11 Predicting future states with Markov analysis

This chapter introduces Markov analysis as a way to predict how a system’s state will evolve using only its current condition and fixed transition probabilities. Leveraging the Markov property (memorylessness), it frames real-world problems—such as market-share churn and credit delinquency—in terms of discrete states and the likelihood of moving between them. The chapter’s goals are to build intuition for states and their probabilities, construct and interpret transition matrices, make short- and long-horizon forecasts, reason about long-run equilibrium, and understand absorbing states, all with both manual calculations and programmatic workflows.

The mechanics center on defining mutually exclusive, collectively exhaustive states; capturing the present distribution as a state-probability vector; and using a square transition matrix whose rows sum to one. One-period forecasts come from multiplying the current state vector by the transition matrix; multi-period forecasts iterate this step or use powers of the matrix to “jump” across multiple periods. A detailed wireless-carrier example shows how churn redistributes market share and how successive forecasts typically exhibit diminishing changes as the system stabilizes. The chapter emphasizes assumptions (constant transition probabilities and an unchanged system) and demonstrates efficient, reliable implementation with Python and NumPy for matrix multiplication and repeated projection.

Long-run behavior is analyzed via equilibrium (steady-state) conditions, where repeated transitions converge to a stable distribution, illustrated with a two-state engine example and a discussion of sensitivity to small changes in transition probabilities. The chapter then treats absorbing states—terminal states that, once entered, cannot be left—using a credit-delinquency model with “current” and “charged off” as absorbing outcomes. It partitions the transition matrix and computes the fundamental matrix (the inverse of identity minus the transient block) to derive absorption probabilities, again automating the workflow in Python. These tools enable practitioners to forecast short- and long-run outcomes, quantify risk, and guide strategy in domains like marketing, finance, and operations.

• Markov analysis is a statistical method used to predict the future states of a system based on its current state and the probabilities of transitioning from one state to another. It relies on the assumption that the future state depends only on the present state and not on the sequence of events that preceded it.
• The vector of state probabilities represents the probabilities of a system being in each possible state at a given period. It is used to describe the distribution of states in the system and is essential for predicting future states using Markov analysis.
• The matrix of transition probabilities is a square matrix where each element represents the probability of transitioning from one state to another in a given period. This matrix is fundamental in Markov analysis for determining the likelihood of moving between states and predicting future state distributions.
• To predict future states in Markov analysis, the vector of state probabilities and the transition probability matrix are used together through the mathematical operation known as the dot product. The vector of state probabilities represents the current distribution of the system across various states. By taking the dot product of this vector with the transition probability matrix, which contains the probabilities of moving from one state to another, we can calculate the vector of state probabilities for the next period. Repeatedly applying the dot product operation allows us to forecast the system's state distribution over multiple future periods, effectively predicting how the system will evolve over time.
• In Markov analysis, equilibrium conditions refer to a state where the system's probabilities stabilize and no longer change significantly over time. At equilibrium, the vector of state probabilities remains constant from one period to the next, indicating that the system has reached a steady state where the transitions between states balance out.
• Absorbing states are states that, once entered, cannot be left. When a system reaches an absorbing state, it remains in that state permanently. These states have a transition probability of 1 for remaining in the same state and 0 for transitioning to any other state.
• Markov analysis has diverse real-world applications across industries. It is otherwise used in finance for credit risk modeling and stock market analysis, healthcare for patient progression modeling, operations management for equipment maintenance and reliability analysis, and cybersecurity for detecting anomalous network behavior and predicting potential security threats. By applying Markov analysis to these domains, businesses and researchers can make data-driven decisions and optimize long-term outcomes.

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