Redundancy (information theory)

Redundancy in information 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, 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, 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


 * $$r = \lim_{n \to \infty} \frac{1}{n} H(M_1, M_2, \dots M_n),$$

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 $$H(M)$$, 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 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


 * $$ D = R - r ,\,$$

the difference between the absolute rate and the rate.

The quantity $$\frac D R$$ is called the relative redundancy and gives the maximum 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 $$R : r$$ 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 the total 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.