What is a Markov Chains?
Markov chains model systems that change condition in phases. Probabilities can define these systems’ overall behaviour, but their details are unpredictable.
Imagine playing a board game where die rolls determine moves. Your chances of advancing forward, backward, or keeping still depend on where you are on the board. Importantly, your future move depends only on where you are now, not on your previous moves. Markov chains work this way.
System “state” is its present state. The “transition” is between states. The “chain” stems from these transitions happening sequentially like links.
Understanding the Markov Property
A important principle of Markov chains is the “Markov property.” This means that the next step in a process depends on the current state, not how you got there.
Imagine trying to forecast tomorrow’s weather. Applying the Markov property means that tomorrow’s weather depends only on today’s weather, not on prior days’ weather. If today is sunny, tomorrow may be sunny, cloudy, or rainy.Markov chains simplify complex processes and make helpful predictions by focussing on the current state.
How Markov Chains Work
Three things are needed to develop a Markov chain model:
- All conceivable system states are listed. Weather states could be “sunny,” “cloudy,” and “rainy.”
- Estimate the probability of relocating to each state (or staying in the same one) for each state.
- A beginning point start from one state and mimic or forecast its evolution.
Run the model to see how the system responds over time after knowing the beginning state and probability of moving between states. A “steady state.” occurs when the process settles into a pattern after enough steps.
Real-World Applications of Markov Chains
Markov chains are useful in many disciplines despite their theoretical nature. Some notable examples:
- Search and PageRank
Google PageRank is a notable Markov chain application. Treat the internet as a network of linked pages. A Markov chain can model a user’s random link clicks to a page. Over time, some pages get greater traffic. The most popular pages score higher in search results. - Weather Prediction
As indicated, weather forecasting models presume tomorrow’s conditions depend primarily on today’s. Markov chains work well with weather systems, especially for short-term forecasting. - Economics/Finance
Economic modelling uses Markov chains to describe market behaviour, consumer spending, and credit ratings. A customer’s current financial situation can predict their loan default risk, not their complete financial history. - Genes and Biology
Genetic DNA sequences often depend on the previous element. Markov chains predict protein folding and genetic sequence evolution. - Language and speech processing
Voice assistants and predictive text use current words to determine the next word. Modelling language flow with Markov chains powers these predictions.
Markov Chain Types
Not all Markov chains function similarly. Some variations:
- Absorbing Chains: After entering one of these Markov chains’ “end” states, the system stays there permanently. Board games that stop when the player wins or loses don’t restart.
- Periodic Chains: State cycles may be fixed in some processes. A manufacturing system may cycle through production, restocking, and maintenance daily. This chain rhythm is regular.
- Ergodic Chains: Chains are more flexible. They let the system go through every condition and settle into a rhythm. Randomness and organisation coexist in natural and social systems.
Decision Making with Markov Chains
Adding choices and outcomes to a Markov chain creates a Markov decision process. Robotics and artificial intelligence employ these to help machines make the best decisions.
A self-driving car must continually decide to turn, stop, or speed. The automobile employs sensors to assess its state and a Markov decision model to decide its next action.
Markov Chains Benefits
Markov chains’ benefits make them popular:
- By focussing on the present, they simplify complex systems to model.
- Even if the process is random, long-term behaviour is generally predictable.
- Their versatility extends to engineering, physics, marketing, and entertainment.
Limitations of Markov chains
Markov chains are helpful yet imperfect. Some restrictions:
Assumption of memorylessness: The past typically affects the future. This is ignored by Markov chains.
Fixed probabilities: The model assumes constant transition probabilities, which may not be accurate in dynamic contexts.
Simplification of reality: Markov chains can oversimplify complex systems, although they are valuable for modelling.
Researchers utilise complex models like Hidden Markov Models or non-Markovian models to account for deeper dependencies.
Conclusion
Markov chains, formerly a mathematical curiosity, are now essential for understanding and predicting change in uncertain systems. Search engines, financial markets, weather forecasts, and DNA analysis help make sense of randomness by focussing on the present.
Their simplicity is powerful. Markov chains simplify system behaviour and allow us to predict its behaviour by assuming that the future depends only on the present.
Knowing how Markov chains work can help you make smarter, data-driven decisions while designing an app, analysing client behaviour, or studying nature.