Entropy Rate Calculator
Enter stationary probabilities and transition matrix. The Entropy Rate Calculator computes the average information per symbol for Markov sources.
About the Entropy Rate Calculator
The Entropy Rate Calculator is a precise, scientifically accurate tool that computes the entropy rate of discrete stochastic processes, particularly first-order Markov chains. This Entropy Rate Calculator implements Shannon's definition H = -Σ π_i Σ p_{ij} log₂ p_{ij}, the long-run average uncertainty per symbol. Proudly supported by Agri Care Hub.
Scientific Formula
H(X | i) = - Σ_{j} p_{ij} log₂ p_{ij}
π = stationary distribution
Why This Calculator Is Essential
Entropy rate is crucial in: • Data compression limits • Speech and language modeling • Cryptography (randomness testing) • Bioinformatics (DNA sequences) • Financial time series • Control theory and prediction It gives the fundamental limit for lossless compression and predictability of stochastic processes.
How to Use
- Select number of states.
- Enter transition probabilities p_{ij} (rows sum to 1).
- Calculator finds stationary distribution π automatically.
- Click “Calculate Entropy Rate”.
- Get entropy rate in bits/symbol.
When Should You Use This Tool?
- Source coding and compression limits
- Markov model analysis
- Language and speech processing
- Biological sequence entropy
- Randomness and predictability studies
- Teaching stochastic processes
Scientific Foundation
Entropy rate was formalized by Claude Shannon in 1948 as the limit of joint entropy per symbol for stationary processes. For Markov chains, it is the weighted conditional entropy. The formula is exact and gives the ultimate compression limit. Full theory at Entropy Rate on Wikipedia and Cover & Thomas’ *Elements of Information Theory*.
Conclusion
The Entropy Rate Calculator brings advanced information theory to your browser — with perfect accuracy and beautiful design. Whether you’re a student exploring Markov sources, a researcher analyzing sequences, or an engineer evaluating compression, this tool delivers precise results every time. For more information theory tools, visit Agri Care Hub.