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"Environmental criminologists begin their study of crime by asking
where and when crimes occur" (Brantingham and Brantingham 1991: 2).
Since criminal phenomena occur when a law, an offender, and a target coincide
in time and space, capturing temporal information with precision and accuracy
is crucial for analytical purposes. The availability of motivated offenders
and suitable targets changes for many locations throughout the day and,
therefore, crime is neither randomly nor uniformly distributed in time
or space. Thus, understanding these patterns can assist criminal justice
personnel in precision focused prevention efforts (Cohen and Felson, 1979;
Brantingham and Brantingham 1984, 1991; Ratcliffe 2000; Rossmo 2001).
Criminal justice databases record temporal information on almost all criminal
events. Sometimes, as with crimes against the person, a precise time is
known. However for many criminal events, such as burglary and automotive
theft, the exact time is not always known. Although some burglaries are
committed in which the offender is apprehended at the scene (or caught
by alarm or surveillance), most do not follow this pattern. What remains
is a time range (or time window) in which the event took place. The temporal
data of these time ranges record only a potential start and end time for
when the event occurred. The problem becomes how to investigate when the
criminal event occurred, having only a start and end time.
Traditional criminological techniques have attempted to get at a more
precise estimate of these ranges using a variety of measures. These measures
include choosing either the start or end time as the best indicator, using
a form of rigid time analysis, or some type of averaging measure. Unfortunately
the results of these traditional measures fail to meet any theoretical
or evidentiary support and therefore cease to be a final solution to the
problem.2 This has led to the statement
that "[the] temporal dimension of crime [has] lagg[ed] behind while
advances in crime location geocoding, mapping technology and user competence
have allowed the spatial element to flourish" (Ratcliffe 2002: 24).
Now at this point the critical thinker may ask: "If it is so difficult
to deal with the time range issue, why not ignore the range data and focus
the analysis on the events that have exact times?" Alternatively,
one could use the average, mode, or median of the known times to predict
the unknown times of other occurrences. These two possibilities are unsatisfactory
for two reasons. First range data may contain information about the criminal
event that exact data do not; therefore, the range data should be incorporated
into the analysis, if possible. And second, using a measure of central
tendency on the known cases for prediction is just as arbitrary as using
the same measure on ranges of the unknown cases.
An alternative to these traditional techniques in criminology, first pioneered
by Ratcliffe and McCullagh (1998), is aoristic analysis. Aoristic analysis
is a technique used when time windows exist that "can provide a temporal
weight and give an indication of the probability that an event occurred
within a defined period" (Ratcliffe 2000: 669). The advantage of
the aoristic technique is that it does not have to exclude any temporal
possibilities. All raw police data, including exact times and range times,
are incorporated into the analysis. This prevents the researcher or investigator
from throwing out a considerable amount of temporal range data gathered
on crimes like burglary and automotive theft (Ratcliffe 2000). If this
measure can solve the time range issue, it becomes a crucial advancement
for dealing with the temporal criminal event. It is, therefore, prudent
to investigate aoristic analysis to see if it can resolve the time range
(window) problem.
After providing a brief example of how aoristic analysis is actually performed,
the properties of aoristic analysis are shown. Then, an alternative methodology,
multinomial logistic regression, is presented to deal with the limitations
of aoristic analysis. An empirical application of both aoristic analysis
and multinomial regression is performed.
AORISTIC ANALYSIS
Figure 1 allows the theoretical conceptualization of the aoristic technique
to be shown. Across the top of Figure 1 is the overall timeline the events
occur within. This timeline can be divided into different minute, hour,
or day boundaries, with the important aspect being that the search block
lengths, as illustrated by search blocks 1 - 4, are of equal duration
(Ratcliffe 2000). Running vertically to the left of Figure 1 are the individual
event time spans (a, b, c, d) that represent the recorded criminal
events. Events a and c span the entire timeline, and therefore
all four search blocks, having a 25 percent probability of occurring within
each individual search block. Event b spans two of the search blocks,
and thus has a 50 percent probability of occurring in each of those blocks.
And event d falls entirely within one search block, so it occurs
with certainty within search block three. The values in each cell are
called the aoristic values.
