# Hierarchical Neural Regression Models for Customer Churn Prediction

Authors: Golshan Mohammadi; Reza Tavakkoli-Moghaddam; Mehrdad Mohammadi
Journal: Journal of Engineering
 (2013)

---

## Abstract

As customers are the main assets of each industry, customer churn prediction is becoming a major task for companies to remain in competition with competitors. In the literature, the better applicability and efficiency of hierarchical data mining techniques has been reported. This paper considers three hierarchical models by combining four different data mining techniques for churn prediction, which are backpropagation artificial neural networks (ANN), self-organizing maps (SOM), alpha-cut fuzzyc-means (α-FCM), and Cox proportional hazards regression model. The hierarchical models are ANN + ANN + Cox, SOM + ANN + Cox, and α-FCM + ANN + Cox. In particular, the first component of the models aims to cluster data in two churner and nonchurner groups and also filter out unrepresentative data or outliers. Then, the clustered data as the outputs are used to assign customers to churner and nonchurner groups by  the second technique. Finally, the correctly classified data are used to create Cox proportional hazards model. To evaluate the performance of the hierarchical models, an Iranian mobile dataset is considered. The experimental results show that the hierarchical models outperform the single Cox regression baseline model in terms of prediction accuracy, Types I and II errors, RMSE, and MAD metrics. In addition, the α-FCM + ANN + Cox model significantly performs better than the two other hierarchical models.


---

## Body

## 1. Introduction

In today’s competitive world, customer churn management (CCM) is an important task for each service provider to build long-term and profitable relationships with specific customers [1, 2]. The service providers in telecommunication industry suffer from attracting valuable customers with competitors; this is known as customer churn. Recently, there have been many changes in the telecommunications industry, such as, loyalty program for more profitable customers [3]. Loyal customers are the most fertile source of data for decision making. This data reflects the customers’ actual behavior and those factors affect their loyalty. The potential value of customers can be evaluated by these data [3], also assessing the risk that they will stop paying their bills, and predicting their future needs [4].Besides, because customer attrition will absolutely result in loss of incomes, customer churn management has received increasing attention in the whole marketing and management literature. Moreover, it has been proven that considerable impact on incomes is occurred by small change in retention rate [5].The effective customer churn management for companies needs building more comprehensive and accurate churn prediction model. Recently, several customer churn prediction models have been presented in a number of domains such as telecommunications [6–8], retail markets [9, 10], subscription management [11, 12], banking service providers [13], and wireless commerce [14]. Among previous studies in the literature, statistical and data mining techniques have been applied to build the prediction models.These techniques include artificial neural networks (ANNs) [7], Bayesian networks [6, 9], decision trees [15, 16], AdaBoosting [13], logistic regression [10, 11, 16, 17], random forest [10, 11], the proportional hazard model [5], and SVMs. Lessmann and Voß [18] gave a detailed review on this topic.Two main tasks of data mining techniques are describing remarkable pattern or relationship in the data and also predicting a conceptual model which data followed up [2].In the literature, it has been proven that hybrid data mining approaches by combining clustering and classification techniques have better performance in comparison with single clustering and classification data mining techniques. Hybrid approaches are particularly combined of two learning stages, in which the first one is preprocessing the data and the second one is the final prediction output [7]. Other hybrid data mining techniques for predicting customer churn model include using well-known metaheuristic algorithms (e.g., genetic algorithm) based on neural network which outperform traditional local search gradient descent/gradient ascent neural networks that use Rumelhart et al. [19] procedure for updating connection weights [20–22].In addition to predicting the customer churn model and determining that which customer belongs to which class (i.e., churned and nonchurned classes), companies are eager to know when, why, and with what probability their customers try to switch their subscription. Having knowledge about those factors which significantly affect customers churn behavior is more important than just knowing classes of customers. These effective factors are needed for companies to plan their long-term strategies for decreasing customer churn rate and above all, scheduling and adopting best marketing strategies based on when and why their customers like to break up their relationship because some companies suffer from marketing expenses in some especial times while they are not aware of what their customers want. On the other hand, having knowledge about effective factors and probability of attrition enables companies to focus on those customers who are more likely to churn. This useful information can be extracted using survival analysis of customers. In order to determine the hazard probability function of the customers and the above-mentioned information, the Cox proportional hazard method is applied as a last part of hierarchical methods because the ANNs are not able to calculate the churn probability of the customers. Another reason for using the Cox regression model is our used data. The customer churn data consists of censored data. Censored data occurs when you know that a measurement exceeds some threshold, but you do not know by how much. So in this study, each customer who has not churned till the end of the experiment is considered as a right censored data. Therefore, the Cox regression model is conducted on the customer data to cope with censored data.However, few papers studied hierarchical data mining techniques for customer churn prediction. Therefore, in this paper, some data mining techniques are presented to create the hierarchical model of customer churn prediction. The hierarchical methods are based on combining clustering, that is, alpha-cut fuzzyc-means (α-FCM), self-organizing maps (SOM), and artificial neural network (ANN), classification techniques, that is, ANN, and survival analysis, that is, Cox proportional hazard regression model, which their combinations are α-FCM + ANN + Cox, SOM + ANN + Cox, and ANN + ANN + Cox. To evaluate the performance of the hierarchical models, an Iranian mobile dataset is considered for comparison between the hierarchical models and the single Cox regression baseline model in terms of prediction accuracy, Types I and II errors, RMSE, and MAD metrics. It also should be mentioned that some other well-known techniques, such as Fuzzy ARTMAP [23] and LLMF [24], were used in designing some other hierarchical methods (e.g., SOM + Fuzzy ARTMAP + Cox, ANN + Fuzzy ARTMAP + Cox, ANN + LLMF + Cox, SOM + LLMF + Cox, and α-FCM + LLMF + Cox), but just the above-mentioned hierarchical techniques are proposed and reported based on their better performance. Finally, some of contributions of this paper are as follows.(i)
Considering nonchurned customer as censored data and using Cox regression model as a first time in the literature in order to determine customers churn prediction.(ii)
Determining important factors affecting the customer churn in the Iranian telecommunication industry.(iii)
Determining the hazard and survival functions of each customer based on effective factors.(iv)
Proposing some new combination of data mining techniques containing ANN, SOM,α-FCM, and Cox regression as hierarchical methods.(v)
Conducting the proposed hierarchical methods on a dataset of Iranian telephony market.(vi)
Comparing different proposed hierarchical methods.The rest of our paper is organized as follows. In Section2, we describe the proposed data mining techniques in this paper. Section 3 describes the research methodology, and Section 4 presents the experimental results. Finally, the conclusion is provided in Section 5.


