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Fraud is a major challenge facing telecommunication industry. A huge amount of revenues are lost to these fraudsters who have developed different techniques and strategies to defraud the service providers. For any service provider to remain in the industry, the expected loss from the activities of these fraudsters should be highly minimized if not eliminated completely. But due to the nature of huge data and millions of subscribers involved, it becomes very difficult to detect this group of people. For this purpose, there is a need for optimal classifier and predictive probability model that can capture both the present and past history of the subscribers and classify them accordingly. In this paper, we have developed some predictive models and an optimal classifier. We simulated a sample of eighty (80) subscribers: their number of calls and the duration of the calls and categorized it into four sub-samples with sample size of twenty (20) each. We obtained the prior and posterior probabilities of the groups. We group these posterior probability distributions into two sample multivariate data with two variates each. We develop linear classifier that discriminates between the genuine subscribers and fraudulent subscribers. The optimal classifier (<i>
β_{A+B</i>}) has a posterior probability of 0.7368, and we classify the subscribers based on this optimal point. This paper focused on domestic subscribers and the parameters of interest were the number of calls per hour and the duration of the calls.

Communication industry has made the world a global village, and among all components of the industry, telecommunication is the most popular and most widely used [

This sector broadly has two types of users―domestic and commercial. There are cases where the connections are bought under domestic categories but the use is on a commercial scale. This causes substantial loss to the sector [

Today, telecommunication market all over the world is facing a severe loss of revenue due to fraudsters [

In this area, some researchers have used different methods to determine both customer churn and fraud detection. Fraud detection and subscribers churn are related in the sense that both are concerned with subscriber’s behavior. Among the models used for data mining for both churn and fraud detection are naïve Bayes model; Gaussian probability distribution; Decision Tree algorithm; logistic regression and artificial neural network (ANN). Data mining is the extraction of vital information from the bulk of data available to the telecommunication industry and using an appropriate predictive model to classify and determine the behavior of subscribers. By refining the data and building an appropriate statistical model, so much hidden information about the subscribers and service providers will be unveiled, see [

Because of the privacy agreement between the service providers and subscribers on one hand and to protect the service providers’ respective businesses on the other hand, the service providers hardly disclose their data. But nevertheless, simulation offers a close substitute for real life data. Hence, in this paper, we simulate data that depict the real life scenario and use it for the study. We simulate data on number of calls per unit time, and the call duration and our interest is on the domestic subscribers only. Eighty (80) sample data points were simulated for the study. The samples were categorized into four (4) with each having twenty (20) observations representing subscribers. The number of calls per subscriber over a period of time was also simulated and these represent real life data and are used for this study. The sample data generated from such process look like real life data drawn from a real system. We employed MINITAB 16.0 for the simulation of the data in this work. A sample of 20 observations each on the average number of calls and rate given as follows; 8 (t = 3), 5 (t = 4), 9 (t = 12), 6 (t = 7), were simulated for the study. The values such as 8, 5,∙∙∙, 7 outside the bracket represent the average number of calls per hour, and the values in bracket represent the average duration of the entire calls in minutes. Our interest is to develop a predictive data mining model for fraud detection in telecommunication industry. The simulated data were categorized into two sample multivariate data groups A and B. Most importantly, service providers determine their customers’ behaviour from the nature of their current calls and their past behaviour.

We need to know the history of these subscribers based on the information available to the network providers (service providers). This information is basically obtained from their call history. For this reason, the appropriate probability model that has a memory and can capture such a past history and relate it to the current history of subscribers’ is the Bayesian statistic model. However Bayesian statistics requires a prior probability. Some researchers make mistake of estimating the prior probability in this type of study using a continuous distribution as though the number of calls belong to a continuous random variable. Actually, the number of calls is a Poisson problem and therefore belongs to a discrete probability distribution. The value of Poisson random variables are the non-negative integers, and any random phenomenon for which a count is of interest can be modeled by assuming a Poisson distribution, provided that the random variables satisfies certain assumptions regarding the distribution [

Λ ( t ) = a t b and ω ( t ) = d Λ ( t ) d t = a b t b − 1 (1)

The parameters of the model are obtained by log linear transformation of the mean value function.

ln Λ ( t ) = ln a + b ln t (2)

and a plot of ln Λ ( t ) against ln(t) will yield the value of ln(a) as the intercept and b as the slope of the linear graph. If the shape parameter b = 1, there is a stationary increment and we have HPP( ω ) but for b > 1, we have NHPP( ω t) [

P n ( t ) = exp { − a b t b − 1 } [ { a b t b − 1 } n ] / n ! ; n = 0 , 1 , ⋯ (3)

[_{n}(t) = the probability of n number of calls at a given time (t) and the other notations retain their usual meaning as defined before.

