# Command: infot¶

## COMMAND¶

infot – calculate information theory measures for Humdrum inputs

## SYNOPSIS¶

 infot -b [-H] [-x regexp] [inputfile ...] infot -n [-H] [-x regexp] [inputfile ...] infot -p [-H] [-x regexp] [inputfile ...] infot -s [-x regexp] [inputfile ...]

## DESCRIPTION¶

The infot command provides a general-purpose tool for measuring the probability relationships between user-selected data tokens. Given a specified input stream, infot can calculate one of several pertinent information-theoretic values. The values may be calculated with reference to an independent repertoire, or may be calculated as so-called “self-information.”

In conjunction with other Humdrum tools (notably the **context** and **humsed** commands), infot permits sophisticated information-theoretic analyses to be carried out, including calculations of information flow, short-term conditional probabilities, and longer-term m-dependency analyses. Alternatively, a simple set of summary statistics can be requested. In most cases, users will want to use infot to generate outputs that are suitable for further processing.

Input to infot is restricted to a single spine. However, the input data tokens may contain multiple-stops representing complex contextual information (such as produced by the **context** command).

For the entire input, infot tabulates the total number of occurrences of each unique data record (hereafter referred to as states’). For the -n, -p and -b options, infot outputs a two-column list where the left column indentifies each unique state and the right column provides one of several corresponding calculated measures. With the -n option, this measure is merely an integer count of the number of occurrences of each corresponding state. With the -p option, this measure is a probability of occurrence for each state. With the -b option, this measure identifies the information content for the corresponding state in bits.

Information content (H) in bits is calculated according to the classic equation devised by Shannon and Weaver (see REFERENCES):

.EQ delim  .EN .EQ H~ =~ sum from i=1 to N ~p sub i ~log sub 2 ~1 over { p sub i } .EN

where $H$ is the average information (in bits), $N$ is the number of possible unique states in the repertoire, and $p sub i$ is the probability of occurrence of state $i$ from the repertoire. .EQ delim off .EN

Note that the outputs produced by infot do not conform to the Humdrum syntax.

## OPTIONS¶

The infot command provides the following options:

 -b output information (in bits) for each unique data token -h displays a help screen summarizing the command syntax -H format output as **humsed** commands -n output frequency count for each unique data token -p output probability value for each unique data token -s output information-related summary statistics -x regexp exclude tokens matching regexp from calculations

Options are specified in the command line.

With the -n option, infot outputs a two-column list where the left column indentifies each unique state present in the input, and the right column provides an integer count indicating the number of occurrences for the corresponding state.

With the -p option, infot outputs a two-column list where probabilities of occurrence are output in the right-hand column, rather than counts.

With the -b option, infot outputs the information (in bits) as calculated according to the Shannon-Weaver equation.

## EXAMPLES¶

The use of infot is illustrated in the following examples. Consider the following input: 

 **foo A B2 C-c A B2 A A B2 C-c A A X Y *-

A simple command invocation would use the -n option to count the number of occurrences of each unique data token (or state):

 infot -n input

The corresponding output is: 

 A 6 B2 3 C-c 2 X Y 1

The tallies indicate that state A’ occurs 6 times, and that the least common state (X Y’) occurs just once. If we had invoked the -p option, the counts would be replaced by probabilities. The command:

 infot -p input

produces the following output: 

 A 0.5 B2 0.25 C-c 0.167 X Y 0.083

Alternatively, the -b option:

 infot -b input

would output information measures for each state, in bits: 

 A 1 B2 2 C-c 2.585 X Y 3.585

In the case of the -s option, summary statistics would be output, rather than a two-column list. For the above input, the following summary statistics would be generated: 

 Total number of unique states in message: 4 Total information of message (in bits): 20.7549 Total possible information for message: 24 Info per state for equi-prob distrib: 2 Average information conveyed per state: 1.72957 Percent redundancy evident in message: 13.5213

The first line of output merely indicates the number of unique states found in the input (in this case just 4). The fifth output line indicates the average information conveyed per state (in bits). The fourth output line indicates the theoretical maximum average information per state that could be communicated by a system having four states. The third line indicates the maximum possible information that could be communicated in a message of the same length as the input – given the theoretical maximum average information. (Since there are 12 data records, this value is simply 12 x 2 bits, or 24 bits.) The second output line gives the actual total information for the given input message. (This is always less-than, or equal-to the maximum theoretical value.) The final line indicates the amount of redundancy – as a percentage. That is, this value contrasts the actual information conveyed with the theoretical maximum.

In general, note that as the probabilities of the input states approach equivalence, the redundancy approaches zero and the average information content approaches the theoretical maximum.