## 2. Proposed Data Mining Techniques

In order to create effective and accurate customer churn prediction models, many data mining techniques have been considered over the past time in the marketing and management literature (e.g., [12, 25]). The proposed data mining techniques are as follows.


### 2.1. Alpha-Cut FuzzyC-Means Clustering

Clustering is an unsupervised learning technique that breaks down a set of patterns into groups (or clusters). Clustering technique refers to the partitioning of a set of data object into clusters. In particular, no predefined classes are assumed [26].Classical clustering partitions each observation is assigned to a single group (cluster), without considering the degree of distinction or similarity of the observation from all the other possible clusters. This type of clustering is often called hard or crisp clustering [5]. Nevertheless, fuzzy clustering methods based on the fuzzy set theory and on the concept of membership functions have been developed. In the fuzzy clustering, observations are allowed to belong to more than one cluster with different degrees of membership.Fuzzy clustering of an observationX into c clusters is characterized by c membership functions μj as follows:
(1)μj:X→[0,1],j=1,…,c,∑j=1cμj(xi)=1,i=1,2,…,n,0<∑i=1nμj(xi)<n,j=1,2,…,c.Membership function is calculated based on the distance of observations from clusters’ center. The well-known method of fuzzy clustering is the fuzzyc-means technique (FCM), initially proposed by Dunn [27]. FCM applies two consecutive steps including (a) calculation of the clusters’ center and (b) assigning the observations to these clusters’ center using specific form of distance, in order to minimize a standard loss function (SLF) as follows:
(2)SLF=∑k=1c∑i=1n[μk(xi)]m∥xi-ck∥2,
where cluster center ck and membership function of observation i in cluster k are calculated by (3) and (4), respectively
(3)ck=∑i[μk(xi)]mxi∑i[μk(xi)]m,(4)μk(xi)=(1/dki)1/(m-1)∑k=1c(1/dki)1/(m-1),
where dki is the distance metric for observation i in cluster k.


### 2.2.  Self-Organizing Maps

A new form of a neural network architecture called self-organizing map (SOM) was proposed by Kohonen [28], which has proved extremely efficient when the high degree of dimensionality and complexity accurses in input data. SOM is used to find out relationships in a dataset and cluster data according to the similarity of data (i.e., similar expression patterns) where the nature of the classification cannot be predicted by the model creators, or there may be more than one method to cluster the characteristics of a dataset [29]. Figure 1 shows an example of a 4 × 4 SOM.Figure 1
A 4× 4 Kohonen’s self-organizing map.


### 2.3. Artificial Neural Network

Classification is one of the commonly used data mining techniques categorizing as supervised learning techniques. It determines the value of some variables and classifies according to results. The common algorithms of classification include decision trees, artificial neural networks (ANNs), and so on [30] in which artificial neural networks are the most recently applied methods in literature.An ANN consists of some nodes and links between them. The ANN takes a number of input data and produces a single output data through an internal weighting system. ANNs can be categorized into single-layer perception or multilayer perception (MLP). The multilayer perception consists of multiple layers of simple, two taste, sigmoid processing nodes, or neurons that act together using internal weighted system. In addition, the neural network consists in one or more several intermediary layers between the input and output layers. Such intermediary layers are called hidden layers and nodes embedded in these layers are called hidden nodes. Figure2 illustrates a multilayer neural network.Figure 2
Multilayer neural network.


### 2.4. Cox Proportional Hazards Model

According to the Cox and Oakes [31], the Cox model is based on a modelling approach in order to analysing survival data. The purpose of the model is to simultaneously explore the effects of several variables on survival. The Cox model is a well-recognised statistical technique for analysing survival data. Survival analysis typically examines the relationship of the survival distribution to covariates. Most commonly, this examination entails the specification of a linear-like model for the log hazard. For example, a parametric model based on the exponential distribution may be written as follows:
(5)loghi(t)=α+β1xi1+β2xi2+⋯+βkxik
or, equivalently,
(6)hi(t)=exp(α+β1xi1+β2xi2+⋯+βkxik).Equation (5) is a linear model for the log-hazard or a multiplicative model for the hazard. In (5), i is a subscript for observation, and the x’s are the covariates. The constant α in this model represents a kind of log-baseline hazard, since loghi(t)=α[orhi(t)=eα] when all of the x’s are zero. Equation (6) is similar to parametric regression models based on the other survival distributions.The Cox model, in contrast, leaves the baseline hazard functionα(t)=logh0(t) unspecified
(7)loghi(t)=α(t)+β1xi1+β2xi2+⋯+βkxik,(8)hi(t)=h0(t)exp(α+β1xi1+β2xi2+⋯+βkxik),
where (8) is a semiparametric because while the baseline hazard can take any form, the covariates enter the model linearly.


