Edge profile models

class pedinf.models.mtanh(radius=None, low_field_side=True)

The standard ‘mtanh’ function used for pedestal fitting. Specifically, the function is:

\[\mathrm{mtanh}(R, \, \underline{\theta}) = L(z) \left(h - b + \frac{awz}{4} \right) + b\]

where

\[z = -4 \frac{R - R_0}{w}, \quad \quad L(x) = \frac{1}{1 + e^{-x}}.\]

The model parameter vector \(\underline{\theta}\) has the following order:

\[\underline{\theta} = \left[ \, R_0, \, h, \, w, \, a, \, b \, \right],\]

where

• \(R_0\) is the radial location of the pedestal.

• \(h\) is the pedestal height.

• \(w\) is the pedestal width.

• \(a\) is the profile gradient beyond the pedestal top.

• \(b\) is the background level.

gradient(radius: ndarray, theta: ndarray) → ndarray

Calculates the gradient (w.r.t. major radius) of the mtanh function. See the documentation for mtanh for details of the function itself.

Parameters

• radius – Radius values at which the gradient is evaluated.

• theta – The model parameters as an array.

Returns

The predicted gradient profile at the given radius values.

jacobian(radius: ndarray, theta: ndarray) → ndarray

Calculates the jacobian of the mtanh model. The jacobian is a matrix where element \(i, j\) is the derivative of the model prediction at the \(i\)’th radial position with respect to the \(j\)’th model parameter. See the documentation for mtanh for details of the model itself.

Parameters

• radius – Radius values at which the jacobian is evaluated.

• theta – The model parameters as an array.

Returns

The jacobian matrix for the given radius values.

prediction(radius: ndarray, theta: ndarray) → ndarray

Calculates the prediction of the mtanh model. See the documentation for mtanh for details of the model itself.

Parameters

• radius – Radius values at which the prediction is evaluated.

• theta – The model parameters as an array.

Returns

The predicted profile at the given radius values.

prediction_and_jacobian(radius: ndarray, theta: ndarray) → Tuple[ndarray, ndarray]

Calculates the prediction and the jacobian of the mtanh model. The jacobian is a matrix where element \(i, j\) is the derivative of the model prediction at the \(i\)’th radial position with respect to the \(j\)’th model parameter. See the documentation for mtanh for details of the model itself.

Parameters

• radius – Radius values at which the prediction and jacobian are evaluated.

• theta – The model parameters as an array.

Returns

The model prediction and the jacobian matrix for the given radius values.

class pedinf.models.lpm(radius=None, low_field_side=True)

A modified version of the ‘mtanh’ function which includes an additional parameter controlling how rapidly the profile decays at the ‘foot’ of the pedestal. Specifically, the function is:

\[\mathrm{lpm}(R, \, \underline{\theta}) = (h - b)\,L^{k}(z + \ln{k}) + \frac{aw}{4}S(z) + b\]

where

\[z = -4 \frac{R - R_0}{w}, \quad \quad L(x) = \frac{1}{1 + e^{-x}}, \quad \quad S(x) = \int_{-\infty}^{x} L(x')\,\mathrm{d}x' = \ln{(1 + e^x)}\]

The model parameter vector \(\underline{\theta}\) has the following order:

\[\underline{\theta} = \left[ \, R_0, \, h, \, w, \, a, \, b, \, k \, \right],\]

where

• \(R_0\) is the radial location of the pedestal.

• \(h\) is the pedestal height.

• \(w\) is the pedestal width.

• \(a\) is the profile gradient beyond the pedestal top.

• \(b\) is the background level.

• \(k\) is a shaping parameter which affects how the profile decays.

gradient(radius: ndarray, theta: ndarray) → ndarray

Calculates the gradient (w.r.t. major radius) of the lpm model. See the documentation for lpm for details of the model itself.

Parameters

• radius – Radius values at which the gradient is evaluated.

• theta – The model parameters as an array.

