PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Probability and Statistics → Stochastic Processes ↓


Markov Chains


Markov chains are an essential concept in the study of stochastic processes, a branch of probability and statistics. Named after Russian mathematician Andrey Markov, Markov chains are used to describe systems that transition from one state to another in a probabilistic manner.

Introduction to stochastic processes

Before delving deeper into Markov chains, let's briefly discuss what stochastic processes are. A stochastic process is essentially a collection of random variables indexed by a parameter, which usually represents time. These processes are used to model systems that evolve over time in a probabilistic manner. Each value or state of the system is not completely determined by any known equation, but rather probabilistically.

Understanding Markov chains

A Markov chain is a specific type of stochastic process where the future state of the process depends only on the current state, not on the sequence of events before it. This property is called the "memoryless" property, formally known as the Markov property. In simple terms, it means that the process has no memory of where it was before.

Discrete-time Markov chain

The most common form of Markov chains is the discrete-time Markov chains. These systems move through states at fixed discrete time intervals. At each step, the system can either remain in the same state or move to a different state with a certain probability.

State location

The set of all possible states in a Markov chain is called the "state space." For example, if we are modeling weather conditions as states, the possible states might be "sunny", "rainy", and "cloudy".

Transition matrix

The probabilities of moving from one state to another are represented in a matrix called the "transition matrix". If there are n states in the state space, the transition matrix will be an n × n matrix in which the elements at i row and j column represent the probability of moving from state i to state j. An example of a transition matrix for a simple weather model with three states might look like this:

P = [ [0.7, 0.2, 0.1],
      [0.3, 0.4, 0.3],
      [0.2, 0.1, 0.7] ]

Make sure that each row of the transition matrix sums to 1, since each row represents the complete set of possible states into which the system can transition from the current state.

Example: Weather model

Let's consider a simple Markov chain that describes weather changes:

  • Conditions: "sunny", "cloudy", "rainy"
  • Transition Matrix: 
        P = [ [0.6, 0.3, 0.1],
          [0.2, 0.5, 0.3],
          [0.3, 0.3, 0.4] ]

This matrix describes the following system: if it is sunny today, there is a 60% chance that it will be sunny tomorrow, a 30% chance that it will be cloudy, and a 10% chance that it will rain.

sunny cloudy Raincoat

Properties of Markov chains

Here are some key properties to better understand the dynamics of Markov chains:

Immutability

A Markov chain is said to be irreversible if it is possible to go from any state to any state. This means that there is a non-zero probability of transitioning from one state to another, either directly or through other states.

Seizure

A state in a Markov chain has a period if it returns to that state in a finite number of steps; the period is the greatest common divisor of the number of steps in which the state returns to itself.

Recurrence and transience

If the probability that the process will eventually return to the same state is 1, the state is called recurrent. If the probability is less than 1, the state is called transient.

Stable distribution

A stationary distribution is a probability distribution that remains unchanged as the process evolves. For a Markov chain with a transition matrix P, a vector π is a stationary distribution if:

πP = π

Using more calculations and the given transition matrix, you can find the stationary distribution for the stochastic process.

Continuous-time Markov chains

While we have focused on discrete-time Markov chains, where the system transitions at identical discrete moments, time can also be continuous. In continuous-time Markov chains, the transition can occur at any continuous moment.

Transition rate metrics

Instead of transition matrices, continuous-time Markov chains use transition rate matrices, often denoted by Q, where each entry qij represents the rate of transition from state i to state j.

Conclusion

Markov chains provide a powerful framework for modeling and analyzing systems that undergo transitions between states over time. With their wide applications in various fields such as economics, genetics, and game theory, Markov chains have become a fundamental topic in probability and statistics.


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Markov Chains

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