The following assumptions must be satisfied by the random variables before we can use Equation (3) above:

Model Assumption:

The stochastic process { N ( t ) , t ≥ 0 } is called a non-homogeneous Poisson Process with rate function { ω ( t ) , t ≥ 0 } if

1) N ( 0 ) = 0 : (The number of events at time zero is equal to zero).

2) { N ( t ) , t ≥ 0 } has independent increment: (The number of events in non-overlapping time interval are independent).

3) lim h → 0 o ( h ) h = 0 : (o(h)―some function of smaller order than h which satisfy the limit).

4) P { N ( t + h ) = k + 1 / N ( t ) = k } = λ ( t ) ⋅ h + 0 ( h ) : (The probability that exactly one event will occur in a small interval of length t + h approximately equal to λ ( t ) ⋅ h + 0 ( h ) ).

5) P { N ( t + h ) = k / N ( t ) = k } = 1 − λ ( t ) ⋅ h + 0 ( h ) : (The probability that no event occur in a small interval of length t + h).

6) P { N ( t + h ) = k + j / N ( t ) = k } = o ( h ) ; j ≥ 2 : (The probability that more than one event will occur in a small interval of length t + h).

7) The events must occur at random [

Bayesian statistics model is adapted for the posterior distribution since it has the attribute of capturing the prior behaviour of these subscribers to determine their current behaviour. Hence, the predictive statistical model for this study is

P ( ψ = ζ / ω ) = P ( ζ / ω ) P ( ω ) ∑ ω = 0 n P ( ζ / ω ) P ( ω ) (4)

where P ( ψ = ζ / ω ) = the conditional probability that the random variable ψ assumes a specific value ζ given that its prior probability was ω . Note that ω is now a random variable. ∑ ω = 0 n P ( ζ / ω ) P ( ω ) = the joint probability distribution of the subscribers [

Our interest is to classify the subscribers as either genuine or fraudulent. Hence, this is a classification problem and linear discriminant analysis will be employed to classify the subscribers where they belong. This classification will enable service providers to determine the measures to take against these fraudsters. The discriminant analysis will discriminate between the legitimate subscribers and fraudulent ones within the network. The idea of discriminant analysis is a search for the differences in two or more groups that consist of multivariate measurements. One (or more) linear function(s) which maximally differentiate(s) between these groups are constructed. These functions are then used to classify new member of similar group into the appropriate group they belong and differentiate them from the group they do not belong to [

β = X T S − 1 ( μ ( 1 ) − μ ( 2 ) ) (5)

where X T = ( X 1 ⋯ , X p ) ; S − 1 is the inverse of the dispersion (variance-covariance) matrix and μ ( 1 ) − μ ( 2 ) is the difference in the mean vectors between the two multivariate samples and β is the linear discriminant function. We established the optimal classifier of the discriminant function and finally classify the sample data accordingly based on their posterior probability distributions. Two multivariate sample data with two variates will be derived from Equation (4). The two sample multivariate data with two variates each are the posterior probabilities of each group. Then, we shall classify the samples as belonging to either genuine or fraudulent subscribers based on the optimal classifier ( β A + B ). Our classification rule will be: classify the subscribers in group A into “A_{1}; A_{2}”, where A_{1} is the fraudulent subscribers and A_{2} is the genuine subscribers. Similarly, we do the same for group B designated by “B_{1}; B_{2}”. Fraud subscribers tend to make use of the services much more than the genuine subscribers and should therefore have higher probability distributions.

Definitions:

Subsc = subscribers.

n-call = the number of calls per subscriber per hour.

t (min) = the time spent on the calls.

Prop.n (n/N) = fraction of the number of calls in relation to the total number of calls.

Pr.of Prio = the probability of priors.

joint prb = the joint probabilities.

Posterior = the posterior probabilities.

Churn = the defection of subscribers from one network to another.

