Probabilistic Approach to Estimate the Risk of Being a Cybercrime Victim

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1 Applied Mathematical Sciences, Vol. 9, 2015, no. 125, HIKARI Ltd, Probabilistic Approach to Estimate the Risk of Being a Cybercrime Victim Youssef Bentaleb EECOMAS-LAb, National School of Applied Sciences Ibn Tofail University, Kenitra, Morocco Abdallah Abarda EECOMAS-LAb, National School of Applied Sciences Ibn Tofail University, Kenitra, Morocco Hassan Mharzi EECOMAS-LAb, National School of Applied Sciences Ibn Tofail University, Kenitra, Morocco Said El Hajji LabMIA, Faculty of sciences Mohamed V University, Rabat, Morocco Copyright c 2015 Youssef Bentaleb, Abdallah Abarda, Hassan Mharzi and Said El Hajji. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The identification and the assessment of population classes who risk being cybercrime victims is a major problem especially when these classes are latent and unobservable. For this reason, we suggest a solution based on the probabilistic approach which derives from the Bayes theory. We adapt the latent class analysis to identify unobservable classes of victims and we propose a measure of risk of cybercrime based on conditional probabilities resulting from LCA.

2 6234 Youssef Bentaleb, Abdallah Abarda, Hassan Mharzi and Said El Hajji Mathematics Subject Classification: 62-07, 62H30, 62P25 Keywords: probabilistic approach, latent class analysis, Bayes theory, cybercrime 1 Introduction The latent class analysis (LCA) [1] was developed using two approaches. The first is that of Newton [2], it uses the log-linear models to describe the relationship between the manifest and the latent variables. The second approach [3] is based on the estimates by the maximum likelihood method, this approach was subsequently developed by using the EM algorithm [4]. This method is already used in identifying latent classes in many researches: Coffman and al. [5] used latent class analysis to examine the relationship between different patterns of drinking motivations and behaviors and suggest four classes of drinking motivations. Kiersten and al. [6] examined concordant result of drug use assessments in adults with schizophrenia and identified characteristics differentiating participants across classes. Lange and al. [7] evaluated a new screening instrument for personality disorders, and applied LCA to identify different classes of personality disorder severity. Dix [8] identified a ranges of student mental health using LCA and proposed a new composite measure of student mental health status. This method is also implemented in other health research [9] where the problem is the diagnosis of the hardest diseases (unobservable) from the symptoms (observable). We consider by analogy, that the disease is cybercrime and symptoms are the behaviors related to the use of the Internet and the security measures. Note that the LCA has never been used for this purpose. Indeed, to give an exact estimation of population victims of fraud or of cybercrime act is a kind of difficult. The first, is related to the form of the questions. Especially how to put precise questions about a critical subject like cybercrime. In other words, a direct question to individuals if they were victims of a cybercrime attack makes the results false because some individuals do not really know if they were victims. The second difficulty is in the nature of the phenomena, which is latent and unobservable. From these difficulties comes the idea to adopt the latent classes analysis [1]. This paper present LCA for identifying the classes who risk being cybercrime victims. And propose a measure of the degree of the risk of cybercrime based on the conditional probabilities.

3 Probabilistic approach to estimate the risk of being a cybercrime victim LCA to identify unobservable classes of victims 2.1 Model Description The latent class models are mainly based on the Bayes theory. To introduce the theory of these models, we will adopt the following notations [10] : k : index of the latent class, (k = 1,..., K) k N. Among these classes, we assume that there is a class of unobservable population who risk being cybercrime victims. v : index of the manifest variable, (v = 1,..., V ). V is the number of behaviors selected for assessing the risk of cybercrime. s : response patterns or outcome vector, (s = 1,..., S). s represents the responses to a list of behavioral interview questions s(v) : selected category of the variable or behavior v, (s(v) = 1,..., I v ) We also note p s the probability of outcome vector s and can be written as p s = K p s,k (1) k=1 p s,k denotes the unobserved probabilities of falling simultaneously in the categories denoted by vector s and the latent class k. The probability of outcome vector s conditional on latent class k is denoted by p s/k. According to Bayes formula and assuming conditional independence, p s,k can be written as p s can be written as follows p s,k = p k p s/k = p k V v p v,s(v)/k (2) p s = K k=1 p k V v p v,s(v)/k (3) 2.2 Parameter estimation The parameters to be estimated are p k and p v,s(v)/k. These parameters are estimated using the EM algorithm [4], it is made up of two important steps: the expectation step denoted by E and the likelihood maximization step denoted by M. The first step consists of calculating the expectation of the log-likelihood assuming that we have the information about classes. The second step consists of the maximization of log-likelihood function. Under the assumption of a multinomial distribution, the kernel of maximum

