Redundancy (information theory)

In Information theory, redundancy measures the fractional difference between the entropy H(X) of an ensemble X, and its maximum possible value ${\displaystyle \log(|{\mathcal {A}}_{X}|)}$.[1][2] Informally, it is the amount of wasted "space" used to transmit certain data. Data compression is a way to reduce or eliminate unwanted redundancy, while checksums are a way of adding desired redundancy for purposes of error detection when communicating over a noisy channel of limited capacity.

Quantitative definition

In describing the redundancy of raw data, the rate of a source of information is the average entropy per symbol. For memoryless sources, this is merely the entropy of each symbol, while, in the most general case of a stochastic process, it is

${\displaystyle r=\lim _{n\to \infty }{\frac {1}{n}}H(M_{1},M_{2},\dots M_{n}),}$

the limit, as n goes to infinity, of the joint entropy of the first n symbols divided by n. It is common in information theory to speak of the "rate" or "entropy" of a language. This is appropriate, for example, when the source of information is English prose. The rate of a memoryless source is simply ${\displaystyle H(M)}$, since by definition there is no interdependence of the successive messages of a memoryless source.

The absolute rate of a language or source is simply

${\displaystyle R=\log |\mathbb {M} |,\,}$

the logarithm of the cardinality of the message space, or alphabet. (This formula is sometimes called the Hartley function.) This is the maximum possible rate of information that can be transmitted with that alphabet. (The logarithm should be taken to a base appropriate for the unit of measurement in use.) The absolute rate is equal to the actual rate if the source is memoryless and has a uniform distribution.

The absolute redundancy can then be defined as

${\displaystyle D=R-r,\,}$

the difference between the absolute rate and the rate.

The quantity ${\displaystyle {\frac {D}{R}}}$ is called the relative redundancy and gives the maximum possible data compression ratio, when expressed as the percentage by which a file size can be decreased. (When expressed as a ratio of original file size to compressed file size, the quantity ${\displaystyle R:r}$ gives the maximum compression ratio that can be achieved.) Complementary to the concept of relative redundancy is efficiency, defined as ${\displaystyle {\frac {r}{R}},}$ so that ${\displaystyle {\frac {r}{R}}+{\frac {D}{R}}=1}$. A memoryless source with a uniform distribution has zero redundancy (and thus 100% efficiency), and cannot be compressed.

Other notions

A measure of redundancy between two variables is the mutual information or a normalized variant. A measure of redundancy among many variables is given by the total correlation.

Redundancy of compressed data refers to the difference between the expected compressed data length of ${\displaystyle n}$ messages ${\displaystyle L(M^{n})\,\!}$ (or expected data rate ${\displaystyle L(M^{n})/n\,\!}$) and the entropy ${\displaystyle nr\,\!}$ (or entropy rate ${\displaystyle r\,\!}$). (Here we assume the data is ergodic and stationary, e.g., a memoryless source.) Although the rate difference ${\displaystyle L(M^{n})/n-r\,\!}$ can be arbitrarily small as ${\displaystyle n\,\!}$ increased, the actual difference ${\displaystyle L(M^{n})-nr\,\!}$, cannot, although it can be theoretically upper-bounded by 1 in the case of finite-entropy memoryless sources.

See also

• Minimum redundancy coding
  • Huffman encoding
• Data compression
• Hartley function
• Negentropy
• Source coding theorem
• Overcompleteness

References

1. Here it is assumed ${\displaystyle {\mathcal {A}}_{X}}$ are the sets on which the probability distributions are defined.
2. MacKay, David J.C. (2003). "2.4 Definition of entropy and related functions". Information Theory, Inference, and Learning Algorithms. Cambridge University Press. p. 33. ISBN 0-521-64298-1. The redundancy measures the fractional difference between H(X) and its maximum possible value, ${\displaystyle |\log(|{\mathcal {A}}_{X}|)}$

• Reza, Fazlollah M. (1994) [1961]. An Introduction to Information Theory. New York: Dover [McGraw-Hill]. ISBN 0-486-68210-2.
• Schneier, Bruce (1996). Applied Cryptography: Protocols, Algorithms, and Source Code in C. New York: John Wiley & Sons, Inc. ISBN 0-471-12845-7.
• Auffarth, B; Lopez-Sanchez, M.; Cerquides, J. (2010). "Comparison of Redundancy and Relevance Measures for Feature Selection in Tissue Classification of CT images". Advances in Data Mining. Applications and Theoretical Aspects. Springer. pp. 248–262. CiteSeerX 10.1.1.170.1528.