Technological advances, increased safety standards and a greater importance of risk management led to a decreasing number of airplane accidents over the last decades. Despite these improved conditions, however, accidents still occur, be it due to human error, mechanical failure or further causes.

Inspired by the seminal work of Clarke (1946) on the distribution of V-2 rockets strikes on London during World War II, this blog post seeks to identify the spatial pattern of airplane accidents that happened in Florida in 2014 by employing a basic point pattern analysis. By visualizing and analyzing the accidents’ geographical distribution, it is assessed whether spatial patterns pointing towards risk-inducing contextual factors, e.g., a high traffic density or unfavourable environmental conditions, were present.

Data and Methods

Data on all airplane accidents for the subsequent spatial analysis were obtained from the National Transportation Safety Board (NTSB) aviation accident database. Generally speaking, the NTSB compiles data on both commercial and general aviation accidents for the United States and its territories from 1962 onwards along with contextual information on, e.g., the number of injuries and fatalities or weather conditions. The precise locations of the accidents are identified by their respective latitude and longitude coordinates in degrees and decimal degrees.

Information on geographical borders for 2014 were obtained from the Census Bureau’s MAF/TIGER geographic database which provides cartographic boundary files on various administrative levels. For both the visualizations and all spatial analyses, the nation-based shapefiles for the state-level were chosen and cropped to the geographic extent of Continental US and Florida, respectively.

The analysis proceeds as follows. In the first step, all airplane accidents in Florida are visualized as two-dimensional maps. In the second step, a point pattern analysis is conducted. Following the recommendations of Baddeley et al. (2015), the PPA starts with analyzing the first-order properties of the point pattern, i.e., the spatial distribution of the events under scrutiny. For this purpose, the density of the observed accidents is computed and visualized by employing quadrat counting methods and kernel density estimation to account for the non-constant intensity of the point process. Subsequently, the second-order properties of the point pattern, i.e., the spatial interactions between events, are examined. In doing so, the observed point pattern is compared with both random and regular samples to determine whether the spatial pattern resulted from a complete spatial randomness (CSR) process, that is, if the observed events within the study area are distributed at random. Moreover, Ripley’s reduced second moment function K(r) is computed to formally assess whether there are interactions between accidents, given the inhomogeneity of the point process.

All spatial analyses and visualizations were carried out in the R software environment for statistical computing and graphics (version 3.2.3). Replication data and R scripts can be retrieved from the author’s GitHub page.

Results

In order to visualize the spatial pattern, all airplane accidents were coded and stored as spatial points. Several accidents occurred close to but not within the state borders of Florida and were excluded from the analyses by taking a geographic subset of all points lying inside the boundary of the Florida polygon. For aesthetic reasons, an unprojected map using latitude and longitude coordinates was chosen.

As can be seen in figure 1, a total number of N=76 airplane accidents occurred. By looking at the map, accidents seem to cluster along the East Coast and in Central Florida, respectively, whereas fewer accidents seem to happen in North and South Florida.


Figure 1: Accidents seem to cluster along the East Coast and in Central Florida.

While visualizations provide a first insight into the spatial distribution of events, they do not suffice for (dis-)confirming spatial clustering hypotheses due to the so-called clustering illusion, i.e., the tendency to erroneously perceive random events as clustered (Gilovich et al. 1985).

To address this problem, a point pattern analysis (PPA) was employed which required two modifications of the spatial data. First, the projection was changed from unprojected longitude and latitude coordinates to an equal-area projection (Albers equal-area conic) to minimize distortions. Second, the spatial points were transformed into a two-dimensional spatial point pattern object containing information about the observation window.

A basic step of PPA is to investigate the intensity (λ) of the spatial point pattern, i.e., the average number of events per unit. λ can either be homogeneous or inhomogeneous, depending on whether the number of events varies across the area being studied. When dealing with airplane accidents, contextual factors leading to variations in the spatial distribution of these accidents will most likely be present, be it demographic, environmental or regulatory characteristics. Hence, the intensity of accidents will not be constant across space but rather vary spatially.

To assess whether the intensity of accidents is indeed (non-)constant, the so-called quadrat counting technique can be applied (see, e.g., Diggle 2014: 29-32). The rationale behind quadrat counting is to partition the study area into a certain number of equal-sized quadrats and count the number of events being located within each quadrat. If the intensity is homogeneous, the number of events within each quadrat should, approximately, be the same.


Figure 2: The intensity of the point process is inhomogeneous.