## 2.1. Alpha-Cut FuzzyC-Means Clustering

Clustering is an unsupervised learning technique that breaks down a set of patterns into groups (or clusters). Clustering technique refers to the partitioning of a set of data object into clusters. In particular, no predefined classes are assumed [26].Classical clustering partitions each observation is assigned to a single group (cluster), without considering the degree of distinction or similarity of the observation from all the other possible clusters. This type of clustering is often called hard or crisp clustering [5]. Nevertheless, fuzzy clustering methods based on the fuzzy set theory and on the concept of membership functions have been developed. In the fuzzy clustering, observations are allowed to belong to more than one cluster with different degrees of membership.Fuzzy clustering of an observationX into c clusters is characterized by c membership functions μj as follows:
(1)μj:X→[0,1],j=1,…,c,∑j=1cμj(xi)=1,i=1,2,…,n,0<∑i=1nμj(xi)<n,j=1,2,…,c.Membership function is calculated based on the distance of observations from clusters’ center. The well-known method of fuzzy clustering is the fuzzyc-means technique (FCM), initially proposed by Dunn [27]. FCM applies two consecutive steps including (a) calculation of the clusters’ center and (b) assigning the observations to these clusters’ center using specific form of distance, in order to minimize a standard loss function (SLF) as follows:
(2)SLF=∑k=1c∑i=1n[μk(xi)]m∥xi-ck∥2,
where cluster center ck and membership function of observation i in cluster k are calculated by (3) and (4), respectively
(3)ck=∑i[μk(xi)]mxi∑i[μk(xi)]m,(4)μk(xi)=(1/dki)1/(m-1)∑k=1c(1/dki)1/(m-1),
where dki is the distance metric for observation i in cluster k.


## 2.2.  Self-Organizing Maps

A new form of a neural network architecture called self-organizing map (SOM) was proposed by Kohonen [28], which has proved extremely efficient when the high degree of dimensionality and complexity accurses in input data. SOM is used to find out relationships in a dataset and cluster data according to the similarity of data (i.e., similar expression patterns) where the nature of the classification cannot be predicted by the model creators, or there may be more than one method to cluster the characteristics of a dataset [29]. Figure 1 shows an example of a 4 × 4 SOM.Figure 1
A 4× 4 Kohonen’s self-organizing map.


## 2.3. Artificial Neural Network

Classification is one of the commonly used data mining techniques categorizing as supervised learning techniques. It determines the value of some variables and classifies according to results. The common algorithms of classification include decision trees, artificial neural networks (ANNs), and so on [30] in which artificial neural networks are the most recently applied methods in literature.An ANN consists of some nodes and links between them. The ANN takes a number of input data and produces a single output data through an internal weighting system. ANNs can be categorized into single-layer perception or multilayer perception (MLP). The multilayer perception consists of multiple layers of simple, two taste, sigmoid processing nodes, or neurons that act together using internal weighted system. In addition, the neural network consists in one or more several intermediary layers between the input and output layers. Such intermediary layers are called hidden layers and nodes embedded in these layers are called hidden nodes. Figure2 illustrates a multilayer neural network.Figure 2
Multilayer neural network.


## 2.4. Cox Proportional Hazards Model

According to the Cox and Oakes [31], the Cox model is based on a modelling approach in order to analysing survival data. The purpose of the model is to simultaneously explore the effects of several variables on survival. The Cox model is a well-recognised statistical technique for analysing survival data. Survival analysis typically examines the relationship of the survival distribution to covariates. Most commonly, this examination entails the specification of a linear-like model for the log hazard. For example, a parametric model based on the exponential distribution may be written as follows:
(5)loghi(t)=α+β1xi1+β2xi2+⋯+βkxik
or, equivalently,
(6)hi(t)=exp(α+β1xi1+β2xi2+⋯+βkxik).Equation (5) is a linear model for the log-hazard or a multiplicative model for the hazard. In (5), i is a subscript for observation, and the x’s are the covariates. The constant α in this model represents a kind of log-baseline hazard, since loghi(t)=α[orhi(t)=eα] when all of the x’s are zero. Equation (6) is similar to parametric regression models based on the other survival distributions.The Cox model, in contrast, leaves the baseline hazard functionα(t)=logh0(t) unspecified
(7)loghi(t)=α(t)+β1xi1+β2xi2+⋯+βkxik,(8)hi(t)=h0(t)exp(α+β1xi1+β2xi2+⋯+βkxik),
where (8) is a semiparametric because while the baseline hazard can take any form, the covariates enter the model linearly.


## 3. Research Methodology

### 3.1. Data Set

For the purpose of this paper, we consider a CRM data set provided by an Iranian mobile operator. Specifically, the dataset contains 3,150 subscribers, including 495 churners and 2,655 nonchurners, from September 2008 to August 2009. In addition, the subscribers have to be mature customers who were with the mobile operator for at least 2 months. Churn was then calculated based on whether the subscriber left the company during the 10 remained months. Churned customer is defined as a customer who has not made any contact with the operator (e.g., making a call, charging a credit, changing subscription, etc.).


### 3.2.  Model Development

#### 3.2.1. The Baseline

As the last part of all proposed hierarchical methods is Cox regression method and also the final aim of these hierarchical methods is determining better hazard and survival functions for customer churn prediction, therefore, we use the original dataset to create a Cox proportional hazards regression model as the baseline Cox model for comparison.


#### 3.2.2. ANN + ANN + Cox

The first hierarchical model is based on combining two ANN models and Cox regression, in which the first ANN performs the data reduction task and the second ANN for churn classification and the last Cox regression for hazard function prediction. As there is no 100% accuracy, there are a number of correctly and incorrectly predicted data from the training set by the first ANN model. Consequently, the incorrectly predicted data can be regarded as outliers since the ANN model cannot predict them accurately. Then, the correctly predicted data by the first ANN model are used to train the second ANN model as the classification model. Finally, the corrected classified data from second ANN are used by Cox regression to predict hazard function.


