**Author: Dr. Nawaf Alotaibi**

** Introduction**

From Tobler’s First Law of Geography in which “everything is related to everything else, but near things are more related than distant things” (Tobler, 1970, p.236), we can also apply the meaning of this law to the spatio-temporal aspect. Thus, we can assume that event that occurred today is more strongly related to event that will occur tomorrow than to those events that will happen next week, and events that occur throughout the day are strongly related to each other, rather than to the event that will happen tomorrow. Moreover, events that occur at the same period of time during the day are more strongly related to each other than to the event that occurred at another period of time of the same day, and so on. On the other hand, if event occurrences show a high concentration at a specific time and this concentration at this time is clustered at specific places, it can be said that there is an interaction between space and time. Hence, event that occurs closely in time and space is most likely to be associated with the same factors that contribute to the occurrences.

Spatial-temporal analysis tools such as kernel density, mean centre, standard deviational ellipse can provide us with useful information on how an incidents (events) change over time and space across the study region. For example, according to the visualization analysis, one can determine if crime increased during a certain time period or is lower in others. However, to effectively examine space and time clustering in an accurate manner, and maximise the power of the spatial and temporal analysis, the use of statistical techniques is critical to determine whether incidents that are clustering throughout the study region are also clustering throughout periods of time, and vice versa.

The following section will attempt to give an overview of the main techniques that have been used in different areas such as epidemiology and crime studies to detect spatial-temporal interaction.

** Methods of statistical spatiotemporal analysis**

A number of methods that have been proposed to detect the spatio-temporal interaction of phenomena are widely adapted in epidemiology research, such as the Knox test (Knox, 1964) Mantel test (Mantel, 1967) and Kuldorff’s space–time scan (Kulldorff et al., 2005). On the other hand, few crime studies have been carried out to detect spatio-temporal interaction; instead, these crime studies have, in general, conducted spatial and temporal aspects separately (Grubesic and Mack, 2008).

One of the most common statistical tests used to determine whether there is interaction between space and time is the Knox test (Zhang et al., 2015, Rey et al., 2012, Ward and Carpenter, 2000). The Knox test has been developed to identify interaction between space and time in medical epidemiology research(Knox, 1964), but it has also been adapted for criminology research (Zhang et al., 2015, Grubesic and Mack, 2008). Knox’s test works by providing an understanding of the possible clustering of examined events over specified spatial and temporal distances, such as 500 metres and within one week. In other words, it examines to what extent the occurrences of events are close to each other in space and time(Knox, 1964). However, the results of Knox’s method might be influenced by the user’s choice for specifying distances for time and space (Ward and Carpenter, 2000). The Knox test can also be adapted within the CrimeStat program.

Another method used in detecting interaction between space and time is Mantel’s test which is also widely used in the space-time clustering of outbreaks of diseases (Mantel, 1967). The Mantel test has been modified from the Knox test; the main difference between them is that Mantel’s test does not require the user to specify critical space and time distance thresholds but it does include spatial and temporal distances between all event pairs in the study area (Malizia et al., 2011, Manley, 2003). However, the critical drawback of the Mantel test is its aim, since it can only detect a linear relationship between space and time; this means that it overlooks any non-linear association between time and space that may exist over the study region (Jacquez, 1996). The Mantel test is available in R and CrimeStat software.

The output both of the Knox and Mantel tests can be visualized to see where the interaction between space and time of events is over the study region (Rey et al., 2012). However, these tests do not detect local changes over time for events across the study region but are only used as a global measurement for space-time clustering (Rey et al., 2012).

Another approach is the space-time K function which is different from the tests described earlier in that it can detect graphically the interaction between space and time within different spatial and temporal scales (Diggle et al., 1995). The basic idea of the space-time K function is that the number of expected events that occur within spatial distance (d) and time interval (t) from randomly selected events is compared to the mean number of events per unit area and per unit time(Si et al., 2008). The advantage of the space-time K function over other global space-time statistics is that it identifies the space-time interactions of events using different scales(Rushton et al., 2007). The space-time K function can also be used by R software.

