Which mathematical model is most commonly used to predict customer purchasing patterns over time when calculating Customer Lifetime Value in retail?

Customer Lifetime Value (CLV) is a critical metric for retail businesses, helping them understand the total worth of a customer over their entire relationship. Mathematics plays a crucial role in calculating and predicting CLV. Various statistical and probabilistic models are employed to forecast future purchasing behavior and estimate the long-term value of customer relationships. How well do you understand the mathematical foundations behind these important retail analytics?

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Markov Chain models, which estimate transition probabilities between different customer states (active, inactive, churned)
Lagrangian multipliers, which optimize customer value functions subject to budget constraints
Fourier transform analysis, which decomposes purchasing patterns into frequency components
Euclidean vector space models, which map customers to n-dimensional purchase propensity spaces