The average number of calls and time spent in each call are presented in

A plot of ln(t) against ln( ω t ) is presented in

From

The implication of the shape parameter being 1 indicate that the intensity function has stationery increment, through the PLP transformation; hence, this distribution follows HPP(ω) and the prior probability distribution of Equation (3) becomes

P n ( t ) = exp { − 9.0250 } [ { 9.0250 } n ] / n ! (6)

Equation (6) is presented in

Subsc | n-call | t(min) | ω t | ln(t) | ln( ω t ) |
---|---|---|---|---|---|

1 | 11 | 6 | 48 | 1.791759 | 3.871201 |

2 | 9 | 2 | 16 | 0.693147 | 2.772589 |

3 | 10 | 1 | 8 | 0 | 2.079442 |

4 | 11 | 1 | 8 | 0 | 2.079442 |

5 | 11 | 3 | 24 | 1.098612 | 3.178054 |

6 | 5 | 3 | 24 | 1.098612 | 3.178054 |

7 | 5 | 1 | 8 | 0 | 2.079442 |

8 | 7 | 3 | 24 | 1.098612 | 3.178054 |

9 | 11 | 2 | 16 | 0.693147 | 2.772589 |

10 | 6 | 4 | 32 | 1.386294 | 3.465736 |

11 | 9 | 2 | 16 | 0.693147 | 2.772589 |

12 | 8 | 1 | 8 | 0 | 2.079442 |

13 | 11 | 4 | 32 | 1.386294 | 3.465736 |

14 | 9 | 1 | 8 | 0 | 2.079442 |

15 | 8 | 2 | 16 | 0.693147 | 2.772589 |

16 | 8 | 2 | 16 | 0.693147 | 2.772589 |

17 | 10 | 4 | 32 | 1.386294 | 3.465736 |

18 | 8 | 1 | 8 | 0 | 2.079442 |

19 | 9 | 1 | 8 | 0 | 2.079442 |

20 | 7 | 4 | 32 | 1.386294 | 3.465736 |

n-call | (a) | Exp(-a) | a^n | Prob. | Prop. N | Prior Pr. | Joint pr. | Poste. Pr |
---|---|---|---|---|---|---|---|---|

11 | 9.025 | 0.00012 | 3.24E+10 | 0.097556 | 0.063584 | 0.006203 | 0.112189 | 0.055291 |

9 | 9.025 | 0.00012 | 3.97E+08 | 0.131354 | 0.052023 | 0.006833 | 0.112189 | 0.06091 |

10 | 9.025 | 0.00012 | 3.58E+09 | 0.118547 | 0.057803 | 0.006852 | 0.112189 | 0.061079 |

11 | 9.025 | 0.00012 | 3.24E+10 | 0.097262 | 0.063584 | 0.006184 | 0.112189 | 0.055124 |

11 | 9.025 | 0.00012 | 3.24E+10 | 0.097262 | 0.063584 | 0.006184 | 0.112189 | 0.055124 |

5 | 9.025 | 0.00012 | 59873.69 | 0.059874 | 0.028902 | 0.00173 | 0.112189 | 0.015424 |

5 | 9.025 | 0.00012 | 59873.69 | 0.059874 | 0.028902 | 0.00173 | 0.112189 | 0.015424 |

7 | 9.025 | 0.00012 | 4876750 | 0.116113 | 0.040462 | 0.004698 | 0.112189 | 0.041878 |

11 | 9.025 | 0.00012 | 3.24E+10 | 0.097262 | 0.063584 | 0.006184 | 0.112189 | 0.055124 |

6 | 9.025 | 0.00012 | 540360.1 | 0.09006 | 0.034682 | 0.003123 | 0.112189 | 0.027841 |

9 | 9.025 | 0.00012 | 3.97E+08 | 0.131354 | 0.052023 | 0.006833 | 0.112189 | 0.06091 |

8 | 9.025 | 0.00012 | 44012667 | 0.13099 | 0.046243 | 0.006057 | 0.112189 | 0.053992 |

11 | 9.025 | 0.00012 | 3.24E+10 | 0.097262 | 0.063584 | 0.006184 | 0.112189 | 0.055124 |

9 | 9.025 | 0.00012 | 3.97E+08 | 0.131354 | 0.052023 | 0.006833 | 0.112189 | 0.06091 |

8 | 9.025 | 0.00012 | 44012667 | 0.13099 | 0.046243 | 0.006057 | 0.112189 | 0.053992 |

8 | 9.025 | 0.00012 | 44012667 | 0.13099 | 0.046243 | 0.006057 | 0.112189 | 0.053992 |

10 | 9.025 | 0.00012 | 3.58E+09 | 0.118547 | 0.057803 | 0.006852 | 0.112189 | 0.061079 |

8 | 9.025 | 0.00012 | 44012667 | 0.13099 | 0.046243 | 0.006057 | 0.112189 | 0.053992 |

