Mathematical Definition of Model

In Stem-ML, a ``tag'' is a tone template, along with a few parameters that describe the scope of the template and how the template interacts with its environment. It corresponds to the mathematical description of an intonation event (e.g., a tone or an accent). Variables in the equations below are defined in Table 2.2.3. Tags have a parameter, type, which controls whether errors in the shape or average value of the pitch curve are most important. In this work, the targets, y, consist of an tone component riding on top of the phrase curve, p.

In order to efficiently solve the optimization problem, and calculate the surface realization of prosody, we write simple approximations to G and R so that the model can be solved efficiently as a set of linear equations:

$$\sum_{t}^{} \dot{e}_{t}^{2} \pi \ddot{e}_{t}^{2}$$ (1)

$$\sum_{k \in \rm tags}^{}$$ (2)

$$\sum_{t\in {\rm tag} k}^{} \pi \pi \bar{e_k} \bar{y_k}$$ (3)

where

$$\bar{e_k} {\frac{\sum_{t\in {\rm tag} k} e_t}{\sum_{t\in {\rm tag} k} 1}}$$ (4)

and

$$\bar{y_k} {\frac{\sum_{t\in {\rm tag} k} y_t}{\sum_{t\in {\rm tag} k} 1}}$$ (5)

and

Finally, f₀ is e, scaled to the speaker's pitch range:

$$\hat{f}_{0}^{} \it add$$ (6)

so that p and e are dimensionless quantities, typically between 0 and 1. The function g() handles linear (add = 1) or Fujisaki (add = 0) scaling: g(e, 1) = e for any e, and also g(0,$\it add$) = 0 and g(1,$\it add$) = 1 for any add.

Table: Definitions of parameters and variables used in this paper. Daggers denote parameters defined more fully in [8].

Symbol	Location	Meaning
add	Eq. 6	Controls the mapping between e and f0. See g().
adroop	Eq. 1	Rate at which e droops toward the phrase curve in the absence of a tag.
base	Eq. 6	The speaker's relaxed f0.
smooth	Eq. 1	Response time of muscles.
type	Eq. 3	Is tone defined by its shape (0) or f0 value (1).
atype	Eq. 7	Controls how the amplitude of the template depends on the strength of a word.
f0	many places	Measured pitch.
$\hat{f}_{0}^{}$	Eq. 6	Modeled pitch.
e , et	§2.2.3	Emphasis, i.e., $\hat{f}_{0}^{}$ relative to the speaker's range.
$\bar{e}$	Eqs. 3, 4	Mean emphasis over the scope of a tag.
y , yt	§2.2.3	Tone template.
$\bar{y}$	Eqs. 3, 5	Mean value of a tone template.
G	Eq. 1	Effort expended in realizing the pitch contour.
ri	Eq. 3	The summed error for word i between the template and the realized pitch.
R	Eq. 2	The summed error for an utterance between the ideal templates and the realized pitch contour.
g()	Eq. 6	Function to map between subjective emphasis (e) and objective f0.