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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.IEEE TRANSACTIONS ON DEPEDABLE AND SECURE COMPUTING4.3. Spending Profile of CardholdersThe spending profile of a cardholder suggests his normal spending behavior. Cardholders can bebroadly categorized into three groups based on their spending habits, namely, high spending (hs)group, medium spending (ms) group and low spending (ls) group. Cardholders who belong to thehigh spending group, normally use their credit cards for buying high-priced items. Similardefinition applies to the other two categories also.Spending profiles of cardholders are determined at the end of the clustering step. Let p ibe the percentage of total number of transactions of the cardholder that belong to cluster withmean c i . Then, the spending profile (SP) of the cardholder u is determined as follows:SP (u)= (10)arg max (p i )iThus, spending profile denotes the cluster number to which most of the transactions of thecardholder belong. In the example of Table III, the spending profile of the cardholder is 2, i.e. mand hence the cardholder belongs to the medium spending group.4.4. Model Parameter Estimation and TrainingWe use Baum-Welch algorithm to estimate the HMM parameters for each cardholder. Thealgorithm starts with an initial estimate of HMM parameters A, B and and converges to thenearest local maximum of the likelihood function. Initial state probability distribution isconsidered to be uniform, i.e., if there are N states, then the initial probability of each state is 1/N.Initial guess of transition and observation probability distributions can also be considered to beuniform. However, to make the initial guess of observation symbol probabilities more accurate,spending profile of the cardholder as determined in Section 4.3 is taken into account. We make15
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.IEEE TRANSACTIONS ON DEPEDABLE AND SECURE COMPUTINGthree sets of initial probability for observation symbol generation for three spending groups - ls,ms and hs. Based on the cardholder’s spending profile, we choose the corresponding set of initialobservation probabilities. The initial estimate of symbol generation probabilities using thismethod leads to accurate learning of the model. Since there is no a priori knowledge about thestate transition probabilities, we consider the initial guesses to be uniform. In case of acollaborative work between an acquiring bank and an issuing bank, we can have better initialguess about state transition probabilities as well.We now start training the Hidden Markov Model. The training algorithm has thefollowing steps (1) Initialization of HMM parameters, (2) Forward procedure and (3) Backwardprocedure. Details of these steps can be found in [18]. For training the HMM, we convert thecardholder’s transaction amount into observation symbols and form sequences out of them. Atthe end of the training phase, we get an HMM corresponding to each cardholder. Since this stepis done off-line, it does not affect the credit card transaction processing performance, whichneeds on-line response.4.5. Fraud DetectionAfter the HMM parameters are learnt, we take the symbols from a cardholder's training data andform an initial sequence of symbols. Let O 1 , O 2 , …O R be one such sequence of length R. Thisrecorded sequence is formed from the cardholder's transactions up to time t. We input thissequence to the HMM and compute the probability of acceptance by the HMM. Let theprobability be 1 , which can be written as follows: 1 = P(O 1 , O 2 , O 3 ,…O R | ) (11)16
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