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"paper_id": "P07-1014",
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"title": "Redundancy Ratio: An Invariant Property of the Consonant Inventories of the World's Languages",
"authors": [
{
"first": "Animesh",
"middle": [],
"last": "Mukherjee",
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"institution": "Indian Institute of Technology",
"location": {
"settlement": "Kharagpur"
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"email": "animeshm@cse.iitkgp.ernet.in"
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{
"first": "Monojit",
"middle": [],
"last": "Choudhury",
"suffix": "",
"affiliation": {
"laboratory": "",
"institution": "Indian Institute of Technology",
"location": {
"settlement": "Kharagpur"
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"email": "monojit@cse.iitkgp.ernet.in"
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{
"first": "Anupam",
"middle": [],
"last": "Basu",
"suffix": "",
"affiliation": {
"laboratory": "",
"institution": "Indian Institute of Technology",
"location": {
"settlement": "Kharagpur"
}
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"email": "anupam@cse.iitkgp.ernet.in"
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{
"first": "Niloy",
"middle": [],
"last": "Ganguly",
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"affiliation": {
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"institution": "Indian Institute of Technology",
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"settlement": "Kharagpur"
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"abstract": "In this paper, we put forward an information theoretic definition of the redundancy that is observed across the sound inventories of the world's languages. Through rigorous statistical analysis, we find that this redundancy is an invariant property of the consonant inventories. The statistical analysis further unfolds that the vowel inventories do not exhibit any such property, which in turn points to the fact that the organizing principles of the vowel and the consonant inventories are quite different in nature.",
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"text": "In this paper, we put forward an information theoretic definition of the redundancy that is observed across the sound inventories of the world's languages. Through rigorous statistical analysis, we find that this redundancy is an invariant property of the consonant inventories. The statistical analysis further unfolds that the vowel inventories do not exhibit any such property, which in turn points to the fact that the organizing principles of the vowel and the consonant inventories are quite different in nature.",
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"text": "Redundancy is a strikingly common phenomenon that is observed across many natural systems. This redundancy is present mainly to reduce the risk of the complete loss of information that might occur due to accidental errors (Krakauer and Plotkin, 2002) . Moreover, redundancy is found in every level of granularity of a system. For instance, in biological systems we find redundancy in the codons (Lesk, 2002) , in the genes (Woollard, 2005) and as well in the proteins (Gatlin, 1974) . A linguistic system is also not an exception. There is for example, a number of words with the same meaning (synonyms) in almost every language of the world. Similarly, the basic unit of language, the human speech sounds or the phonemes, is also expected to exhibit some sort of a redundancy in the information that it encodes.",
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"start": 222,
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"text": "(Krakauer and Plotkin, 2002)",
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"start": 395,
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"text": "(Lesk, 2002)",
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"start": 423,
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"text": "(Woollard, 2005)",
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"section": "Introduction",
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"text": "In this work, we attempt to mathematically capture the redundancy observed across the sound (more specifically the consonant) inventories of the world's languages.",
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"text": "For this purpose, we present an information theoretic definition of redundancy, which is calculated based on the set of features 1 (Trubetzkoy, 1931) that are used to express the consonants. An interesting observation is that this quantitative feature-based measure of redundancy is almost an invariance over the consonant inventories of the world's languages. The observation is important since it can shed enough light on the organization of the consonant inventories, which unlike the vowel inventories, lack a complete and holistic explanation. The invariance of our measure implies that every inventory tries to be similar in terms of the measure, which leads us to argue that redundancy plays a very important role in shaping the structure of the consonant inventories. In order to validate this argument we determine the possibility of observing such an invariance if the consonant inventories had evolved by random chance. We find that the redundancy observed across the randomly generated inventories is substantially different from their real counterparts, which leads us to conclude that the invariance is not just \"by-chance\" and the measure that we define, indeed, largely governs the organizing principles of the consonant inventories.",
