. T. , 2015, “ Entropy Analysis of Industrial Accident Data Series,” ASME J. Comput. Nonlinear Dyn., 11(3), p. 031006. [CrossRef]
Asgarani, S. , and Mirza, B. , 2015, “ Two-Parameter Entropies, Sk,r, and Their Dualities,” Phys. A, 417, pp. 185–192. [CrossRef]
Khinchin, A. I. , 1957, Mathematical Foundations of Information Theory, Dover, Mineola, NY.
Kenneth, M. , and Ross, B. , 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
Machado, J. , Lopes, A. , Duarte, F. , Ortigueira, M. , and Rato, R. , 2014, “ Rhapsody in Fractional,” Fract. Calculus Appl. Anal., 17(4), pp. 1188–1214.
Machado, J. T. , 2014, “ Fractional Order Generalized Information,” Entropy, 16(4), pp. 2350–2361. [CrossRef]
Borg, I. , and Groenen, P. J. , 2005, Modern Multidimensional Scaling: Theory and Applications, Springer, Berlin.
Lopes, A. M. , and Machado, J. T. , 2014, “ Analysis of Temperature Time-Series: Embedding Dynamics Into the MDS Method,” Commun. Nonlinear Sci. Numer. Simul., 19(4), pp. 851–871. [CrossRef]
Lopes, A. M. , Machado, J. T. , Pinto, C. M. , and Galhano, A. M. , 2013, “ Fractional Dynamics and MDS Visualization of Earthquake Phenomena,” Comput. Math. Appl., 66(5), pp. 647–658. [CrossRef]
Cha, S.-H. , 2007, “ Comprehensive Survey on Distance/Similarity Measures Between Probability Density Functions,” Int. J. Mathe. Models Meth. App. Sci., 4(1), pp. 300–307.
Rousseeuw, P. J. , 1987, “ Silhouettes: A Graphical Aid to the Interpretation and Validation of Cluster Analysis,” J. Comput. Appl. Math., 20, pp. 53–65. [CrossRef]
Draper, N. R. , Smith, H. , and Pownell, E. , 2014, Applied Regression Analysis, Wiley, New York.

Figures

Grahic Jump Location
Fig. 1

Voronoi tessellation for K = 20 and geographical location of the events (dots) for the period 1984–2013

Grahic Jump Location
Fig. 2

Cross-correlation functions between FF and weather variables temperature rT (smooth line) and precipitation rP (noisy line) versus time delay, for cell V1, during period 1984–2013

Grahic Jump Location
Fig. 3

Time delays for the maximum cross correlation between FF and temperature τT (top values) and FF and precipitation τP (bottom values) for the 20 Voronoi cells

Grahic Jump Location
Fig. 4

Characterization of cell V1: FF data series x1(t) (bottom line); entropy s1(k) for α = 0.7 (top line)

Grahic Jump Location
Fig. 5

The three-dim MDS map for the Canberra index Cij

Grahic Jump Location
Fig. 6

Assessing the quality of the MDS for the Canberra index Cij: (a) Shepard plot for the three-dim plot and (b) stress plot

Grahic Jump Location
Fig. 7

The three-dim MDS map with the superimposed object, for the Canberra index Cij

Grahic Jump Location
Fig. 8

Original series, si(k) (filled circles), model ŝi(k) (continuous line), and forecast values (nonfilled circles), for the period 1984–2018 and clusters: (a) A and (b) B

Grahic Jump Location
Fig. 9

Original series, s1(k) (filled circles), model ŝ1(k) (continuous line), and forecast values (nonfilled circles) for cell V1 and period 1984–2018

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