Assessing the contagiousness of mass shootings with nonparametric Hawkes processes
AAssessing the contagiousness of mass shootings withnonparametric Hawkes processes
Peter Boyd, James MolyneuxOregon State University, Department of StatisticsSeptember 18, 2020
Abstract
Gun violence and mass shootings are high-profile epidemiological issues facing the UnitedStates with questions regarding their contagiousness gaining prevalence in news media. Throughthe use of nonparametric Hawkes processes, we examine the evidence for the existence of con-tagiousness within a catalog of mass shootings and highlight the broader benefits of using suchnonparametric point process models in modeling the occurrence of such events.
Gun violence in the United States is a national public heath crisis [Bauchner et al., 2017] withfirearm homicide rates 19.5 times that of other high-income countries [Grinshteyn and Hemenway,2011]. Mass shootings in particular represent a phenomenon of interest in that these high-profileevents with multiple, and occasionally numerous, victims generate large amounts of media cover-age. Such media coverage may lead to both a contagion effect that may incite others to carry outsimilar acts as well as an imitation effect that may allow mass shooters to learn from those thatpreceded them [Meindl and Ivy, 2017]. Though the term mass shooting lacks a specific, rigorousdefinition, the number of gun related incidences with multiple victims has become so common inthe past two decades that research of these events has become a necessary component of publichealth studies in the United States [Dzau and Leshner, 2018]. From 2000 to 2018, the US Federal1 a r X i v : . [ s t a t . A P ] S e p ureau of Investigation (FBI) recorded 277 active shooter incidents in which an individual shootsand kills (or attempts to kill) others in a public space, resulting in 2430 casualties [Federal Bureauof Investigation, 2016]. The FBI further notes that the number of such incidents is on the rise, with69% of these incidents occurring between 2010 and 2018. The need to address the contagion factorof these events, whereby a single mass shooting event inspires or is correlated with future massshooting events, represents a fundamental question in the underlying mass shooting phenomenon.Previous research by Jetter and Walker [2018] proposed that the ideation and implementation ofmass shootings are linked to media coverage of such events. A contagion factor was also previouslyfound by Towers et al. [2015] which used a self-excitation contagion model to quantify the degree towhich previous events inspired future events. In their work, Towers et al. [2015] model the increasedprobability of a mass shooting event occurring on day t j given a previous event occurred on day t i , t i < t j , and the average duration of the contagion process T excite using an exponential probabilitydistribution. That is, the probability of a new mass shooting event occurring sometime in the 24hours of day t j is expressed as P ( t j | t i , T excite ) = (cid:90) t j − t i t j − t i − dx e − x/T excite T excite . Towers et al. [2015] then couple this probability model with a non-contagion related baseline numberof events, N ( t ) , and a total number of expected secondary events, N secondary , to compute anexpected number of events, N exp , on day t n expressed as N exp ( t n ) = N ( t n ) + N secondary (cid:88) i : t i A point process is a random collection of points { τ , τ , . . . } occurring in some metric space [Daleyand Vere-Jones, 2004]. These points often occur in some temporal or spatio-temporal window where t i ∈ R represents the temporal dimension of the i th point and s i ∈ R n represents the spatial coordi-nates of the i th point. In practice, R n is often taken to be R or R so that the spatial coordinatesland on some two-dimensional plane or three-dimensional space where the third dimension can betaken to be the depth of the point. For our purposes, we consider the occurrence of mass shootingevents to be a collection of n marked spatio-temporal points, { ( t i , x i , y i , m i ) : i = 1 , , . . . , n } , such3hat t i ∈ [0 , T ] represents the time the event occurred with 0 and T taken to be the start and end ofthe temporal window, respectively, and ( x i , y i ) ∈ [ −∞ , ∞ ] × [ −∞ , ∞ ] represents the spatial locationof the event. The mark, m i , of point i is then some additional covariate information which we taketo be the number of victims, excluding the perpetrator, of the i th mass shooting event. For themarks of the process, we define the number of victims to be the number of individuals either killedor injured during the shooting. In defining the marks in this way, we intend to measure how eventswith different numbers of victims impacts the ability of an event to incite future events.In general, point processes are typically modeled via their conditional intensity function, λ ( t ) or λ ( s, t ) for time and space-time point processes, respectively. The conditional intensity is defined asthe infinitesimal expected rate at which points occur given the history of the processes, H t . Thatis, we model the occurrence of points in time as λ ( t |H t ) = lim ∆ t → E [ N ([ t, t + ∆ t )) |H t ]∆ t or in space-time as λ ( s, t |H t ) = lim ∆ s, ∆ t → E [ N (( s, s + ∆ s ) × ( t, t + ∆ t )) |H t ]∆ s ∆ t where N ( · ) is taken to be a counting measure [Daley and Vere-Jones, 2004].In what follows, we introduce the self-exciting, or Hawkes, point process and then elaboratefurther on the estimation and evaluation of the nonparametric version of the processes. When the occurrence of a point causes the temporary elevation in the occurrence of future pointsnearby in time or space and time, we refer to such a process as a self-exciting point process.Foundational work in self-exciting point processes was done by Hawkes [1971] who defined theconditional intensity as λ ( t |H t ) = µ + (cid:88) i : t i The Brady Campaign ( ) is a nonprofit group advocating for guncontrol and striving to end gun violence. The organization is named after James Brady, a cabinetmember during the Ronald Reagan presidency who was shot during the assassination attempt onthe president. Brady, left permanently disabled from the gunshot, became an advocate for guncontrol. The group has compiled data including incidents in which at least three people wereshot or injured, but not necessarily killed. The data spans from February 2005 to January 2013,containing a total of 477 incidents. The Brady Campaign data set used in this article is alsoused in the Towers et al. [2015] analysis to allow for comparison of results. Data can be accessedhere: https://journals.plos.org/plosone/article/file?type=supplementary&id=info:doi/10.1371/journal.pone.0117259.s002 . The Stanford Mass Shootings in America data was compiled in an effort to create a comprehensivecollection of mass shooting data in the United States. Incidents included involve three or morepeople shot, but not necessarily killed. The data ranges from August 1966 to June 2016 whenmaintenance and updates to the database were halted. We utilize data beginning in January 1999,with Columbine happening months later on April 20, 1999, to study the occurrence of mass shootingsas a more modern phenomenon. The data originally contained 335 observations, but was reduced to262 to reflect the altered starting date. Data can be accessed here: https://library.stanford. du/projects/mass-shootings-america . Gun Violence Archive (GVA) ( ) is a nonprofit group thatcompiles records of gun related incidents in the United States. Incidents recorded involved four ormore people shot but not necessarily killed. New records are updated in near real time, with dataranging from January 2012 until the present. While some data sets exclude events such as gangviolence, GVA does not set any limiting terms to their definition of a mass shooting other thanthe number of individuals shot and killed, leading to a data set that contains a greater numberof events. For events in which the perpetrator is killed or commits suicide during the shooting,GVA also differs from the other data sets in that the perpetrator is included in the number of totalvictims. Mother Jones is an investigative journalism organization that has compiled a collection of massshootings under stricter criterion than others. With data ranging from 1982 until the present,Mother Jones initially recorded only incidents in which four or more people were killed. Whenthe United States government redefined a mass killing to involve three or more people, MotherJones followed suit, redefining the criterion for the database. For this analysis, the data will bereduced to events taking place on or after January 1, 1999. Data can be accessed here: .Dataset Beginning Date End Date Definition ObservationsBrady February 2005 January 2013 3+ killed 477Stanford January 1999 June 2016 3+ shot 262Mother Jones January 1999 February 2020 3+ killed 92GVA January 2012 December 2019 4+ shot 2024Table 1: Summaries for each data set used in the analysis including the time window of data used inthe analyses, definition of what constitutes a mass shooting, and the number of observations fallingin the time window. 11 Year Brady Year GVA Year Mother Jones Year Stanford N u m be r o f E v en t s pe r M on t h Figure 1: Monthly totals of the number of mass shootings for each data set. Mother Jones StanfordBrady GVA − − − − 10 11 − 15 16 − 20 21 − 50 50 + − − − − 10 11 − 15 16 − 20 21 − 50 50 + Number of victims P r opo r t i on o f da t a s e t Figure 2: Distribution of the number of victims for events in each data set. Nonparametric Hawkes processes were fit to each data set listed in Section 3 using the MISD algo-rithm to estimate their conditional intensity functions. Initially, the spatial triggering componentwas included in the conditional intensity function but was later dropped as the spatial triggeringcomponent was found to have little to no effect in triggering subsequent events. The remainingresults focus on the triggering of the temporal and mark components, g ( t ) and k ( m ) respectively.Intervals for the temporal triggering function were chosen to reflect natural breaks in the inter-event12ime differences, i.e. 2 weeks, 3 months, 6 months, +1 year, while the intervals for the marks trig-gering function were selected using quantiles to roughly allocate an equal number of events into eachinterval based on the number of victims. With discrete mark values, an exactly uniform division ofevents into quantiles could not be realized as certain values