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"text": "Interestingly, this redundancy, when measured for the vowel inventories, does not exhibit any similar invariance. This immediately reveals that the principles that govern the formation of these two types of inventories are quite different in nature. Such an observation is significant since whether or not these principles are similar/different for the two inventories had been a question giving rise to perennial debate among the past researchers (Trubetzkoy, 1969 (Trubetzkoy, /1939 Lindblom and Maddieson, 1988; Boersma, 1998; Clements, 2004) . A possible reason for the observed dichotomy in the behavior of the vowel and consonant inventories with respect to redundancy can be as follows: while the organization of the vowel inventories is known to be governed by a single force -the maximal perceptual contrast (Jakobson, 1941; Liljencrants and Lindblom, 1972; de Boer, 2000) ), consonant inventories are shaped by a complex interplay of several forces (Mukherjee et al., 2006) . The invariance of redundancy, perhaps, reflects some sort of an equilibrium that arises from the interaction of these divergent forces.",
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"text": "The rest of the paper is structured as follows. In section 2 we briefly discuss the earlier works in connection to the sound inventories and then systematically build up the quantitative definition of redundancy from the linguistic theories that are already available in the literature. Section 3 details out the data source necessary for the experiments, describes the baseline for the experiments, reports the experiments performed, and presents the results obtained each time comparing the same with the baseline results. Finally we conclude in section 4 by summarizing our contributions, pointing out some of the implications of the current work and indicating the possible future directions.",
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"text": "Linguistic research has documented a wide range of regularities across the sound systems of the world's languages. It has been postulated earlier by functional phonologists that such regularities are the consequences of certain general principles like maximal perceptual contrast (Liljencrants and Lindblom, 1972) , which is desirable between the phonemes of a language for proper perception of each individ-ual phoneme in a noisy environment, ease of articulation (Lindblom and Maddieson, 1988; de Boer, 2000) , which requires that the sound systems of all languages are formed of certain universal (and highly frequent) sounds, and ease of learnability (de Boer, 2000) , which is necessary for a speaker to learn the sounds of a language with minimum effort. In fact, the organization of the vowel inventories (especially those with a smaller size) across languages has been satisfactorily explained in terms of the single principle of maximal perceptual contrast (Jakobson, 1941; Liljencrants and Lindblom, 1972; de Boer, 2000) .",
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"text": "On the other hand, in spite of several attempts (Lindblom and Maddieson, 1988; Boersma, 1998; Clements, 2004) the organization of the consonant inventories lacks a satisfactory explanation. However, one of the earliest observations about the consonant inventories has been that consonants tend to occur in pairs that exhibit strong correlation in terms of their features (Trubetzkoy, 1931) . In order to explain these trends, feature economy was proposed as the organizing principle of the consonant inventories (Martinet, 1955) . According to this principle, languages tend to maximize the combinatorial possibilities of a few distinctive features to generate a large number of consonants. Stated differently, a given consonant will have a higher than expected chance of occurrence in inventories in which all of its features have distinctively occurred in other consonants. The idea is illustrated, with an example, through Table 1. Various attempts have been made in the past to explain the aforementioned trends through linguistic insights (Boersma, 1998; Clements, 2004) mainly establishing their statistical significance. On the contrary, there has been very little work pertaining to the quantification of feature economy except in (Clements, 2004) , where the author defines economy index, which is the ratio of the size of an inventory to the number of features that characterizes the inventory. However, this definition does not take into account the complexity that is involved in communicating the information about the inventory in terms of its constituent features.",
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"text": "Inspired by the aforementioned studies and the concepts of information theory (Shannon and Weaver, 1949) we try to quantitatively capture the amount of redundancy found across the consonant plosive voiced voiceless dental /d/ /t/ bilabial /b/ /p/ Table 1: The table shows four plosives. If a language has in its consonant inventory any three of the four phonemes listed in this table, then there is a higher than average chance that it will also have the fourth phoneme of the table in its inventory.",
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"text": "Table 1: The table shows four",