#### 3.2.3. SOM + ANN + Cox

For the second hierarchical model, a self-organizing map (SOM), which is a clustering technique, is used for the data reduction task. Then, the corrected clustered data are used to train the second model based on ANN. Finally, the classification result is used to hazard function prediction. To develop the SOM, the map size is set by 2* 2, 3 * 3, 4 * 4, 5 * 5, and 6 * 6, respectively, in order to obtain the highest rate of prediction accuracy. Then, two clusters of SOM which contain the highest proportion of the churner and nonchurner groups, respectively, are selected as the clustering result.


#### 3.2.4.α-FCM + ANN + Cox

In the third hierarchical model,α-FCM, which is a clustering approach, is used for data reduction task. In the fuzzy c-means (FCM) clustering algorithm, almost none of the data points have a membership value of 1. Besides, noise and outliers may cause difficulties in obtaining appropriate clustering results from the FCM algorithm. Therefore, many studies have been done about the FCM algorithm in the literature [32]. Furthermore, studies about FCM can be divided into two categories. One is to extend the dissimilarity (or distance) measure d(xj,ai) between the data point xj and the cluster center ai in the FCM objective function by replacing the Euclidean distance with other types of metric measures [33]. The other category is to extend the FCM objective function by adding a penalty term [34].One of the best methods for assigning a data point to exactly one cluster is that if the membership valueμij of the data point xj in the ith cluster is larger than a given value α, then the point xj will exactly belong to the ith cluster with membership value of 1 and μi’j = 0 for all i≠i′.     In order to guarantee that no two of these c cluster cores will overlap, the value of α is set to interval [0.5, 1] [35]. The cluster cores generated by FCMα can be calculated by (9)
(9)μij=d(xj,ai)-1/(m-1)∑k=1cd(xj,ak)-1/(m-1)>α
which is equivalent to
(10)d(xj,ai)-1/(m-1)<1α∑k=1cd(xj,ak)-1/(m-1),
where mis the fuzziness index so its value is considered as 2. Interesting readers are referred to [35] for more detail. Then, the corrected clustered data are used to train second ANN model in order to customer classification. Finally, the hazard function using Cox regression is predicted based on the corrected classified result from ANN model.


#### 3.2.5. Evaluation Method

To evaluate the proposed churn prediction models, prediction accuracy, and the Type I and II errors are considered. They can be measured by a confusion matrix shown in Table1. The rate of prediction accuracy is defined as (a+d)/(a+b+c+d).Table 1
Confusion matrix.
Actual
Nonchurners
Churners
Predicted
Nonchurners
a
b
(
II:
S
*
>
α
)
Churners
c
(
I:
H
*
*
>
α
)
d
*Probability of survival.
**Probability of hazard.The Type I error is the error of not rejecting a null hypothesis when the alternative hypothesis is the true state of nature. In this paper, it means that the customer is not churned when the model has predicted that the hazard function of that customer is more thanα (i.e., α is the alpha cut in fuzzy c-means clustering method). On the other hand, the Type II error is defined as the error of rejecting a null hypothesis when it is the true state of nature. It means that the customer is churned when the model has predicted that the survival function of that customer is more than α.We also compare the performance of the proposed model with pure Cox proportional hazards model in predicting the churn or survival probability of the customers. The observed outcome for each customer in the sample is either churn or survival (i.e., still active) by the end of the study period. We compute the deviation between observed and predicted outcomes (i.e., the probability of churn or survival as predicted by the model) for both proposed and pure Cox model. The Root Mean Squared Error (RMSE) and Mean Absolute Deviation (MAD) are calculated for comparing both models as follows:(11)RMSE=∑i=1Nch(Schi-0)2Nch+∑j=1NNch(1-SNchj)2NNch,MAD=1Nch∑i=1Nch|echi-e-ch|+1NNch∑j=1NNch|eNchj-e-Nch|,
where SchiandSNchj are the survival probability of churned customer i and non-churned customer j, respectively; NchandNNch are the number of churned and nonchurned customer, respectively; echi is the deviation of churned customer i from zero (i.e., echi=(Schi-0)) and eNchj is  the deviation of non-churned customer j from one (i.e., eNchj=(1-SNchj)), and e-chande-Nch are the mean of the deviation of churned and non-churned customers, respectively.


## 3.1. Data Set

For the purpose of this paper, we consider a CRM data set provided by an Iranian mobile operator. Specifically, the dataset contains 3,150 subscribers, including 495 churners and 2,655 nonchurners, from September 2008 to August 2009. In addition, the subscribers have to be mature customers who were with the mobile operator for at least 2 months. Churn was then calculated based on whether the subscriber left the company during the 10 remained months. Churned customer is defined as a customer who has not made any contact with the operator (e.g., making a call, charging a credit, changing subscription, etc.).


## 3.2.  Model Development

### 3.2.1. The Baseline

As the last part of all proposed hierarchical methods is Cox regression method and also the final aim of these hierarchical methods is determining better hazard and survival functions for customer churn prediction, therefore, we use the original dataset to create a Cox proportional hazards regression model as the baseline Cox model for comparison.


### 3.2.2. ANN + ANN + Cox

The first hierarchical model is based on combining two ANN models and Cox regression, in which the first ANN performs the data reduction task and the second ANN for churn classification and the last Cox regression for hazard function prediction. As there is no 100% accuracy, there are a number of correctly and incorrectly predicted data from the training set by the first ANN model. Consequently, the incorrectly predicted data can be regarded as outliers since the ANN model cannot predict them accurately. Then, the correctly predicted data by the first ANN model are used to train the second ANN model as the classification model. Finally, the corrected classified data from second ANN are used by Cox regression to predict hazard function.


