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Entropy

Module by: Anders Gjendemsjø, Behnaam Aazhang. E-mail the authors

Summary: Entropy

The self information gives the information in a single outcome. In most cases, e.g in data compression, it is much more interesting to know the average information content of a source. This average is given by the expected value of the self information with respect to the source's probability distribution. This average of self information is called the source entropy.

Definition of entropy

Definition 1: Entropy
1. The entropy (average self information) of a discrete random variable XX is a function of its probability mass function and is defined as
HX= i =1N p X x i log p X x i H X i 1 N p X x i p X x i
(1)
where NN is the number of possible values of XX and P X x i =PrX= x i P X x i X x i . If log is base 2 then the unit of entropy is bits per (source)symbol. Entropy is a measure of uncertainty in a random variable and a measure of information it can reveal.
2. If symbol has zero probability, which means it never occurs, it should not affect the entropy. Letting 0×log0=0 0 0 0 , we have dealt with that.

In texts you will find that the argument to the entropy function may vary. The two most common are HX H X and Hp H p . We calculate the entropy of a source X, but the entropy is, strictly speaking, a function of the source's probabilty function p. So both notations are justified.

Calculating the binary logarithm

Most calculators does not allow you to directly calculate the logarithm with base 2, so we have to use a logarithm base that most calculators support. Fortunately it is easy to convert between different bases.

Assume you want to calculate log2x 2 x , where x>0 x 0 . Then log2x=y 2 x y implies that 2y=x 2 y x . Taking the natural logarithm on both sides we obtain

Logarithm conversion:

log2x=lnxln2 2 x x 2

Examples

Example 1

When throwing a dice, one may ask for the average information conveyed in a single throw. Using the formula for entropy we get HX=i=16pXxilogpXxi=log6 bits/symbol H X i 1 6 pX xi pX xi 6 bits/symbol

Example 2

If a soure produces binary information 01 0 1 with probabilities pp and 1p 1 p . The entropy of the source is

HX=((plog 2 p))(1p)log 2 (1p) H X p 2 p 1 p 2 1 p
(2)
If p=0 p 0 then HX=0 H X 0 , if p=1 p 1 then HX=0 H X 0 , if p=1/2 p 12 then HX=1 H X 1 . The source has its largest entropy if p=1/2 p 12 and the source provides no new information if p=0 p 0 or p=1 p 1 .

Figure 1
Figure 1 (entropy_plot.png)

Example 3

An analog source is modeled as a continuous-time random process with power spectral density bandlimited to the band between 0 and 4000 Hz. The signal is sampled at the Nyquist rate. The sequence of random variables, as a result of sampling, are assumed to be independent. The samples are quantized to 5 levels -2-1012 -2 -1 0 1 2 . The probability of the samples taking the quantized values are 121418116116 1 2 1 4 1 8 1 16 1 16 , respectively. The entropy of the random variables are

HX=i=15pXxilogpXxi=12+12+38+14+14=158bits/sample H X i 1 5 pX xi pX xi 1 2 1 2 3 8 1 4 1 4 15 8 bits/sample
(3)
There are 8000 samples per second. Therefore, the source produces 8000×158=15000 8000 15 8 15000 bits/sec of information.

Entropy is closely tied to source coding. The extent to which a source can be compressed is related to its entropy. There are many interpretations possible for the entropy of a random variable, including

  • (Average)Self information in a random variable
  • Minimum number of bits per source symbol required to describe the random variable without loss
  • Description complexity
  • Measure of uncertainty in a random variable

References

  • Øien, G.E. and Lundheim,L. (2003) Information Theory, Coding and Compression, Trondheim: Tapir Akademisk forlag.

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