Figure 2 indicates that the intensity of the point process is inhomogeneous since the number of airplane accidents varies spatially. With regard to the spatial distribution of the depicted quadrats, accidents seem to occur more often in Central Florida, whereas fewer accidents happen in Northern Florida. However, a major drawback of the quadrat counting technique is its sensitivity to the number and size of the plotted quadrats which are user-defined and, thus, arbitrarily chosen.

Nonparametric kernel density estimation offers a fruitful solution to this problem. For estimating the intensity of the point process by means of kernel smoothing, an isotropic Gaussian kernel was used. As with the quadrat counts approach, one drawback of this method is that the kernel estimator is sensitive to the selected bandwidth. Following Diggle (1985), the bandwidth was chosen to minimise the mean square error of the kernel smoother. As indicated by the lighter areas in figure 3, there are several accident hotspots in Central Florida and along the eastern coastline of Florida.


Figure 3: There are several accident hotspots in Central Florida and along the eastern coastline of Florida.

After assessing the first-order properties, PPA proceed with testing whether events tend to cluster spatially. By employing nonparametric distance functions, it can be analyzed if the observed spatial pattern is the result of a CSR process, which disconfirms clustering hypotheses. In other words, if spatial clustering of accidents is present, they did not occur (geographically) at random. However, while spatial variations of airplane accidents are to be expected, spatial clustering of airplane accidents should not necessarily be present in the data as the occurrence of an airplane accident – at least in most cases – neither increases nor decreases the propability of other accidents happening nearby.


Figure 4: From Left to Right: (a) Observed points, (b) Random points, (c) Regular points.

One common testing procedure is to plot the observed points, i.e., accidents, within the study alongside sampled random and regular points (see, e.g., Bivand et al. 2008). For the sampled point locations, the sample size was chosen to match the number of observed points (N=76). If spatial clustering were present in the accident data, substantial differences should be visible when comparing the observed pattern to both random and regular point patterns. As can be seen in figure 4, the observed point pattern does not resemble a regular pattern. Still, clustering cannot be ruled out (or confirmed, for that matter) by means of these visual inspections only.

Distance functions, on the other hand, provide a more formal approach to testing whether spatial clustering occurs or not. The rationale is to compare the observed empirical values to theoretically expected values under CSR. Since conventional distance functions (i.e., the nearest neighbour distance distribution function, the empty space function or Ripley’s reduced second moment function) assume homogeneity of the point process and are estimated under the assumption of the intensity of the events being constant across the study area, a generalisation of Ripley’s K function (1977) for inhomogeneous point patterns was computed.


Figure 5: A statistically significant clustering of airplane accidents occurred only at smaller distances.

The plotted K function in figure 5 shows that a statistically significant clustering of airplane accidents occurred only at smaller distances, whereas the empirical values Kˆ(r) at greater distances lie close to the expected values K(r) under CSR within the (simulated) envelope. Generally speaking, empirical values above the envelope would indicate a clustered distribution of the spatial points, while empirical values below the envelope would indicate a regular, i.e., dispersed, pattern. Thus, at best, only a marginal clustering of airplane accidents seems to be present, which is line with the preceding results.

Conclusion

The main objective of this blog post was to identify the spatial pattern of airplane accidents that occurred in Florida in 2014. In a nutshell, the results of the spatial analyses indicate that while the average number of airplane accidents per unit area varies spatially, only a marginal clustering of accidents was found with hotspots being in Central Florida and along the East Coast, respectively. However, these empirical findings should only serve as a point of departure for more advanced future analyses taking both covariates (e.g., causes of accidents, traffic volumes or air networks) and the temporal dimension of airplane accidents into account.

Literature

Baddeley, A., Rubak, E., & Turner, R. (2015). Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.

Bivand, R. S., Pebesma, E., & Gómez-Rubio, V. (2008). Applied Spatial Data Analysis with R. 2nd edition. New York: Springer.

Clarke, R. D. (1946). An application of the Poisson distribution. Journal of the Institute of Actuaries, 72, 481.

Diggle, P. J. (1985). A kernel method for smoothing point process data. Applied Statis- tics, 34(2), 138-147.

Diggle, P. J. (2014). Statistical Analysis of Spatial and Spatio-Temporal Point Patterns. 3rd edition. Boca Raton: CRC Press.

Gilovich, T., Vallone, R., & Tversky, A. (1985). The hot hand in basketball: On the misperception of random sequences. Cognitive Psychology, 17(3), 295-314.

Ripley, B. D. (1977). Modelling spatial patterns. Journal of the Royal Statistical Society, Series B (Methodological), 39(2), 172-212.