Shannon Entropy Calculator
Developed from Claude Shannon’s groundbreaking work at Bell Labs in 1948, this shannon entropy calculator implements his mathematical framework for measuring the inherent unpredictability in data sequences.
For analyzing a text message containing repeated characters like “AAAAAAA“. The calculator would reveal extremely low entropy due to high predictability. A diverse message like “Hello1#@” exhibits higher entropy because it contains varied, less predictable characters, demonstrating greater information richness.
Shannon Entropy Formula
The equation for Shannon entropy (H) is:
H = -Σ (pi *log2(pi))Key components include:
- H represents the entropy measured in bits
- pi denotes the probability of each symbol occurring
- Σ indicates summation across all possible symbols
- log2 represents the logarithm base 2
Analyzing the word “HELLO“:
- Total length: 5 characters
- H appears once (probability = 1/5)
- E appears once (probability = 1/5)
- L appears twice (probability = 2/5)
- O appears once (probability = 1/5)
- Binary String Analysis Examining “1100”:
- Probability(0) = 2/4 = 0.5
- Probability(1) = 2/4 = 0.5 Entropy = -(0.5 log2(0.5) + 0.5 log2(0.5)) = 1 bit
- Message Content For “aab”:
- Probability(a) = 2/3
- Probability(b) = 1/3 Entropy = -(2/3 log2(2/3) + 1/3 log2(1/3)) ≈ 0.918 bits
- Die Roll Distribution For a fair six-sided die:
- Probability(each number) = 1/6 Entropy = -(6 (1/6 log2(1/6))) ≈ 2.58 bits
How is Shannon Entropy Calculated?
- Symbol identification within the dataset
- Probability calculation for each unique symbol
- Logarithmic multiplication of probabilities
- Final summation and negative conversion
“MISSISSIPPI” analysis:
- Total letters: 11
- M: 1/11, I: 4/11, S: 4/11, P: 2/11 Final calculation: -Σ(pi * log2(pi)) for each probability value
What is the Shannon entropy of DNA
DNA sequences present fascinating entropy patterns due to their four-nucleotide alphabet (A, T, G, C). The maximum theoretical entropy reaches 2 bits per base position (log2(4) = 2). However, actual DNA sequences typically show lower entropy due to biological constraints and non-random patterns.
Example analyzing “ATGCATGC”:
Each nucleotide appears twice (probability = 0.25) Entropy = -(4 0.25 log2(0.25)) = 2 bits
What is the Shannon’s entropy index
The entropy index provides a normalized measure of diversity or uncertainty within systems. Values range from 0 (complete certainty) to 1 (maximum uncertainty), enabling meaningful comparisons across different datasets.
Species diversity example: Three species with populations [50%, 30%, 20%]: Entropy = -(0.5log2(0.5) + 0.3log2(0.3) + 0.2log2(0.2)) *Normalized by dividing by log2(3)What is Shannon Entropy?
Shannon entropy quantifies the unpredictability and information content within systems. This revolutionary concept parallels thermodynamic entropy but applies to information theory rather than physical states. Higher entropy indicates greater uncertainty or information richness, while lower entropy suggests predictability or redundancy.