Aoristic analysis uses this temporal information to calculate both an
aoristic sum and an aoristic probability. The aoristic sum is simply the
sum of all the probabilities (aoristic values) that occurred within each
search block. For example, the aoristic sum for search block one would
be 0.25 + 0.25 = 0.5. The aoristic probability divides the aoristic sum
by the number of criminal events to derive the probability distribution,
0.5/4 = 0.125 for search block one. These two values are represented in
the histogram in Figure 1 and the calculation is elaborated on below-for
a complete theoretical account of aoristic analysis, see Ratcliffe (2000;
2002).
Figure 1. Aoristic Analysis.
Source: Adapted from Ratcliffe (2000).
As with most theoretical representations it is useful to
include a concrete example of how data actually fits into the analysis-Figure
2 illustrates in a pragmatic fashion how the analysis is done. The timeline
represents a 24-hour day and four search blocks are established, each
with a duration of 6 hours (12 am - 6 am, 6 am - 12 pm, 12 pm - 6 pm,
and 6 pm - 12 am). There are four events listed vertically on the left
hand side of Figure 2, each with a start and end time.
Figure 2. Aoristic Analysis, a Concrete Example.
Source: Adapted from Ratcliffe (2000).
The first event occurred between 1 am and 10 pm, a total
duration of twenty-one hours. Since five of the twenty-one hours fall
within the first search block, it is assigned a value of 5/21. The second
and third search blocks are spanned completely, giving 6/21 for each.
The four remaining hours for the event fall in the fourth search block,
4/21. This process is continued for the entire data set. At the conclusion
of this process the cells are summed vertically in each of the search
blocks to produce an aoristic sum and divided by the number of events
to produce the aoristic probability. The use of this approach allows the
researcher to incorporate all criminal event data whether they are specific
times, or more often, range data.
PROPERTIES OF THE AORISTIC
TECHNIQUE
For simplicity, we abstract from events such as b and only deal
with criminal events that fall completely within one search block or cover
all search blocks in the analysis. It should be noted, however, that the
following results hold in a more complicated analysis; a simplified example
is provided for a clearer exposition of the properties of aoristic analysis.3
As stated above, each bar in the histogram of Figure 1 represents the
sum of the aoristic values for all of the criminal events used in the
analysis. In order to better illustrate the properties of aoristic analysis,
the probability distribution, which shows the probability that the criminal
event occurred in a particular search block, will be used-similar results
hold for the aoristic sum histogram.
Using only the criminal events that fall within one search block and the
criminal events that span all the search blocks, the aoristic probability
that a criminal event occurred during a particular search block can be
represented by:
(1)
The fractions u/n and ki/n
represent the proportions of unknown and known cases, respectively, such
that u/n + ki/n = 1. In the case of Figure 1, i =
4.
Suppose the number of unknown criminal events remains constant while the
number of criminal events that occur within one search block increases.
Beginning with one criminal event of each classification, we have u/n
= 1/2 and k/n = 1/2. Then, add one known case in order to have a total
number of cases of three, and u/n = 1/3 and k/n = 2/3. Continuing
in this manner, we can see that the proportion of unknown cases approaches
zero while the proportion of known cases approaches one.
Mathematically speaking, in the limit the effect of the
unknown cases in the aoristic probability becomes zero:
 (2)
Therefore, the probability of a criminal event occurring
in a particular search block is merely the proportion of total criminal
offences that occur within that particular search block i:
 (3)
and in the example above, we have a probability distribution
reflecting the proportion of criminal events within each search block.
Figure 3. Probability Distribution, No Unknown Events.
Conversely, suppose the number of known criminal events
remains constant, while the number of unknown criminal events increases.
Beginning again with one criminal event of each classification, we would
again have u/n = 1/2 and k/n = 1/2. However, if we add one
unknown case, we would have u/n = 2/3 and k/n = 1/3. Invoking
the concept of a limit again, the effect of the unknown events approaches
one, while the effect of the known criminal events approaches zero:
 (4)
such that each search block will have the same probability,
equal to w:
 (5)
This result gives the uniform distribution as the aoristic
probability distribution.