Consider now an example where a large number of messages from a repertoire (dubbed repertoire) is passed to infot:

 infot -b repertoire

Suppose that the following output is produced: 

 ABC 3.124 BAC 1.306 C C D 1.95 X 5.075 XYZ 19.334

This result indicates that, although there might have been hundreds of data tokens processed in the repertoire, only five different unique states were present. The greatest information content (lowest probability) is associated with the state XYZ (19.334 bits), whereas the lowest information content (highest probability) is associated with the state BAC (1.306 bits). Notice that the multiple-stop C C D is treated as a single state.

Now imagine we had another message presumed to belong to the same repertoire as our input. We would like to trace how the information increases and decreases over the course of this new message’. This goal involves a two-part operation. First, we re-invoke infot adding the -H option, and redirect the output to a file replace:

 infot -bH repertoire > replace

This causes infot to produce as output a set of **humsed** commands. Given the identical repertoire input, the following output is sent to the file replace:

 s/^ABC$/3.124/g; s/^ABC /3.124/g; s/ ABC$/3.124/g; s/ ABC /3.124/g s/^BAC$/1.306/g; s/^BAC /1.306/g; s/ BAC$/1.306/g; s/ BAC /1.306/g s/^C C D$/1.95/g; s/^C C D /1.95/g; s/ C C D$/1.95/g; s/ C C D /1.95/g s/^X$/5.075/g; s/^X /5.075/g; s/ X$/5.075/g; s/ X /5.075/g s/^XYZ$/19.334/g; s/^XYZ /19.334/g; s/ XYZ$/19.334/g; s/ XYZ /19.334/g

Although these commands may appear somewhat cryptic, they merely instruct the Humdrum stream editor (**humsed**) to replace all occurrences of the five data tokens (in any input file) by the corresponding numerical values – in this case, values that represent the number of bits of information.

### The following file (called input) contains the message of interest:¶

 **bar BAC BAC C C D . = * C C D XYZ X ABC BAC *-

This file can be transformed so that the data tokens are replaced by corresponding information values as determined from the original repertoire. This is done by invoking the **humsed** command, and providing it with the substitution commands held in the file replace:

 humsed -f replace input > output

The resulting output file would be as follows: 

 **bar 1.306 1.306 1.950 . = * 1.950 19.334 5.075 3.124 1.306 *-

Notice that input data tokens which do not appear in the probability list (such as the equals-signs) remain unmodified.

Several interpretations may be made about this message. For example, the above passage appears to show a pattern of initially low information that increases and then decreases toward the end of the passage. This suggests that the beginning and ending of this passage are more highly constrained or stereotypic than the middle part of the passage.

Summing together the individual information values for this passage, the total message conveys 35.35 bits. For five states, the maximum average information is 2.322 bits per state, and so the expected maximum for a message consisting of 8 items would be 8 x 2.322 or 18.58 bits. This suggests that this message is considerably less banal, (less redundant or more unique) than a typical message from the original repertoire. In particular, the occurrence of the state `XYZ’ has a low probability of occurrence – and is likely to be a distinctive feature of this passage.

In the above examples, only simple (zeroth-order) probabilities have been examined. Higher-order and m-dependency probabilities may be measured by reformulating the input using the context command.

## PORTABILITY¶

DOS 2.0 and up, with the MKS Toolkit. OS/2 with the MKS Toolkit. UNIX systems supporting the Korn shell or Bourne shell command interpreters, and revised awk (1985).

**context** (4), **humsed** (4), **patt** (4), **pattern** (4), **simil** (4)

## REFERENCES¶

Knopoff, L. & Hutchinson, W. “Entropy as a measure of style: The influence of sample length.” Journal of Music Theory, Vol. 27 (1983) pp. 75-97.

Moles, A. Information Theory and Esthetic Perception, Urbana: University of Illinois Press, 1968.

Pinkerton, R.C. “Information theory and melody.” Scientific American, Vol. 194 (1956) pp. 77-86.

Shannon, C. E., & Weaver, W. The Mathematical Theory of Communication. Urbana: University of Illinois Press, 1949.

Snyder, J.L. “Entropy as a measure of musical style: The influence of a priori assumptions.” Music Theory Spectrum, Vol. 12, No. 1 (1990) pp. 121-160.

Wong, A. K. C., & Ghahraman, D. A statistical analysis of interdependence in character sequences. Information Sciences, Vol. 8 (1975) pp. 173-188.

Youngblood, J.E. “Style as information.” Journal of Music Theory, Vol. 7 (1962) pp. 137-162.