## About the function¶

Functional load is a measure of the “work” that any particular contrast does in a language, as compared to other contrasts (e.g., [Hockett1955], [Hockett1966]; [Kucera1963]; [King1967]; [Surendran2003]). Two contrasts in a language, such as [d] / [t] vs. [ð] / [θ] in English, may have very different functional loads. The difference between [d] and [t] is used to distinguish between many different lexical items, so it has a high functional load; there are, on the other hand, very few lexical items that hinge on the distinction between [ð] and [θ], so its functional load is much lower. One of the primary claims about functional load is that it is related to sounds’ propensity to merge over time, with pairs of sounds that have higher functional loads being less likely to merge than pairs of sounds with lower functional loads (e.g., [Wedel2013], [Todd2012]). The average functional load of a particular sound has also been claimed to affect its likelihood of being used as an epenthetic vowel [Hume2013]. Functional load has also been illustrated to affect the perceived similarity of sounds [Hall2014a].

## Method of calculation¶

There are two primary ways of calculating functional load that are provided as part of the PCT package. One is based on the change of entropy in a system upon merger of a segment pair or set of segment pairs (cf. [Surendran2003]); the other is based on simply counting up the number of minimal pairs (differing in only the target segment pair or pairs) that occur in the corpus.

### Change in entropy¶

The calculation based on change in entropy is described in detail in [Surendran2003]. Entropy is an Information-Theoretic measure of the amount of uncertainty in a system [Shannon1949], and is calculated using the formula in (1); it will also be used for the calculation of predictability of distribution (see Method of calculation). For every symbol i in some inventory (e.g., every phoneme in the phoneme inventory, or every word in the lexicon), one multiplies the probability of i by the $$log_{2}$$ of the probability of i; the entropy is the sum of the products for all symbols in the inventory.

Entropy:

$$H = -\sum_{i \in N} p_{i} * log_{2}(p_{i})$$

The functional load of any pair of sounds in the system, then, can be calculated by first calculating the entropy of the system at some level of structure (e.g., words, syllables) with all sounds included, then merging the pair of sounds in question and re-calculating the entropy of the new system. That is, the functional load is the amount of uncertainty (entropy) that is lost by the merger. If the pair has a functional load of 0, then nothing has changed when the two are merged, and $$H_{1}$$ will equal $$H_{2}$$. If the pair has a non-zero functional load, then the total inventory has become smaller through the conflating of pairs of symbols that were distinguished only through the given pair of sounds.

Functional load as change in entropy:

$$\Delta H = H_{1} - H_{2}$$

Consider a toy example, in which the following corpus is assumed (note that, generally speaking, there is no “type frequency” column in a PCT corpus, as it is assumed that each row in the corpus represents 1 type; it is included here for clarity):

Consider a toy example, in which the following corpus is assumed (note that, generally speaking, there is no “type frequency” column in a PCT corpus, as it is assumed that each row in the corpus represents 1 type; it is included here for clarity):

Word Original Under [h] / [ŋ] merger Under [t] / [d] merger
Trans. Type Freq. Token Freq. Trans. Type Freq. Token Freq. Trans. Type Freq. Token Freq.
hot [hɑt] 1 2 [Xɑt] 1 2 [hɑX] 1 2
song [sɑŋ] 1 4 [sɑX] 1 4 [sɑŋ] 1 4
hat [hæt] 1 1 [Xæt] 1 1 [hæX] 1 1
sing [sɪŋ] 1 6 [sɪX] 1 6 [sɪŋ] 1 6
tot [tɑt] 1 3 [tɑt] 1 3 [XɑX] 1 8
dot [dɑt] 1 5 [dɑt] 1 5 [XɑX]
hip [hɪp] 1 2 [Xɪp] 1 2 [hɪp] 1 2
hid [hɪd] 1 7 [Xɪd] 1 7 [hɪX] 1 7
team [tim] 1 5 [tim] 1 5 [Xim] 1 10
deem [dim] 1 5 [dim] 1 5 [Xim]
toot [tut] 1 9 [tut] 1 9 [XuX] 1 11
dude [dud] 1 2 [dud] 1 2 [XuX]
hiss [hɪs] 1 3 [Xɪs] 1 3 [hɪs] 1 3
his [hɪz] 1 5 [Xɪz] 1 5 [hɪz] 1 5
sizzle [sɪzəl] 1 4 [sɪzəl] 1 4 [sɪzəl] 1 4
dizzy [dɪzi] 1 3 [dɪzi] 1 3 [Xɪzi] 1 7
tizzy [tɪzi] 1 4 [tɪzi] 1 4 [Xɪzi]
Total 17 70   17 70   13 70

