A latent Markov model for detecting patterns of criminal activity

The paper investigates the problem of determining patterns of criminal behaviour from official criminal histories, concentrating on the variety and type of offending convictions. The analysis is carried out on the basis of a multivariate latent Markov model which allows for discrete covariates affecting the initial and the transition probabilities of the latent process. We also show some simplifications which reduce the number of parameters substantially; we include a Rasch-like parameterization of the conditional distribution of the response variables given the latent process and a constraint of partial homogeneity of the latent Markov chain. For the maximum likelihood estimation of the model we outline an EM algorithm based on recursions known in the hidden Markov literature, which make the estimation feasible also when the number of time occasions is large. Through this model, we analyse the conviction histories of a cohort of offenders who were born in England and Wales in 1953. The final model identifies five latent classes and specifies common transition probabilities for males and females between 5-year age periods, but with different initial probabilities.

[1]  Lee Manning Wiggins,et al.  Panel analysis : Latent probability models for attitude and behavior processes , 1974 .

[2]  G. Thomas,et al.  Patterns of Drug Use among White Institutionalized Delinquents in Georgia: Evidence from a Latent Class Analysis , 1995, Journal of Drug Education.

[3]  Kathryn Roeder,et al.  Modeling Uncertainty in Latent Class Membership: A Case Study in Criminology , 1999 .

[4]  R. Langeheine,et al.  A Unifying Framework for Markov Modeling in Discrete Space and Discrete Time , 1990 .

[5]  L. A. Goodman.  Exploratory latent structure analysis using both identifiable and unidentifiable models , 1974 .

[6]  Paul F. Lazarsfeld,et al.  Latent Structure Analysis. , 1969 .

[7]  Jeroen K. Vermunt,et al.  Longitudinal data analysis using log-linear path models with latent variables , 2002 .

[8]  Brian Francis,et al.  Sex Offenders: Specialists, Generalists—or Both? , 2000 .

[9]  Alex R. Piquero,et al.  On the Development of Different Kinds of Criminal Activity , 2001 .

[10]  Neil Henry.  Latent structure analysis , 1969 .

[11]  Lain L. MacDonald,et al.  Hidden Markov and Other Models for Discrete- valued Time Series , 1997 .

[12]  G. Rasch.  On General Laws and the Meaning of Measurement in Psychology , 1961 .

[13]  Brian Francis,et al.  Patterns of offending behaviour : a new approach (Full report) , 2002 .

[14]  Julian Stander,et al.  MARKOV CHAIN ANALYSIS AND SPECIALIZATION IN CRIMINAL CAREERS , 1989 .

[15]  Steven Buyske,et al.  Adolescence-limited versus persistent delinquency: Extending Moffitt's hypothesis into adulthood. , 2001 .

[16]  Ab Mooijaart,et al.  Latent Markov models for catagorical variables and time-dependent covariates , 1998 .

[17]  L. Collins,et al.  Latent Class Models for Stage-Sequential Dynamic Latent Variables , 1992 .

[18]  K. Land,et al.  AGE, CRIMINAL CAREERS, AND POPULATION HETEROGENEITY: SPECIFICATION AND ESTIMATION OF A NONPARAMETRIC, MIXED POISSON MODEL* , 1993 .

[19]  Brian Francis,et al.  SEX OFFENDERS: SPECIALISTS, GENERALISTS—OR BOTH? A 32-year Criminological Study , 2000 .

[20]  B. Lindsay,et al.  Semiparametric Estimation in the Rasch Model and Related Exponential Response Models, Including a Simple Latent Class Model for Item Analysis , 1991 .

[21]  Z. D. Feng,et al.  Using Bootstrap Likelihood Ratio in Finite Mixture Models , 1994 .

[22]  Rolf Langeheine,et al.  Latent variables Markov models. , 1994 .

[23]  K. Land,et al.  How Many Latent Classes of Delinquent/ Criminal Careers? Results from Mixed Poisson Regression Analyses1 , 1998, American Journal of Sociology.

[24]  F. Bartolucci.  Likelihood inference for a class of latent Markov models under linear hypotheses on the transition probabilities , 2006 .

[25]  Brian Francis,et al.  Identifying Patterns and Pathways of Offending Behaviour , 2004 .

[26]  Brian Francis,et al.  Patterns of offending behaviour: a new approach , 2002 .

[27]  D S Nagin,et al.  Analyzing developmental trajectories of distinct but related behaviors: a group-based method. , 2001, Psychological methods.

[28]  Jan de Leeuw,et al.  A Latent Markov Model to Correct for Measurement Error , 1986 .

[29]  T. Moffitt.  Adolescence-limited and life-course-persistent antisocial behavior: a developmental taxonomy. , 1993, Psychological review.

[30]  L. Wasserman,et al.  Computing Bayes Factors by Combining Simulation and Asymptotic Approximations , 1997 .

[31]  Angela Dale,et al.  Discrete-Time Mixed Markov Latent Class Models , 1994 .

[32]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.

[33]  Jay Magidson,et al.  Latent Class Factor and Cluster Models, Bi-Plots, and Related Graphical Displays , 2001 .

[34]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[35]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[36]  T. Moffitt,et al.  LIFE-COURSE TRAJECTORIES OF DIFFERENT TYPES OF OFFENDERS* , 1995 .

[37]  J. Vermunt,et al.  Discrete-Time Discrete-State Latent Markov Models with Time-Constant and Time-Varying Covariates , 1999 .

[38]  F. V. D. Pol,et al.  MIXED MARKOV LATENT CLASS MODELS , 1990 .

[39]  Rolf Langeheine,et al.  Manifest and Latent Markov Chain Models for Categorical Panel Data , 1988 .

[40]  A. Blumstein,et al.  Assessing the Impact of Exposure Time and Incapacitation on Longitudinal Trajectories of Criminal Offending , 2001 .

[41]  Ab Mooijaart,et al.  Latent Markov Modelling of Recidivism Data , 2003 .

[42]  L. R. Rabiner,et al.  An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition , 1983, The Bell System Technical Journal.