src.variants.metrics.area_metrics module

This module includes area validation metrics.

It includes several types of area validation metrics. See details in their own documentations. The functions are mainly based on numpy arrays for fast vectorized operations.

The general theory can be found, e.g., in [1, Ch. 12.8.2].

Literature: [1] W. L. Oberkampf and C. J. Roy, Verification and Validation in Scientific Computing, Cambridge, Cambridge University Press, 2010, ISBN: 9780511760396.

Contact person: Stefan Riedmaier Creation date: 03.06.2020 Python version: 3.8

src.variants.metrics.area_metrics.avm(pbox_y_model, pbox_y_system, pbox_x_model_list, pbox_x_system_list, axis=-1)

This function calculates the area validation metric (AVM).

The theory can be found, e.g., in [1, Ch. 12.8.2].

Literature: [1] W. L. Oberkampf and C. J. Roy, Verification and Validation in Scientific Computing, Cambridge, Cambridge University Press, 2010, ISBN: 9780511760396.

Parameters:

  • pbox_y_model (np.ndarray) – y vector of the model

  • pbox_y_system (np.ndarray) – y vector of the system

  • pbox_x_model_list (list[np.ndarray, np.ndarray]) – 2-element list with left and right x vector of the model

  • pbox_x_system_list (list[np.ndarray, np.ndarray]) – 2-element list with left and right x vector of the system

  • axis (int) – (optional) along this axis the operation is performed

Returns:

area validation metric

Rtpye:

np.ndarray

src.variants.metrics.area_metrics.calc_areas(pbox_y_model, pbox_y_system, pbox_x_model_list, pbox_x_system_list, axis=-1)

This function determines the different areas.

It calls the calc_left_area and calc_right_area functions to determine the areas.

Parameters:

  • pbox_y_model (np.ndarray) – y vector of the model

  • pbox_y_system (np.ndarray) – y vector of the system

  • pbox_x_model_list (list[np.ndarray, np.ndarray]) – 2-element list with left and right x vector of the model

  • pbox_x_system_list (list[np.ndarray, np.ndarray]) – 2-element list with left and right x vector of the system

  • axis (int) – (optional) along this axis the operation is performed

Returns:

arrays of left, right, worst case left and worst case right areas

Return type:

tuple(np.ndarray, np.ndarray, np.ndarray, np.ndarray)

src.variants.metrics.area_metrics.calc_left_area(y, x1, x2, axis=-1)

This function calculates the areas where x2 is on the left side of x1.

The areas can be interpreted as rectangles.

Parameters:

  • y (np.ndarray) – joint y vector

  • x1 (np.ndarray) – first x vector

  • x2 (np.ndarray) – second x vector

  • axis (int) – (optional) along this axis the operation is performed

Returns:

calculated areas

Return type:

np.ndarray

src.variants.metrics.area_metrics.calc_right_area(y, x1, x2, axis=-1)

This function calculates the area where x2 is on the right side of x1.

The areas can be interpreted as rectangles.

Parameters:

  • y (np.ndarray) – joint y vector

  • x1 (np.ndarray) – first x vector

  • x2 (np.ndarray) – second x vector

  • axis (int) – (optional) along this axis the operation is performed

Returns:

calculated areas

Return type:

np.ndarray

src.variants.metrics.area_metrics.iavm(pbox_y_model, pbox_y_system, pbox_x_model_list, pbox_x_system_list, axis=-1)

This function calculates the interval area validation metric (IAVM).

d_upper_plus / area_right: system upper limit function / left pbox edge > model ECDF / right pbox edge d_upper_minus / area_left_wc: system upper limit function / left pbox edge < model ECDF / left pbox edge d_lower_plus / area_right_wc: system lower limit function / right pbox edge > model ECDF / right pbox edge d_lower_minus / area_left: system lower limit function / right pbox edge < model ECDF / left pbox edge

Literature: N. Wang, W. Yao, Y. Zhao, X. Chen, X. Zhang and L. Li, „A New Interval Area Metric for Model Validation With Limited Experimental Data,“ Journal of Mechanical Design, vol. 140, no. 6, 2018.

Parameters:

  • pbox_y_model (np.ndarray) – y vector of the model

  • pbox_y_system (np.ndarray) – y vector of the system

  • pbox_x_model_list (list[np.ndarray, np.ndarray]) – 2-element list with left and right x vector of the model

  • pbox_x_system_list (list[np.ndarray, np.ndarray]) – 2-element list with left and right x vector of the system

  • axis (int) – (optional) along this axis the operation is performed

Returns:

interval area validation metric

Rtpye:

np.ndarray

src.variants.metrics.area_metrics.mavm(pbox_y_model, pbox_y_system, pbox_x_model_list, pbox_x_system_list, axis=-1, f0=4.0, f1=1.25)

This function calculates the modified area validation metric (MAVM).

Literature: I. T. Voyles and C. J. Roy, „Evaluation of Model Validation Techniques in the Presence of Aleatory and Epistemic Input Uncertainties,“ in 17th AIAA Non-Deterministic Approaches Conference, 2015, ISBN: 978-1-62410-347-6.

Parameters:

  • pbox_y_model (np.ndarray) – y vector of the model

  • pbox_y_system (np.ndarray) – y vector of the system

  • pbox_x_model_list (list[np.ndarray, np.ndarray]) – 2-element list with left and right x vector of the model

  • pbox_x_system_list (list[np.ndarray, np.ndarray]) – 2-element list with left and right x vector of the system

  • axis (int) – (optional) along this axis the operation is performed

  • f0 (float) – (optional) f0 value of the safety factor fs

  • f1 (float) – (optional) f1 value of the safety factor fs

Returns:

modified area validation metric

Rtpye:

np.ndarray

src.variants.metrics.area_metrics.merge_ecdf_functions(ay, by, ax_list=None, bx_list=None, axis=-1)

This function merges ecdf functions to get a joint y vector and the corresponding x vectors.

This is important for the rectangular area calculations, since both ecdfs need to have the same number of steps.

If the ax_list resp. the bx_list is provided, the resulting x vectors are returned. Otherwise, the index vectors are returned to index x vectors.

Parameters:

  • ay (np.ndarray) – first y vector

  • by (np.ndarray) – second y vector

  • ax_list (list[np.ndarray, np.ndarray]) – (optional) list of first x vectors

  • bx_list (list[np.ndarray, np.ndarray]) – (optional) list of second x vectors

  • axis (int) – (optional) along this axis the operation is performed

Returns:

joint y vector and the corresponding x vectors resp. the indices to index them

Return type:

tuple(np.ndarray, list[np.ndarray], list[np.ndarray])

src.variants.metrics.area_metrics.polygon_area(x, y)

This function calculates the area of a simple polygon with Shoelace formula.

https://stackoverflow.com/questions/24467972/calculate-area-of-polygon-given-x-y-coordinates

Parameters:

  • x – input vector

  • y – output vector

Returns:

area