Available Functions and Distributions

Output Distributions

The distributions are provided by Scipy’s stats module. Instead of calling the .rvs() method, however, we use the .ppf() (percent point function) method to obtain realizations given the error terms.

For the parameters with described range, we can specify correction.

Distribution	Parameters	Range
`bernoulli`	`p_`: success probability	\([0, 1]\)
`uniform`	`mu_`: center point: (a+b)/2	\(\mathbb{R}\)
	`diff_`: range: b-a	\(\mathbb{R}\)
`normal`	`mu_`: mean	\(\mathbb{R}\)
	`sigma_`: standard deviation	\(\mathbb{R}_+\)
`lognormal`	`mu_`: mean	\(\mathbb{R}\)
	`sigma_`: standard deviation	\(\mathbb{R}_+\)
`poisson`	`lambda_`: rate	\(\mathbb{N}_0\)
`Exponential`	`lambda_`: rate	\(\mathbb{R}_+\)

Edge Functions

Read more about each edge function by clicking the link in the table title.

Identity

Formula

\[{z^{*}}_i = z_i\]

PARCS Code

pyparcs.api.mapping_functions.py

    return array

Parameters

identity() edge function has no parameters.

Sigmoid

Formula

\[z^*_i = \sigma\left( (-1)^{\gamma} \alpha (z_i - \beta)^{\tau} \right), \quad \sigma(a) = \frac{1}{1+e^{-a}}\]

PARCS Code

pyparcs.api.mapping_functions.py

    """Sigmoid edge function"""
    parcs_assert(gamma in [0, 1] and tau % 2 == 1,
                 EdgeFunctionError,
                 f'error in gamma={gamma} or tau={tau} values')

    expon = (-1) ** gamma * alpha * ((array - beta) ** tau)
    return expit(expon)


def edge_gaussian_rbf(array: np.ndarray,

Parameters

`sigmoid()` parameters
Parameter	Description	Default
\(\alpha\): `alpha`	Scale of the Sigmoid. Takes positive values. The reasonable range is approx. \([1, 3]\).	`2`
\(\beta\): `beta`	Offset of the Sigmoid. Takes real values. The reasonable range is approx. \([-2, 2]\).	`0`
\(\gamma\): `gamma`	Mirroring on x-axis. Takes \(\{0, 1\}\).	`0`
\(\tau\): `tau`	Induces a dwelling region in the center of the function. Takes odd values \(\{1, 3, \dots\}\).	`1`

The figures below depict the effect of each parameter on the sigmoid transformation:

Gaussian RBF

Formula

\[\begin{align} z^*_i = \gamma + (-1)^\gamma . \exp\big(-\alpha \|z_i - \beta\|^\tau\big), \end{align}\]

PARCS Code

pyparcs.api.mapping_functions.py

    parcs_assert(gamma in [0, 1] and tau % 2 == 0,
                 EdgeFunctionError,
                 f'error in gamma={gamma} or tau={tau} values')

    expon = -alpha * ((array - beta) ** tau)
    return gamma + ((-1) ** gamma) * np.exp(expon)


def edge_arctan(array: np.ndarray,
                alpha: float = 2, beta: float = 0, gamma: float = 0) -> np.ndarray:

Parameters

`gaussian_rbf()` parameters
Parameter	Description	Default
\(\alpha\): `alpha`	Scale of the RBF. Takes positive values. The reasonable range is approx. \([1, 3]\).	`2`
\(\beta\): `beta`	Offset of the RBF. Takes real values. The reasonable range is approx. \([-2, 2]\).	`0`
\(\gamma\): `gamma`	Mirroring on y-axis. Takes \(\{0, 1\}\).	`0`
\(\tau\): `tau`	Induces a dwelling region in the center of the function. Takes even values \(\{2, 4, \dots\}\).	`2`

The figures below depict the effect of each parameter on the Gaussian RBF transformation:

Arctan

Formula

\[\begin{align} z^*_i = (-1)^{\gamma} . \arctan{(\alpha(z_i-\beta))}, \end{align}\]

PARCS Code

pyparcs.api.mapping_functions.py

                 f'error in gamma={gamma} value')

    return (-1) ** gamma * np.arctan(alpha * (array - beta))


EDGE_FUNCTIONS = {
    'identity': edge_identity,
    'sigmoid': edge_sigmoid,

Parameters

`edge_arctan()`
Parameter	Description	Default
\(\alpha\): `alpha`	Scale of the RBF. Takes positive values. The reasonable range is approx. \([1, 3]\)	`2`
\(\beta\): `beta`	Offset of the RBF. Takes real values. The reasonable range is approx. \([-2, 2]\)	`0`
\(\gamma\): `gamma`	Mirroring on y-axis. Takes \(\{0, 1\}\)	`0`

The figures below depict the effect of each parameter on the arctan transformation:

Corrections

Sigmoid Correction

`SigmoidCorrection()`
Parameter	Description	type	default (yml/py)
`lower, upper`	Lower and upper range of the corrected values.	float	0, 1
`to_center`	If the raw values must be centered (mean = 0) before sigmoid correction.	bool	false/False
`target_mean`	Specify this option if you want to fix the mean of corrected values. Must be in range of (lower, upper)	float	null/None

Edge Correction

`EdgeCorrection()`
Parameter	Description	type	default (yml/py)