Normalization
dynax.normalization_coefficients(std_y, std_v=None, std_a=None, w_y=1.0, w_v=1.0, w_a=1.0, tol=1e-06, maxiter=50, verbosity=1)
Compute normalization factors for signals and their derivatives while preserving derivative consistency.
Consider a trajectory \(y(t) \in \mathbb{R}^n\) together with its derivatives \(v(t) = \partial_t \, y(t),\; a(t) = \partial_{tt} \, y(t)\). Normalizing \(y, v, a\) independently (e.g. to unit variance) breaks the derivative relationships: the derivative of the normalized trajectory \(\partial_t \tilde y(t)\) would no longer match the normalized velocity \(\tilde v(t)\).
To maintain consistency, the same scaling factor \(\alpha_i\) must be applied to \(y_i, v_i, a_i\). Additionally, we allow a rescaling of time, \(\tilde t = \tau t\), to gain more flexibility. With this, the rescaled signals are $$ \tilde t = \tau t, \quad \tilde y_i = \alpha_i y_i, \quad \tilde v_i = \tau \alpha_i v_i, \quad \tilde a_i = \tau^2 \alpha_i a_i. $$
This function finds the scaling factors \(\alpha \in \mathbb{R}^n\) and \(\tau \in \mathbb{R}\) that make the standard deviations of \(\tilde y, \tilde v\) and \(\tilde a\) as close as possible to one, while preserving derivative consistency: $$ \partial_{\tilde t} \, \tilde y_i(\tilde t) = \tilde v_i(\tilde t), \quad \partial_{\tilde t \tilde t} \, \tilde y_i(\tilde t) = \tilde a_i(\tilde t). $$
This is done via the following optimization problem: Given standard deviations \(\sigma(y), \sigma(v), \sigma(a)\), minimize $$ \sum_i \big[ w_y (\sigma(\tilde y_i) - 1)^2 \;+\; w_v (\sigma(\tilde v_i) - 1)^2 \;+\; w_a (\sigma(\tilde a_i) - 1)^2 \big], $$ where \(w_y, w_v, w_a\) control the relative importance of each term.
Implementation Notes
The problem reduces to a one-dimensional optimization over \(\tau\), with
$$
\alpha_i(\tau) = \frac{N_i(\tau)}{D_i(\tau)}, \quad
\tau^\star = \underset{\tau}{\operatorname{argmin}}
\sum_i -\frac{N_i(\tau)^2}{D_i(\tau)}.
$$
This function solves the reformulated problem using scipy.optimize.minimize
with the BFGS method. Additionally:
- Optimization is performed over \(\log(\tau)\) to enforce \(\tau > 0\).
- The initial guess \(\tau_0\) is computed heuristically from \(\sigma(y), \sigma(v), \sigma(a)\).
| PARAMETER | DESCRIPTION |
|---|---|
std_y
|
Standard deviation of the signal \(y(t)\).
TYPE:
|
std_v
|
Standard deviation of the signal \(v(t) = \partial_t y(t)\). Optinoal. Defaults to None.
TYPE:
|
std_a
|
Standard deviation of the signal \(a(t) = \partial_{tt} y(t)\). Optinoal. Defaults to None.
TYPE:
|
w_y
|
Optimization weight for signal \(y(t)\). Defaults to 1.0.
TYPE:
|
w_v
|
Optimization weight for signal \(v(t)\). Defaults to 1.0.
TYPE:
|
w_a
|
Optimization weight for signal \(a(t)\). Defaults to 1.0.
TYPE:
|
tol
|
Relative error in solution for \(\tau\) acceptable for convergence. Defaults to 1e-6.
TYPE:
|
maxiter
|
Maximum number of optimization iterations to perform. Defaults to 50.
TYPE:
|
verbosity
|
If non-zero, print messages.
TYPE:
|
| RETURNS | DESCRIPTION |
|---|---|
tuple[Float[Array, 'n'], Scalar]
|
Tuple containing \(\alpha_i\) and \(\tau\). |