Figure 4. Probability Distribution, All Unknown Events.
This analysis has shown that the method of aoristic analysis
needs to be used with caution when disproportional amounts of extreme
time range data are held within the criminal justice database. Police
need to be sensitive to this issue and take appropriate steps to ensure
their findings are accurate. However, the utility for police to incorporate
range data into their analysis-without discarding it-far outweighs any
cautions that may or may not need to be employed. The technique is also
a tremendous step forward from other techniques and continues to highlight
academic relevance in dealing with real world police and community problems.
Once patterns in range data for a particular place are known, the scholar
may feel compelled to look for similarities, along with differences, with
other places. In order to facilitate these comparisons, statistical inference
may be used to test hypothetical relationships. We acknowledge that the
purpose of aoristic analysis is not statistical inference, but a smoothing
algorithm. Without statistical inference, however, the possible uses for
the method are limited. Second, for computational ease, the method of
calculating the aoristic value assumes a linear relationship between the
probability of the criminal event and time. Previous research has shown
that there is no reason to believe this relationship is linear, because
criminal opportunities are unevenly distributed over time and space (Brantingham
and Brantingham 1984).
Without a theoretical framework, and corresponding explanatory variables,
standard hypothesis testing is precluded and the analysis can only give
an indication of when the event is happening but cannot tell us why it
is happening at that particular time. Therefore, a technique is needed
that does not impose a randomly or uniformly distributed temporal aspect
of crime and instead incorporates a nonlinear relationship that allows
for hypothesis testing of those variables that are having the effect.
An alternative technique is suggested, multinomial logistic regression,
which incorporates the advantages of aoristic analysis and addresses the
issues of theory and testing.
MULTINOMIAL LOGISTIC REGRESSION
The use of regression analysis can incorporate a nonlinear relationship
between the criminal event and time through functional form and test the
hypotheses of various theoretical models. The alternative technique put
forth for analyzing the temporal aspect of criminal events is multinomial
logistic regression (MLR).
Multinomial logistic regression, developed by McFadden (1981),
belongs to a class of statistical techniques called categorical data analysis
that specify the dependent variable qualitatively (categories), such as
the search blocks used in aoristic analysis, so the unspecified criminal
event can be investigated. However, before we discuss the multinomial
logistic regression model, it will be instructive to discuss the standard
logistic model and its advantages over ordinary least squares regression.
The use of qualitative, or categorical, dependent variables causes problems
with ordinary least squares (OLS) regression models. In the case of the
binary choice model-2 categories-the dependent variable can be defined
as a 0-1 dummy variable, with the variable equal to 1 if a particular
choice is made. Upon estimation, the resulting predicted value of the
dependent variable can be interpreted as the probability of that choice,
such as the probability of a burglary occurring at a particular time (range).
The main problem that arises using OLS-the linear probability model in
this context-is that it is possible to produce predicted values outside
the 0-1 range (Kennedy 1998). Categorical variables also produce heteroskedasticity,
which invalidates statistical inference. However, the form of heteroskedasticity
is known and can be modeled.4 The predicted
values outside the 0-1 range are not a desirable property of a probability
since it implies deterministic behavior in the model. This difficulty
can be avoided using the nonlinear logistic function-see Figure 5. The
result of estimation using the logistic function is a ``true'' probability
because it is bounded by 0 and 1.
Figure 5. Logistic versus Linear Probability Model.
The probability that an event/choice occurs (Y =
1) and an event/choice does not occur (Y = 0) are given by:
 (6)
where e is the natural exponential function; X
is the matrix of independent variables thought to affect the choice of
search block, based on criminological theory; and  is
the vector of estimated parameters. It should be noted that given the
nonlinear nature of the probability function, cannot
have its OLS interpretation of the marginal effect of X on Y.
If a marginal effect is desired, the difference in probabilities when
changing a variable xi should be calculated,
but interpreted with caution as the probability difference will not remain
constant with different starting values for xi
(Greene 2000).
The method of calculation used for obtaining the  vector
is not the same as OLS and, therefore, there is no  to
measure goodness of fit: rather than minimizing the squared errors (least
squares) through the choice of ,
the logistic regression maximizes a likelihood function by choosing .