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"text": "inventories in terms of their constituent features. Let us assume that we want to communicate the information about an inventory of size N over a transmission channel. Ideally, one should require log N bits to do the same (where the logarithm is with respect to base 2). However, since every natural system is to some extent redundant and languages are no exceptions, the number of bits actually used to encode the information is more than log N . If we assume that the features are boolean in nature, then we can compute the number of bits used by a language to encode the information about its inventory by measuring the entropy as follows. For an inventory of size N let there be p f consonants for which a particular feature f (where f is assumed to be boolean in nature) is present and q f other consonants for which the same is absent. Thus the probability that a particular consonant chosen uniformly at random from this inventory has the feature f is",
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"text": "N and the probability that the consonant lacks the feature f is",
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"text": "q f N (=1- p f N",
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"text": "). If F is the set of all features present in the consonants forming the inventory, then feature entropy F E can be expressed as",
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"text": "EQUATION",
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"raw_str": "F E = f \u2208F (\u2212 p f N log p f N \u2212 q f N log q f N )",
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"text": "F E is therefore the measure of the minimum number of bits that is required to communicate the information about the entire inventory through the transmission channel. The lower the value of F E the better it is in terms of the information transmission overhead. In order to capture the redundancy involved in the encoding we define the term redundancy ratio as follows,",
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"text": "EQUATION",
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"text": "which expresses the excess number of bits that is used by the constituent consonants of the inventory Figure 1 : The process of computing RR for a hypothetical inventory.",
"cite_spans": [],
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"text": "Figure 1",
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"text": "in terms of a ratio. The process of computing the value of RR for a hypothetical consonant inventory is illustrated in Figure 1 . In the following section, we present the experimental setup and also report the experiments which we perform based on the above definition of redundancy. We subsequently show that redundancy ratio is invariant across the consonant inventories whereas the same is not true in the case of the vowel inventories.",
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"text": "In this section we discuss the data source necessary for the experiments, describe the baseline for the experiments, report the experiments performed, and present the results obtained each time comparing the same with the baseline results.",
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"section": "Experiments and Results",
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"text": "Many typological studies (Ladefoged and Maddieson, 1996; Lindblom and Maddieson, 1988) of segmental inventories have been carried out in past on the UCLA Phonological Segment Inventory Database (UPSID) (Maddieson, 1984) . UPSID gathers phonological systems of languages from all over the world, sampling more or less uniformly all the linguistic families. In this work we have used UPSID comprising of 317 languages and 541 consonants found across them, for our experiments.",
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"text": "In this section we measure the redundancy ratio (described earlier) of the consonant inventories of the languages recorded in UPSID. Figure 2 shows the scatter-plot of the redundancy ratio RR of each of the consonant inventories (y-axis) versus the inventory size (x-axis). The plot immediately reveals that the measure (i.e., RR) is almost invariant across the consonant inventories with respect to the inventory size. In fact, we can fit the scatter-plot with a straight line (by means of least square regression), which as depicted in Figure 2 , has a negligible slope (m = -0.018) and this in turn further confirms the above fact that RR is an invariant property of the consonant inventories with regard to their size. It is important to mention here that in this experiment we report the redundancy ratio of all the inventories of size less than or equal to 40. We neglect the inventories of the size greater than 40 since they are extremely rare (less than 0.5% of the languages of UP-SID), and therefore, cannot provide us with statistically meaningful estimates. The same convention has been followed in all the subsequent experiments. Nevertheless, we have also computed the values of RR for larger inventories, whereby we have found that for an inventory size \u2264 60 the results are similar to those reported here. It is interesting to note that the largest of the consonant inventories Ga (size = 173) has an RR = 1.9, which is lower than all the other inventories. The aforementioned claim that RR is an invariant across consonant inventories can be validated by performing a standard test of hypothesis. For this purpose, we randomly construct language inventories, as discussed later, and formulate a null hypothesis based on them. Null Hypothesis: The invariance in the distribution of RRs observed across the real consonant inventories is also prevalent across the randomly generated inventories.",
