- Redundancy (information theory)
**Redundancy**ininformation theory is the number of bits used to transmit a message minus the number of bits of actual information in the message. Informally, it is the amount of wasted "space" used to transmit certain data.Data compression is a way to reduce or eliminate unwanted redundancy, whilechecksum s are a way of adding desired redundancy for purposes oferror detection when communicating over a noisy channel of limited capacity.**Quantitative definition**In describing the redundancy of raw data, recall that 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:$r\; =\; lim\_\{n\; o\; infty\}\; frac\{1\}\{n\}\; H(M\_1,\; M\_2,\; dots\; M\_n),$

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 $H(M)$, 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:$R\; =\; log\; |mathbb\; M|\; ,,$

the

logarithm of thecardinality of the message space, or alphabet. (This formula is sometimes called theHartley 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:$D\; =\; R\; -\; r\; ,,$

the difference between the absolute rate and the rate.

The quantity $frac\; D\; R$ is called the

**relative redundancy**and gives the maximum possibledata 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 $R\; :\; r$ gives the maximum compression ratio that can be achieved.) Complementary to the concept of relative redundancy is**efficiency**, defined as $frac\; r\; R\; ,$ so that $frac\; r\; R\; +\; frac\; D\; R\; =\; 1$. A memoryless source with a uniform distribution has zero redundancy (and thus 100% efficiency), and cannot be compressed.**Other notions of redundancy**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 thetotal correlation .Redundancy of compressed data refers to the difference between the expected compressed data length of $n$ messages $L(M^n)\; ,!$ (or expected data rate $L(M^n)/n\; ,!$) and the entropy $nr\; ,!$ (or entropy rate $r\; ,!$). (Here we assume the data is ergodic and

stationary , e.g., a memoryless source.) Although the rate difference $L(M^n)/n-r\; ,!$ can be arbitrarily small as $n\; ,!$ increased, the actual difference $L(M^n)-nr\; ,!$, cannot, although it can be theoretically upper-bounded by 1 in the case of finite-entropy memoryless sources.**ee also***

Data compression

*Exformation

*Negentropy

*Source coding theorem **References*** Fazlollah M. Reza. "An Introduction to Information Theory". New York: McGraw-Hill 1961. New York: Dover 1994. ISBN 0-486-68210-2

* B. Schneier, "Applied Cryptography: Protocols, Algorithms, and Source Code in C". New York: John Wiley & Sons, Inc. 1996. ISBN 0-471-12845-7

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