9 | 9.025 | 0.00012 | 3.97E+08 | 0.131354 | 0.052023 | 0.006833 | 0.112189 | 0.06091 |

7 | 9.025 | 0.00012 | 4,876,750 | 0.116113 | 0.040462 | 0.004698 | 0.112189 | 0.041878 |

The average number of calls and time spent in each call are presented in

A plot of ln(t) against ln( ω t ) is presented in

From

The prior probability distribution in Equation (3) becomes

P n ( t ) = exp { − 5.4739 } [ { 5.4739 } n ] / n ! (7)

Equation (7) is presented in

The average number of calls and time spent in each call are presented in

A plot of ln(t) against ln( ω t ) is presented in

From

The prior probability distribution in Equation (3) becomes

P n ( t ) = exp { − 8.1662 } [ { 8.1662 } n ] / n ! (8)

Equation (8) is presented in

The average number of calls and time spent in each call are presented in

A plot of ln(t) against ln( ω t ) is presented in

From

The prior probability distribution in Equation (3) becomes

P n ( t ) = exp { − 4.8550 } [ { 4.8550 } n ] / n ! (9)

Equation (9) is presented in

Subsc | n-call | t(min) | ω t | ln(t) | ln( ω t ) |
---|---|---|---|---|---|

1 | 8 | 5 | 25 | 1.609438 | 3.218876 |

2 | 6 | 5 | 25 | 1.609438 | 3.218876 |

3 | 9 | 3 | 15 | 1.098612 | 2.70805 |

4 | 7 | 1 | 5 | 0 | 1.609438 |

5 | 9 | 5 | 25 | 1.609438 | 3.218876 |

6 | 7 | 2 | 10 | 0.693147 | 2.302585 |

7 | 7 | 3 | 15 | 1.098612 | 2.70805 |

8 | 4 | 1 | 5 | 0 | 1.609438 |

9 | 7 | 3 | 15 | 1.098612 | 2.70805 |

10 | 5 | 5 | 25 | 1.609438 | 3.218876 |

11 | 5 | 6 | 30 | 1.791759 | 3.401197 |

12 | 7 | 1 | 5 | 0 | 1.609438 |

13 | 6 | 4 | 20 | 1.386294 | 2.995732 |

14 | 7 | 2 | 10 | 0.693147 | 2.302585 |

15 | 3 | 4 | 20 | 1.386294 | 2.995732 |

16 | 4 | 4 | 20 | 1.386294 | 2.995732 |

17 | 6 | 2 | 10 | 0.693147 | 2.302585 |

18 | 4 | 3 | 15 | 1.098612 | 2.70805 |

19 | 5 | 5 | 25 | 1.609438 | 3.218876 |

20 | 1 | 5 | 25 | 1.609438 | 3.218876 |

n-call | (a) | Exp(−a) | a^n | Prob. | Prop. N | Prior Pr. | Joint pr. | Poste. Pr |
---|---|---|---|---|---|---|---|---|

8 | 5.4739 | 0.004195 | 806,073.9 | 0.083863 | 0.068376 | 0.005734 | 0.122948 | 0.046639 |

6 | 5.4739 | 0.004195 | 26,901.79 | 0.156734 | 0.051282 | 0.008038 | 0.122948 | 0.065374 |

9 | 5.4739 | 0.004195 | 4,412,368 | 0.051006 | 0.076923 | 0.003924 | 0.122948 | 0.031912 |

7 | 5.4739 | 0.004195 | 147,257.7 | 0.122564 | 0.059829 | 0.007333 | 0.122948 | 0.059642 |

9 | 5.4739 | 0.004195 | 4,412,368 | 0.051006 | 0.076923 | 0.003924 | 0.122948 | 0.031912 |

7 | 5.4739 | 0.004195 | 147,257.7 | 0.122564 | 0.059829 | 0.007333 | 0.122948 | 0.059642 |

7 | 5.4739 | 0.004195 | 147,257.7 | 0.122564 | 0.059829 | 0.007333 | 0.122948 | 0.059642 |

4 | 5.4739 | 0.004195 | 897.8162 | 0.156925 | 0.034188 | 0.005365 | 0.122948 | 0.043636 |

7 | 5.4739 | 0.004195 | 147,257.7 | 0.122564 | 0.059829 | 0.007333 | 0.122948 | 0.059642 |

5 | 5.4739 | 0.004195 | 4914.556 | 0.171798 | 0.042735 | 0.007342 | 0.122948 | 0.059715 |

