• Portfolio, Churn & Customer Value
  • Abstract
  • 1 Introduction
    • 1.1 How to define a customer portfolio?
    • 1.2 What is attrition?
    • 1.3 What does customer value mean?
  • 2 Literature Review
    • 2.1 On customer portfolio
    • 2.2 On attrition
    • 2.3 On customer value
      • 2.3.1 Recency Frequency Monetary models
      • 2.3.2 NBP-Pareto model
      • 2.3.3 Econometric models
  • 3 Duration models
    • 3.1 Definition
    • 3.2 Censoring and Truncation
      • 3.2.1 Censoring mechanisms
      • 3.2.2 Selection bias
    • 3.3 Probabilistic concepts
      • 3.3.1 Survival function
      • 3.3.2 Hazard and Cumulative Hazard functions
    • 3.4 Nonparametric models
      • 3.4.1 Notations
      • 3.4.2 Hazard function estimator
      • 3.4.3 Kaplan-Meier estimator
      • 3.4.4 Nelson-Aalen estimator
    • 3.5 Parametric models
      • 3.5.1 Constant hazard (exponential model)
      • 3.5.2 Monotone hazard
      • 3.5.3 Concave and convex hazard
    • 3.6 Semi-parametric estimation
      • 3.6.1 Proportional Hazards models
      • 3.6.2 Cox PH model
    • 3.7 Machine Learning for Survival Data
      • 3.7.1 Survival Trees
      • 3.7.2 Random Survival Forests (RSF)
      • 3.7.3 Cox Boosting
    • 3.8 Performance metrics
      • 3.8.1 Concordance index (C-index)
      • 3.8.2 Brier score
  • 4 Data Mining methods
    • 4.1 Mutliple Correspondence Analysis (MCA)
      • 4.1.1 Definition
      • 4.1.2 Complete disjunctive table
      • 4.1.3 Distances
      • 4.1.4 Algorithm
    • 4.2 Unsupervised classification
      • 4.2.1 Hierarchical Clustering on Principal Components (HCPC)
      • 4.2.2 Agglomerative Hierarchical Clustering (AHC)
      • 4.2.3 The k-means algorithm
  • 5 Data
    • 5.1 General Overview
    • 5.2 Churn_Value and Tenure_Months
      • Demographic data
      • Data on services subscribed
      • Customer account data
    • 5.3 Churn, duration and price
  • 6 Estimation techniques
    • 6.1 Feature selection
    • 6.2 Portfolio segmentation
      • 6.2.1 Transforming qualitative variables into principal axes
      • 6.2.2 Hierarchical clustering on principal components
    • 6.3 Churn analysis
      • 6.3.1 The Cox model
      • 6.3.2 Other survival models
    • 6.4 Portfolio value estimation
      • 6.4.1 The model
      • 6.4.2 Customer Lifetime Raw Value
      • 6.4.3 Cluster contribution to the portfolio value
      • 6.4.4 Simulations
  • Conclusion
  • Appendix
    • Hazard function
    • Link between cumulative hazard and survivor functions
    • Contribution to the partial likelihood function in PH models
    • Partial likelihood function in PH models
    • Multiple correspondence analysis
    • Hierarchical clustering on principal components
      • Cluster visualisation
  • References
  • Get the source code on Github.

Portfolio, Churn & Customer Value

Link between cumulative hazard and survivor functions

\[\begin{equation} \begin{aligned} & \Lambda(t) = \int_{0}^{t} \lambda(s)ds \\\\ \iff & \Lambda(t) = \int_{0}^{t} \frac{f(s)}{S(s)}ds \\\\ \iff & \Lambda(t) = -\ln \big(S(t)\big) \\\\ \iff & S(t) = \exp \big(-\Lambda(t)\big) \end{aligned} \tag{6.4} \end{equation}\]