accounted for a large proportion of thedata that would otherwise have spanned several quantiles, specifically in the GVA data in whichroughly 55% of incidents involved four victims.Dataset Diagonal Mass Background Rate Number Offspring Number 13 Day OffspringBrady 10.39% 0.0168 0.8980 0.1913Stanford 28.42% 0.0117 0.7186 0.4140GVA 34.87% 0.3484 0.6516 0.6146Mother Jones 54.58% 0.0065 0.4592 0.0043Table 2: Numeric summaries of implementing the MISD algorithm for each data set. Diagonalmass indicates the percent of the probability matrix P that lies on the main diagonal. Backgroundrate is the estimated background rate of the data catalog. Number of offspring is the estimatednumber of events that are triggered offspring of previous events, and number of 13 day offspring isthe estimated number of offspring occurring within 13 days of an event.The diagonal mass of the probability matrix P , estimated background rate, average numberof offspring events, and average number of offspring events occurring in the first two weeks aredisplayed in Table 2. For most data sets, the majority of events are probabilistically treated as trig-gered events, with background events making up roughly 10% to 55% of observed mass shootings.The estimated background rate for the Brady, Stanford and Mother Jones data sets are estimatedto be between 0.007 to 0.017 mass shooting events per day while the background rate for GVA issubstantially larger with an estimated daily rate of mass shootings of 0.35.For the Brady data set, the model estimated the expected number of offspring per mass shootingevent to be roughly 0.90 events with 0.19 of those events, occurring in the first two weeks. Thisthen implies that for an event in the Brady data, 21% of the offspring events occur in the firsttwo weeks with the remaining 79% of events occurring sometime afterward. The Stanford datahad an estimated expected number of offspring per event of 0.72 with just over half, 0.41, of theseevents occurring in the first two weeks. The GVA data set had a slightly smaller overall expected13umber of offspring per event than Brady or Stanford with an expected number of 0.65 child events.However, the overwhelming majority, approximately 94%, of the offspring events occurred in thefirst two weeks. Meanwhile, the Mother Jones data had the smallest expected number of offspringper events, 0.46 child events per mass shooting, yet 99% of the child events occurred more than twoweeks after the initial mass shooting.The estimated histogram estimators for the triggering functions of each data set are shown inFigures 3 - 6. For each plot, the estimated constants of the histogram estimator step functionsare shown as a horizontal line spanning the time or mark sub-interval for which the constant wasestimated. The grey vertical bars then represent ± standard errors for each estimated constantof the histogram estimator. The standard error bars are truncated at zero to reflect only valuesthat plausibly represent the phenomenon of interest. The temporal triggering functions, g ( t ) , aredensities and thus the areas underneath the step function represent the probabilities of child eventoccurring over some time-span. The marks triggering functions, k ( m ) , represent productivity mul-tipliers which increase or decrease the rate of triggered events based on the number of victimsimpacted in prior mass shootings. The x -axes of the temporal triggering functions are truncated asthe functions tended towards zero as t j − t i , for j > i , grew larger; x -axes of the marks triggeringfunctions are truncated shortly after the final sub-interval as shown graphically.In general, with the exception of Mother Jones, the value of the temporal triggering function, g ( t ) , monotonically decreases as t increases to each subsequent time bin. For the Brady data, thedecrease in the temporal triggering decreases more smoothly from roughly 0.0152 to 0.0078, to0.0018 down to 0. For the Stanford and GVA data, the decay in the temporal triggering decreasesmuch more drastically; from 0.41 down to 0.0054 down to zero for the first three time intervals inthe Stanford data and from 0.067 down to essentially zero in the first two time intervals in the GVAdata. For the Mother Jones data, the temporal estimates of the triggering are more volatile withestimates starting around 0.0007 and 0.008 for the first and second time interval, rises to around0.0034 in the third and fourth intervals, then finally falls to zero. The Mother Jones data is alsounique in that the estimated constants of the triggering function are much smaller in value thanthe other data sets. 14or the estimated triggering functions of the marks for the Brady data, k ( m ) had an estimatedproductivity of around 0.71 for the initial interval, and then increased to 1.43 for five victims, beforefalling to 0.91 for 6-8 victims and 0.57 for nine or more victims. The estimated mark triggeringfunctions for Stanford and GVA contain the same pattern of an initial increase followed by twodescending values. Stanford has an estimate of 0.99 for the initial bin and then jumped to 1.18 forfive victims, before falling to 0.41 for 6-7 victims and 0.14 beyond 7 victims. GVA begins with at0.55, increasing to 0.83 for 5 victims, then falls to 0.80 and 0.21 for 6-9 and 10+ victims, respectively.For the Mother Jones data, k ( m ) also followed a less consistent form, with the highest value of 1.24in the first bin before falling to 0.19 for 7 - 10 victims and 0.0007 for 11 - 17 victims before risingto 0.28 for 18 or more victims. The Stanford data yielded an estimated k ( m ) that did not followa monotone pattern, beginning at 0.99 in the first bin, increasing to 1.18 for five victims, thendecreasing to 0.41 for six or seven victims, and 0.14 for larger numbers of victims. t (time in days) g ( t ) m (number of victims) k ( m ) Figure 3: Brady Campaign triggering functions. In the figure on the left, values of the temporaltriggering function are plotted over time, with the time bins used in the analysis shown on the x axis. In the figure on the right, values of the marks triggering function are plotted over the marks(number of people injured). Standard error regions are shown in gray, and latter time bins with g ( t ) ≈ and the final mark bin is truncated in the figure.Figures 7 - 10 show the observed number of monthly mass shootings for each data source alongwith the estimated number of monthly shootings based on the models. The estimated values arecomputed by taking the median conditional intensity for each month and multiplying it by thelength of the month. The models appear to fit the data fairly well in that the estimated number of15 .000.010.020.030.04 0 14 93 t (time in days) g ( t ) m (number of victims) k ( m ) Figure 4: Stanford triggering functions. In the figure on the left, values of the temporal triggeringfunction are plotted over time, with the time bins used in the analysis shown on the x axis. In thefigure on the right, values of the marks triggering function are plotted over the marks (number ofpeople injured). Standard error regions are shown in gray, and latter time bins with g ( t ) ≈ andthe final mark bin is truncated in the figure. t (time in days) g ( t ) m (number of victims) k ( m ) Figure 5: Mother Jones triggering functions. In the figure on the left, values of the temporaltriggering function are plotted over time, with the time bins used in the analysis shown on the xaxis. In the figure on the right, values of the marks triggering function are plotted over the marks(number of people injured). Standard error regions are shown in gray, and latter time bins with g ( t ) ≈ and the final mark bin is truncated in the figure.monthly mass shootings tends to follow the trends in the the observed values. The Mother Jonesand Stanford data sets, Figures 8 and 9 respectively, contain instances where no mass shootingevents occurred over a sequence of consecutive months. For these months, the models tended to16 .000.020.040.06 0 14 93 t (time in days) g ( t ) m (magnitude) k ( m ) Figure 6: GVA triggering functions. In the figure on the left, values of the temporal triggeringfunction are plotted over time, with the time bins used in the analysis shown on the x axis. In thefigure on the right, values of the marks triggering function are plotted over the marks (number ofpeople injured). Standard error regions are shown in gray, and latter time bins with g ( t ) ≈ andthe final mark bin is truncated in the figure.over-estimate the number of events as the model assumes a constant background rate. Date Number of Monthly Events ObservedEstimated Figure 7: Brady Campaign conditional intensity plot. The number of monthly mass shootings isplotted (solid line) over time. The median value of the estimated conditional intensity of the observedpoints is calculated for each month, multiplied by the number of days in each corresponding month,and plotted (dashed line) over time.Super-thinning was implemented to evaluate each model’s fit to the individual data sets with17 Date Number of Monthly Events ObservedEstimated Figure 8: Stanford conditional intensity plot. The number of monthly mass shootings is plotted(solid line) over time. The median value of the estimated conditional intensity of the observedpoints is calculated for each month, multiplied by the number of days in each corresponding month,and plotted (dashed line) over time. Date Number of Monthly Events ObservedEstimated Figure 9: Mother Jones conditional intensity plot. The number of monthly mass shootings is plotted(solid line) over time. The median value of the estimated conditional intensity of the observed pointsis calculated for each month, multiplied by the number of days in each corresponding month, andplotted (dashed line) over time.tuning parameter, b , set to the median estimated conditional intensity for each source. To assess theoverall fit of the model, the residual process for each data set is displayed as histograms in Figures 1118 Date Number of Monthly Events ObservedEstimated Figure 10: GVA conditional intensity plot. The number of monthly mass shootings is plotted (solidline) over time. The median value of the estimated conditional intensity of the observed points iscalculated for each month, multiplied by the number of days in each corresponding month, andplotted (dashed line) over time.