Outputs#
There are several different outputs given by UQPCE. The below section will discuss these outputs.
Output Files#
There are four output files generated in addition to the probability box plot, error distribution plots, and the optional graphs.
output.dat#
This file contains information about the individual run as well as the mean, variance, and confidence interval of the responses.
Example#
### UQPCE v0.3.0 Output
### Analysis of case: None
### Analysis started: 2021-05-26 14:16:49.558007
### Analysis finished: 2021-05-26 14:17:28.128840
### Total compute time: 0:00:38.570833
--------------------------------------------------------------------------------
Mean of response 9.8
Variance of response 6.6741
Mean error of surrogate 2.5171e-14
Signal to noise ratio 2.6515e+14
95.0% Confidence Interval on Response [5.5907 , 15.514]
Shapiro-Wilks test statistic is 0.97742, P-value is 0.44888
Insufficient evidence to infer errors are not from a normal distribution
The probability curve did not converge. 25000 samples were used.
--------------------------------------------------------------------------------
Mean error between model and verification 4.8672e-14
The ratio of verification error to surrogate model error is 1.9337
PRESS statistic: 2.8799e-24
R^2: 1.0
R^2 adjusted: 1.0
average of mean error: 2.1223e-14 variance of mean error: 3.9617e-29
average of mean: 9.8 variance of mean: 9.4285e-29
average of variance: 6.6741 variance of variance: 6.7392e-27
95.0% Confidence Interval on the mean [9.8, 9.8]
95.0% Confidence Interval on the variance [6.6741, 6.6741]
--------------------------------------------------------------------------------
The settings used to generate these results are:
input_file: analytical/input.yaml
matrix_file: analytical/run_matrix.dat
results_file: analytical/results.dat
verification_results_file: analytical/verification_results.dat
verification_matrix_file: analytical/verification_run_matrix.dat
output_directory: outputs_analytical_order2
case: None
significance: 0.05
order: [2]
user_func: None
over_samp_ratio: 2.380952380952381
verify_over_samp_ratio: 0.5
adapt_samp_terms: 4
backend: TkAgg
aleat_sub_samp_size: 5000
epist_sub_samp_size: 25
conv_threshold_percent: 0.0005
epist_samp_size: 125
aleat_samp_size: 25000
var_limit: None
mean_LB: None
mean_UB: None
init_term_count: 4
deg_free: 1
max_iter_count: 30
sob_thresh: 0.0001
sob_final_thresh: 0.005
coeff_thresh: 1e-15
var_thresh: None
var_diff_thresh: None
analytical: False
square_penalty: False
version: True
verbose: True
verify: True
plot: True
plot_stand: True
track_convergence_off: False
adaptive_sampling: False
generate_samples: False
model_conf_int: True
stats: True
report: False
seed: False
opt_design: False
bound_limits: None
resp_order: 2
--------------------------------------------------------------------------------
The input file used is:
Variable 0:
distribution: normal
mean: 1
stdev: 0.5
Variable 1:
distribution: uniform
interval_low: 1.75
interval_high: 2.25
Variable 2:
distribution: exponential
lambda: 3
Variable 3:
distribution: beta
alpha: 0.5
beta: 2.0
Variable 4:
distribution: gamma
alpha: 1.0
theta: 0.5
Settings:
order: 2
version: true
verbose: true
plot: true
plot_stand: true
model_conf_int: true
stats: true
verify: true
Labeled information about the case, data, and model are displayed.
sobol.dat#
This file contains the Sobol indices for the model parameters. These indices represent how sensitive the model is to each parameter. These values will always sum together to equal one.
Note that there are two different total Sobols output. “Total Sobols” is the sum of all Sobol terms that include the given variable. “Rescaled Total Sobols” is these total Sobols divided by the sum of all total Sobols such that the values sum to 1.