The starting entropy, assuming word types as the relative unit of structure and counting, is:

$$H_{1 - types} = -[(\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17}))\\ + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17}))\\ + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17})) + (\frac{1}{17} log_{2}(\frac{1}{17}))] =4.087$$

The starting entropy, assuming word tokens, is:

$$H_{1 - tokens} = -[(\frac{2}{70} log_{2}(\frac{2}{70})) + (\frac{4}{70} log_{2}(\frac{4}{70})) + (\frac{1}{70} log_{2}(\frac{1}{70})) + (\frac{6}{70} log_{2}(\frac{6}{70})) + (\frac{3}{70} log_{2}(\frac{3}{70}))\\ + (\frac{5}{70} log_{2}(\frac{5}{70})) + (\frac{2}{70} log_{2}(\frac{2}{70})) + (\frac{7}{70} log_{2}(\frac{7}{70})) + (\frac{5}{70} log_{2}(\frac{5}{70})) + (\frac{5}{70} log_{2}(\frac{5}{70})) + (\frac{9}{70} log_{2}(\frac{9}{70}))\\ + (\frac{2}{70} log_{2}(\frac{2}{70})) + (\frac{3}{70} log_{2}(\frac{3}{70})) + (\frac{5}{70} log_{2}(\frac{5}{70})) + (\frac{4}{70} log_{2}(\frac{4}{70})) + (\frac{3}{70} log_{2}(\frac{3}{70})) + (\frac{4}{70} log_{2}(\frac{4}{70}))] = 3.924$$

Upon merger of [h] and [ŋ], there is no change in the number of unique words; there are still 17 unique words with all their same token frequencies. Thus, the entropy after an [h] / [ŋ] merger will be the same as it was before the merger. The functional load, then would be 0, as the pre-merger and post-merger entropies are identical.

Upon merger of [t] and [d], on the other hand, four pairs of words have been collapsed. E.g., the difference between team and deem no longer exists; there is now just one word, [Xim], where [X] represents the result of the merger. Thus, there are only 13 unique words, and while the total token frequency count remains the same, at 70, those 70 occurrences are divided among only 13 unique words instead of 17.

Thus, the entropy after a [t] / [d] merger, assuming word types, is:

$$H_{1 - types} = -[(\frac{1}{13} log_{2}(\frac{1}{13})) + (\frac{1}{13} log_{2}(\frac{1}{13})) + (\frac{1}{13} log_{2}(\frac{1}{13})) + (\frac{1}{13} log_{2}(\frac{1}{13})) + (\frac{1}{13} log_{2}(\frac{1}{13}))\\ + (\frac{1}{13} log_{2}(\frac{1}{13})) + (\frac{1}{13} log_{2}(\frac{1}{13})) + (\frac{1}{13} log_{2}(\frac{1}{13})) + (\frac{1}{13} log_{2}(\frac{1}{13})) + (\frac{1}{13} log_{2}(\frac{1}{13})) + (\frac{1}{13} log_{2}(\frac{1}{13}))\\ + (\frac{1}{13} log_{2}(\frac{1}{13})) + (\frac{1}{13} log_{2}(\frac{1}{13}))] = 3.700$$

And the entropy after a [t] / [d] merger, assuming word tokens, is:

$$H_{1 - tokens} = -[(\frac{2}{70} log_{2}(\frac{2}{70})) + (\frac{4}{70} log_{2}(\frac{4}{70})) + (\frac{1}{70} log_{2}(\frac{1}{70})) + (\frac{6}{70} log_{2}(\frac{6}{70})) + (\frac{8}{70} log_{2}(\frac{8}{70}))\\ + (\frac{2}{70} log_{2}(\frac{2}{70})) + (\frac{7}{70} log_{2}(\frac{7}{70})) + (\frac{10}{70} log_{2}(\frac{10}{70})) + (\frac{11}{70} log_{2}(\frac{11}{70})) + (\frac{3}{70} log_{2}(\frac{3}{70})) + (\frac{5}{70} log_{2}(\frac{5}{70}))\\ + (\frac{4}{70} log_{2}(\frac{4}{70})) + (\frac{7}{70} log_{2}(\frac{7}{70}))] = 3.466$$

$$\Delta H = H_{1-types} - H_{2-types} = 4.087– 3.700 = 0.387$$

And the functional load of [t] / [d] based on word tokens is:

$$\Delta H = H_{1-tokens} - H_{2-tokens} = 3.924– 3.466 = 0.458$$

### (Relative) Minimal Pair Counts¶

The second means of calculating functional load that is included in PCT is a straight count of minimal pairs, which can be relativized to the number of words in the corpus that are potential minimal pairs—i.e. the number of words in the corpus with at least one of the target segments.

