Phoneme frequency

1. Problem and history

The problem is to find a rank-frequency distribution for the phoneme of a text or of a corpus. Sometimes letters or even sounds are counted, which is fully justified. In the same way one could count e.g. the syllables of the Japanese katakana or hiragana. The number of examinations is enormous, some of them give the absolute frequencies other ones merely the proportions.

The counting began in the 19th century (Förstemann 1852; Bourdon 1892) and developed quickly on practical grounds since stenographers, printers, constructors of typewriters, decoders etc., needed urgently the frequency of letters for their own purposes.

The first who considered phonemes from the frequency point of view and set up hypotheses was G.K. Zipf (1929, 1935, 1949). Afterwards a great number of works appeared using phoneme frequencies for finding other interrelations. The first empirical model for a distribution, namely the geometric (and the right truncated geometric) distribution, was proposed by Sigurd (1968). Good (1969) brought a partial-sums distribution whose modelling was revived in word length ($\rightarrow$) research. Altmann (1993) used the synergetic way of modelling and derived a special function for this purpose. Martindale, Gusein-Zade, Mckenzie and Borodovsky (1996) compared several curves (functions) and many data in order to find the “best” model. Altmann and Lehfeldt (1980) and Zörnig, Altmann (1983, 1984) developed hypotheses on the entropy ($\rightarrow$) and the repeat rate ($\rightarrow$) of phonemes, Kubáček (1994) derived the formula for the necessary size of the phoneme count in order to attain confident counts. Naranan and Balasubrahmanyan (2000) developed a theory from which different curves for phoneme frequencies are derivable.

Not all arguments holding for word frequencies are valid in this domain.

2. Hypothesis

The ranked frequencies of phonemes follow a regular probability function or a regular monotone decreasing function.

The result depends on whether one considers the ranked frequencies as a discrete distribution (normalized) or merely a regular series approached by a continuous function (not normalized).

3. Derivation

The formulas used up to now can be derived from different approaches.

3.1. Tuldava´s approach (1988)

This approach can be represented by the simple differential equation

(1) $y' = \frac{b}{x}$

to obtain

(2)$A y = a + b \ln x \quad$

This curve is frequently used in other domains, too (cf. also Martindale et al. 1996).

3.2. Derivations related to the unified theory (→) are

(a) Zipf´s law (zeta distribution)

When formula (2) of the unified theory is used with $a_0 = a_2 = a_3 = ... = 0, a_1 = -b \quad$, this yields

(3) $\frac{dy}{y} = -\frac{b}{x}dx$

resulting in

(4)$y = Ax^{-b}\quad$.

This is, perhaps, the most disseminated formula in linguistics.

(b) Yule´s species/genera function

When formula (2) of the unified theory is used with $a_0 = c, a_1 = b, a_2 = a_3 = ... = 0\quad$, this yields

(5)$\frac{dy}{y}= \left(c- \frac{b}{x} \right)dx$

resulting in

(6)$y= ae^{cx}x^{-b}= ad^x b^{-b}\quad$ .

(c) Naranan and Balasubrahmanyan´s (1992a,b, 2000) function

When formula (2) of the unified theory is used with $a_0 = 0, a_3 = a_4 = ... = 0\quad$, this yields

(7)$\frac{dy}{y}\left(- \frac{a_1}{x}{a_2}{x^2} \right)dx$

resulting in

(8)$y= Ce^{-a_2/x}x^{-a_1}$.

(d) Altmann´s ranking function (1993)

Using formula (11) of the unified theory, which can be written as

(9)$y_x = \left(1-a_0 + \frac{a_1}{(x-b_1)^{c_1}} + \frac{a_2}{(x-b_2)^{c_2}} \right)y_{x-1}$ ,

and reparametrizing $a_i = 0 (i = 0,2,3,...), c_1 = 1$, yields

(10)$y_x = \left(1+ \frac{a_1}{x-b_1} \right)y_{x-1}$.

Upon setting $b_1 = -a$, $a_1 - b_1 = b\quad$, this results in

(11)$y_x = \frac{\begin{pmatrix} b+x \\ x-1 v \end{pmatrix}}{\begin{pmatrix} a+x \\ x-1 \end{pmatrix}}y_1 \quad, x = 1,2,3,...$

All these formulas can be transformed in distributions by appropriate normalizing.

(e) Geometric distribution

Sigurd (1968) used simply the geometric distribution. It can be obtained from formula (10) of the unified theory setting $a_i = 0 (i = 1,2,3,...)\quad$, which yields

(12)$y_{x+1}= (1+a_0)y_x\quad$.

For$- < a_ < 0, 1+a_ = q, 1-q = p, y_ = Px\quad$tains the usual (1-displaced) geometric distribution

(13)$P_x = pq^{x-1}, \quad x = 1,2,3,...$

The same result was proposed also by Orlov, Boroda, Nadarejšvili (1982). Treating directly the relative frequencies one can write (13) as

(14)$y_x = y_1 q^{x-1}, \quad x=1,2,3,...$

3.3. Partial-sums distributions (Good 1969)

Good (1969) introduced a new distribution, mentioned in Martindale et al. (1996). It is a so-called partial-sums distribution, namely a “sterred” discrete uniform distribution (cf. Wimmer, Altmann 1999). Their provenience is shown in the chapter on Word frequency ($\rightarrow$)has the form

(15)$P_x = \frac{1}{n}\sum_{i=x}^n \frac{1}{i},\quad x=1,2,...,n$.

Example: Frequency of phonemes in Hawaiian

In Table 1 and Fig. 1 one can find the fitting of the above formulas to the relative frequencies of Hawaiian phonemes. If functions are used, normalizing is not necessary.

Except for the geometric distribution, all of them yield in this case a good – approximately equal – fitting. In Fig. 1, only fitting (11) is shown.

Fig. 1. Fitting function (11) to Hawaiian phoneme frequencies

4. Authors: G. Altmann

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