- 14. If the model fits the data well, then we would expect the histograms to demonstrate a roughlyuniform distribution throughout the entire time window. Of the four data sets, the estimated modelfor the Mother Jones data appears the least uniform in shape with substantial deviations throughoutthe time-window. The residual process for the GVA data appears the most uniform overall, thoughalso with some deviations. The distributions of the Brady and Stanford deviation are somewherein the middle with many time intervals appearing roughly uniform with some systematic deviationsfor certain time periods. The residual process for the Stanford data appears to have, in general,lower values prior to 2005 and slightly higher values in the years following, while the Brady residualprocess exhibits more of a unimodal distribution with a peak in values from 2008 - 2010. In this article, we investigate the contagiousness of mass shootings by treating the data as a markedself-exciting point process and analyze it through nonparametric Hawkes procedures. The conta-giousness of mass shootings was previously studied by Towers et al. [2015], reporting that eachmass shooting will incite at least 0.30 new events brought on by an increase in probability of eventsthat lasts for 13 days after an event. The self-excitation contagion model utilized in the Towers19 Date f r equen cy Figure 11: Brady Campaign histogram of super-thinned process. After super-thinning is imple-mented, the data are plotted over time, displaying the distribution of the super-thinned process. Date f r equen cy Figure 12: Stanford histogram of super-thinned process. After super-thinning is implemented, thedata are plotted over time, displaying the distribution of the super-thinned process.analysis requires several parametric assumptions including assuming a distribution for the decayof contagiousness, a constant number of secondary events, and the duration of contagion process.With little research on the contagiousness of mass shootings, circumventing the reliance on para-metric assumptions through a nonparametric modeling framework is an important contribution tothe study of this devastating phenomenon. 20 .02.55.07.5 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 Date f r equen cy Figure 13: Mother Jones histogram of super-thinned process. After super-thinning is implemented,the data are plotted over time, displaying the distribution of the super-thinned process. Date f r equen cy Figure 14: GVA histogram of super-thinned process. After super-thinning is implemented, thedata are plotted over time, displaying the distribution of the super-thinned process.Through our nonparametric approach, we see evidence that events may produce higher numbersof offspring than previous results, with estimated number of offspring ranging from 0.59 to 0.86,as much as almost 3 times the value reported by Towers et al. [2015] when using the same BradyCampaign data set. We also note that a contagion effect exists after 13 days with expected number21f offspring ranging from 0.03, based on the Mother Jones data, up to 0.60, with the GVA dataset and 0.18 events in the Brady data. The mean of these four values is 0.29, yielding an expectednumber of offspring within a 13 day period similar to the 0.30 reported by Towers et al. [2015].Also, similar to the results found in the Towers article, we noted no substantial spatial effect usingthe nonparametric framework.In Figures 3 - 6, the temporal histogram estimators tended to agree that the initial two-weekperiod after a mass shooting event tended to have larger contagion effects compared to time periodsafter the initial two weeks, save for Mother Jones which had a temporal histogram estimator whichwas much more volatile. This volatility might not be entirely unexpected given that the MotherJones data set had slightly more than one-third of the total number of observations compared tothe next smallest data set but featured the longest time window of all the data sets. These factorsthen imply that very few of the pairwise time differences between events in the Mother Jones datafall in the shorter time intervals. The GVA data meanwhile is by far the largest data source withthe shortest time-window and, as seen in Figure 6, shows that nearly all of the contagion factoroccurs in the first two-weeks. This is likely due to many of the pairwise inter-event time differencesoccurring relatively quickly after previous events.The triggering functions for the marks show much less consistency between the data sets butdemonstrates the benefit of allowing the expected number of secondary events to vary depending onthe size of the marks. The histogram estimator for the number of victims for the Brady campaign,Figure 3, demonstrates that mass shootings with larger numbers of victims increases the produc-tivity of those events in spurring future events. The Stanford data meanwhile, Figure 4, shows thatevents with between four to seven victims were more productive than larger events with greaterthan seven victims. A similar result was seen in the GVA data, Figure 6. Again, the histogramestimator for the Mother Jones data, Figure 5, is drastically different compared to the rest in thatsmaller events are more productive than larger events with high victim numbers. Furthermore,while the model appears to be finding some signal in regards to how the number of victims impactsthe productivity of mass shooting events to spur future events, it should be noted that there’s aconsiderable