There is, however, a variant of for
logistic regression provided by McFadden (1974), the likelihood ratio
index:
 (7)
where ln L is the log-likelihood function with all
the model parameters from the model and  is
the log-likelihood function only including a constant term. As with the
measure of , this index
is bounded between 0 and 1. An alternative to measure the goodness of
fit is to provide the percentage of successful predictions, which is bounded
between 0 and 1.
The binary choice logistic model is easily extended to
many choices/categories: the multinomial logistic regression (MLR) model.
Similar to the binary choice logistic model, the estimation of the MLR
model provides the probability of a given search block being chosen. Since
there are multiple alternatives, estimation provides a vector
for each alternative in order to calculate their respective probabilities:
 (8)
where J + 1 the number of alternatives. The end result
of estimation is the ability to calculate the probability of each alternative
occurring, with those probabilities changing depending on the values of
the independent variables.
These resulting probabilities can be interpreted as representing the utility,
or gain, from the choices. In other words, it is the probability that
a particular choice will have the greatest gains over the other choices:
maximizes the returns from the criminal event, or equivalently, minimizes
being apprehended. This framework is usually referred to as the random
utility model. In this type of model, the choices made by individuals
will depend on the individual-specific attributes, choice- or category-specific
attributes, or both. The selection of the model employed depends on theory
and available data (Kennedy 1998).5
A COMPARISON OF THE TECHNIQUES
In order to facilitate a comparison of aoristic analysis and multinomial
logistic regression, the crime of break and enter, hereafter referred
to as burglary, in the City of Burnaby, British Columbia, Canada is investigated.
Since these two methods of analysis are not directly comparable (aoristic
analysis is descriptive and multinomial logistic regression is inferential)
the resulting output from these techniques is of the greatest interest.
The final output of aoristic analysis is a probability distribution, and
a similar probability distribution can be calculated from the multinomial
logistic regression results.6
Aoristic analysis uses the actual crime data and, therefore, is an appropriate
representation of the time distribution of the criminal event under study.
If multinomial logistic regression is to be proven useful, its resulting
probability distribution should be similar to that of aoristic analysis.
Its benefit over aoristic analysis is then judged by the underlying explanation
of why the probabilities are changing.
Burglary and Aoristic Analysis
The distribution of time range data is shown in Table 1. Just over one-third
(37.5 percent) of the burglaries occurred at a known time.7
However, a large portion of the burglary data exhibit significant time
ranges. A further 56.3 percent of the burglary data (totaling 93.8 percent
of all burglaries in the range data set) have a time range of up to 10
hours. This should be no surprise since the majority of burglaries occur
during the day when people are at work-an 8-hour workday plus commuting
time is generally between 9 and 10 hours. With such a large proportion
of the burglary data represented by time ranges, the utility of aoristic
analysis is quite apparent.
Table 1. Distribution of Time Ranges for Burglary.
The aoristic probability output is presented in Figure 6.
As expected according to routine activity theory (Cohen and Felson 1979),
due to a lack of guardianship when people are at work, the highest probability
of a burglary is during the early afternoon, with a spike in the probability
occurring just after midnight. With one-third of the burglary data recorded
with exact times, the probability distribution does not exhibit the tendency
towards the uniform distribution, shown above. In fact, aside from the
spike occurring after midnight, the probability distribution appears to
be normally distributed, centered at 1 pm.
Figure 6. Burglary and Aoristic Analysis.
Burglary and Multinomial Logistic Regression
In order to ease estimation, two-hour time blocks are used for the multinomial
logistic regression. The variables used for the inferential analysis are
the unemployment rate, ethnic heterogeneity, the standard deviation of
income, and the average dwelling value; all variables are measured at
the census tract level in which each burglary occurred. These variables
are chosen to represent social disorganization theory-see Sun et al. (2004)
for a recent analysis employing this theory.