In the above example, the number of minimal pairs that hinge on [h] vs. [ŋ] is of course 0, so the functional load of [h] / [ŋ] is 0. The number of minimal pairs that hinge on [t] / [d] is 3, and the number of words with either [t] or [d] is 11; the functional load as a relativized minimal pair count would therefore be 3/11 = 0.273. Note that here, a relatively loose definition of minimal pair is used; specifically, two words are considered to be a minimal pair hinging on sounds A and B if, upon merger of A and B into a single symbol X, the words are identical. Thus, toot and dude are considered a minimal pair on this definition, because they both become [XuX] upon merger of [t] and [d].

The resulting calculations of functional load are thus quite similar between the two measures, but the units are entirely different. Functional load based on change in entropy is measured in bits, while functional load based on relativized minimal pair counts is simply a percentage. Also note that functional load based on minimal pairs is only based on type frequency; the frequency of the usage of the words is not used as a weighting factor, the way it can be under the calculation of functional load as change in entropy.

## Implementing the functional load function in the GUI¶

As with most analysis functions, a corpus must first be loaded (see Loading in corpora). Once a corpus is loaded, use the following steps.

1. Getting started: Choose “Analysis” / “Calculate functional load...” from the top menu bar.
2. Sound selection: First, select which two sounds you want the functional load to be calculated for. Do this by clicking on “Add pair of sounds”; the “Select segment pair” dialogue box will open. The segment choices that are available will automatically correspond to all of the unique transcribed characters in your corpus. The order of the sounds is irrelevant; picking [i] first and [u] second will yield the same results as picking [u] first and [i] second. Once a pair of sounds has been selected, click “Add.” They will appear in the “Functional load” dialogue box. Multiple pairs of sounds can be selected and added to the list for calculation simultaneously. To do this without going back to the “Functional Load” dialogue box first, click “Add and create another.” When multiple pairs are selected, they can be treated in two different ways, listed under “Options” on the right-hand side of the “Functional Load” dialogue box under “Multiple segment pair behaviour”:
1. All segment pairs together: This option allows for the calculation of the functional load of featural contrasts. E.g., if the pairs [e]/[i] and [o]/[u] are chosen, PCT will calculate the functional load from both pairs at the same time. This option is useful for investigating the functional load of featural contrasts: e.g., if the above pairs are the ONLY pairs of sounds in the corpus that differ by exactly the single feature [high], then this option will allow you to calculate the functional load of the [high] contrast. Note that the results table will list “[e], [o]” as “sound 1” and “[i], [u]” as “sound 2” in this scenario, to remind you that you are getting a single functional load value. Note too that this does not collapse all four sounds to a single sound (which would erroneously also neutralize [e]/[o], [e]/[u], [i]/[o], [i]/[u]), but rather collapses each pair of segments and only then checks for any minimal pairs or drop in entropy.
2. Each segment pair individually: This option cycles through the list of pairs and gives the functional load of each pair individually from the corpus. E.g., if the pairs [e]/[i] and [o]/[u] are chosen, you will get results showing first the functional load of [e]/[i] in the corpus and then the functional load of [o]/[u] in the corpus, independently.
3. Functional load algorithm: Select which of the two methods of calculation you want to use—i.e., minimal pairs or change in entropy. (See discussion above for details of each.)
4. Tier: Select which tier the functional load should be calculated from. The default is the “transcription” tier, i.e., looking at the entire word transcriptions. If another tier has been created (see Creating new tiers in the corpus), functional load can be calculated on the basis of that tier. For example, if a vowel tier has been created, then “minimal pairs” will be entries that are identical except for one entry in the vowels only, entirely independently of consonants. Thus, the words [mapotik] and [ʃɹaɡefli] would be treated as a minimal pair, given that their vowel-tier representations are [aoi] and [aei].
5. Minimum frequency: It is possible to set a minimum token frequency for words in the corpus in order to be included in the calculation. This allows easy exclusion of rare words; for example, if one were calculating the functional load of [s] vs. [ʃ] in English and didn’t set a minimum frequency, words such as santy (vs. shanty) might be included, which might not be a particularly accurate reflection of the phonological knowledge of speakers. To include all words in the corpus, regardless of their token frequency, set the the minimum frequency to 0.
6. Additional parameters for minimal pairs: If minimal pairs serve as the means of calculation, there are two additional parameters can be set.
1. Raw vs. relative count: First, PCT can report only the raw count of minimal pairs that hinge on the contrast in the corpus, if you just want to know the scope of the contrast. On the other hand, the default is to relativize the raw count to the corpus size, by dividing the raw number by the number of lexical entries that include at least one instance of any of the target segments.
2. Include vs. ignore homophones: Second, PCT can either include homophones or ignore them. For example, if the corpus includes separate entries for the words sock (n.), sock (v.), shock (n.), and shock (v.), this would count as four minimal pairs if homophones are included, but only one if homophones are ignored. The default is to ignore homophones.
7. Additional parameters for change in entropy: If you are calculating functional load using change in entropy, one additional parameter can be set.
1. Type or token frequency: As described in Change in entropy, entropy can be calculated using either type or token frequencies. This option determines which to use.