In order to identify the location of specific clusters in a study area at certain time, space–time scan can be used (Kulldorff, 1997). Kulldorff’s space-time scan has been developed for detecting space-time disease clusters as other space-time statistics (Kulldorff, 2006). It compares the spatial and temporal locations of an observed number of events with the number of expected events if they randomly occurred when there is no interaction between space and time (Kulldorff, 2006). The main aim of this test is not to find the interaction between space and time for individual cases but to detect a cluster of events in a specific geographical area at a specific time (Law et al., 2014). When clusters are detected, the Monte Carlo test can be used to examine statistically the significance of these clusters(Kulldorff, 2006).The space-time scan statistic can be used with SatScan software and is also available in R.

It can be concluded, that there is a need to do further spatial-temporal analysis to statistically examine whether incidents (events) that cluster in space are also clustered in time in order to detect if there is interaction between space and time in incidents.

**References**

BERNASCO, W. & ELFFERS, H. 2010. Statistical analysis of spatial crime data. Handbook of quantitative criminology. Springer.

BRANTINGHAM, P. & BRANTINGHAM, P. 1993. Environment, routine and situation: Toward a pattern theory of crime. Advances in criminological theory, 5, 259-294.

CLARKE, R. V. & WEISBURD, D. 1994. Diffusion of crime control benefits: Observations on the reverse of displacement. Crime prevention studies, 2, 165-184.

DIGGLE, P. J., CHETWYND, A. G., HÄGGKVIST, R. & MORRIS, S. E. 1995. Second-order analysis of space-time clustering. Statistical methods in medical research, 4, 124-136.

ECK, J. E. The threat of crime displacement. Criminal Justice Abstracts, 1993. Springer-Verlag, 527-546.

ECK, J. E., GERSH, J. S. & TAYLOR, C. 2000. Finding crime hot spots through repeat address mapping.

GRUBESIC, T. H. & MACK, E. A. 2008. Spatio-temporal interaction of urban crime. Journal of Quantitative Criminology, 24, 285-306.

JACQUEZ, G. M. 1996. A k nearest neighbour test for space–time interaction. Statistics in medicine, 15, 1935-1949.

KNOX, E. 1964. The detection of space-time interactions. Applied Statistics, 25-30.

KULLDORFF, M. 1997. A spatial scan statistic. Communications in Statistics-Theory and methods, 26, 1481-1496.

KULLDORFF, M. 2006. SaTScanTM User Guide.

KULLDORFF, M., HEFFERNAN, R., HARTMAN, J., ASSUNÇAO, R. & MOSTASHARI, F. 2005. A space–time permutation scan statistic for disease outbreak detection. PLoS medicine, 2, e59.

LAW, J., QUICK, M. & CHAN, P. 2014. Bayesian spatio-temporal modeling for analysing local patterns of crime over time at the small-area level. Journal of quantitative criminology, 30, 57-78.

MALIZIA, N., MACK, E. A. & REY, S. J. Measuring Population Shift Bias in Tests of Space-Time Interaction. Proceedings of the 11th International Conference on Geocomputation. , 2011 London, UK.

MANLEY, D. K. 2003. Enabling analytical and Modeling Tools for Enhanced Disease Surveillance. Sandia National Labs., Albuquerque, NM (US); Sandia National Labs., Livermore, CA (US).

MANTEL, N. 1967. The detection of disease clustering and a generalized regression approach. Cancer research, 27, 209-220.

REPPETTO, T. A. 1976. Crime prevention and the displacement phenomenon. Crime & Delinquency, 22, 166-177.

REY, S. J., MACK, E. A. & KOSCHINSKY, J. 2012. Exploratory space–time analysis of burglary patterns. Journal of Quantitative Criminology, 28, 509-531.

RUSHTON, S., GOODFELLOW, M., O’DONNELL, A. & MAGEE, J. 2007. The epidemiology of atypical mycobacterial diseases in northern England: a space–time clustering and Generalized Linear Modelling approach. Epidemiology and infection, 135, 765-774.

SI, Y., DEBBA, P., SKIDMORE, A., TOXOPEUS, A. & LI, L. Spatial and temporal patterns of global H5N1 outbreaks. 2008. International Society for Photogrammetry and Remote Sensing.

TOBLER, W. R. 1970. A computer movie simulating urban growth in the Detroit region. Economic geography, 46, 234-240.

WARD, M. P. & CARPENTER, T. E. 2000. Analysis of time–space clustering in veterinary epidemiology. Preventive Veterinary Medicine, 43, 225-237.

ZHANG, Y., ZHAO, J., REN, L. & HOOVER, L. 2015. Space–Time Clustering of Crime Events and Neighborhood Characteristics in Houston. Criminal justice review, 0734016815573309.