4 6236 Youssef Bentaleb, Abdallah Abarda, Hassan Mharzi and Said El Hajji likelihood can be simplified to: L = S s K k p N s,k s,k (4) where N s,k is the number of individuals in the sample who choose pattern s and which belong to the class k, this quantity is unknown. However, we know the value of n s (the number of individuals who have chosen the response pattern s). In step E, we give conditional expectation of N s,k, Bayes formula gives : p k/s = p k p s/k t=k t=1 p tp s/t (5) The probability p k/s can be estimated by p k/s = N s,k N s. The conditional expectation is : p k p s/k N s,k = n s p k/s = N s t=k t=1 p (6) tp s/t In step M, we maximize the log-likelihood function log(l) = S s K N s,k log(p k k V v p v,s(v)/k ) (7) The number of parameter is equal to p = K V v=1 (I v 1) + K determining the number of classes To determine the number of classes, we use the information criteria AIC [11] and BIC [12]. The AIC (Akaike Information Criterion) is one of the most known from information criteria. AIC = 2log(L) + 2p (8) The BIC (Bayesian information criterion) weights in a different ways the number of parameters. In the case of latent classes, it is equal to BIC = 2log(L) + Log(N)p (9) To decide the number of classes, we choose the model that minimizes these information criteria.

5 Probabilistic approach to estimate the risk of being a cybercrime victim Measuring the degree of the risk of cybercrime The conditional probabilities resulting from the LCA is the probability to respond to a behavior for each class. These probabilities are used to interpret the relationship between behaviors and latent class. For LCA models with categorical outcomes, each probabilities provide information on the probability of an individual in that class to endorse the item [11]. Figure 1: Relationship between latent classes and manifest variables Assuming that we have V manifest variables and K latent class. We also assume that the specific modality s (v) for each variable v the variable contributes to the construction of the given latent class k, then the degree of the risk of cybercrime denoted by ERC for each class k is the average of the conditional probabilities p v,s(v)/k. And we can write ERC k = v=v v=1 w vp v,s (v)/k v=v v=1 w v (10) Were w v is the importance of the variable v. The classes of individuals with high ERC, are the most exposed to risks of cybercrime.

6 6238 Youssef Bentaleb, Abdallah Abarda, Hassan Mharzi and Said El Hajji 2.5 Application procedure To address the problem of identifying the class of the population who risk being cybercrime victims and to measure risk levels, we follow the procedure described below Figure 2: LCA to identify unobservable classes of victims 3 Conclusion In this paper, we used a probabilistic approach to estimate the proportion of population who risk being cybercrime victims. And we proposed a measure of the degree of risk of cybercrime, this measure is resulting from conditional probabilities. By using LCA, we can solve the problem of identifying unobservable classes. This technique remains a very effective means for data mining and the risk assessment related to Cybercrime. Acknowledgements. The authors would like to thank the CMRPI (Moroccan Centre polytechnic research and innovation: for its support.

7 Probabilistic approach to estimate the risk of being a cybercrime victim 6239 References [1] P. Lazarsfeld, N. Henry, Latent Structure Analysis, Houghton-Mifflin, New York, [2] S. J. Haberman, A stabilized Newton-Raphson algorithm for log-linear models for frequency tables derived by indirect observation, Sociological Methodology, 18 (1988), [3] L. A. Goodman, Exploratory Latent Structure Analysis Using Both Identifiable and Unidentifiable Models, Biometrika, 61 (1974), [4] A.P. Dempster, N.M. Laird, D.B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. R. Stat. Soc., Ser. B, 57 (1977), [5] Donna L. Coffman, Megan E. Patrick, Lori Ann Palen, Brittany L. Rhoades, Alison K. Ventura, Why Do High School Seniors Drink? Implications for a Targeted Approach to Intervention, Prevention Science, 8 (2007), [6] Kiersten L. Johnson, Sarah L. Desmarais, Marvin S. Swartz, Richard A. Van Dorn, Latent class analysis of discordance between results of drug use assessments in the CATIE data, Schizophrenia Research, 161 (2015), [7] J. Lange, C. Geiser, K. H. Wiedl, H. Schöttke, Screening for personality disorders: A new questionnaire and its validation using Latent Class Analysis, Psychological Test and Assessment Modeling, 54 (2012), [8] Katherine L. Dix, Identifying Ranges of Student Mental Health Using Latent Class Analysis, Flinders University, [9] Linda M. Collins, Stephanie T. Lanza, Latent Class and Latent Transition Analysis: With Applications in the Social, Behavioral, and Health Sciences, The Pennsylvania State University, Wiley, [10] Peter G. M. Van Der Heijden, Ab Mooijaart, The EM Algorithm for latent class analysis with equality constraints, Psychometrika, 57 (1992),

8 6240 Youssef Bentaleb, Abdallah Abarda, Hassan Mharzi and Said El Hajji [11] Hirotugu Akaike, Information Theory and an Extension of the Maximum Likelihood Principle, Second International Symposium on Information Theory, [12] G. Schwarz, Estimating the dimension of a model, Annals of Statistics, 6 (1978), Received: August 15, 2015; Published: October 12, 2015