### 3.2.3. SOM + ANN + Cox

For the second hierarchical model, a self-organizing map (SOM), which is a clustering technique, is used for the data reduction task. Then, the corrected clustered data are used to train the second model based on ANN. Finally, the classification result is used to hazard function prediction. To develop the SOM, the map size is set by 2* 2, 3 * 3, 4 * 4, 5 * 5, and 6 * 6, respectively, in order to obtain the highest rate of prediction accuracy. Then, two clusters of SOM which contain the highest proportion of the churner and nonchurner groups, respectively, are selected as the clustering result.


### 3.2.4.α-FCM + ANN + Cox

In the third hierarchical model,α-FCM, which is a clustering approach, is used for data reduction task. In the fuzzy c-means (FCM) clustering algorithm, almost none of the data points have a membership value of 1. Besides, noise and outliers may cause difficulties in obtaining appropriate clustering results from the FCM algorithm. Therefore, many studies have been done about the FCM algorithm in the literature [32]. Furthermore, studies about FCM can be divided into two categories. One is to extend the dissimilarity (or distance) measure d(xj,ai) between the data point xj and the cluster center ai in the FCM objective function by replacing the Euclidean distance with other types of metric measures [33]. The other category is to extend the FCM objective function by adding a penalty term [34].One of the best methods for assigning a data point to exactly one cluster is that if the membership valueμij of the data point xj in the ith cluster is larger than a given value α, then the point xj will exactly belong to the ith cluster with membership value of 1 and μi’j = 0 for all i≠i′.     In order to guarantee that no two of these c cluster cores will overlap, the value of α is set to interval [0.5, 1] [35]. The cluster cores generated by FCMα can be calculated by (9)
(9)μij=d(xj,ai)-1/(m-1)∑k=1cd(xj,ak)-1/(m-1)>α
which is equivalent to
(10)d(xj,ai)-1/(m-1)<1α∑k=1cd(xj,ak)-1/(m-1),
where mis the fuzziness index so its value is considered as 2. Interesting readers are referred to [35] for more detail. Then, the corrected clustered data are used to train second ANN model in order to customer classification. Finally, the hazard function using Cox regression is predicted based on the corrected classified result from ANN model.


### 3.2.5. Evaluation Method

To evaluate the proposed churn prediction models, prediction accuracy, and the Type I and II errors are considered. They can be measured by a confusion matrix shown in Table1. The rate of prediction accuracy is defined as (a+d)/(a+b+c+d).Table 1
Confusion matrix.
Actual
Nonchurners
Churners
Predicted
Nonchurners
a
b
(
II:
S
*
>
α
)
Churners
c
(
I:
H
*
*
>
α
)
d
*Probability of survival.
**Probability of hazard.The Type I error is the error of not rejecting a null hypothesis when the alternative hypothesis is the true state of nature. In this paper, it means that the customer is not churned when the model has predicted that the hazard function of that customer is more thanα (i.e., α is the alpha cut in fuzzy c-means clustering method). On the other hand, the Type II error is defined as the error of rejecting a null hypothesis when it is the true state of nature. It means that the customer is churned when the model has predicted that the survival function of that customer is more than α.We also compare the performance of the proposed model with pure Cox proportional hazards model in predicting the churn or survival probability of the customers. The observed outcome for each customer in the sample is either churn or survival (i.e., still active) by the end of the study period. We compute the deviation between observed and predicted outcomes (i.e., the probability of churn or survival as predicted by the model) for both proposed and pure Cox model. The Root Mean Squared Error (RMSE) and Mean Absolute Deviation (MAD) are calculated for comparing both models as follows:(11)RMSE=∑i=1Nch(Schi-0)2Nch+∑j=1NNch(1-SNchj)2NNch,MAD=1Nch∑i=1Nch|echi-e-ch|+1NNch∑j=1NNch|eNchj-e-Nch|,
where SchiandSNchj are the survival probability of churned customer i and non-churned customer j, respectively; NchandNNch are the number of churned and nonchurned customer, respectively; echi is the deviation of churned customer i from zero (i.e., echi=(Schi-0)) and eNchj is  the deviation of non-churned customer j from one (i.e., eNchj=(1-SNchj)), and e-chande-Nch are the mean of the deviation of churned and non-churned customers, respectively.


## 3.2.1. The Baseline

As the last part of all proposed hierarchical methods is Cox regression method and also the final aim of these hierarchical methods is determining better hazard and survival functions for customer churn prediction, therefore, we use the original dataset to create a Cox proportional hazards regression model as the baseline Cox model for comparison.


## 3.2.2. ANN + ANN + Cox

The first hierarchical model is based on combining two ANN models and Cox regression, in which the first ANN performs the data reduction task and the second ANN for churn classification and the last Cox regression for hazard function prediction. As there is no 100% accuracy, there are a number of correctly and incorrectly predicted data from the training set by the first ANN model. Consequently, the incorrectly predicted data can be regarded as outliers since the ANN model cannot predict them accurately. Then, the correctly predicted data by the first ANN model are used to train the second ANN model as the classification model. Finally, the corrected classified data from second ANN are used by Cox regression to predict hazard function.


## 3.2.3. SOM + ANN + Cox

For the second hierarchical model, a self-organizing map (SOM), which is a clustering technique, is used for the data reduction task. Then, the corrected clustered data are used to train the second model based on ANN. Finally, the classification result is used to hazard function prediction. To develop the SOM, the map size is set by 2* 2, 3 * 3, 4 * 4, 5 * 5, and 6 * 6, respectively, in order to obtain the highest rate of prediction accuracy. Then, two clusters of SOM which contain the highest proportion of the churner and nonchurner groups, respectively, are selected as the clustering result.