Returns

The predicted gradient profile at the given radius values.

jacobian(radius: ndarray, theta: ndarray) → ndarray

Calculates the jacobian of the lpm model. The jacobian is a matrix where element \(i, j\) is the derivative of the model prediction at the \(i\)’th radial position with respect to the \(j\)’th model parameter. See the documentation for lpm for details of the model itself.

Parameters

• radius – Radius values at which the jacobian is evaluated.

• theta – The model parameters as an array.

Returns

The jacobian matrix for the given radius values.

prediction(radius: ndarray, theta: ndarray) → ndarray

Calculates the prediction of the lpm model. See the documentation for lpm for details of the model itself.

Parameters

• radius – Radius values at which the prediction is evaluated.

• theta – The model parameters as an array.

Returns

The predicted profile at the given radius values.

prediction_and_jacobian(radius: ndarray, theta: ndarray) → Tuple[ndarray, ndarray]

Calculates the prediction and the jacobian of the lpm model. The jacobian is a matrix where element \(i, j\) is the derivative of the model prediction at the \(i\)’th radial position with respect to the \(j\)’th model parameter. See the documentation for lpm for details of the model itself.

Parameters

• radius – Radius values at which the prediction and jacobian are evaluated.

• theta – The model parameters as an array.

Returns

The model prediction and the jacobian matrix for the given radius values.

class pedinf.models.exspline(knots: ndarray, radius=None, low_field_side=True)

‘exspline’ uses a combination of exponentiated b-splines and a logistic function to model the profile shape. The logistic function is combined with the exponentiated splines multiplicatively, and so models the pedestal as a localised fractional decrease in the background profile. The size of the fractional decrease is controlled by the parameter \(f \in [0, 1]\).

\[\mathrm{exspline}(R, \, \underline{\theta}) = \exp{\left[\sum_{i=1}^{n} a_i \phi_i(R)\right]} \left((1 - f) L(z) + f\right)\]

where

\[z = -4 \frac{R - R_0}{w}, \quad \quad L(x) = \frac{1}{1 + e^{-x}}\]

The model parameter vector \(\underline{\theta}\) has the following order:

\[\underline{\theta} = \left[ \, R_0, \, f, \, w, \, a_1 \, a_2, \, \ldots, \, a_n \, \right],\]

where

• \(R_0\) is the logistic function location.

• \(f\) is the logistic function ‘floor’.

• \(w\) is the logistic function width.

• \(a_i\) is the weight for the \(i\)’th b-spline basis function.

gradient(radius: ndarray, theta: ndarray) → ndarray

Calculates the gradient (w.r.t. major radius) of the exspline model. See the documentation for exspline for details of the model itself.

Parameters

• radius – Radius values at which the gradient is evaluated.

• theta – The model parameters as an array.

Returns

The predicted gradient profile at the given radius values.

jacobian(radius: ndarray, theta: ndarray) → ndarray

Calculates the jacobian of the exspline model. The jacobian is a matrix where element \(i, j\) is the derivative of the model prediction at the \(i\)’th radial position with respect to the \(j\)’th model parameter. See the documentation for exspline for details of the model itself.

Parameters

• radius – Radius values at which the jacobian is evaluated.

• theta – The model parameters as an array.

Returns

The jacobian matrix for the given radius values.

prediction(radius: ndarray, theta: ndarray) → ndarray

Calculates the prediction of the exspline model. See the documentation for exspline for details of the model itself.

Parameters

• radius – Radius values at which the prediction is evaluated.

• theta – The model parameters as an array.

Returns

The predicted profile at the given radius values.

prediction_and_jacobian(radius: ndarray, theta: ndarray) → Tuple[ndarray, ndarray]

Calculates the prediction and the jacobian of the exspline model. The jacobian is a matrix where element \(i, j\) is the derivative of the model prediction at the \(i\)’th radial position with respect to the \(j\)’th model parameter. See the documentation for exspline for details of the model itself.

Parameters

• radius – Radius values at which the prediction and jacobian are evaluated.

• theta – The model parameters as an array.

Returns

The model prediction and the jacobian matrix for the given radius values.