Here is an example of selecting [m] and [n], with functional load to be calculated on the basis of minimal pairs, only including words with a token frequency of at least 1, from the built-in example corpus:

1. Results table: Once all parameters have been set, click one of the two “Calculate functional load” buttons. If this is the first calculation, the option to “start new results table” should be selected. For subsequent calculations, the calculation can be added to the already started table, for direct comparison, or a new table can be started. [Note that if a table is closed, new calculations will not be added to the previously open table; a new table must be started.] Either way, the results table will have the following columns, with one row per calculation: segment 1, segment 2, which tier was used, which measurement method was selected, the resulting functional load, what the minimum frequency was, and for calculations using minimal pairs, whether the count is absolute or relative and whether homophones were ignored or not. (For calculations using change in entropy, “N/A” values are entered into the latter two columns.)
2. Saving results: Once a results table has been generated for at least one pair, the table can be saved by clicking on “Save to file” at the bottom of the table to open a system dialogue box and save the results at a user-designated location.

(Note that in the above screen shot, not all columns are visible; they are visible only by scrolling over to the right, due to constraints on the window size. All columns would be saved to the results file.)

To return to the function dialogue box with your most recently used selections, click on “Reopen function dialog.” Otherwise, the results table can be closed and you will be returned to your corpus view.

## Implementing the functional load function on the command line¶

In order to perform this analysis on the command line, you must enter a command in the following format into your Terminal:

pct_funcload CORPUSFILE ARG2


...where CORPUSFILE is the name of your *.corpus file and ARG2 is either the transcription character(s) of a single segment (if calculating relative functional load) or the name of your segment pair(s) file (if calculating a single functional load value). The segment pairs file must list the pairs of segments whose functional load you wish to calculate, with each pair separated by a tab (t) and one pair on each line. You may also use command line options to change various parameters of your functional load calculations. Descriptions of these arguments can be viewed by running pct_funcload –h or pct_funcload --help. The help text from this command is copied below, augmented with specifications of default values:

Positional arguments:

corpus_file_name

Name of corpus file

pairs_file_name_or_segment

Name of file with segment pairs (or target segment if relative fl is True)

Optional arguments:

-h
--help

Show help message and exit

-a ALGORITHM
--algorithm ALGORITHM

Algorithm to use for calculating functional load: “minpair” for minimal pair count or “deltah” for change in entropy. Defaults to minpair.

-f FREQUENCY_CUTOFF
--frequency_cutoff FREQUENCY_CUTOFF

Minimum frequency of words to consider as possible minimal pairs or contributing to lexicon entropy.

-d DISTINGUISH_HOMOPHONES
--distinguish_homophones DISTINGUISH_HOMOPHONES

For minimal pair FL: if False, then you’ll count sock~shock (sock=clothing) and sock~shock (sock=punch) as just one minimal pair; but if True, you’ll overcount alternative spellings of the same word, e.g. axel~actual and axle~actual. False is the value used by Wedel et al.

-t TYPE_OR_TOKEN
--type_or_token TYPE_OR_TOKEN

For change in entropy FL: specifies whether entropy is based on type or token frequency.

-e RELATIVE_FL
--relative_fl RELATIVE_FL

If True, calculate the relative FL of a single segment by averaging across the functional loads of it and all other segments.

-s SEQUENCE_TYPE
--sequence_type SEQUENCE_TYPE

The attribute of Words to calculate FL over. Normally this will be the transcription, but it can also be the spelling or a user-specified tier.

-o OUTFILE
--outfile OUTFILE

Name of output file

EXAMPLE 1: If your corpus file is example.corpus and you want to calculate the minimal pair functional load of the segments [m] and [n] using defaults for all optional arguments, you first need to create a text file that contains the text “mtn” (where t is a tab; no quotes in the file). Let us call this file pairs.txt. You would then run the following command in your terminal window:

pct_funcload example.corpus pairs.txt


EXAMPLE 2: Suppose you want to calculate the relative (average) functional load of the segment [m]. Your corpus file is again example.corpus. You want to use the change in entropy measure of functional load rather than the minimal pairs measure, and you also want to use type frequency instead of (the default value of) token frequency. In addition, you want the script to produce an output file called output.txt. You would need to run the following command:

pct_funcload example.corpus m -a deltah -t type -o output.txt