The resulting probabilities are divided by two and shown at one-hour
intervals in Figure 7. It should be quite apparent that the probability
distribution is very similar to that of aoristic analysis, with the highest
probability being in the early afternoon and the presence of a spike in
probability shortly after midnight-the spike is not as pronounced as aoristic
analysis, however. The most notable difference between the two probability
distributions is that the multinomial logistic regression probability
distribution gives much less weight to the time ranges outside of the
early afternoon; from 1 pm to 5 pm, the aoristic total probability of
a burglary is 17.7 percent, whereas the multinomial logistic regression
probability for the same time period is 38.2 percent-a significant difference
that is likely due to some smoothing of the aoristic probabilities resulting
from the use of time ranges.
Figure 7. Burglary and Multinomial Logistic Regression.
Although the purpose of this comparison is not to test any particular
theory, moderate support for social disorganization theory is found-the
results of estimation are shown in Table 2. The magnitude and significance
levels of each variable, not surprisingly, vary for different time periods,
but are all significant for at least one of the time ranges. Therefore,
social disorganization theory not only proves useful in a spatial analysis
of crime, but a spatio-temporal analysis of crime as well.
Table 2. Multinomial Regression Results.
Note: Hour 1 - 2 is the base category.
CONCLUSIONS
Until recently, the temporal component of the criminal event had been
neglected in criminological studies while the spatial component had burgeoned.
Since environmental criminologists study both where and when the
criminal event occurs, the temporal component must advance along with
the spatial component to further the understanding of crime. Aoristic
analysis is such an advancement in the temporal component that attempts
to resolve the time range (window) problem that previous criminological
techniques could not deal with satisfactorily.
However, the current development of aoristic analysis is not the final
resolution to the temporal component of the criminal event. The aoristic
technique needs to be used with caution when a large proportion of the
crime data are extreme time ranges. Unfortunately, this limits the extent
to which the measure can be relied upon, but is a tremendous step forward
from other techniques and continues to highlight academic relevance in
dealing with real world police and community problems. Also, although
the technique, by definition, recognizes the importance of the temporal
aspect of the criminal event, aoristic analysis does not incorporate a
major component of environmental criminological understanding: crime is
neither randomly nor uniformly distributed over time and space. Only when
theoretical suppositions are successfully combined with robust statistical
measures will the temporal analysis of crime thrive alongside the spatial
component. A promising statistical measure is multinomial logistic regression.
The multinomial logistic regression technique has three advantages over
aoristic analysis that allow the academic and the practitioner to both
find greater depth in the empirical results. First, multinomial logistic
regression does not impose a linear relationship on the criminal event
and time. Second, multinomial logistic regression allows for the researcher
to use theoretical suppositions, such as social disorganization and routine
activity theories, to predict when the criminal event occurred. And finally,
in order to learn which theoretical suppositions are having the greater
affect on when the criminal event occurs, variables representing theory
can be isolated and tested.
The multinomial logistic regression methodology outlined in this paper
provides an alternative to aoristic analysis for investigating the temporal
criminal event, which allows for statistical inference. An empirical application
shows that both aoristic analysis and multinomial logistic regression
produce similar probability distributions, but also that there are theoretical
reasons (and corresponding statistical verification) for why crime happens
at particular times-lending support for an inferential analysis of the
spatio-temporal criminal event. Although we present an alternative methodology
to aoristic analysis, we do not presume to replace this form of analysis
in every instance. Practitioners in the field of criminal science need
a pragmatic approach for analyzing crime. They do not have the luxury
of waiting for data or techniques that are a perfect fit because investigations
run in real time, not having the benefit of academic hindsight.
ENDNOTES
1. We would like to thank Patricia L. Brantingham, Paul J. Brantingham,
and Seminar participants at the Western Society of Criminology, 30th Annual
Conference for comments. back
2. See Ratcliffe and McCullagh (1998) for a more detailed discussion.
back
3. In the examples below, the extremes of all exact times and all time
ranges will be discussed. Clearly, if the researcher were only dealing
with exact times, aoristic analysis would not be employed. These extreme
examples, however, are instructive to understanding the technique. back
4. These problems are not associated with aoristic analysis. back
5. The estimation of both binary and multinomial regression models is
provided in most statistical packages. back
6. Both analyses are performed at the city level. back
7. A time is considered "known" if the start time and end time
of the burglary are in the same hour: 5:00am - 5:59am, for example. back
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