amount of uncertainty in these estimates, as represented by the standard error bars,especially for the Stanford and GVA data. 22igures 11 - 14 show the results of super-thinning the point process models for the different datacatalogs. By considering the uniformity of the super-thinned residual processes we can evaluate theoverall fit of the models in that models that fit the data well should have a uniform appearance inthe histograms. In Figure 11, we observe that the residual process for the Brady model has a uni-modal appearance rather than the desired uniform distribution. Examining Figure 15, which showsthe composition of the points for the super-thinned residual process for the Brady data, allows usto further investigate the unimodal distribution. The simulated lines at the top of the plot showsthe points which were superposed while the retained lines show the points of the original processwhich were retained after thinning. The points which were thinned are then shown at the bottomof the plots. From the figure, it is evident that the super-thinned process simulates events in areasof low intensity and removes events from areas of high intensity, but by simultaneously analyzingFigures 11 and 15, we see that a lack of sufficient thinning spurs departures from uniformity in thehistogram. This lack of thinning then indicates that the model was not able to capture the fullcontagion effect present in the data. thinnedretainedsimulated 2006 2008 2010 2012 Year Figure 15: Brady Campaign super-thinning plot. After super-thinning is implemented, events areplotted by their classification type over time, indicating events that were removed (thinned) fromthe data, events that were not removed (retained), and simulated events were superposed into thedata (simulated).In Figure 12, we observe an approximately uniform distribution, save a few spikes and falls, most23otably at the end of 2010. Throughout the Stanford catalog, super-thinning appears to be perform-ing as expected, despite the abrupt increase in the number of shootings that can be seen in Figure 8.Figure 14 also displays an approximately uniform distribution after super-thinning the GVA catalog.Figure 13 shows a non-uniform distribution of super-thinned residuals for the Mother Jones data.With so few events recorded in the Mother Jones data set, well-fitting models are more challengingto realize without adding further complexity to the model. In Figure 9, the frequency of observedevents appears to vary considerably over time, with 40% of events occurring in only the last fiveyears of the catalog. With such disparity in the frequency of events, fitting a single background ratefor the entire process may oversimplify trends in the data; employing a nonconstant backgroundrate may allow for a stronger representation of the data.Varying data sets and definitions of mass shootings lead to seemingly inconsistent trends andresults across analyses; more conclusive findings may be obtained with a more consistent definitionof such events and better data collection methodologies. Comparisons of results across data sets canbe difficult with data sources providing wildly different estimates; the Gun Violence Archive reports2024 mass shootings over eight years, while the original Mother Jones data reports 118 incidentsover nearly thirty-eight years. Although Brady and Mother Jones both define mass shootings asevents in which three or more individuals are killed, the number of events in each data set arestarkly different. The Stanford data set offers the well-fitting model but excludes data post 2016.The GVA and Mother Jones data, as shown in Figure 1, have an upward trend in the numberof mass shootings in later years; this trend may have also been evident in the Stanford data sethad data collection been continued, potentially offering a broader understanding of mass shootingcontagion, especially in later years.Despite wildly different data and definitions, results are consistent in that a large percentage ofmass shootings are probabilistically treated to be triggered events through the application of theMISD algorithm. Such findings support previously studied assertions that mass shootings may bemotivated by a contagion effect spread through media.24 Conclusion In this article, we assess the the contagiousness of mass shootings using a nonparametric Hawkesprocess framework for a variety of data sources. This framework relies on fewer parametric as-sumptions than previous studies and detects a contagion effect which varies over both time and thenumber of victims. We also find that the level of contagion is contingent upon the data source usedas no definitive catalog of data for mass shootings yet exists.Although the estimated conditional intensity for each process appears to closely mirror the truedata process, more complex models with additional features may yield better fitting models in thefuture. Specifically, adapting a nonconstant background rate over time and/or a productivity func-tion which is allowed to vary over time would allow future models to capture temporal changes tothese two components. More complex models might also allow for the incorporation of meaningfulspatial attributes or additional relevant covariates. 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