5 | 5.4739 | 0.004195 | 4914.556 | 0.171798 | 0.042735 | 0.007342 | 0.122948 | 0.059715 |

7 | 5.4739 | 0.004195 | 147,257.7 | 0.122564 | 0.059829 | 0.007333 | 0.122948 | 0.059642 |

6 | 5.4739 | 0.004195 | 26,901.79 | 0.156734 | 0.051282 | 0.008038 | 0.122948 | 0.065374 |

7 | 5.4739 | 0.004195 | 147,257.7 | 0.122564 | 0.059829 | 0.007333 | 0.122948 | 0.059642 |

3 | 5.4739 | 0.004195 | 164.0176 | 0.114671 | 0.025641 | 0.00294 | 0.122948 | 0.023915 |

4 | 5.4739 | 0.004195 | 897.8162 | 0.156925 | 0.034188 | 0.005365 | 0.122948 | 0.043636 |

6 | 5.4739 | 0.004195 | 26,901.79 | 0.156734 | 0.051282 | 0.008038 | 0.122948 | 0.065374 |

4 | 5.4739 | 0.004195 | 897.8162 | 0.156925 | 0.034188 | 0.005365 | 0.122948 | 0.043636 |

5 | 5.4739 | 0.004195 | 4914.556 | 0.171798 | 0.042735 | 0.007342 | 0.122948 | 0.059715 |

1 | 5.4739 | 0.004195 | 5.4739 | 0.022962 | 0.008547 | 0.000196 | 0.122948 | 0.001596 |

Subsc | n-call | t(min) | ω t | ln(t) | ln( ω t ) |
---|---|---|---|---|---|

1 | 11 | 14 | 126 | 2.639057 | 4.836282 |

2 | 11 | 15 | 135 | 2.70805 | 4.905275 |

3 | 12 | 10 | 90 | 2.302585 | 4.49981 |

4 | 13 | 10 | 90 | 2.302585 | 4.49981 |

5 | 8 | 7 | 63 | 1.94591 | 4.143135 |

6 | 12 | 9 | 81 | 2.197225 | 4.394449 |

7 | 12 | 15 | 135 | 2.70805 | 4.905275 |

8 | 10 | 5 | 45 | 1.609438 | 3.806662 |

9 | 3 | 11 | 99 | 2.397895 | 4.59512 |

10 | 4 | 14 | 126 | 2.639057 | 4.836282 |

11 | 14 | 17 | 153 | 2.833213 | 5.030438 |

12 | 12 | 16 | 144 | 2.772589 | 4.969813 |

13 | 6 | 4 | 36 | 1.386294 | 3.583519 |

14 | 14 | 6 | 54 | 1.791759 | 3.988984 |

15 | 8 | 12 | 108 | 2.484907 | 4.682131 |

16 | 9 | 12 | 108 | 2.484907 | 4.682131 |

17 | 5 | 16 | 144 | 2.772589 | 4.969813 |

18 | 12 | 7 | 63 | 1.94591 | 4.143135 |

19 | 14 | 9 | 81 | 2.197225 | 4.394449 |

20 | 8 | 16 | 144 | 2.772589 | 4.969813 |

n-call | (a) | Exp(−a) | a^n | Prob. | Prop. N | Prior Pr. | Joint pr. | Poste. Pr |
---|---|---|---|---|---|---|---|---|

11 | 8.1662 | 0.000284 | 1.08E+10 | 0.076653 | 0.055556 | 0.004258 | 0.065554 | 0.064962 |

11 | 8.1662 | 0.000284 | 1.08E+10 | 0.076653 | 0.055556 | 0.004258 | 0.065554 | 0.064962 |

12 | 8.1662 | 0.000284 | 8.80E+10 | 0.052164 | 0.060606 | 0.003161 | 0.065554 | 0.048226 |

13 | 8.1662 | 0.000284 | 7.18E+11 | 0.032768 | 0.065657 | 0.002151 | 0.065554 | 0.032819 |

8 | 8.1662 | 0.000284 | 19776986 | 0.139349 | 0.040404 | 0.00563 | 0.065554 | 0.085887 |

12 | 8.1662 | 0.000284 | 8.80E+10 | 0.052164 | 0.060606 | 0.003161 | 0.065554 | 0.048226 |

12 | 8.1662 | 0.000284 | 8.80E+10 | 0.052164 | 0.060606 | 0.003161 | 0.065554 | 0.048226 |

10 | 8.1662 | 0.000284 | 1.32E+09 | 0.103253 | 0.050505 | 0.005215 | 0.065554 | 0.079549 |

3 | 8.1662 | 0.000284 | 544.5779 | 0.025785 | 0.015152 | 0.000391 | 0.065554 | 0.00596 |

