# MFred: Python functions for best estimate calculation in a sensor network in the presence of redundancy¶

In this section some method for analysing redundant measurement data is presented. Redundancy means that there is more than one way to derive the value of the measurand Y from the values of the sensor data X_i. Following main cases are considered in the module:

1. Redundant measurement of the measurand Y by independent sensors directly measuring Y

2. Redundant measurement of the measurand Y by correlated sensors directly measuring Y

3. Redundant measurement of the measurand Y by correlated sensors X_i indirectly measuring Y, with a linear relationship y = a + A * x between the vector x of sensor values and the vector y containing the various (redundant) estimates of the measurand Y, where a is a vector and A a matrix both of appropriate size.

Details of the different modules are presented in the next sections.

## Details of the main module `redundancy1`¶

The module `redundancy1` implements methods for analysing redundant estimates provided by redundant measurement data from a sensor network.

The main functions included in the file redundancy1.py are:

1. `calc_consistent_estimates_no_corr()`: Calculation of n_rows of best estimates for n_rows of sets of

independent estimates with associated standard uncertainty.

2. `calc_best_estimate()`: Calculation of the best estimate for a given set of estimates with associated uncertainty

matrix.

3. `calc_lcs()`: Calculation of the largest subset of consistent estimates of a measurand.

4. `calc_lcss()`: Calculation of the largest subset of sensor values that yield consistent estimates of a measurand

linked to the sensor values by a linear system of equations.

G. Kok and P. Harris, “Uncertainty Evaluation for Metrologically Redundant Industrial Sensor Networks,” 2020 IEEE International Workshop on Metrology for Industry 4.0 & IoT, Roma, Italy, 2020, pp. 84-88, doi: 10.1109/MetroInd4.0IoT48571.2020.9138297.

exception Met4FoF_redundancy.MFred.redundancy1.AllColumnsZeroSum[source]

Custom exception to handle the case when all columns have zero sum

exception Met4FoF_redundancy.MFred.redundancy1.ColumnNotZeroError[source]

Custom exception to handle the case when a redundant column has not been reduced to zero

exception Met4FoF_redundancy.MFred.redundancy1.SensorsNotLinearlyIndependentError[source]

Custom exception to handle the case when sensor results are not linearly independent

exception Met4FoF_redundancy.MFred.redundancy1.SystemMatrixNotReducibleError[source]

Custom exception to handle the case when the system matrix A is not reducible

Met4FoF_redundancy.MFred.redundancy1.calc_best_est_lin_sys(a_arr, a_arr2d, x_arr, vx_arr2d, problim)[source]

Function to calculate the best estimate of a linear system y = a + A * x and determines if the inputs are consistent in view of problim.

Parameters
• a_arr (np.ndarray of shape (n_estimates)) – vector a of linear system y = a + A * x

• a_arr2d (np.ndarray of shape (n_estimates, n_sensors)) – matrix A of linear system y = a + A * x

• x_arr (np.ndarray of shape (n_sensors)) – vector with sensor values vector x of linear system y = a + A * x

• vx_arr2d (np.ndarray of shape (n_sensors, n_sensors)) – uncertainty matrix associated with vector x_arr

• problim (float) – probability limit used for consistency evaluation. Typically 0.95.

Returns

• isconsist (bool) – indicator whether provided estimates are consistent in view of problim

• ybest (float) – best estimate

• uybest (float) – standard uncertainty of best estimate

• chi2obs (float) – observed chi-squared value

Met4FoF_redundancy.MFred.redundancy1.calc_best_estimate(y_arr, vy_arr2d, problim)[source]

Calculate the best estimate for a set of estimates with associated uncertainty matrix, and determine if the set of estimates are consistent using a provided limit probability.

Parameters
• y_arr (np.ndarray of shape (n)) – vector of estimates of a measurand Y

• vy_arr2d (np.ndarray of shape (n, n)) – uncertainty matrix associated with y_arr

• problim (float) – probability limit used for assessing the consistency of the estimates. Typically, problim equals 0.95.

Returns

• isconsist (bool) – indicator whether provided estimates are consistent in view of problim

• ybest (float) – best estimate of measurand

• uybest (float) – uncertainty associated with ybest

• chi2obs (float) – observed value of chi-squared, used for consistency evaluation

Met4FoF_redundancy.MFred.redundancy1.calc_consistent_estimates_no_corr(y_arr2d, uy_arr2d, prob_lim)[source]

Calculation of consistent estimate for n_sets of estimates y_ij (contained in y_arr2d) of a quantity Y, where each set contains n_estims estimates. The uncertainties are assumed to be independent and given in uy_arr2d. The consistency test is using limit probability limit prob_lim. For each set of estimates, the best estimate, uncertainty, observed chi-2 value and a flag if the provided estimates were consistent given the model are given as output.

