N1904addons - Feature: morph_entr_norm

Feature group	Feature type	Data type	Available for node types	Feature status
`statistic`	`Node`	`int`	`word`	✅

Feature short description

Normalized entropy of a morph(-tag of a word)* as predictor of its parent phrase function (like Subject, Object, etc.).

*) For the calculation of entropy the dialect tag (like -ATT) was ignored.

Feature values

A number stored as integer representing the entropy normalized to a range from 0 to 1000 (inclusive).

Feature detailed description

In practical terms, entropy reflects how predictable an element’s syntactic behavior is predictable. This feature expresses how consistently a given lemma (as stored in the morph feature) maps to a particular phrase function.

In this context these phrase functions are derived from feature function and expanded with several additional categories (to view all details expand the item below).

Parent phrase function details

In the N1904-TF dataset, not all words belong to phrases with well-defined syntactic functions such as Subject or Object. For instance, conjunctions like δὲ or καὶ typically do not form part of syntactic phrases in the strict sense.

To ensure that every word can still be assigned a functional label, the following Python script was developed. This script prioritizes assigning the canonical phrase function where available, but also supplements gaps with a set of extended categories.

The table below distinguishes between these two types of categories and shows the number of word nodes mapped to each one.

Source	Value	Description	Frequency
From feature `function` (6 classes)	Cmpl	Complement	35442
	Pred	Predicate	25138
	Subj	Subject	21567
	Objc	Object	19371
	PreC	Predicate-Complement	9595
	Adv	Adverbial	5367
Augmented pseudo classes (5 classes)	Conj	Conjunction	16316
	Unkn	Unknown	2076
	Intj	Interjection	1470
	Aux	Auxiliar	1136
	Appo	Apposition	301

The "Unkn" (unknown) category accounts for approximately 1.5% of all mappings, slightly raising both the absolute and normalized entropy.

High entropy values indicate that a form is ambiguous, as it appears in multiple syntactic functions with similar probabilities. In contrast, low entropy values signify that a form is strongly associated with a single syntactic function, making it a reliable indicator of that role within the parent phrase.

Detailed mathematic description

Definition

Entropy is a measure from information theory that quantifies uncertainty or unpredictability in a probability distribution. It is defined as:

$$H(X) = -\sum_i P(x_i) \log_2 P(x_i)$$

Where:

The part $ P(x_i) $ is the probability of the $ i-th $ outcome.
The part $ log_2 $ ensures the result is expressed in bits.
It is assumed in this context that $ log_2(0)=0 $.

Entropy measures the uncertainty associated with a probability distribution. It reaches its maximum when all outcomes are equally likely (i.e., maximum uncertainty), and its minimum (zero) when one outcome is certain.

Application

In the context of the N1904-FT dataset, we apply this principle to estimate the uncertainty of syntactic function prediction based on linguistic features.

Let an element $e \in D $, where $ D = \{ \text{lemma}, \text{morph}, \text{text} \} $, represent a linguistic feature. If this element is associated with $ n $ different phrase functions $ f $, then the entropy $ H(e \mid f) $ in bits is calculated as:

$$H(e|f) = -\sum_{i=1}^{n} p_i \log_2(p_i)$$

where $ p_i $ is the probability that element $ e $ corresponds to the $ i-th $ function.

If the distribution is uniform (i.e., all $ p_i = \frac{1}{n} $ ), the entropy reaches its maximum:

$$H(e|f) = -n \cdot \frac{1}{n} \cdot \log_2\left(\frac{1}{n}\right) = \log_2(n)$$

In the mapping used for calculating this feature, there are $ n = 11 $ phrase function categories. Thus, the theoretical maximum entropy for a given datatype $ D $ is:

$$H_{\text{max}}(D) = \log_2(11) \approx 3.459 \text{ bits}$$

This value represents the upper bound of uncertainty when a linguistic feature provides no predictive information about phrase function.

To obtain a normalized entropy, where values for $ H(e|f) $ are in the range 0 to 1 (inclusive), the following formula can be applied for each datatype $ D $:

$$H_{\text{norm}}(D) = \frac{H(D)}{H_{\text{max}}(D)}$$

The following table provides some statistic key metrics for this feature counted over the total of 1055 unique morphs:

=== morph ===
Count:   1018
Min:     0000
25%ile:  0000
Median:  0000
75%ile:  0254
Max:     0788
Mean:    0125
StdDev:  0168

This indicates that most morphs are highly predictable in terms of their syntactic roles, while a small subset show high entropy due to usage in multiple phrase functions.

The following plot illustrates both the absolute and normalized entropy distribution for all 1055 unique morph tag in the N1904-TF dataset:

Entropy distribution morph -> phrase function

Theoretical example

To better understand the significance of entropy values, the following plot illustrates the absolute entropy associated with several synthetic probability distributions.

Entropy examples

The information can also be presented in a table augmented with the values for normalized entropy. In the example above, we assumed a total of 8 possible classes. Therefore, to compute the normalized entropy, we can simply divide each absolute entropy by 3, since since $ \log_2(8) $ = 3.

Distribution	Absolute entropy (bits)	Normalized entropy (8 classes)
Uniform (2 outcomes)	1.0000	0,3333
Skewed (0.9, 0.1)	0.4689	0.1563
Heavy skewed (0.99, 0.01)	0.0808	0.1563
Moderate skewed (0.7, 0.3)	0.8813	0.2938
Uniform (4 outcomes)	2.0000	0.667
Uniform (8 outcomes)	3.0000	1.0000
Mixed (0.6, 0.2, 0.1, 0.1)	~1.6855	~0.5618

This highlight the following key properties of entropy:

Entropy increases with the number of equally likely outcomes.
Entropy decreases when one outcome dominates the distribution.

References

Shannon, C. E. “A Mathematical Theory of Communication” in Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656, July, October, 1948.
Manning, C. D., Schütze, H. Foundations of Statistical Natural Language Processing (Cambridge, MA: MIT Press, 1999), esp. page 61 and further.

Data source

Github repository.