## 3.2.4.α-FCM + ANN + Cox

In the third hierarchical model,α-FCM, which is a clustering approach, is used for data reduction task. In the fuzzy c-means (FCM) clustering algorithm, almost none of the data points have a membership value of 1. Besides, noise and outliers may cause difficulties in obtaining appropriate clustering results from the FCM algorithm. Therefore, many studies have been done about the FCM algorithm in the literature [32]. Furthermore, studies about FCM can be divided into two categories. One is to extend the dissimilarity (or distance) measure d(xj,ai) between the data point xj and the cluster center ai in the FCM objective function by replacing the Euclidean distance with other types of metric measures [33]. The other category is to extend the FCM objective function by adding a penalty term [34].One of the best methods for assigning a data point to exactly one cluster is that if the membership valueμij of the data point xj in the ith cluster is larger than a given value α, then the point xj will exactly belong to the ith cluster with membership value of 1 and μi’j = 0 for all i≠i′.     In order to guarantee that no two of these c cluster cores will overlap, the value of α is set to interval [0.5, 1] [35]. The cluster cores generated by FCMα can be calculated by (9)
(9)μij=d(xj,ai)-1/(m-1)∑k=1cd(xj,ak)-1/(m-1)>α
which is equivalent to
(10)d(xj,ai)-1/(m-1)<1α∑k=1cd(xj,ak)-1/(m-1),
where mis the fuzziness index so its value is considered as 2. Interesting readers are referred to [35] for more detail. Then, the corrected clustered data are used to train second ANN model in order to customer classification. Finally, the hazard function using Cox regression is predicted based on the corrected classified result from ANN model.


## 3.2.5. Evaluation Method

To evaluate the proposed churn prediction models, prediction accuracy, and the Type I and II errors are considered. They can be measured by a confusion matrix shown in Table1. The rate of prediction accuracy is defined as (a+d)/(a+b+c+d).Table 1
Confusion matrix.
Actual
Nonchurners
Churners
Predicted
Nonchurners
a
b
(
II:
S
*
>
α
)
Churners
c
(
I:
H
*
*
>
α
)
d
*Probability of survival.
**Probability of hazard.The Type I error is the error of not rejecting a null hypothesis when the alternative hypothesis is the true state of nature. In this paper, it means that the customer is not churned when the model has predicted that the hazard function of that customer is more thanα (i.e., α is the alpha cut in fuzzy c-means clustering method). On the other hand, the Type II error is defined as the error of rejecting a null hypothesis when it is the true state of nature. It means that the customer is churned when the model has predicted that the survival function of that customer is more than α.We also compare the performance of the proposed model with pure Cox proportional hazards model in predicting the churn or survival probability of the customers. The observed outcome for each customer in the sample is either churn or survival (i.e., still active) by the end of the study period. We compute the deviation between observed and predicted outcomes (i.e., the probability of churn or survival as predicted by the model) for both proposed and pure Cox model. The Root Mean Squared Error (RMSE) and Mean Absolute Deviation (MAD) are calculated for comparing both models as follows:(11)RMSE=∑i=1Nch(Schi-0)2Nch+∑j=1NNch(1-SNchj)2NNch,MAD=1Nch∑i=1Nch|echi-e-ch|+1NNch∑j=1NNch|eNchj-e-Nch|,
where SchiandSNchj are the survival probability of churned customer i and non-churned customer j, respectively; NchandNNch are the number of churned and nonchurned customer, respectively; echi is the deviation of churned customer i from zero (i.e., echi=(Schi-0)) and eNchj is  the deviation of non-churned customer j from one (i.e., eNchj=(1-SNchj)), and e-chande-Nch are the mean of the deviation of churned and non-churned customers, respectively.


## 4. Experimental Results

### 4.1. The Baseline

In order to create the Cox model, 2350 and remained 800 numbers of data are used for training and testing the Cox model, respectively. Table2 shows the prediction performance of the baseline Cox proportional hazards model based on type I and II errors, accuracy, RMSE, and MAD metrics. On average, the baseline Cox proportional hazards model provides about 84% accuracy meaning that in 128 cases of data, the Cox model was unable to correctly predict the survival and hazard probability based on value of alpha-cut 0.7. The type I and II errors were equal to 87 and 41 cases of incorrectly predicted data. The baseline Cox model also provides 0.083 and 0.098 as the RMSE and MAD error metrics, respectively.Table 2
The prediction performance of the baseline Cox proportional hazards model.
Performance metrics
Accuracy
Error type I
Error type II
RMSE
MAD
Value
84.79%
87
35
0.091
0.098


### 4.2. ANN + ANN + Cox

For the first hierarchical model based on combining two ANN models and Cox regression, the first ANN model performs the data reduction task. Therefore, we run the ANN model by a set of different hidden layer and learning epochs. The result of different combination of hidden layer and learning epochs is as Table3 in which an ANN models with 16 and 12 hidden layer and 100 and 300 learning epochs are considered for two ANN model, respectively. Finally, the accuracy and other performance metrics for hierarchical ANN + ANN + Cox model are shown in Table 4.Table 3
Prediction performance of ANN + ANN hierarchical models.
Hidden layer
Learning epochs
Clustering ANN
Classification ANN
50
100
200
300
50
100
200
300
8
87.90
89.34
90.66
91.17
85.65
87.50
89.59
91.84
12
83.50
87.42
90.11
90.87
85.23
88.25
91.90
92.49
16
90.25
92.69
88.45
90.77
91.18
88.43
88.57
84.76
20
88.54
90.21
85.80
91.06
90.54
87.92
84.28
86.08
24
91.55
91.85
89.29
86.46
86.85
88.41
86.80
83.46
28
91.61
90.21
86.02
91.29
91.69
90.55
89.37
89.95
32
90.78
89.35
88.13
86.11
91.28
86.55
84.80
90.39Table 4
Performance metrics of ANN + ANN + Cox hierarchical models.
Performance metrics
Accuracy
Error type I
Error type II
RMSE
MAD
Value
88.42%
61
32
0.071
0.088