Parameters
• y_arr2d (np.ndarray of size (n_rows, n_estimates)) – each row contains m=n_estimates independent estimates of a measurand

• uy_arr2d (np.ndarray of size (n_rows, n_estimates)) – each row contains the standard uncertainty u(y_ij) of y_ij = y_arr2d[i,j]

• prob_lim (limit probability used in consistency test. Typically 0.95.) –

Returns

• isconsist_arr (bool array of shape (n_rows)) – indicates for each row if the n_estimates are consistent or not

• ybest_arr (np.ndarray of shape (n_rows)) – contains the best estimate for each row of individual estimates

• uybest_arr (np.ndarray of shape (n_rows)) – contains the uncertainty associated with each best estimate for each row of y_arr2d

• chi2obs_arr (observed chi-squared value for each row)

Met4FoF_redundancy.MFred.redundancy1.calc_lcs(y_arr, vy_arr2d, problim)[source]

Function to calculate the best estimate of a measurand based on individual estimates of the measurand with associated uncertainty matrix.

Parameters
• y_arr (np.ndarray of shape (n)) – vector with estimates of the measurand

• vy_arr2d (np.ndarray of shape (n, n)) – uncertainty matrix of the vector y_arr

• problim (float) – limit probability used in the consistency evaluation. Typically 0.95.

Met4FoF_redundancy.MFred.redundancy1.calc_lcss(a_arr, a_arr2d, x_arr, vx_arr2d, problim)[source]

Calculation of the largest consistent subset of sensor values and the implied best estimate.

Parameters
• x_arr (np.ndarray of shape (n_sensors)) –

• vx_arr2d (np.ndarray of shape (n_sensors, n_sensors)) –

• a_arr (np.ndarray of shape (n_estimates)) –

• a_arr2d (np.ndarray of shape (n_estimates, n_sensors)) –

• problim (float) –

• a_arr – vector a of linear system y = a + A * x

• a_arr2d – matrix A of linear system y = a + A * x

• x_arr – vector with sensor values vector x of linear system y = a + A * x

• vx_arr2d – uncertainty matrix associated with vector x_arr

• problim – probability limit used for consistency evaluation. Typically 0.95.

Returns

• isconsist (bool) – indicator whether provided estimates are consistent in view of problim

• ybest (float) – best estimate

• uybest (float) – standard uncertainty of best estimate

• chi2obs (float) – observed chi-squared value

Met4FoF_redundancy.MFred.redundancy1.print_output_cbe(isconsist_arr, ybest_arr, uybest_arr, chi2obs_arr)[source]

Function to print the full output of calc_best_estimate.

Parameters
• isconsist_arr (bool array of shape (n_rows)) – indicates for each row if the n_estimates are consistent or not

• ybest_arr (np.ndarray of shape (n_rows)) – contains the best estimate for each row of individual estimates

• uybest_arr (np.ndarray of shape (n_rows)) – contains the uncertainty associated with each best estimate for each row of y_arr2d

• chi2obs_arr (observed chi-squared value for each row) –

Met4FoF_redundancy.MFred.redundancy1.print_output_lcs(n_sols, ybest, uybest, chi2obs, indkeep, y_arr)[source]

Method to print the output of the method `calc_lcs()`.

Parameters
• n_sols (int) – number of best solutions

• ybest (float or np.ndarray of shape (n_sols)) – best estimate or vector of best estimates

• uybest (float or np.ndarray of shape (n_sols)) – standard uncertainty of best estimate or vector with standard uncertainty of best estimates

• chi2obs (float) – observed chi-squared value of all best solutions

• indkeep (np.ndarary of shape (n) or (n_sols, n)) – indices of retained estimates of y_arr for the calculation of the best estimate ybest

• y_arr (np.ndarray of shape (n)) – individual estimates of measurand

Met4FoF_redundancy.MFred.redundancy1.print_output_single(isconsist, ybest, uybest, chi2obs)[source]

Function to print the output of a single row of the calculate_best_estimate function.

Parameters
• isconsist (bool) – Indicates if provided estimates were consistent

• ybest (float) – best estimate

• uybest (float) – uncertainty of best estimate

• chi2obs (float) – observed value of chi-squared

## Details of the test module `test1`¶

The module `test_MFred.py` calls all test functions which are implemented in the module `redundancy1`. These test functions are:

test_redundancy.test_MFred.test_calc_best_estimate()[source]

Test function for calc_best_estimate.

test_redundancy.test_MFred.test_calc_consistent_estimates_no_corr()[source]

Test function for calc_consistent_estimates_no_corr(), implementing two test cases.

test_redundancy.test_MFred.test_calc_lcs()[source]

Test function for `calc_lcs()`. Implements 4 test cases.

test_redundancy.test_MFred.test_calc_lcss()[source]

Test function for method `calc_lcss()`. Implements 4 test cases.