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"start": 133,
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"text": "Having formulated the null hypothesis we now systematically attempt to reject the same with a very high probability. For this purpose we first construct random inventories and then perform a two sample t-test (Cohen, 1995) comparing the RRs of the real and the random inventories. The results show that Figure 2 : The scatter-plot of the redundancy ratio RR of each of the consonant inventories (y-axis) versus the inventory size (x-axis). The straight linefit is also depicted by the bold line in the figure. indeed the null hypothesis can be rejected with a very high probability. We proceed as follows.",
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{
"start": 209,
"end": 222,
"text": "(Cohen, 1995)",
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"start": 303,
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"section": "Redundancy Ratio across the Consonant Inventories",
"sec_num": "3.2"
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"text": "We employ two different models to generate the random inventories. In the first model the inventories are filled uniformly at random from the pool of 541 consonants. In the second model we assume that the distribution of the occurrence of the consonants over languages is known a priori. Note that in both of these cases, the size of the random inventories is same as its real counterpart. The results show that the distribution of RRs obtained from the second model has a closer match with the real inventories than that of the first model. This indicates that the occurrence frequency to some extent governs the law of organization of the consonant inventories. The detail of each of the models follow.",
"cite_spans": [],
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"section": "Construction of Random Inventories",
"sec_num": "3.2.1"
},
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"text": "In this model we assume that the distribution of the consonant inventory size is known a priori. For each language inventory L let the size recorded in UPSID be denoted by s L . Let there be 317 bins corresponding to each consonant inventory L. A bin corresponding to an inventory L is packed with s L consonants chosen uniformly at random (without repetition) from the pool of 541 available consonants. Thus the consonant inventories of the 317 languages corresponding to the bins are generated. The method is summarized Pack the consonant c into the bin so chosen if it has not been already packed into this bin earlier; end end Algorithm 2: Algorithm to construct random inventories using Model II",
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"section": "Model I -Purely Random Model:",
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"text": "In this section we enumerate the results obtained by computing the RRs of the randomly generated inventories using Model I and Model II respectively. We compare the results with those of the real inven- tories and in each case show that the null hypothesis can be rejected with a significantly high probability.",
"cite_spans": [],
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"section": "Results Obtained from the Random Models",
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"text": "Results from Model I: Figure 3 illustrates, for all the inventories obtained from 100 different simulation runs of Algorithm 1, the average redundancy ratio exhibited by the inventories of a particular size (y-axis), versus the inventory size (x-axis). The term \"redundancy ratio exhibited by the inventories of a particular size\" actually means the following. Let there be n consonant inventories of a particular inventory-size k. The average redundancy ratio of the inventories of size k is therefore given by 1 n n i=1 RR i where RR i signifies the redundancy ratio of the i th inventory of size k. In Figure 3 we also present the same curve for the real consonant inventories appearing in UPSID. In these curves we further depict the error bars spanning the entire range of values starting from the minimum RR to the maximum RR for a given inventory size. The curves show that in case of real inventories the error bars span a very small range as compared to that of the randomly constructed ones. Moreover, the slopes of the curves are also significantly different. In order to test whether this difference is significant, we perform a t-test comparing the distribution of the values of RR that gives rise to such curves for the real and the random inventories. The results of the test are noted in Table 2 . These statistics clearly shows that the distribution of RRs for the real and the random inventories are significantly different in nature. Stated differently, we can reject the null hypothesis with (100 -9.29e-15)% confidence.",
"cite_spans": [],
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{
"start": 22,
"end": 30,
"text": "Figure 3",
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{
"start": 605,
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"section": "Results Obtained from the Random Models",