### 4.3. SOM + ANN + Cox

To construct the second hierarchical model by combination of SOM, ANN, and Cox regression, 2* 2, 3 * 3, 4 * 4, 5 * 5, and 6 * 6 SOMs are used to cluster the data at first. We found that 4 * 4 SOM performs the best which can provide the highest rate of accuracy for two clusters, that is, churner and nonchurner clusters. Then, the accurate clustered data are used for training classifier ANN and the result of different hidden layer and learning epochs is as Table 5 in which an ANN model with 16 hidden layer and 200 epochs is considered as classifier model. Finally, the accuracy and other performance metrics for hierarchical SOM + ANN + Cox model are shown in Table 6.Table 5
Prediction performance of ANN in SOM + ANN + Cox hierarchical models.
Hidden layer
Learning epochs
50
100
200
300
8
83.40
85.30
89.30
90.67
12
84.31
86.46
89.31
89.77
16
87.45
90.28
93.88
91.18
20
90.30
90.29
91.87
84.06
24
85.76
89.44
83.37
93.87
28
90.99
91.57
87.59
90.20
32
91.46
83.28
85.71
89.60Table 6
Performance metrics of SOM + ANN + Cox hierarchical models.
Performance metrics
Accuracy
Error type I
Error type II
RMSE
MAD
Value
86.31%
76
34
0.084
0.091


### 4.4.α-FCM + ANN + Cox

To construct the third hierarchical model based on alpha-cut fuzzyc-means, classifier ANN model and Cox proportional hazards model, the alpha-cut fuzzy c-means with alpha-cut equal to 0.7, and ANN with 16 hidden layer and 100 learning epochs were found regarding to the best reported accuracy. The α-FCM, used for data reduction task, results in two clusters including 2210 churners and 940 nonchurners. Then, 2350 and remained 800 numbers of data are, respectively, used for training and testing the classification ANN model where the prediction performance of ANN model is shown in Table 7. Finally, Table 8 shows the performance metrics of  α-FCM + ANN + Cox  hierarchical models.Table 7
Prediction performance of ANN inα-FCM + ANN + Cox hierarchical models.
Hidden layer
Learning epochs
50
100
200
300
8
84.32
85.31
87.96
89.21
12
84.71
87.76
86.63
86.34
16
87.35
93.20
92.67
90.78
20
90.35
83.81
91.01
89.59
24
88.11
86.21
88.78
88.33
28
83.72
86.54
89.98
87.53
32
85.69
87.91
85.41
87.15Table 8
Performance metrics ofα-FCM + ANN + Cox hierarchical models.
Performance metrics
Accuracy
Error type I
Error type II
RMSE
MAD
Value
95.86%
21
12
0.031
0.042On average, theα-FCM + ANN + Cox hierarchical model provides about 95.49% accuracy based on alpha-cut equal to 0.7 and ANN with 16 hidden layer and 100 learning epochs. The type I and II errors are equal to 21 and 12 cases of incorrectly predicted data. The α-FCM + ANN + Cox hierarchical model also provides 0.031 and 0.042 as the RMSE and MAD error metrics, respectively.In order to show the high performance ofα-FCM + ANN + Cox hierarchical model, the accuracy, errors type I and II, RMSE, and MAD metrics are illustrated in Figures 3, 4, and 5, respectively.Figure 3
The accuracy of hierarchical models.Figure 4
The Type I and II errors of hierarchical models.Figure 5
The RMSE and MAD errors of hierarchical models.


## 4.1. The Baseline

In order to create the Cox model, 2350 and remained 800 numbers of data are used for training and testing the Cox model, respectively. Table2 shows the prediction performance of the baseline Cox proportional hazards model based on type I and II errors, accuracy, RMSE, and MAD metrics. On average, the baseline Cox proportional hazards model provides about 84% accuracy meaning that in 128 cases of data, the Cox model was unable to correctly predict the survival and hazard probability based on value of alpha-cut 0.7. The type I and II errors were equal to 87 and 41 cases of incorrectly predicted data. The baseline Cox model also provides 0.083 and 0.098 as the RMSE and MAD error metrics, respectively.Table 2
The prediction performance of the baseline Cox proportional hazards model.
Performance metrics
Accuracy
Error type I
Error type II
RMSE
MAD
Value
84.79%
87
35
0.091
0.098


## 4.2. ANN + ANN + Cox

For the first hierarchical model based on combining two ANN models and Cox regression, the first ANN model performs the data reduction task. Therefore, we run the ANN model by a set of different hidden layer and learning epochs. The result of different combination of hidden layer and learning epochs is as Table3 in which an ANN models with 16 and 12 hidden layer and 100 and 300 learning epochs are considered for two ANN model, respectively. Finally, the accuracy and other performance metrics for hierarchical ANN + ANN + Cox model are shown in Table 4.Table 3
Prediction performance of ANN + ANN hierarchical models.
Hidden layer
Learning epochs
Clustering ANN
Classification ANN
50
100
200
300
50
100
200
300
8
87.90
89.34
90.66
91.17
85.65
87.50
89.59
91.84
12
83.50
87.42
90.11
90.87
85.23
88.25
91.90
92.49
16
90.25
92.69
88.45
90.77
91.18
88.43
88.57
84.76
20
88.54
90.21
85.80
91.06
90.54
87.92
84.28
86.08
24
91.55
91.85
89.29
86.46
86.85
88.41
86.80
83.46
28
91.61
90.21
86.02
91.29
91.69
90.55
89.37
89.95
32
90.78
89.35
88.13
86.11
91.28
86.55
84.80
90.39Table 4
Performance metrics of ANN + ANN + Cox hierarchical models.
Performance metrics
Accuracy
Error type I
Error type II
RMSE
MAD
Value
88.42%
61
32
0.071
0.088