4 | 8.1662 | 0.000284 | 4447.132 | 0.052642 | 0.020202 | 0.001063 | 0.065554 | 0.016223 |

14 | 8.1662 | 0.000284 | 5.87E+12 | 0.019113 | 0.070707 | 0.001351 | 0.065554 | 0.020616 |

12 | 8.1662 | 0.000284 | 8.80E+10 | 0.052164 | 0.060606 | 0.003161 | 0.065554 | 0.048226 |

6 | 8.1662 | 0.000284 | 296565.1 | 0.117018 | 0.030303 | 0.003546 | 0.065554 | 0.054093 |

14 | 8.1662 | 0.000284 | 5.87E+12 | 0.019113 | 0.070707 | 0.001351 | 0.065554 | 0.020616 |

8 | 8.1662 | 0.000284 | 19776986 | 0.139349 | 0.040404 | 0.00563 | 0.065554 | 0.085887 |

9 | 8.1662 | 0.000284 | 1.62E+08 | 0.126439 | 0.045455 | 0.005747 | 0.065554 | 0.087672 |

5 | 8.1662 | 0.000284 | 36316.17 | 0.085977 | 0.025253 | 0.002171 | 0.065554 | 0.03312 |

12 | 8.1662 | 0.000284 | 8.80E+10 | 0.052164 | 0.060606 | 0.003161 | 0.065554 | 0.048226 |

14 | 8.1662 | 0.000284 | 5.87E+12 | 0.019113 | 0.070707 | 0.001351 | 0.065554 | 0.020616 |

8 | 8.1662 | 0.000284 | 19776986 | 0.139349 | 0.040404 | 0.00563 | 0.065554 | 0.085887 |

Subsc | n-call | t(min) | ω t | ln(t) | ln( ω t ) |
---|---|---|---|---|---|

1 | 4 | 2 | 12 | 0.693147 | 2.484907 |

2 | 4 | 10 | 60 | 2.302585 | 4.094345 |

3 | 13 | 7 | 42 | 1.94591 | 3.73767 |

4 | 8 | 5 | 30 | 1.609438 | 3.401197 |

5 | 3 | 7 | 42 | 1.94591 | 3.73767 |

6 | 5 | 10 | 60 | 2.302585 | 4.094345 |

7 | 5 | 4 | 24 | 1.386294 | 3.178054 |

8 | 5 | 5 | 30 | 1.609438 | 3.401197 |

9 | 2 | 1 | 6 | 0 | 1.791759 |

10 | 4 | 8 | 48 | 2.079442 | 3.871201 |

11 | 5 | 10 | 60 | 2.302585 | 4.094345 |

12 | 6 | 6 | 36 | 1.791759 | 3.583519 |

13 | 11 | 10 | 60 | 2.302585 | 4.094345 |

14 | 5 | 7 | 42 | 1.94591 | 3.73767 |

15 | 7 | 7 | 42 | 1.94591 | 3.73767 |

16 | 8 | 2 | 12 | 0.693147 | 2.484907 |

17 | 6 | 7 | 42 | 1.94591 | 3.73767 |

18 | 7 | 6 | 36 | 1.791759 | 3.583519 |

19 | 4 | 5 | 30 | 1.609438 | 3.401197 |

20 | 2 | 4 | 24 | 1.386294 | 3.178054 |

n-call | (a) | Exp(-a) | a^n | Prob. | Prop. N | Prior Pr. | joint pr. | Poste. Pr |
---|---|---|---|---|---|---|---|---|

4 | 4.855 | 0.007789 | 555.5932 | 0.180321 | 0.035088 | 0.006327 | 0.088983 | 0.071104 |

4 | 4.855 | 0.007789 | 555.5932 | 0.180321 | 0.035088 | 0.006327 | 0.088983 | 0.071105 |

13 | 4.855 | 0.007789 | 8.33E+08 | 0.001042 | 0.035088 | 3.65E-05 | 0.088983 | 0.000411 |

8 | 4.855 | 0.007789 | 308683.8 | 0.059634 | 0.035088 | 0.002092 | 0.088983 | 0.023515 |

3 | 4.855 | 0.007789 | 114.4373 | 0.148565 | 0.035088 | 0.005213 | 0.088983 | 0.058583 |

