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:

G = $\displaystyle \sum_{t}^{}$$\displaystyle \dot{e}_{t}^{2}$ + ($\displaystyle \pi$ . smooth/2)2$\displaystyle \ddot{e}_{t}^{2}$ + adroop2 . et2 (1)

R = $\displaystyle \sum_{k \in \rm tags}^{}$sk2rk (2)

rk = $\displaystyle \sum_{t\in {\rm tag} k}^{}$cos(type . $\displaystyle \pi$/2)(et - yk, t)2 + sin(type . $\displaystyle \pi$/2)($\displaystyle \bar{e_k}$ - $\displaystyle \bar{y_k}$)2, (3)

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

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


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

$\displaystyle \hat{f}_{0}^{}$ = g(e,$\displaystyle \it add$) . range + base (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\dag Eq. 6 Controls the mapping between e and f0. See g().
adroop\dag Eq. 1 Rate at which e droops toward the phrase curve in the absence of a tag.
base\dag Eq. 6 The speaker's relaxed f0.
smooth\dag Eq. 1 Response time of muscles.
type\dag 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\dag, et §2.2.3 Emphasis, i.e., $ \hat{f}_{0}^{}$ relative to the speaker's range.
$ \bar{e}$\dag Eqs. 3, 4 Mean emphasis over the scope of a tag.
y\dag, yt §2.2.3 Tone template.
$ \bar{y}$\dag Eqs. 3, 5 Mean value of a tone template.
G\dag 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\dag Eq. 2 The summed error for an utterance between the ideal templates and the realized pitch contour.
g()\dag Eq. 6 Function to map between subjective emphasis (e) and objective f0.

Greg Kochanski, Chilin Shih 2002-08-03