## 4.3. SOM + ANN + Cox

To construct the second hierarchical model by combination of SOM, ANN, and Cox regression, 2* 2, 3 * 3, 4 * 4, 5 * 5, and 6 * 6 SOMs are used to cluster the data at first. We found that 4 * 4 SOM performs the best which can provide the highest rate of accuracy for two clusters, that is, churner and nonchurner clusters. Then, the accurate clustered data are used for training classifier ANN and the result of different hidden layer and learning epochs is as Table 5 in which an ANN model with 16 hidden layer and 200 epochs is considered as classifier model. Finally, the accuracy and other performance metrics for hierarchical SOM + ANN + Cox model are shown in Table 6.Table 5
Prediction performance of ANN in SOM + ANN + Cox hierarchical models.
Hidden layer
Learning epochs
50
100
200
300
8
83.40
85.30
89.30
90.67
12
84.31
86.46
89.31
89.77
16
87.45
90.28
93.88
91.18
20
90.30
90.29
91.87
84.06
24
85.76
89.44
83.37
93.87
28
90.99
91.57
87.59
90.20
32
91.46
83.28
85.71
89.60Table 6
Performance metrics of SOM + ANN + Cox hierarchical models.
Performance metrics
Accuracy
Error type I
Error type II
RMSE
MAD
Value
86.31%
76
34
0.084
0.091


## 4.4.α-FCM + ANN + Cox

To construct the third hierarchical model based on alpha-cut fuzzyc-means, classifier ANN model and Cox proportional hazards model, the alpha-cut fuzzy c-means with alpha-cut equal to 0.7, and ANN with 16 hidden layer and 100 learning epochs were found regarding to the best reported accuracy. The α-FCM, used for data reduction task, results in two clusters including 2210 churners and 940 nonchurners. Then, 2350 and remained 800 numbers of data are, respectively, used for training and testing the classification ANN model where the prediction performance of ANN model is shown in Table 7. Finally, Table 8 shows the performance metrics of  α-FCM + ANN + Cox  hierarchical models.Table 7
Prediction performance of ANN inα-FCM + ANN + Cox hierarchical models.
Hidden layer
Learning epochs
50
100
200
300
8
84.32
85.31
87.96
89.21
12
84.71
87.76
86.63
86.34
16
87.35
93.20
92.67
90.78
20
90.35
83.81
91.01
89.59
24
88.11
86.21
88.78
88.33
28
83.72
86.54
89.98
87.53
32
85.69
87.91
85.41
87.15Table 8
Performance metrics ofα-FCM + ANN + Cox hierarchical models.
Performance metrics
Accuracy
Error type I
Error type II
RMSE
MAD
Value
95.86%
21
12
0.031
0.042On average, theα-FCM + ANN + Cox hierarchical model provides about 95.49% accuracy based on alpha-cut equal to 0.7 and ANN with 16 hidden layer and 100 learning epochs. The type I and II errors are equal to 21 and 12 cases of incorrectly predicted data. The α-FCM + ANN + Cox hierarchical model also provides 0.031 and 0.042 as the RMSE and MAD error metrics, respectively.In order to show the high performance ofα-FCM + ANN + Cox hierarchical model, the accuracy, errors type I and II, RMSE, and MAD metrics are illustrated in Figures 3, 4, and 5, respectively.Figure 3
The accuracy of hierarchical models.Figure 4
The Type I and II errors of hierarchical models.Figure 5
The RMSE and MAD errors of hierarchical models.


## 5. Conclusion

As customers are the main competitive advantage of each industry, customer churn prediction is becoming a major task for companies to remain in competition with other industries. Therefore, building an effective customer churn prediction model, which provides an acceptable level of accuracy, has become a research problem for companies in recent years. In the literature, the better applicability and efficiency of hierarchical data mining techniques in order to predict customer attrition by combining two or more techniques has been reported over a number of different domain problems. In this paper, we consider three different hierarchical data mining techniques based on combination of some neural networks and regression model to examine their performances for telecommunication industry. In particular, backpropagation artificial neural networks (ANN), self-organizing maps (SOM), alpha-cut fuzzyc-means, and Cox proportional hazard model are considered. Consequently, ANN+ANN+Cox,SOM+ANN+Cox, and α-FCM + ANN + Cox hierarchical models are developed, in which the first component of the hierarchical models filter out unrepresentative data or outliers. Then, the corrected output clustered data are used to classify customer into churner and nonchurner groups.To evaluate the performance of the hierarchical models, an Iranian mobile dataset is considered. The experimental results show that the hierarchical models outperform the single Cox regression baseline model in terms of prediction accuracy, types I and II errors, RMSE, and MAD metrics. In addition, theα-FCM + ANN + Cox model significantly performs better than the SOM+ANN+Cox and ANN+ANN+Cox models.For future work, other prediction techniques can be applied, such as support vector machines, genetic algorithms, logistic regression, and so forth. Finally, other domain datasets about churn prediction can be used for further comparison.



---

*Source: 543940-2013-02-14.xml*