5 | 4.855 | 0.007789 | 2697.405 | 0.175092 | 0.035088 | 0.006144 | 0.088983 | 0.069043 |

5 | 4.855 | 0.007789 | 2697.405 | 0.175092 | 0.035088 | 0.006144 | 0.088983 | 0.069043 |

5 | 4.855 | 0.007789 | 2697.405 | 0.175092 | 0.035088 | 0.006144 | 0.088983 | 0.069043 |

2 | 4.855 | 0.007789 | 23.57103 | 0.091801 | 0.035088 | 0.003221 | 0.088983 | 0.036199 |

4 | 4.855 | 0.007789 | 555.5932 | 0.180321 | 0.035088 | 0.006327 | 0.088983 | 0.071105 |

5 | 4.855 | 0.007789 | 2697.405 | 0.175092 | 0.035088 | 0.006144 | 0.088983 | 0.069043 |

6 | 4.855 | 0.007789 | 13095.9 | 0.141678 | 0.035088 | 0.004971 | 0.088983 | 0.055867 |

11 | 4.855 | 0.007789 | 35324952 | 0.006893 | 0.035088 | 0.000242 | 0.088983 | 0.002718 |

5 | 4.855 | 0.007789 | 2697.405 | 0.175092 | 0.035088 | 0.006144 | 0.088983 | 0.069043 |

7 | 4.855 | 0.007789 | 63580.6 | 0.098264 | 0.035088 | 0.003448 | 0.088983 | 0.038748 |

8 | 4.855 | 0.007789 | 308683.8 | 0.059634 | 0.035088 | 0.002092 | 0.088983 | 0.023515 |

6 | 4.855 | 0.007789 | 13095.9 | 0.141678 | 0.035088 | 0.004971 | 0.088983 | 0.055867 |

7 | 4.855 | 0.007789 | 63580.6 | 0.098264 | 0.035088 | 0.003448 | 0.088983 | 0.038748 |

4 | 4.855 | 0.007789 | 555.5932 | 0.180321 | 0.035088 | 0.006327 | 0.088983 | 0.071105 |

2 | 4.855 | 0.007789 | 23.57103 | 0.091801 | 0.035088 | 0.003221 | 0.088983 | 0.036199 |

S/No | Variates | |
---|---|---|

1 | 0.055291 | 0.046639 |

2 | 0.06091 | 0.065374 |

3 | 0.061079 | 0.031912 |

4 | 0.055124 | 0.059642 |

5 | 0.055124 | 0.031912 |

6 | 0.015424 | 0.059642 |

7 | 0.015424 | 0.059642 |

8 | 0.041878 | 0.043636 |

9 | 0.055124 | 0.059642 |

10 | 0.027841 | 0.059715 |

11 | 0.06091 | 0.059715 |

12 | 0.053992 | 0.059642 |

13 | 0.055124 | 0.065374 |

14 | 0.06091 | 0.059642 |

15 | 0.053992 | 0.023915 |

16 | 0.053992 | 0.043636 |

17 | 0.061079 | 0.065374 |

18 | 0.053992 | 0.043636 |

19 | 0.06091 | 0.059715 |

20 | 0.041878 | 0.001596 |

S/No | Variates | |
---|---|---|

1 | 0.064962 | 0.071104 |

2 | 0.064962 | 0.071105 |

3 | 0.048226 | 0.000411 |

4 | 0.032819 | 0.023515 |

5 | 0.085887 | 0.058583 |

6 | 0.048226 | 0.069043 |

7 | 0.048226 | 0.069043 |

8 | 0.079549 | 0.069043 |

9 | 0.00596 | 0.036199 |

10 | 0.016223 | 0.071105 |

11 | 0.020616 | 0.069043 |

12 | 0.048226 | 0.055867 |

13 | 0.054093 | 0.002718 |

14 | 0.020616 | 0.069043 |

15 | 0.085887 | 0.038748 |

16 | 0.087672 | 0.023515 |

17 | 0.03312 | 0.055867 |

18 | 0.048226 | 0.038748 |

19 | 0.020616 | 0.071105 |

20 | 0.085887 | 0.036199 |

The variance-covariance matrix with two variates is given as:

( n i − 1 ) S i 2 = [ ∑ x 1 2 − n x ¯ 1 2 ∑ x 1 x 2 − n x ¯ 1 x ¯ 2 ∑ x 1 x 2 − n x ¯ 1 x ¯ 2 ∑ x 2 2 − n x ¯ 2 2 ]

where n_{1} and n_{2} respectively stand for the first and second samples respectively [