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"text": "Results from Model II: Figure 4 illustrates, for all the inventories obtained from 100 different simu- Figure 3 : Curves showing the average redundancy ratio exhibited by the real as well as the random inventories (obtained through Model I) of a particular size (y-axis), versus the inventory size (x-axis). lation runs of Algorithm 2, the average redundancy ratio exhibited by the inventories of a particular size (y-axis), versus the inventory size (x-axis). The figure shows the same curve for the real consonant inventories also. For each of the curve, the error bars span the entire range of values starting from the minimum RR to the maximum RR for a given inventory size. It is quite evident from the figure that the error bars for the curve representing the real inventories are smaller than those of the random ones. The nature of the two curves are also different though the difference is not as pronounced as in case of Model I. This is indicative of the fact that it is not only the occurrence frequency that governs the organization of the consonant inventories and there is a more complex phenomenon that results in such an invariant property. In fact, in this case also, the t-test statistics comparing the distribution of RRs for the real and the random inventories, reported in Table 3 , allows us to reject the null hypothesis with (100-2.55e-3)% confidence.",
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"start": 23,
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"text": "Until now we have been looking into the organizational aspects of the consonant inventories. In this section we show that this organization is largely different from that of the vowel inventories in the sense that there is no such invariance observed across the vowel inventories unlike that of consonants. For this reason we start by computing the RRs of all Figure 4 : Curves showing the average redundancy ratio exhibited by the real as well as the random inventories (obtained through Model II) of a particular size (y-axis), versus the inventory size (x-axis). the vowel inventories appearing in UPSID. Figure 5 shows the scatter plot of the redundancy ratio of each of the vowel inventories (y-axis) versus the inventory size (x-axis). The plot clearly indicates that the measure (i.e., RR) is not invariant across the vowel inventories and in fact, the straight line that fits the distribution has a slope of -0.14, which is around 10 times higher than that of the consonant inventories. Figure 6 illustrates the average redundancy ratio exhibited by the vowel and the consonant inventories of a particular size (y-axis), versus the inventory size (x-axis). The error bars indicating the variability of RR among the inventories of a fixed size also span a much larger range for the vowel inventories than for the consonant inventories.",
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{
"start": 360,
"end": 368,
"text": "Figure 4",
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"start": 608,
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"text": "Figure 5",
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"start": 995,
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"text": "Figure 6",
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"section": "Comparison with Vowel Inventories",
"sec_num": "3.3"
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{
"text": "The significance of the difference in the nature of the distribution of RRs for the vowel and the consonant inventories can be again estimated by performing a t-test. The null hypothesis in this case is as follows. Null Hypothesis: The nature of the distribution of RRs for the vowel and the consonant inventories is same.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Comparison with Vowel Inventories",
"sec_num": "3.3"
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{
"text": "We can now perform the t-test to verify whether we can reject the above hypothesis. ",
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"ref_spans": [],
"eq_spans": [],
"section": "Comparison with Vowel Inventories",
"sec_num": "3.3"
},
{
"text": "In this paper we have mathematically captured the redundancy observed across the sound inventories of the world's languages. We started by systematically defining the term redundancy ratio and measuring the value of the same for the inventories. Some of our important findings are, 1. Redundancy ratio is an invariant property of the consonant inventories with respect to the inventory size. 2. A more complex phenomenon than merely the occurrence frequency results in such an invariance. 3. Unlike the consonant inventories, the vowel inventories are not indicative of such an invariance. Until now we have concentrated on establishing the invariance of the redundancy ratio across the consonant inventories rather than reasoning why it could have emerged. One possible way to answer this question is to look for the error correcting capability of the encoding scheme that nature had employed for characterization of the consonants. Ideally, if redundancy has to be invariant, then this capability should be almost constant. As a proof of concept we randomly select a consonant from inventories of different size and compute its hamming distance from the rest of the consonants in the inventory. Figure 7 shows for a randomly chosen consonant c from an inventory of size 10, 15, 20 and 30 respectively, the number of the consonants at a particular hamming distance from c (y-axis) versus the hamming distance (x-axis). The curve clearly indicates that majority of the consonants are at a hamming distance of 4 from c, which in turn implies that the encoding scheme has almost a fixed error correcting capability of 1 bit. This can be the precise reason behind the invariance of the redundancy ra- Figure 7 : Histograms showing the the number of consonants at a particular hamming distance (y-axis), from a randomly chosen consonant c, versus the hamming distance (x-axis).",
