Prediction of criminal careers through 2-dimensional Extrapolation T.K. Cocx, tcocx@liacs.nl 2/23/2019 W. Kosters et al.

? 2/23/2019 T.K. Cocx, tcocx@liacs.nl

2-Dimensional Extrapolation ? ? 2-Dimensional Extrapolation 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Research Area Criminal Career Study Computer Science Sociology Psychology Criminology Law 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Criminal Careers 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Analysis Goal Analysis 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Practical Factors Nature Frequency Seriousness Duration 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Strategic analysis done on this Paradigm Four factors Distance Measure Strategic analysis done on this Clustering Prediction 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Alignment 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Calculating Distance between Careers Nature Severity 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Results: Clustering and Classification Year 4 Year 3 Year 2 Year 1 2/23/2019 T.K. Cocx, tcocx@liacs.nl

2-Dimensional Extrapolation The ‘Marble in Funnel’ and the ‘Criminal Career Prediction’ are two variants of the same problem: Extrapolation of a time sequence in a plane. Year 1 Year 2 Year 3 Year 4 ? 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Regular Mathematical Extrapolation One variable (usually time, x) is given. One Variable (Value, Temperature, weight, etc, y) is dependant on the given variable 2/23/2019 T.K. Cocx, tcocx@liacs.nl

2-Dimensional Extrapolation One variable (usually time, t) is given. Two variables (x, y) are dependant on the given variable. Sometimes (as in the criminal career prediction) x and y are meaningless. Only the location relative to already placed elements is important. Relatively under-researched area in mathematics. 2/23/2019 T.K. Cocx, tcocx@liacs.nl

2-Dimensional Extrapolation Possible solutions Assume y depends on x Rotate image to optimally Arrange t-order on x-axis Regular second degree extrapolation Same as Left option Regular third degree extrapolation 2-Dimensional Extrapolation Spline interpolate items t and t+1 Extrapolate after tlast Different methods Assume x depends on t and y depends on t separately Extrapolate separately Combine in {x,y}-system 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Spline Extrapolation There are two choices in spline extrapolation: Straight line cont. Polynomial cont. 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Future Class Calculation Select n existing data points closest to extrapolated line. The closer to ‘last known’ point, the more accurate. Calculate expected attributes of individual under consideration with weighted average of the n points. Classify current individual using these attributes. 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Results Using the original Dutch National Criminal Record Database (App. 1 million offenders) 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Comparison High-dim. Extrapolation Speed increase with a factor 100. Accuracy increase with 0.8 percentage points. Multi-dimensional space is rather empty. Classification is based upon a 2-dimensional clustering anyway. 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Conclusion Criminal career analysis can serve as basis for career prediction. Using the concept of 2-dimensional extrapolation on an existing clustering yields the movement in time of an individual from his past to his future Using ‘straight line spline’ extrapolation with the maximum existing elements predicts the future class of an offender with an 88.7% Accuracy. 2/23/2019 T.K. Cocx, tcocx@liacs.nl

Interrogation 2/23/2019 T.K. Cocx, tcocx@liacs.nl