The variances are: X 11 = ∑ 1 20 X 1 2 − n X ¯ 1 2 ; X 22 = ∑ 1 20 X 2 2 − n X ¯ 2 2

The co-variances are: X 12 = X 21 = ∑ 1 20 X 1 X 2 − n X ¯ 1 X ¯ 2 ;

The observed sample multivariate data are the respective posterior probabilities of the four groups, the tendency is that their respective means will be equal thereby making the difference in the means vector to be zero. The reason behind this is that the sum of probabilities is one (1). But we can overcome this by observing the sample data carefully. Any of the sample point that cannot be approximated to two decimal places (2.d.p) with value is regarded as zero, and the sample size adjusted accordingly. Hence from sample A; delete serial number 19 from column 2; therefore, variate 1 has n = 20 and variate 2 has n = 19. Similarly, from sample B; delete serial numbers 3 and 13 from column 2; therefore, variate 1 has n = 20 and variate 2 has n = 18. The adjusted variance-covariance matrix for sample A are:

variances: X 11 = ∑ 1 n X 1 2 − n X ¯ 1 2 ; , X 22 = ∑ 1 n − 1 X 2 2 − ( n − 1 ) X ¯ 2 2

co-variances: X 12 = X 21 = ∑ 1 n − 1 X 1 X 2 − n 1 + n 2 2 X ¯ 1 X ¯ 2 ;

And

The adjusted variance-covariance matrices for sample B are:

Variances: X 11 = ∑ 1 n X 1 2 − n X ¯ 1 2 ; X 22 = ∑ 1 n − 2 X 2 2 − ( n − 2 ) X ¯ 2 2

co-variances: X 12 = X 21 = ∑ 1 n − 2 X 1 X 2 − n 1 + n 2 2 X ¯ 1 X ¯ 2 ;

The above is a symmetric matrix, where the diagonal elements are the variances. The upper and lower entries are the covariance. The values of the matrices are presented below.

From sample A and B, we have:

X ¯ ( A ) = [ 0.05 0.05255 ] ; X ¯ ( B ) = [ 0.05 0.055382 ]

( n A − 1 ) S A 2 = [ 0.0039 − 0.0012 0.0029 ]

From sample B, we have:

( n B − 1 ) S B 2 = [ 0.0129 − 0.0048 0.0053 ]

The pooled sample dispersion matrix is

( n A + n B − 2 ) S A + B = [ 0.0168 − 0.00168 0.0082 ]

The dispersion matrix is

S A + B = [ 0.0004541 − 0.00004541 0.0002216 ]

The inverse of the dispersion matrix is

S A + B − 1 = [ 2248.229 460.70423 4607.0423 ]

The linear discriminant function is

β = X T S A + B − 1 ( x ¯ ( 1 ) − x ¯ ( 2 ) ) T

The differences in the sample mean vector for sample A and B

X ¯ ( A ) − X ¯ ( B ) = [ 0 0.002832 ]

Since the variates are the posterior probabilities which cannot be negative, the difference in the mean vector cannot be negative.

β = X T S A + B − 1 ( X ¯ ( A ) − X ¯ (B) )

β = ( X 1 X 2 ) [ 2248.229 460.70423 4607.0423 ] [ 0 0.002832 ]

β = 1.30471 x 1 + 13.0471 x 2

β A = 1.30471 ( 0.05 ) + 13.0471 ( 0.05255 ) = 0.7509

β B = 1.30471 ( 0.05 ) + 13.0471 ( 0.055382 ) = 0.7226

The optimal classifier for discrimination is

β A + B = 1 2 ( β A + β B )

β A + B = 1 2 ( 0.7509 + 0.7226 ) = 0.7368

Classify subscribers in group A whose posterior probability is 0.7368 and above into group A_{1} and those whose posterior probability falls below 0.7368 into group A_{2}. Also classify subscribers in group B whose posterior probability is 0.7368 and above into group B_{1} and those whose posterior probability falls below 0.7368 into group B_{2}.

The subscribers that belong to A_{1} and B_{1} are the fraudulent subscribers while those that belong to A_{2} and B_{2} are the legitimate subscribers. From the sample observations in

The authors declare no conflicts of interest regarding the publication of this paper.

Amuji, H.O., Chukwuemeka, E. and Ogbuagu, E.M. (2019) Optimal Classifier for Fraud Detection in Telecommunication Industry. Open Journal of Optimization, 8, 15-31. https://doi.org/10.4236/ojop.2019.81002