"cite_spans": [],
"ref_spans": [
{
"start": 1197,
"end": 1205,
"text": "Figure 7",
"ref_id": null
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{
"start": 1698,
"end": 1706,
"text": "Figure 7",
"ref_id": null
}
],
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"section": "Conclusions, Discussion and Future Work",
"sec_num": "4"
},
{
"text": "tio. Initial studies into the vowel inventories show that for a randomly chosen vowel, its hamming distance from the other vowels in the same inventory varies with the inventory size. In other words, the error correcting capability of a vowel inventory seems to be dependent on the size of the inventory.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusions, Discussion and Future Work",
"sec_num": "4"
},
{
"text": "We believe that these results are significant as well as insightful. Nevertheless, one should be aware of the fact that the formulation of RR heavily banks on the set of features that are used to represent the phonemes. Unfortunately, there is no consensus on the set of representative features, even though there are numerous suggestions available in the literature. However, the basic concept of RR and the process of analysis presented here is independent of the choice of the feature set. In the current study we have used the binary features provided in UPSID, which could be very well replaced by other representations, including multi-valued feature systems; we look forward to do the same as a part of our future work.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusions, Discussion and Future Work",
"sec_num": "4"
},
{
"text": "In phonology, features are the elements, which distinguish one phoneme from another. The features that distinguish the consonants can be broadly categorized into three different classes namely the manner of articulation, the place of articulation and phonation. Manner of articulation specifies how the flow of air takes place in the vocal tract during articulation of a consonant, whereas place of articulation specifies the active speech organ and also the place where it acts. Phonation describes the activity regarding the vibration of the vocal cords during the articulation of a consonant.",
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"FIGREF0": {
"text": "Algorithm 1. for I = 1 to 317 do for size = 1 to s L do Choose a consonant c uniformly at random (without repetition) from the pool of 541 available consonants; Pack the consonant c in the bin corresponding to the inventory L; end end Algorithm 1: Algorithm to construct random inventories using Model I Model II -Occurrence Frequency based Random Model: For each consonant c let the frequency of occurrence in UPSID be denoted by f c . Let there be 317 bins each corresponding to a language in UP-SID. f c bins are then chosen uniformly at random and the consonant c is packed into these bins. Thus the consonant inventories of the 317 languages corresponding to the bins are generated. The entire idea is summarized in Algorithm 2. for each consonant c do for i = 1 to f c do Choose one of the 317 bins, corresponding to the languages in UPSID, uniformly at random;",
"num": null,
"type_str": "figure",
"uris": null
},
"FIGREF1": {
"text": "The scatter-plot of the redundancy ratio RR of each of the vowel inventories (y-axis) versus the inventory size (x-axis). The straight line-fit is depicted by the bold line in the figure.",
"num": null,
"type_str": "figure",
"uris": null
},
"FIGREF2": {
"text": "Curves showing the average redundancy ratio exhibited by the vowel as well as the consonant inventories of a particular size (y-axis), versus the inventory size (x-axis).",
"num": null,
"type_str": "figure",
"uris": null
},
"TABREF1": {
"type_str": "table",
"num": null,
"text": "The results of the t-test comparing the distribution of RRs for the real and the random inventories (obtained through Model I). SDV: standard deviation, t: t-value of the test, DF: degrees of freedom, p: residual uncertainty.",
"content": "<table/>",
"html": null
},
"TABREF3": {
"type_str": "table",
"num": null,
"text": "The results of the t-test comparing the distribution of RRs for the real and the random inventories (obtained through Model II).",
"content": "<table/>",
"html": null
},
"TABREF4": {
"type_str": "table",
"num": null,
"text": "presents the results of the test. The statistics immediately confirms that the null hypothesis can be rejected with 99.932% confidence.",
"content": "<table><tr><td>Parameters</td><td>Consonant Inv.</td><td>Vowel Inv.</td></tr><tr><td>Mean</td><td>2.51177</td><td>2.98797</td></tr><tr><td>SDV</td><td>0.209531</td><td>0.726547</td></tr><tr><td>Parameters</td><td/><td>Values</td></tr><tr><td>t</td><td/><td>3.612</td></tr><tr><td>DF</td><td/><td>54</td></tr><tr><td>p</td><td colspan=\"2\">\u2264 0.000683</td></tr></table>",
"html": null
},
"TABREF5": {
"type_str": "table",
"num": null,
"text": "The results of the t-test comparing the distribution of RRs for the consonant and the vowel inventories.",
"content": "<table/>",
"html": null
}
}
}
} |