Example#
x0 0.33712 < 0.33712 < 0.33712
x1 0.028094 < 0.028094 < 0.028094
x2 0.14983 < 0.14983 < 0.14983
x3 0.10959 < 0.10959 < 0.10959
x4 0.33712 < 0.33712 < 0.33712
x0^2 0.0 < 1.4775e-32 < 1.7941e-15
x0*x1 0.00078037 < 0.00078038 < 0.00078038
x0*x2 0.0 < 3.9727e-31 < 3.6297e-15
x0*x3 0.0 < 2.2829e-31 < 1.4906e-15
x0*x4 0.0 < 1.6621e-30 < 1.3982e-15
x1^2 0.0 < 1.3297e-32 < 1.8259e-15
x1*x2 0.0 < 1.0248e-28 < 3.2893e-15
x1*x3 0.0 < 2.9972e-29 < 2.6159e-15
x1*x4 0.0 < 2.8466e-30 < 1.0631e-15
x2^2 0.0 < 2.3908e-29 < 7.3404e-15
x2*x3 0.0 < 3.6526e-30 < 1.5211e-15
x2*x4 0.0 < 2.5425e-29 < 6.6618e-15
x3^2 0.0 < 1.737e-29 < 2.5168e-15
x3*x4 0.0 < 2.2829e-29 < 2.1845e-15
x4^2 0.037458 < 0.037458 < 0.037458
Total Sobols
Total Sobol x0 = 0.3379
Total Sobol x1 = 0.028874
Total Sobol x2 = 0.14983
Total Sobol x3 = 0.10959
Total Sobol x4 = 0.37458
Rescaled Total Sobols
Total Sobol x0 = 0.3376
Total Sobol x1 = 0.028852
Total Sobol x2 = 0.14971
Total Sobol x3 = 0.10951
Total Sobol x4 = 0.37430
The interaction that the Sobol index represents is listed beside each value for clarity. Variable x0 is the first variable in the input file, x1 is the second, and so on. This example file is using the --model-conf-int flag, which the confidence intervals around additional model metrics.
coefficients.dat#
This file contains the intercept of the responses as well as the coefficients from solving the system of equations. These coefficients show how much the responses change based on each of the combinations of input parameters.
Example#
intercept 9.8 +/- 9.4365e-08
x0 1.5 +/- 9.6689e-08
x1 0.75 +/- 1.5593e-07
x2 3.0 +/- 6.1395e-07
x3 4.0 +/- 4.7315e-07
x4 1.5 +/- 1.2658e-07
x0^2 -2.2204e-16 +/- 7.7375e-08
x0*x1 0.125 +/- 2.0786e-07
x0*x2 -4.885e-15 +/- 4.6693e-07
x0*x3 -5.7732e-15 +/- 4.665e-07
x0*x4 3.3307e-15 +/- 9.6601e-08
x1^2 -6.6613e-16 +/- 2.4684e-07
x1*x2 -1.3589e-13 +/- 7.699e-07
x1*x3 1.1458e-13 +/- 1.0704e-06
x1*x4 -7.5495e-15 +/- 1.459e-07
x2^2 5.6843e-14 +/- 9.9602e-07
x2*x3 -6.9278e-14 +/- 1.4137e-06
x2*x4 3.908e-14 +/- 6.3258e-07
x3^2 2.0961e-13 +/- 2.5231e-06
x3*x4 -5.7732e-14 +/- 5.6474e-07
x4^2 0.25 +/- 5.3862e-08
From this output, we have the intercept of the data and the coefficients for the interaction terms in the matrix system. This example file is using the --model-conf-int flag.
convergence_values.dat#
This file is located in the graph directory, and it is created when tracking convergence is left on. For each set of epistemic sample iterations, all of the low and high confidence intervals are generated, and the overall low and high intervals for the set are displayed. This allows the user to view the convergence for both the set of curves.
Example#
low: [87.9223907053667, 87.96646060205886, 88.01792539580406, 87.83992456472066, 87.80174323289735]
high: [128.7802929016811, 128.72131677630222, 128.56207338904005, 128.6577881178519, 128.56207338904005]
Each set of low and high values corresponds to an individual curve as it increases in sample number.
In the case of one or more epistemic variables, the file will contain the values of the overall confidence interval for sets of curves as they converge:
set: [46.835342059289935, 51.323999003168474]
set: [46.799205926776334, 51.423733890432326]
The set of values set represents one set of curves and its lowest limits of the confidence interval.
UQPCEGroup Outputs#
Below will detail the naming convention of the outputs of OpenMDAO-compatible UQPCEGroup class.
Inspired by NASA’s aircraft analysis, design, and optimization tool Aviary, colons are used separate the names of the PCE model outputs from the response name that the model is build on. The output names are given by
Name |
Description |
|---|---|
|
the matrix coefficients corresponding to the terms in the PCE model |
|
the pseudo-responses generated for a Monte Carlo of the PCE model |
|
the mean of the PCE model |
|
the variance of the PCE model |
|
the mean plus the variance |
|
the lower confidence interval; output if tail= |
|
the upper confidence interval; output if tail= |